Finite Group Databasebeta

← #34019242

Subgroups — #34019242

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#22626882C211yesmaximal, characteristic, complemented
2#22626792C211yesmaximal, characteristic, complemented
3#22625565C211yesmaximal, characteristic, complemented
4#22625563C211yesmaximal, characteristic, complemented
5#22625558C211yesmaximal, characteristic, complemented
6#22625551C211yesmaximal, characteristic, complemented
7#22625551C211yesmaximal, characteristic, complemented
8#22625549C211yesmaximal, characteristic, complemented
9#22625515C211yesmaximal, characteristic, complemented
10#22625512C211yesmaximal, characteristic, complemented
11#22625506C211yesmaximal, characteristic, complemented
12#22625284C211yesmaximal, characteristic, complemented
13C2×#17989435C211yesmaximal, characteristic, complemented
14#22625136C211yesmaximal, characteristic, complemented
15#22619981C211yesmaximal, characteristic
16#17989638C2211yescharacteristic, complemented
17Q8×#10596833C2211yescharacteristic, complemented
18#17989621C2211yescharacteristic
19#17989621C2211yescharacteristic
20#17989621C2211yescharacteristic, complemented
21#17989617—21no—
22#17989617—21no—
23#17989616—21no—
24#17989615C2211yescharacteristic
25#17989615C2211yescharacteristic
26#17989615C2211yescharacteristic, complemented
27#17989614—21no—
28#17989612—21no—
29#17989608C2211yescharacteristic
30#17989604C2211yescharacteristic
31#17989594—21no—
32C2×C4⋊#10596833C2211yescharacteristic
33#17989558—21no—
34#17989555—21no—
35C2×C4×#10596833C2211yescharacteristic
36#17989450C2211yescharacteristic, complemented
37#17989450C2211yescharacteristic, complemented
38#17989449C2211yescharacteristic, complemented
39#17989449C2211yescharacteristic, complemented
40Q8⋊#10596833C2211yescharacteristic, complemented
41Q8⋊#10596833C2211yescharacteristic, complemented
42#17989435C2211yescharacteristic
43#17989435C2211yescharacteristic
44#17989435C2211yescharacteristic
45#17989435C2211yescharacteristic
46#17989435C2211yescharacteristic, complemented
47#17989435C2211yescharacteristic, complemented
48#17989429C2211yescharacteristic, complemented
49#17989429C2211yescharacteristic, complemented
50C8×#10596833C2211yescharacteristic, complemented
51C8×#10596833C2211yescharacteristic, complemented
52#17989123C2211yescharacteristic
53C2×#14556553C2211yescharacteristic
54#17988905C411yescharacteristic
55#17988890C411yescharacteristic
56#17988890C411yescharacteristic
57#17988805C411yescharacteristic
58#17988805C411yescharacteristic
59#17988737C411yescharacteristic
60#17988723C411yescharacteristic
61#17988672C2211yescharacteristic
62#17988672C2211yescharacteristic
63#17988669C2211yescharacteristic
64#17988669C2211yescharacteristic
65#17988637C2211yescharacteristic
66#484×C73C411yescharacteristic
67C2×#11924038—21no—
68C2×#11924038—21no—
69C2×#11924038—21no—
70C2×#11924036—21no—
71C2×#11924036—21no—
72C2×#11924036—21no—
73C22×#10596833—21no—
74C22×#10596833—21no—
75#14556658—21no—
76#14556658D411yescharacteristic
77#14556658D411yescharacteristic
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84#14556648—21no—
85#14556648—21no—
86#14556648—21no—
87#14556648—21no—
88#14556648—21no—
89#14556648—21no—
90C22⋊#10596833—21no—
91C22⋊#10596833D411yescharacteristic
92C22⋊#10596833D411yescharacteristic
93#14556646—21no—
94#14556646—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C22×D292C2×C411yescharacteristic
102C22×C4×D73—21no—
103C2×#11924015C2×C411yescharacteristic
104C2×#11924015C2×C411yescharacteristic
105C2×D584C2×C411yescharacteristic
106C2×D584C2×C411yescharacteristic
107#14556586—21no—
108#14556582C2×C411yescharacteristic
109#14556580C2×C411yescharacteristic
110#14556575—21no—
111#14556565C2×C411yescharacteristic
112Q8×Dic73C2311yescharacteristic
113#14556553—21no—
114#14556553—21no—
115#14556553C2×C411yescharacteristic
116#14556553C2×C411yescharacteristic
117#14556553C2311yescharacteristic
118#14556553C2311yescharacteristic
119#14556553C2311yescharacteristic
120#14556551—21no—
121#14556549—21no—
122C2×#11924005—21no—
123C2×C4×Dic73C2311yescharacteristic
124#14556522C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126#14556520C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556513C2×C411yescharacteristic
129#14556513C2×C411yescharacteristic
130#14556511C2×C411yescharacteristic
131#14556511C2×C411yescharacteristic
132#14556511C2×C411yescharacteristic
133#14556511C2×C411yescharacteristic
134C22⋊#10596831C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C22⋊Q8×C73C2×C411yescharacteristic
138C2×C4⋊C4×C73C2×C411yescharacteristic
139#107×C73C2×C411yescharacteristic
140#107×C73C2×C411yescharacteristic
141#103×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143C22⋊C8×C73C2×C411yescharacteristic
144#11924038—81no—
145#11924038—81no—
146#11924038—81no—
147#11924036—81no—
148#11924036—81no—
149#11924036—81no—
150C2×#10596833—24no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—21no—
155C2×#10596833—21no—
156C2×#10596833—21no—
157C2×#10596833—81no—
158C2×#10596833C2×D411yescharacteristic
159C2×#10596833C2×D411yescharacteristic
160C2×#10596833C2×D411yescharacteristic
161C2×#10596833C2×D411yescharacteristic
162C23×D73—21no—
163C2×D292—22no—
164C2×D292—21no—
165C2×D292—41no—
166C2×D292C22×C411yescharacteristic
167C2×D292C22×C411yescharacteristic
168C2×D292C22×C411yescharacteristic
169C2×C22⋊D73—21no—
170C22×Dic73C2×D411yescharacteristic
171C22×Dic73#4111yescharacteristic
172C2×C4×D73—21no—
173C2×C4×D73—21no—
174C2×C4×D73—21no—
175C2×C4×D73—21no—
176C2×C4×D73—21no—
177C2×C4×D73—81no—
178C2×C4×D73—81no—
179#11924015—22no—
180#11924015—41no—
181D584—22no—
182D584—41no—
183C2×#10596831C22×C411yescharacteristic
184C2×#10596831C22×C411yescharacteristic
185C4⋊Dic73—21no—
186C4⋊Dic73—21no—
187C4⋊Dic73C22×C411yescharacteristic
188#11924007—21no—
189#11924005—21no—
190#11924005—21no—
191#11924005—21no—
192#11924005—21no—
193#11924005—21no—
194#11924005—21no—
195#11924005—21no—
196#11924005—21no—
197C4×Dic73—21no—
198C4×Dic73C22×C411yescharacteristic
199C4×Dic73C2×D411yescharacteristic
200C4×Dic73#4111yescharacteristic
201C4×Dic73C2411yescharacteristic
202C2×Q8×C73C22×C411yescharacteristic
203C22×C4×C73—21no—
204C22×C4×C73C22×C411yescharacteristic
205C2×C8×C73C22×C411yescharacteristic
206C2×C8×C73C22×C411yescharacteristic
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73C22×C411yescharacteristic
210C4⋊C4×C73C22×C411yescharacteristic
211C4⋊C4×C73C22×C411yescharacteristic
212C22⋊C4×C73—21no—
213C23×C73C4×D411yescharacteristic
214C22×D73—21no—
215C22×D73—21no—
216C22×D73—21no—
217C22×D73—42no—
218C22×D73—42no—
219C22×D73—81no—
220Q8×C73—21no—
221Q8×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—81no—
229C2×C4×C73C4×D411yescharacteristic
230C2×C4×C73C23×C411yescharacteristic
231C22⋊D73—42no—
232C2×Dic73—22no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73C4×D411yescharacteristic
237C2×Dic73C4×D411yescharacteristic
238C2×Dic73C22×D411yescharacteristic
239C2×Dic73C2×#4111yescharacteristic
240D292—21no—
241D292—21no—
242D292—21no—
243D292—21no—
244D292—21no—
245D292—42no—
246D292#13112yes—
247C4×D73—41no—
248C4×D73—41no—
249C4×D73—41no—
250C4×D73—41no—
251C4×D73—41no—
252C4×D73—41no—
253C4×D73—81no—
254C4×D73—81no—
255#10596833—22no—
256#10596833—22no—
257#10596833—21no—
258#10596833—21no—
259#10596833—21no—
260#10596833—21no—
261#10596833—41no—
262#10596833—41no—
263#10596833—41no—
264#10596833—41no—
265C8×C73—21no—
266C8×C73—21no—
267#10596831—21no—
268#10596831—21no—
269C22×C73—21no—
270C22×C73—21no—
271C22×C73#44312yes—
272C22×C73C2×C4×D411yescharacteristic
273C2×D73—22no—
274C2×D73—41no—
275C2×D73—41no—
276C2×D73—41no—
277C2×D73—41no—
278C2×D73—41no—
279C2×D73—41no—
280C2×D73—41no—
281C2×D73—41no—
282C2×D73—84no—
283C4×C73—21no—
284C4×C73—41no—
285C4×C73—41no—
286C4×C73—41no—
287C4×C73—41no—
288C4×C73C2×#13111yescharacteristic
289C4×C73#56811yescharacteristic
290Dic73—21no—
291Dic73—21no—
292Dic73—21no—
293Dic73—41no—
294Dic73#57611yescharacteristic
295Dic73#57711yescharacteristic
296#18014—731nomaximal, Sylow
297C2×C73—22no—
298C2×C73#293311yescharacteristic
299C2×C73#294611yescharacteristic
300C2×C73C2×#44311yescharacteristic
301D73—42no—
302D73—81no—
303D73—81no—
304#3105—731no—
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#529—731no—
321C2×C4×D4—731no—
322#484#6440811yescharacteristic
323#481—731no—
324#478—731no—
325#478—731no—
326#463—731no—
327#463—731no—
328#450—731no—
329#447—731no—
330#446—731no—
331C8×Q8—731no—
332C4×SD16—731no—
333C4×SD16—731no—
334C4×D8—731no—
335C4×D8—731no—
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#431—731no—
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×D4—731no—
372C23×C4—1461no—
373C2×SD16—731no—
374C2×SD16—731no—
375C2×D8—731no—
376C2×D8—731no—
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—1461no—
386#125—731no—
387#124—731no—
388#123—1461no—
389C22⋊Q8C2×#6440811yescharacteristic
390#121—731no—
391C4×Q8—731no—
392C4×D4—731no—
393C4×D4—731no—
394C4×D4—731no—
395C4×D4—731no—
396C4×D4—731no—
397C4×D4—1461no—
398C4×D4—1461no—
399C4×D4—1461no—
400#117—1461no—
401C2×C4⋊C4C2×#6440811yescharacteristic
402C2×C22⋊C4—1461no—
403C2×C42—731no—
404#107C2×#6440811yescharacteristic
405#107C2×#6440811yescharacteristic
406#106—731no—
407#106—731no—
408C4⋊C8—731no—
409C4⋊C8—731no—
410C4⋊C8—731no—
411C4⋊C8—731no—
412C4⋊C8—731no—
413C4⋊C8—731no—
414C4⋊C8—731no—
415C4⋊C8—731no—
416C4⋊C8—1461no—
417C4⋊C8—1461no—
418C4⋊C8—1461no—
419#103C2×#6440811yescharacteristic
420#103C2×#6440811yescharacteristic
421#102—731no—
422#102—731no—
423#102—731no—
424#102—731no—
425#102—731no—
426#102—731no—
427C22⋊C8—731no—
428C22⋊C8—731no—
429C22⋊C8—731no—
430C22⋊C8—1461no—
431C22⋊C8—1461no—
432C22⋊C8—1461no—
433C22⋊C8—1461no—
434C22⋊C8—1461no—
435C22⋊C8—1461no—
436C22⋊C8—1461no—
437C22⋊C8C2×#6440811yescharacteristic
438#97—1461no—
439#97—1461no—
440C4×C8—731no—
441C4×C8—731no—
442C4×C8—731no—
443C4×C8—731no—
444C4×C8—731no—
445C4×C8—731no—
446C4×C8—1461no—
447C4×C8—1461no—
448C24—1461no—
449C2×Q8C22×#6440811yescharacteristic
450C2×D4—2921no—
451C2×D4—731no—
452C2×D4—731no—
453C2×D4—731no—
454C2×D4—1462no—
455C2×D4—1461no—
456C2×D4—1461no—
457C22×C4—21no—
458C22×C4—5841no—
459C22×C4—5841no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4C22×#6440811yescharacteristic
468SD16—2921no—
469SD16—1462no—
470D8—2921no—
471D8—1462no—
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—731no—
484C2×C8—731no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1464no—
492C2×C8C22×#6440811yescharacteristic
493C2×C8C22×#6440811yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—731no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4C22×#6440811yescharacteristic
501C4⋊C4C22×#6440811yescharacteristic
502C4⋊C4C22×#6440811yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—1461no—
505C22⋊C4—1461no—
506C22⋊C4—1461no—
507C22⋊C4—1461no—
508C22⋊C4—1461no—
509C22⋊C4—1461no—
510C22⋊C4—1461no—
511C22⋊C4—1461no—
512C42—731no—
513C42—731no—
514C42—731no—
515C42—731no—
516C42—1461no—
517C23—2922no—
518C23—2922no—
519C23—5841no—
520C23—1461no—
521C23—1461no—
522C23—1461no—
523C23D4×#6440811yescharacteristic
524Q8—21no—
525Q8—21no—
526D4—2922no—
527D4—2922no—
528D4—732no—
529D4—1461no—
530D4—1461no—
531D4—1461no—
532D4—1461no—
533D4—1461no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—81no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—5841no—
548C2×C4—5841no—
549C2×C4—731no—
550C2×C4—731no—
551C2×C4—731no—
552C2×C4—731no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1462no—
557C2×C4D4×#6440811yescharacteristic
558C2×C4C23×#6440811yescharacteristic
559C8—21no—
560C8—21no—
561C8—2921no—
562C8—2921no—
563C8—2921no—
564C8—2921no—
565C8—1461no—
566C8—1461no—
567C8—1461no—
568C8—1461no—
569C8—1462no—
570C8—1462no—
571C8—1461no—
572C8—1461no—
573C22—21no—
574C22—21no—
575C22—2921no—
576C22—2921no—
577C22—2921no—
578C22—2921no—
579C22—2921no—
580C22—2921no—
581C22—2921no—
582C22—2921no—
583C22—5844no—
584C22—1462no—
585C22#1798963112yes—
586C22C2×D4×#6440811yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—2921no—
593C4—731no—
594C4—731no—
595C4—1461no—
596C4—1461no—
597C4—1461no—
598C4D73⋊#13111yescharacteristic
599C4#1798966611yescharacteristic
600C2—22no—
601C2—2922no—
602C2—5841no—
603C2—5841no—
604C2#2262666511yescharacteristic
605C2#2262668811yescharacteristic
606C2C2×#1798963111yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#484×C73C4
socleC22×C73C2×C4×D4
prime radical#17988723C4
cosocleC4×Dic73C24
hypercenter#484#64408
hypocenterC73#18014
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #22626882C2semidirect
2. #22626792C2semidirect
3. #22625565C2semidirect
4. #22625563C2semidirect
5. #22625558C2semidirect
6. #22625551C2semidirect
7. #22625551C2semidirect
8. #22625549C2semidirect
9. #22625515C2semidirect
10. #22625512C2semidirect
11. #22625506C2semidirect
12. #22625284C2semidirect
13. C2×#17989435C2semidirect
14. #22625136C2semidirect
16. #17989638C22semidirect
17. Q8×#10596833C22semidirect
20. #17989621C22semidirect
26. #17989615C22semidirect
36. #17989450C22semidirect
37. #17989450C22semidirect
38. #17989449C22semidirect
39. #17989449C22semidirect
40. Q8⋊#10596833C22semidirect
41. Q8⋊#10596833C22semidirect
46. #17989435C22semidirect
47. #17989435C22semidirect
48. #17989429C22semidirect
49. #17989429C22semidirect
50. C8×#10596833C22semidirect
51. C8×#10596833C22semidirect
319. C73#18014semidirect

Maximal subgroups

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

subgroupindex
#226268822
#226267922
#226255652
#226255632
#226255582
#226255512
#226255492
#226255152
#226255122
#226255062
#226252842
C2×#179894352
#226251362
#226199812
#1801473