Finite Group Databasebeta

← #33057142

Subgroups — #33057142

Order 18688.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619091C211yesmaximal, characteristic, complemented
2#22618966C211yesmaximal, characteristic, complemented
3#22608822C211yesmaximal, characteristic, complemented
4#22608816C211yesmaximal, characteristic, complemented
5#22608810C211yesmaximal, characteristic, complemented
6#22608796C211yesmaximal, characteristic, complemented
7#22608796C211yesmaximal, characteristic, complemented
8#22608789C211yesmaximal, characteristic, complemented
9SD16×#10596831C211yesmaximal, characteristic, complemented
10D8×#10596831C211yesmaximal, characteristic, complemented
11#22608716C211yesmaximal, characteristic, complemented
12#22608391C211yesmaximal, characteristic, complemented
13C2×D4⋊#10596831C211yesmaximal, characteristic, complemented
14#22608215C211yesmaximal, characteristic, complemented
15C73×#3105C211yesmaximal, characteristic
16#17988718C2211yescharacteristic, complemented
17Q8×#10596831C2211yescharacteristic, complemented
18#17988665C2211yescharacteristic
19#17988665C2211yescharacteristic
20#17988665C2211yescharacteristic, complemented
21#17988659—21no—
22#17988659—21no—
23#17988659—21no—
24D4×#10596831—21no—
25D4×#10596831C2211yescharacteristic
26D4×#10596831C2211yescharacteristic
27D4×#10596831C2211yescharacteristic, complemented
28#17988648—21no—
29#17988644C2211yescharacteristic
30#17988640C2211yescharacteristic
31#17988614—21no—
32C2×C4⋊#10596831C2211yescharacteristic
33#17988551—21no—
34C2×C22⋊#10596831—21no—
35C2×C4×#10596831C2211yescharacteristic
36#17988340C2211yescharacteristic, complemented
37#17988340C2211yescharacteristic, complemented
38#17988338C2211yescharacteristic, complemented
39#17988338C2211yescharacteristic, complemented
40Q8⋊#10596831C2211yescharacteristic, complemented
41Q8⋊#10596831C2211yescharacteristic, complemented
42D4⋊#10596831C2211yescharacteristic
43D4⋊#10596831C2211yescharacteristic
44D4⋊#10596831C2211yescharacteristic
45D4⋊#10596831C2211yescharacteristic
46D4⋊#10596831C2211yescharacteristic, complemented
47D4⋊#10596831C2211yescharacteristic, complemented
48#17988309C2211yescharacteristic, complemented
49#17988309C2211yescharacteristic, complemented
50C8×#10596831C2211yescharacteristic, complemented
51C8×#10596831C2211yescharacteristic, complemented
52#529×C73C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54#484×C73C411yescharacteristic
55#481×C73C411yescharacteristic
56#478×C73C411yescharacteristic
57#478×C73C411yescharacteristic
58#463×C73C411yescharacteristic
59#463×C73C411yescharacteristic
60#450×C73C411yescharacteristic
61#447×C73C411yescharacteristic
62C4×SD16×C73C2211yescharacteristic
63C4×SD16×C73C2211yescharacteristic
64C4×D8×C73C2211yescharacteristic
65C4×D8×C73C2211yescharacteristic
66#431×C73C2211yescharacteristic
67C2×#11924012—21no—
68C2×#11924012—21no—
69C2×#11924012—21no—
70C22×#10596831—21no—
71C22×#10596831—21no—
72C22×#10596831—21no—
73C22×#10596831—21no—
74C22×#10596831—21no—
75C4⋊#10596831—21no—
76C4⋊#10596831—21no—
77C4⋊#10596831—21no—
78C4⋊#10596831D411yescharacteristic
79C4⋊#10596831D411yescharacteristic
80C4⋊#10596831C2311yescharacteristic
81C4⋊#10596831C2311yescharacteristic
82C4⋊#10596831C2311yescharacteristic
83C4⋊#10596831C2311yescharacteristic
84C22⋊#10596831—21no—
85C22⋊#10596831—21no—
86C22⋊#10596831—21no—
87C22⋊#10596831—21no—
88C22⋊#10596831—21no—
89C22⋊#10596831—21no—
90C22⋊#10596831—21no—
91C22⋊#10596831D411yescharacteristic
92C22⋊#10596831D411yescharacteristic
93#14556502—21no—
94#14556502—21no—
95C4×#10596831—21no—
96C4×#10596831—21no—
97C4×#10596831C2311yescharacteristic
98C4×#10596831C2311yescharacteristic
99C4×#10596831C2311yescharacteristic
100C4×#10596831C2311yescharacteristic
101C22×D4×C73C2×C411yescharacteristic
102C23×C4×C73—21no—
103C2×SD16×C73C2×C411yescharacteristic
104C2×SD16×C73C2×C411yescharacteristic
105C2×D8×C73C2×C411yescharacteristic
106C2×D8×C73C2×C411yescharacteristic
107#126×C73—21no—
108#125×C73C2×C411yescharacteristic
109#124×C73C2×C411yescharacteristic
110#123×C73—21no—
111C22⋊Q8×C73C2×C411yescharacteristic
112#121×C73C2×C411yescharacteristic
113C4×Q8×C73C2311yescharacteristic
114C4×D4×C73—21no—
115C4×D4×C73—21no—
116C4×D4×C73—21no—
117C4×D4×C73C2×C411yescharacteristic
118C4×D4×C73C2×C411yescharacteristic
119C4×D4×C73C2311yescharacteristic
120C4×D4×C73C2311yescharacteristic
121C4×D4×C73C2311yescharacteristic
122#117×C73—21no—
123C2×C4⋊C4×C73C2×C411yescharacteristic
124C2×C22⋊C4×C73—21no—
125C2×C42×C73C2311yescharacteristic
126#107×C73C2×C411yescharacteristic
127#107×C73C2×C411yescharacteristic
128#106×C73C2×C411yescharacteristic
129#106×C73C2×C411yescharacteristic
130C4⋊C8×C73C2×C411yescharacteristic
131C4⋊C8×C73C2×C411yescharacteristic
132#103×C73C2×C411yescharacteristic
133#103×C73C2×C411yescharacteristic
134#102×C73C2×C411yescharacteristic
135#102×C73C2×C411yescharacteristic
136#102×C73C2×C411yescharacteristic
137#102×C73C2×C411yescharacteristic
138#102×C73C2×C411yescharacteristic
139#102×C73C2×C411yescharacteristic
140C22⋊C8×C73C2×C411yescharacteristic
141C22⋊C8×C73C2×C411yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144#11924012—81no—
145#11924012—81no—
146#11924012—81no—
147C2×#10596831—24no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—81no—
155C2×#10596831—81no—
156C2×#10596831—81no—
157C2×#10596831—81no—
158C2×#10596831C2×D411yescharacteristic
159C2×#10596831C2×D411yescharacteristic
160C2×#10596831C2×D411yescharacteristic
161C2×#10596831C2×D411yescharacteristic
162C24×C73—21no—
163C2×Q8×C73C22×C411yescharacteristic
164C2×D4×C73—22no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—41no—
168C2×D4×C73C22×C411yescharacteristic
169C2×D4×C73C22×C411yescharacteristic
170C2×D4×C73C22×C411yescharacteristic
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—81no—
178C22×C4×C73—81no—
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73#4111yescharacteristic
182SD16×C73—22no—
183SD16×C73—41no—
184D8×C73—22no—
185D8×C73—41no—
186C2×C8×C73C22×C411yescharacteristic
187C2×C8×C73C22×C411yescharacteristic
188C2×C8×C73C22×C411yescharacteristic
189C2×C8×C73C22×C411yescharacteristic
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73C22×C411yescharacteristic
196C4⋊C4×C73C22×C411yescharacteristic
197C4⋊C4×C73C22×C411yescharacteristic
198C4⋊C4×C73C22×C411yescharacteristic
199C22⋊C4×C73—21no—
200C22⋊C4×C73—21no—
201C22⋊C4×C73—21no—
202C22⋊C4×C73—21no—
203C22⋊C4×C73—21no—
204C22⋊C4×C73—21no—
205C22⋊C4×C73—21no—
206C22⋊C4×C73—21no—
207C22⋊C4×C73—21no—
208C42×C73—21no—
209C42×C73C22×C411yescharacteristic
210C42×C73C2×D411yescharacteristic
211C42×C73#4111yescharacteristic
212C42×C73C2411yescharacteristic
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—42no—
217C23×C73—42no—
218C23×C73—81no—
219C23×C73C4×D411yescharacteristic
220Q8×C73—21no—
221Q8×C73—21no—
222D4×C73—21no—
223D4×C73—21no—
224D4×C73—21no—
225D4×C73—21no—
226D4×C73—21no—
227D4×C73—42no—
228D4×C73—42no—
229D4×C73#13112yes—
230C2×C4×C73—22no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—81no—
247C2×C4×C73—81no—
248C2×C4×C73—81no—
249C2×C4×C73C4×D411yescharacteristic
250C2×C4×C73C4×D411yescharacteristic
251C2×C4×C73C4×D411yescharacteristic
252C2×C4×C73C23×C411yescharacteristic
253C2×C4×C73C22×D411yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C8×C73—21no—
256C8×C73—21no—
257C8×C73—21no—
258C8×C73—21no—
259#10596831—22no—
260#10596831—22no—
261#10596831—21no—
262#10596831—21no—
263#10596831—21no—
264#10596831—21no—
265#10596831—41no—
266#10596831—41no—
267#10596831—41no—
268#10596831—41no—
269C22×C73—21no—
270C22×C73—21no—
271C22×C73—22no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—84no—
281C22×C73#44312yes—
282C22×C73C2×C4×D411yescharacteristic
283C4×C73—21no—
284C4×C73—21no—
285C4×C73—21no—
286C4×C73—21no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—41no—
291C4×C73—41no—
292C4×C73C2×#13111yescharacteristic
293C4×C73#56811yescharacteristic
294C4×C73#57611yescharacteristic
295C4×C73#57711yescharacteristic
296#18014—731nomaximal, Sylow
297C2×C73—22no—
298C2×C73—42no—
299C2×C73—81no—
300C2×C73—81no—
301C2×C73#293311yescharacteristic
302C2×C73#294611yescharacteristic
303C2×C73C2×#44311yescharacteristic
304#3105D7311yescharacteristic
305#2994—731no—
306#2974—731no—
307#1590—731no—
308#1589—731no—
309#1588—731no—
310#1586—731no—
311#1586—731no—
312#1585—731no—
313C8×SD16—731no—
314C8×D8—731no—
315#1573—731no—
316#1494—731no—
317C2×D4⋊C8—731no—
318#1450—731no—
319C73#1801411yescharacteristic, complemented, Sylow
320#529C2×D7311yescharacteristic
321C2×C4×D4C2×D7311yescharacteristic
322#484Dic7311yescharacteristic
323#481Dic7311yescharacteristic
324#478Dic7311yescharacteristic
325#478Dic7311yescharacteristic
326#463Dic7311yescharacteristic
327#463Dic7311yescharacteristic
328#450Dic7311yescharacteristic
329#447Dic7311yescharacteristic
330#446—731no—
331C8×Q8—731no—
332C4×SD16C2×D7311yescharacteristic
333C4×SD16C2×D7311yescharacteristic
334C4×D8C2×D7311yescharacteristic
335C4×D8C2×D7311yescharacteristic
336#436—731no—
337#436—731no—
338#436—731no—
339#435—1461no—
340#435—1461no—
341#435—1461no—
342C8×D4—731no—
343C8×D4—731no—
344C8×D4—731no—
345C8×D4—1461no—
346#433—1461no—
347#432—731no—
348#431—731no—
349#431C2×D7311yescharacteristic
350#424—1461no—
351C2×C4⋊C8—731no—
352#407—1461no—
353C2×C22⋊C8—1461no—
354C2×C4×C8—731no—
355#335—731no—
356#335—731no—
357#334—731no—
358#334—731no—
359Q8⋊C8—731no—
360Q8⋊C8—731no—
361D4⋊C8—731no—
362D4⋊C8—731no—
363D4⋊C8—731no—
364D4⋊C8—731no—
365D4⋊C8—731no—
366D4⋊C8—731no—
367#322—731no—
368#322—731no—
369C82—731no—
370C82—731no—
371C22×D4C2×Dic7311yescharacteristic
372C23×C4—21no—
373C2×SD16C2×Dic7311yescharacteristic
374C2×SD16C2×Dic7311yescharacteristic
375C2×D8C2×Dic7311yescharacteristic
376C2×D8C2×Dic7311yescharacteristic
377C2×M16—1461no—
378C2×M16—1461no—
379C2×M16—1461no—
380C22×C8—1461no—
381C22×C8—1461no—
382C22×C8—1461no—
383C22×C8—1461no—
384C22×C8—1461no—
385#126—21no—
386#125C2×Dic7311yescharacteristic
387#124C2×Dic7311yescharacteristic
388#123—21no—
389C22⋊Q8C2×Dic7311yescharacteristic
390#121C2×Dic7311yescharacteristic
391C4×Q8C22×D7311yescharacteristic
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4C2×Dic7311yescharacteristic
396C4×D4C2×Dic7311yescharacteristic
397C4×D4C22×D7311yescharacteristic
398C4×D4C22×D7311yescharacteristic
399C4×D4C22×D7311yescharacteristic
400#117—21no—
401C2×C4⋊C4C2×Dic7311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C42C22×D7311yescharacteristic
404#107C2×Dic7311yescharacteristic
405#107C2×Dic7311yescharacteristic
406#106C2×Dic7311yescharacteristic
407#106C2×Dic7311yescharacteristic
408C4⋊C8—731no—
409C4⋊C8—731no—
410C4⋊C8—731no—
411C4⋊C8—731no—
412C4⋊C8—731no—
413C4⋊C8—731no—
414C4⋊C8—1461no—
415C4⋊C8—1461no—
416C4⋊C8—1461no—
417C4⋊C8C2×Dic7311yescharacteristic
418C4⋊C8C2×Dic7311yescharacteristic
419#103C2×Dic7311yescharacteristic
420#103C2×Dic7311yescharacteristic
421#102C2×Dic7311yescharacteristic
422#102C2×Dic7311yescharacteristic
423#102C2×Dic7311yescharacteristic
424#102C2×Dic7311yescharacteristic
425#102C2×Dic7311yescharacteristic
426#102C2×Dic7311yescharacteristic
427C22⋊C8—731no—
428C22⋊C8—731no—
429C22⋊C8—1461no—
430C22⋊C8—1461no—
431C22⋊C8—1461no—
432C22⋊C8—1461no—
433C22⋊C8—1461no—
434C22⋊C8—1461no—
435C22⋊C8—1461no—
436C22⋊C8C2×Dic7311yescharacteristic
437C22⋊C8C2×Dic7311yescharacteristic
438#97—1461no—
439#97—1461no—
440C4×C8—731no—
441C4×C8—731no—
442C4×C8—731no—
443C4×C8—731no—
444C4×C8—1461no—
445C4×C8—1461no—
446C4×C8C22×D7311yescharacteristic
447C4×C8C22×D7311yescharacteristic
448C24—21no—
449C2×Q8C22×Dic7311yescharacteristic
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—41no—
454C2×D4C22×Dic7311yescharacteristic
455C2×D4C22×Dic7311yescharacteristic
456C2×D4C22×Dic7311yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4C22×Dic7311yescharacteristic
466C22×C4D4×D7311yescharacteristic
467C22×C4#1192402711yescharacteristic
468SD16—22no—
469SD16—41no—
470D8—22no—
471D8—41no—
472M16—5841no—
473M16—5841no—
474M16—5841no—
475C2×C8—5841no—
476C2×C8—5841no—
477C2×C8—5841no—
478C2×C8—5841no—
479C2×C8—731no—
480C2×C8—731no—
481C2×C8—731no—
482C2×C8—731no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1464no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8C22×Dic7311yescharacteristic
491C2×C8C22×Dic7311yescharacteristic
492C2×C8C22×Dic7311yescharacteristic
493C2×C8C22×Dic7311yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4C22×Dic7311yescharacteristic
500C4⋊C4C22×Dic7311yescharacteristic
501C4⋊C4C22×Dic7311yescharacteristic
502C4⋊C4C22×Dic7311yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C42—21no—
513C42C22×Dic7311yescharacteristic
514C42D4×D7311yescharacteristic
515C42#1192402711yescharacteristic
516C42C23×D7311yescharacteristic
517C23—21no—
518C23—21no—
519C23—21no—
520C23—42no—
521C23—42no—
522C23—81no—
523C23D4×Dic7311yescharacteristic
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—21no—
531D4—42no—
532D4—42no—
533D4#1455659812yes—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4D4×Dic7311yescharacteristic
554C2×C4D4×Dic7311yescharacteristic
555C2×C4D4×Dic7311yescharacteristic
556C2×C4C23×Dic7311yescharacteristic
557C2×C4C2×D4×D7311yescharacteristic
558C2×C4C2×#1192402711yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—2921no—
564C8—2921no—
565C8—2921no—
566C8—2921no—
567C8—1462no—
568C8—1462no—
569C8—1461no—
570C8—1461no—
571C8—1461no—
572C8—1461no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—84no—
585C22#1798870412yes—
586C22C2×D4×Dic7311yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4C2×#1455659811yescharacteristic
597C4#1798935411yescharacteristic
598C4#1798939111yescharacteristic
599C4#1798939711yescharacteristic
600C2—22no—
601C2—42no—
602C2—81no—
603C2—81no—
604C2#2261865811yescharacteristic
605C2#2261876411yescharacteristic
606C2C2×#1798870411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#3105C2
socleC22×C73C2×C4×D4
prime radical#447×C73C4
cosocleC42×C73C24
hypercenter#3105D73
hypocenterC73#18014
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3105

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22619091C2semidirect
2. #22618966C2semidirect
3. #22608822C2semidirect
4. #22608816C2semidirect
5. #22608810C2semidirect
6. #22608796C2semidirect
7. #22608796C2semidirect
8. #22608789C2semidirect
9. SD16×#10596831C2semidirect
10. D8×#10596831C2semidirect
11. #22608716C2semidirect
12. #22608391C2semidirect
13. C2×D4⋊#10596831C2semidirect
14. #22608215C2semidirect
16. #17988718C22semidirect
17. Q8×#10596831C22semidirect
20. #17988665C22semidirect
27. D4×#10596831C22semidirect
36. #17988340C22semidirect
37. #17988340C22semidirect
38. #17988338C22semidirect
39. #17988338C22semidirect
40. Q8⋊#10596831C22semidirect
41. Q8⋊#10596831C22semidirect
46. D4⋊#10596831C22semidirect
47. D4⋊#10596831C22semidirect
48. #17988309C22semidirect
49. #17988309C22semidirect
50. C8×#10596831C22semidirect
51. C8×#10596831C22semidirect
319. C73#18014semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226190912
#226189662
#226088222
#226088162
#226088102
#226087962
#226087892
SD16×#105968312
D8×#105968312
#226087162
#226083912
C2×D4⋊#105968312
#226082152
C73×#31052
#1801473