Finite Group Databasebeta

← #33096595

Subgroups — #33096595

Order 18688.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612554C211yesmaximal, characteristic
2#22612540C211yesmaximal, characteristic
3#22610701C211yesmaximal, characteristic, complemented
4#22610701C211yesmaximal, characteristic, complemented
5#22610695C211yesmaximal, characteristic, complemented
6#22610690C211yesmaximal, characteristic, complemented
7#22610132C211yesmaximal, characteristic
8#22610132C211yesmaximal, characteristic
9#22610126C211yesmaximal, characteristic
10#22610121C211yesmaximal, characteristic
11#22610099C211yesmaximal, characteristic, complemented
12#22610095C211yesmaximal, characteristic, complemented
13#22610065C211yesmaximal, characteristic, complemented
14#22610059C211yesmaximal, characteristic, complemented
15C2×#433×C73C211yesmaximal, characteristic
16C2×#14556550C2211yescharacteristic
17C22×C4⋊Dic73C2211yescharacteristic
18C2×#14556525C2211yescharacteristic
19C2×#14556525C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556523C2211yescharacteristic
26#17988486—21no—
27#17988486—21no—
28#17988480—21no—
29#17988480—21no—
30#17988473—21no—
31#17988473—21no—
32#17988462—21no—
33#17988462—21no—
34#17988457C2211yescharacteristic
35#17988457C2211yescharacteristic
36#17988457C2211yescharacteristic
37#17988453C2211yescharacteristic
38#17988453C2211yescharacteristic
39#17988453C2211yescharacteristic
40#17988447C2211yescharacteristic
41#17988447C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43C4×C4⋊Dic73C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52C2×#117×C73C2211yescharacteristic
53#433×C73C418yescomplemented
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×C4⋊C8×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57C2×C22⋊C8×C73C2211yescharacteristic
58C2×#97×C73C2211yescharacteristic
59C2×C4×C8×C73C2211yescharacteristic
60C23×Dic73—21no—
61#14556550—41no—
62#14556550—41no—
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—41no—
71C2×C4⋊Dic73—41no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86#14556525—41no—
87#14556525—41no—
88#14556525—41no—
89#14556525—41no—
90#14556523—41no—
91#14556523—41no—
92#14556523—41no—
93#14556523—41no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104C23×C4×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109#117×C73C2×C414yes—
110#117×C73Q814yes—
111C2×C4⋊C4×C73D411yescharacteristic
112C2×C4⋊C4×C73D411yescharacteristic
113C2×C22⋊C4×C73D411yescharacteristic
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C42×C73C2311yescharacteristic
116C2×C42×C73C2311yescharacteristic
117C4⋊C8×C73C2×C414yescomplemented
118C4⋊C8×C73C2×C414yescomplemented
119C22⋊C8×C73C2×C414yes—
120C22⋊C8×C73C2×C414yes—
121#97×C73C2×C414yescomplemented
122C4×C8×C73C2×C414yescomplemented
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—42no—
132C22×Dic73—81no—
133C22×Dic73—81no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—42no—
151C4⋊Dic73—81no—
152C4⋊Dic73—81no—
153C4⋊Dic73—81no—
154C4⋊Dic73—81no—
155C4⋊Dic73—81no—
156C4⋊Dic73—81no—
157C22⋊Dic73—42no—
158C22⋊Dic73—81no—
159C22⋊Dic73—81no—
160C4×Dic73—42no—
161C4×Dic73—81no—
162C4×Dic73—81no—
163C24×C73C2×D411yescharacteristic
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73C22×C412yes—
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×Q812yes—
175C22×C4×C73C2411yescharacteristic
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73C22×C412yes—
185C2×C8×C73C22×C412yes—
186C2×C8×C73C22×C412yes—
187C2×C8×C73C22×C412yes—
188C4⋊C4×C73C4⋊C414yes—
189C4⋊C4×C73C4⋊C414yes—
190C22⋊C4×C73C4⋊C414yes—
191C22⋊C4×C73C4⋊C414yes—
192C42×C73C22×C412yes—
193C42×C73C22×C412yes—
194C42×C73C2×Q812yes—
195C42×C73C2×Q812yes—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73C2×C4⋊C412yes—
203C23×C73C22×D411yescharacteristic
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73C2×C4⋊C412yes—
219C2×C4×C73C2×C4⋊C412yes—
220C2×C4×C73C2×C4⋊C412yes—
221C2×C4×C73C2×C4⋊C412yes—
222C2×C4×C73C2×C4⋊C412yes—
223C2×C4×C73C23×C411yescharacteristic
224C2×C4×C73C22×Q811yescharacteristic
225C2×C4×C73H(5,𝔽2)11yescharacteristic
226C2×C4×C73H(5,𝔽2)11yescharacteristic
227C2×C4×C732−1+411yescharacteristic
228C2×C4×C732−1+411yescharacteristic
229C2×Dic73—41no—
230C2×Dic73—41no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
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—84no—
254C8×C73—22no—
255C8×C73—22no—
256C8×C73—22no—
257C8×C73—22no—
258C22×C73—22no—
259C22×C73—22no—
260C22×C73—22no—
261C22×C73—41no—
262C22×C73—41no—
263C22×C73—41no—
264C22×C73—41no—
265C22×C73C22×C4⋊C411yescharacteristic
266C22×C73#53511yescharacteristic
267C22×C73#53611yescharacteristic
268C22×C73#57611yescharacteristic
269C22×C73#57711yescharacteristic
270C22×C73#57711yescharacteristic
271C22×C73#57811yescharacteristic
272C4×C73—22no—
273C4×C73—22no—
274C4×C73—22no—
275C4×C73—22no—
276C4×C73—22no—
277C4×C73#52011yescharacteristic
278C4×C73#52011yescharacteristic
279C4×C73#54411yescharacteristic
280C4×C73#54411yescharacteristic
281Dic73—81no—
282Dic73—81no—
283Dic73—81no—
284Dic73—81no—
285Dic73—81no—
286Dic73—81no—
287Dic73—81no—
288Dic73—81no—
289#21992—731nomaximal, Sylow
290C2×C73—42no—
291C2×C73#231711yescharacteristic
292C2×C73#291211yescharacteristic
293C2×C73#291311yescharacteristic
294C2×C73#322411yescharacteristic
295C2×C73#322511yescharacteristic
296C2×C73#324111yescharacteristic
297C2×C73#324211yescharacteristic
298C2×#433D7311yescharacteristic
299#2317—731no—
300#2314—731no—
301#1949—731no—
302#1949—731no—
303#1948—731no—
304#1947—731no—
305#1854—731no—
306#1854—731no—
307#1853—731no—
308#1852—731no—
309#1847—731no—
310#1846—731no—
311#1836—731no—
312#1834—731no—
313C73#2199211yescharacteristic, complemented, Sylow
314C2×#117—731no—
315C2×#117C2×D7311yescharacteristic
316C22×C4⋊C4—731no—
317#433Dic7318yescomplemented
318C2×#107—731no—
319C2×#107—731no—
320C2×#107—731no—
321C2×#107—731no—
322C2×#106—731no—
323C2×#106—731no—
324C2×#106—731no—
325C2×#106—731no—
326C2×C4⋊C8C2×D7311yescharacteristic
327C2×C4⋊C8C2×D7311yescharacteristic
328C2×C22⋊C8C2×D7311yescharacteristic
329C2×C22⋊C8C2×D7311yescharacteristic
330C2×#97C2×D7311yescharacteristic
331C2×C4×C8C2×D7311yescharacteristic
332#389—1461no—
333#389—1461no—
334#388—1461no—
335#388—1461no—
336#387—1461no—
337#387—1461no—
338#385—1461no—
339#385—1461no—
340#384—731no—
341#384—731no—
342#384—731no—
343#382—731no—
344#382—731no—
345#382—731no—
346#380—731no—
347#380—731no—
348C4×C4⋊C4—731no—
349C4×C4⋊C4—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358C23×C4—1461no—
359C23×C4C22×D7311yescharacteristic
360C22×C8C22×D7311yescharacteristic
361C22×C8C22×D7311yescharacteristic
362C22×C8C22×D7311yescharacteristic
363C22×C8C22×D7311yescharacteristic
364#117—2921no—
365#117—2921no—
366#117Dic14614yes—
367#117C2×Dic7314yes—
368C2×C4⋊C4—2921no—
369C2×C4⋊C4—2921no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4D29211yescharacteristic
386C2×C4⋊C4D29211yescharacteristic
387C2×C22⋊C4—1461no—
388C2×C22⋊C4—1461no—
389C2×C22⋊C4—1461no—
390C2×C22⋊C4D29211yescharacteristic
391C2×C22⋊C4D29211yescharacteristic
392C2×C42—1461no—
393C2×C42—1461no—
394C2×C42—1461no—
395C2×C42C22×D7311yescharacteristic
396C2×C42C22×D7311yescharacteristic
397#107—2921no—
398#107—2921no—
399#107—2921no—
400#107—2921no—
401#106—2921no—
402#106—2921no—
403#106—2921no—
404#106—2921no—
405C4⋊C8C2×Dic7314yescomplemented
406C4⋊C8C2×Dic7314yescomplemented
407C22⋊C8C2×Dic7314yes—
408C22⋊C8C2×Dic7314yes—
409#97C2×Dic7314yescomplemented
410C4×C8C2×Dic7314yescomplemented
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421C24C2×D29211yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—2922no—
427C22×C4—5841no—
428C22×C4—5841no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4C22×Dic7312yes—
438C22×C4C2×D29211yescharacteristic
439C22×C4C2×D29211yescharacteristic
440C22×C4C2×D29211yescharacteristic
441C22×C4C2×D29211yescharacteristic
442C22×C4C2×D29211yescharacteristic
443C22×C4C2×Dic14612yes—
444C22×C4C23×D7311yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8C22×Dic7312yes—
454C2×C8C22×Dic7312yes—
455C2×C8C22×Dic7312yes—
456C2×C8C22×Dic7312yes—
457C4⋊C4—2922no—
458C4⋊C4—5841no—
459C4⋊C4—5841no—
460C4⋊C4—5841no—
461C4⋊C4—5841no—
462C4⋊C4—5841no—
463C4⋊C4—5841no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4C4⋊Dic7314yes—
481C4⋊C4C4⋊Dic7314yes—
482C22⋊C4—2922no—
483C22⋊C4—5841no—
484C22⋊C4—5841no—
485C22⋊C4C4⋊Dic7314yes—
486C22⋊C4C4⋊Dic7314yes—
487C42—2922no—
488C42—5841no—
489C42—5841no—
490C42C22×Dic7312yes—
491C42C22×Dic7312yes—
492C42C2×Dic14612yes—
493C42C2×Dic14612yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C2×C4⋊Dic7312yes—
501C23C22×D29211yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—5844no—
541C2×C4C2×C4⋊Dic7312yes—
542C2×C4C2×C4⋊Dic7312yes—
543C2×C4C2×C4⋊Dic7312yes—
544C2×C4C2×C4⋊Dic7312yes—
545C2×C4C2×C4⋊Dic7312yes—
546C2×C4C23×Dic7311yescharacteristic
547C2×C4C22×Dic14611yescharacteristic
548C2×C4#1455663711yescharacteristic
549C2×C4#1455663711yescharacteristic
550C2×C4#1455663811yescharacteristic
551C2×C4#1455663811yescharacteristic
552C8—22no—
553C8—22no—
554C8—22no—
555C8—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C22×C4⋊Dic7311yescharacteristic
564C22#1798915611yescharacteristic
565C22#1798916311yescharacteristic
566C22#1798939211yescharacteristic
567C22#1798939911yescharacteristic
568C22#1798939911yescharacteristic
569C22#1798940411yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—5841no—
576C4—5841no—
577C4—5841no—
578C4—5841no—
579C4—5841no—
580C4—5841no—
581C4—5841no—
582C4—5841no—
583C4#1798907011yescharacteristic
584C4#1798907011yescharacteristic
585C4#1798921511yescharacteristic
586C4#1798921511yescharacteristic
587C2—42no—
588C2#2261255411yescharacteristic
589C2#2261852611yescharacteristic
590C2#2261853111yescharacteristic
591C2#2262102711yescharacteristic
592C2#2262104211yescharacteristic
593C2#2262118611yescharacteristic
594C2#2262120111yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#433×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenterC2×#433D73
hypocenterC73#21992
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. #22610701C2semidirect
4. #22610701C2semidirect
5. #22610695C2semidirect
6. #22610690C2semidirect
11. #22610099C2semidirect
12. #22610095C2semidirect
13. #22610065C2semidirect
14. #22610059C2semidirect
53. #433×C73C4semidirect
117. C4⋊C8×C73C2×C4semidirect
118. C4⋊C8×C73C2×C4semidirect
121. #97×C73C2×C4semidirect
122. C4×C8×C73C2×C4semidirect
313. C73#21992semidirect
317. #433Dic73semidirect
405. C4⋊C8C2×Dic73semidirect
406. C4⋊C8C2×Dic73semidirect
409. #97C2×Dic73semidirect
410. C4×C8C2×Dic73semidirect

Maximal subgroups

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

subgroupindex
#226125542
#226125402
#226107012
#226106952
#226106902
#226101322
#226101262
#226101212
#226100992
#226100952
#226100652
#226100592
C2×#433×C732
#2199273