Finite Group Databasebeta

← #33144841

Subgroups — #33144841

Order 18688.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988996C211yesmaximal, characteristic, complemented
2C2×#17988629C211yesmaximal, characteristic
3#22613644C211yesmaximal, characteristic
4#22613633C211yesmaximal, characteristic
5#22611409C211yesmaximal, characteristic, complemented
6#22611404C211yesmaximal, characteristic, complemented
7#22611270C211yesmaximal, characteristic, complemented
8#22611265C211yesmaximal, characteristic, complemented
9#22610669C211yesmaximal, characteristic, complemented
10#22610664C211yesmaximal, characteristic, complemented
11#22610132C211yesmaximal, characteristic
12#22610126C211yesmaximal, characteristic
13#22610088C211yesmaximal, characteristic, complemented
14#22609964C211yesmaximal, characteristic
15C73×#1753C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17C2×#14556583C2211yescharacteristic
18C2×#14556568—21no—
19C2×#14556550C2211yescharacteristic
20C22×C4⋊Dic73C2211yescharacteristic
21#17988996—24no—
22#17988629—24no—
23C2×#14556525C2211yescharacteristic
24C2×#14556525C2211yescharacteristic
25C2×#14556525C2211yescharacteristic
26C2×#14556525C2211yescharacteristic
27C2×#14556523C2211yescharacteristic
28C2×#14556523C2211yescharacteristic
29C2×#14556523C2211yescharacteristic
30C2×#14556523C2211yescharacteristic
31C2×#14556523C2211yescharacteristic
32C2×#14556523C2211yescharacteristic
33#17988523—21no—
34#17988519—21no—
35#17988510—21no—
36#17988508—21no—
37#17988485C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988485C2211yescharacteristic
40#17988485C2211yescharacteristic
41#17988484—21no—
42#17988470—21no—
43#17988462—21no—
44#17988457C2211yescharacteristic
45#17988457C2211yescharacteristic
46#17988447C2211yescharacteristic
47#17988447C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988350C2211yescharacteristic
54C22×M16×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56C2×#97×C73C2211yescharacteristic
57C2×C4×C8×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60#336×C73C2211yescharacteristic
61C22×Dic146—21no—
62C23×Dic73—21no—
63#14556591—22no—
64#14556591—41no—
65#14556583—22no—
66#14556583—41no—
67#14556568—44no—
68#14556550—22no—
69#14556550—41no—
70C2×C4⋊Dic73—22no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73—41no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×C4⋊Dic73C2311yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×#11924007—21no—
87C2×#11924007—21no—
88C2×#11924007—21no—
89C2×#11924007—21no—
90C2×#11924007—21no—
91C2×#11924007—21no—
92C2×C22⋊Dic73—21no—
93C2×C22⋊Dic73—21no—
94C2×C22⋊Dic73—21no—
95C2×C4×Dic73—21no—
96#14556525—22no—
97#14556525—22no—
98#14556525—22no—
99#14556525D414yes—
100#14556523—22no—
101#14556523—22no—
102#14556523—22no—
103#14556523—22no—
104#14556523—41no—
105#14556523—41no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112C23×C4×C73C2311yescharacteristic
113C2×M16×C73—22no—
114C2×M16×C73—22no—
115C2×M16×C73Q814yescomplemented
116C22×C8×C73C2311yescharacteristic
117C22×C8×C73C2311yescharacteristic
118C22×C8×C73C2311yescharacteristic
119C22×C8×C73C2311yescharacteristic
120C2×C22⋊C4×C73D411yescharacteristic
121C2×C22⋊C4×C73D411yescharacteristic
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124C22⋊C8×C73Q814yescomplemented
125#97×C73—22no—
126C4×C8×C73—22no—
127#95×C73D411yescharacteristic
128#95×C73D411yescharacteristic
129C2×Dic146—42no—
130C2×Dic146—42no—
131C2×Dic146—81no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—42no—
141C22×Dic73—42no—
142C22×Dic73—81no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—42no—
158C4⋊Dic73—42no—
159C4⋊Dic73—42no—
160C4⋊Dic73—42no—
161C4⋊Dic73—81no—
162C4⋊Dic73—81no—
163C4⋊Dic73C2×D412yes—
164C4⋊Dic73C2×D412yes—
165#11924007—42no—
166#11924007—42no—
167#11924007—42no—
168#11924007—42no—
169#11924007—42no—
170#11924007—81no—
171C22⋊Dic73—42no—
172C22⋊Dic73—42no—
173C22⋊Dic73—81no—
174C4×Dic73—42no—
175C24×C73C2×D411yescharacteristic
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×Q812yes—
186C22×C4×C73C2×Q812yes—
187C22×C4×C73C2411yescharacteristic
188M16×C73—28no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73C2×D412yes—
196C2×C8×C73C2×Q812yes—
197C2×C8×C73C2×Q812yes—
198C2×C8×C73C2×Q812yes—
199C2×C8×C73C2×Q812yes—
200C2×C8×C73#4112yes—
201C22⋊C4×C73—22no—
202C22⋊C4×C73—22no—
203C42×C73—21no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73C4⋊Q812yes—
214C23×C73C22×D411yescharacteristic
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73C4⋊Q812yes—
232C2×C4×C73AGL(1,ℤ/8)12yes—
233C2×C4×C73#13712yes—
234C2×C4×C73C22×D411yescharacteristic
235C2×C4×C73C22×Q811yescharacteristic
236C2×C4×C73C22×Q811yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73H(5,𝔽2)11yescharacteristic
239C2×C4×C732−1+411yescharacteristic
240C2×Dic73—22no—
241C2×Dic73—22no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—84no—
265Dic146—84no—
266C8×C73—22no—
267C8×C73—22no—
268C8×C73—22no—
269C8×C73C22⋊Q814yescomplemented
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73C2×C4⋊Q811yescharacteristic
278C22×C73D4211yescharacteristic
279C22×C73#54711yescharacteristic
280C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C73C2×#13711yescharacteristic
282C22×C73#57711yescharacteristic
283C22×C73#57711yescharacteristic
284C4×C73—22no—
285C4×C73—22no—
286C4×C73—22no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73C2×C22⋊Q811yescharacteristic
290C4×C73#54411yescharacteristic
291C4×C73D4×Q811yescharacteristic
292C4×C73#55411yescharacteristic
293Dic73—42no—
294Dic73—42no—
295Dic73—81no—
296Dic73—81no—
297Dic73—81no—
298Dic73—81no—
299Dic73—81no—
300Dic73—81no—
301#26078—731nomaximal, Sylow
302C2×C73—42no—
303C2×C73#244011yescharacteristic
304C2×C73#316311yescharacteristic
305C2×C73#316511yescharacteristic
306C2×C73#327211yescharacteristic
307C2×C73#327811yescharacteristic
308C2×C73#334411yescharacteristic
309C2×C73#334511yescharacteristic
310C2×#501—731no—
311C2×#428—731no—
312#2441—731no—
313#2440—731no—
314#2084—731no—
315#2083—731no—
316#2059—731no—
317#2058—731no—
318#1943—731no—
319#1942—731no—
320#1854—731no—
321#1853—731no—
322#1844—731no—
323#1809—731no—
324#1753D7311yescharacteristic
325C73#2607811yescharacteristic, complemented, Sylow
326C22×M16C2×D7311yescharacteristic
327C2×C4⋊Q8—731no—
328C2×#125—731no—
329C2×C22⋊Q8—1461no—
330C2×#117—731no—
331C22×C4⋊C4—731no—
332#501—1464no—
333#428—1464no—
334C2×#107—731no—
335C2×#107—731no—
336C2×#107—731no—
337C2×#107—731no—
338C2×#106—731no—
339C2×#106—731no—
340C2×#106—731no—
341C2×#106—731no—
342C2×#106—731no—
343C2×#106—731no—
344C2×C22⋊C8C2×D7311yescharacteristic
345C2×#97C2×D7311yescharacteristic
346C2×C4×C8C2×D7311yescharacteristic
347#399—1461no—
348#398—1461no—
349#396—1461no—
350#395—1461no—
351#389—731no—
352#389—731no—
353#389—731no—
354#389—731no—
355#389—1461no—
356#387—1461no—
357#385—1461no—
358#384—731no—
359#384—731no—
360#380—731no—
361#380—731no—
362C4×C22⋊C4C2×D7311yescharacteristic
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#336C2×D7311yescharacteristic
370#336C2×D7311yescharacteristic
371C22×Q8—1461no—
372C23×C4—1461no—
373C23×C4C22×D7311yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16Dic14614yescomplemented
377C22×C8C22×D7311yescharacteristic
378C22×C8C22×D7311yescharacteristic
379C22×C8C22×D7311yescharacteristic
380C22×C8C22×D7311yescharacteristic
381C4⋊Q8—2921no—
382C4⋊Q8—1462no—
383#125—2921no—
384#125—1462no—
385C22⋊Q8—2924no—
386#117—2921no—
387#117—1462no—
388C2×C4⋊C4—2921no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4—1462no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4—1461no—
410C2×C22⋊C4—1461no—
411C2×C22⋊C4—1461no—
412C2×C22⋊C4—1461no—
413C2×C22⋊C4D29211yescharacteristic
414C2×C22⋊C4D29211yescharacteristic
415C2×C42—1461no—
416C2×C42C22×D7311yescharacteristic
417C2×C42C22×D7311yescharacteristic
418#107—734no—
419#107—1462no—
420#107—1462no—
421#107—1462no—
422#106—2921no—
423#106—2921no—
424#106—1462no—
425#106—1462no—
426#106—1462no—
427#106—1462no—
428C22⋊C8Dic14614yescomplemented
429#97—22no—
430C4×C8—22no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95D29211yescharacteristic
438#95D29211yescharacteristic
439C24C2×D29211yescharacteristic
440C2×Q8—2922no—
441C2×Q8—2922no—
442C2×Q8—5841no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—2922no—
448C22×C4—2922no—
449C22×C4—5841no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4C2×D29211yescharacteristic
459C22×C4C2×D29211yescharacteristic
460C22×C4C2×D29211yescharacteristic
461C22×C4C2×D29211yescharacteristic
462C22×C4C2×D29211yescharacteristic
463C22×C4C2×Dic14612yes—
464C22×C4C2×Dic14612yes—
465C22×C4C23×D7311yescharacteristic
466M16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8D4×D7312yes—
474C2×C8C2×Dic14612yes—
475C2×C8C2×Dic14612yes—
476C2×C8C2×Dic14612yes—
477C2×C8C2×Dic14612yes—
478C2×C8#1192402712yes—
479C4⋊C4—2922no—
480C4⋊C4—2922no—
481C4⋊C4—2922no—
482C4⋊C4—2922no—
483C4⋊C4—2922no—
484C4⋊C4—2922no—
485C4⋊C4—2922no—
486C4⋊C4—2922no—
487C4⋊C4—2922no—
488C4⋊C4—5841no—
489C4⋊C4—5841no—
490C4⋊C4—5841no—
491C4⋊C4—732no—
492C4⋊C4—732no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—2922no—
510C22⋊C4—2922no—
511C22⋊C4—5841no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—2922no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1455659112yes—
524C23C22×D29211yescharacteristic
525Q8—5844no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—5844no—
565C2×C4—1462no—
566C2×C4—1462no—
567C2×C4#1455659112yes—
568C2×C4#1455661912yes—
569C2×C4#1455662412yes—
570C2×C4C2×D4×D7311yescharacteristic
571C2×C4C22×Dic14611yescharacteristic
572C2×C4C22×Dic14611yescharacteristic
573C2×C4C2×#1192402711yescharacteristic
574C2×C4#1455663711yescharacteristic
575C2×C4#1455663811yescharacteristic
576C8—22no—
577C8—22no—
578C8—22no—
579C8#1455656814yescomplemented
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1455659111yescharacteristic
588C22D4×D29211yescharacteristic
589C22#1798924111yescharacteristic
590C22C2×#1455661911yescharacteristic
591C22C2×#1455662411yescharacteristic
592C22#1798939911yescharacteristic
593C22#1798939911yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—2922no—
600C4—2922no—
601C4—5841no—
602C4—5841no—
603C4—5841no—
604C4—5841no—
605C4—5841no—
606C4—5841no—
607C4C2×#1455656811yescharacteristic
608C4#1798921511yescharacteristic
609C4D4×Dic14611yescharacteristic
610C4#1798929411yescharacteristic
611C2—42no—
612C2#2261363311yescharacteristic
613C2#2262046211yescharacteristic
614C2#2262047711yescharacteristic
615C2#2262153311yescharacteristic
616C2#2262159911yescharacteristic
617C2#2262238811yescharacteristic
618C2#2262240311yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1753C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1753D73
hypocenterC73#26078
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#17988996C2semidirect
5. #22611409C2semidirect
6. #22611404C2semidirect
7. #22611270C2semidirect
8. #22611265C2semidirect
9. #22610669C2semidirect
10. #22610664C2semidirect
13. #22610088C2semidirect
115. C2×M16×C73Q8semidirect
124. C22⋊C8×C73Q8semidirect
269. C8×C73C22⋊Q8semidirect
325. C73#26078semidirect
376. C2×M16Dic146semidirect
428. C22⋊C8Dic146semidirect
579. C8#14556568semidirect

Maximal subgroups

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

subgroupindex
C2×#179889962
C2×#179886292
#226136442
#226136332
#226114092
#226114042
#226112702
#226112652
#226106692
#226106642
#226101322
#226101262
#226100882
#226099642
C73×#17532
#2607873