Finite Group Databasebeta

← #33242811

Subgroups — #33242811

Order 18688.

622 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22622320C211yesmaximal, characteristic, complemented
2D8×C22⋊D73C211yesmaximal, characteristic
3#22612153C211yesmaximal, characteristic, complemented
4#22612117C211yesmaximal, characteristic, complemented
5#22612115C211yesmaximal, characteristic, complemented
6#22612076C211yesmaximal, characteristic, complemented
7#22612029C211yesmaximal, characteristic, complemented
8#22612024C211yesmaximal, characteristic, complemented
9#22611969C211yesmaximal, characteristic, complemented
10#22611858C211yesmaximal, characteristic, complemented
11SD32×Dic73C211yesmaximal, characteristic, complemented
12#22611748C211yesmaximal, characteristic, complemented
13C2×#17988388C211yesmaximal, characteristic, complemented
14#357×D73C211yesmaximal, characteristic, complemented
15C73×#2187C211yesmaximal, characteristic, complemented
16C2×D8×D73C2211yescharacteristic
17#17989295—21no—
18D4×C22⋊D73—21no—
19C2×#14556534C2211yescharacteristic, complemented
20C22×#11924003C2211yescharacteristic, complemented
21C2×C16×D73C2211yescharacteristic, complemented
22#17988991C2211yescharacteristic, complemented
23#17988962C2211yescharacteristic
24#17988921—21no—
25#17988920—21no—
26#17988880—21no—
27#17988825C2211yescharacteristic, complemented
28#17988823C2211yescharacteristic
29#17988820C2211yescharacteristic
30#17988780—21no—
31#17988721—21no—
32#17988719—21no—
33Q16×Dic73C2211yescharacteristic, complemented
34D8×Dic73C2211yescharacteristic
35C8×C22⋊D73C2211yescharacteristic
36C2×#14556525C2211yescharacteristic, complemented
37#107×D73C2211yescharacteristic, complemented
38#17988415C2211yescharacteristic, complemented
39#17988415C2211yescharacteristic, complemented
40#17988414C2211yescharacteristic, complemented
41#17988405C2211yescharacteristic, complemented
42#17988391C2211yescharacteristic, complemented
43#17988389C2211yescharacteristic, complemented
44#17988389C2211yescharacteristic, complemented
45#17988388C2211yescharacteristic
46#17988388C2211yescharacteristic
47#17988388C2211yescharacteristic, complemented
48#17988388C2211yescharacteristic, complemented
49#17988387C2211yescharacteristic, complemented
50#17988366C2211yescharacteristic, complemented
51C16×Dic73C2211yescharacteristic, complemented
52C22×D8×C73C2211yescharacteristic
53C2×SD32×C73C2211yescharacteristic
54C2×SD32×C73C2211yescharacteristic, complemented
55#467×C73C2211yescharacteristic, complemented
56#357×C73C2211yescharacteristic
57#357×C73C2211yescharacteristic, complemented
58C22⋊C16×C73C2211yescharacteristic, complemented
59C22×C22⋊D73—41no—
60C2×D4×D73—21no—
61C2×D584C2311yescharacteristic
62C2×#11924013—21no—
63C2×#11924013—21no—
64D8×D73—22no—
65D8×D73—41no—
66C2×C8×D73C2311yescharacteristic
67#14556590—21no—
68#14556587—21no—
69#14556581—41no—
70#14556571—41no—
71#14556568—41no—
72#14556567—21no—
73#14556565—21no—
74#14556563—41no—
75#14556562—21no—
76#14556560—41no—
77#14556559—41no—
78Q8×Dic73—21no—
79D4×Dic73—21no—
80C4×C22⋊D73D411yescharacteristic
81#14556551—41no—
82C2×C4⋊Dic73—21no—
83C4⋊C4×D73—21no—
84#14556534—41no—
85C2×#11924003—41no—
86C2×#11924003C2311yescharacteristic
87C2×#11924003C2311yescharacteristic
88C16×D73—41no—
89#14556525—21no—
90#14556525C2311yescharacteristic
91#14556525C2311yescharacteristic
92#14556525C2311yescharacteristic
93#14556524—21no—
94#14556524—21no—
95#14556520D411yescharacteristic
96#14556515—21no—
97#14556515—21no—
98#14556513—21no—
99#14556512—21no—
100#14556512—21no—
101C22⋊#10596831D411yescharacteristic
102#14556503D411yescharacteristic
103C8×Dic73C2311yescharacteristic
104C22×D4×C73—21no—
105C2×Q16×C73C2311yescharacteristic
106C2×D8×C73—22no—
107C2×D8×C73—21no—
108C2×D8×C73—41no—
109C2×D8×C73C2311yescharacteristic
110C2×D8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C22⋊Q8×C73—21no—
113SD32×C73—22no—
114SD32×C73—41no—
115C2×C16×C73C2311yescharacteristic
116C2×C16×C73C2311yescharacteristic
117#107×C73C2311yescharacteristic
118#103×C73—21no—
119C23×D73—41no—
120C2×Dic146—41no—
121C2×D292—21no—
122C2×C22⋊D73—41no—
123C2×C22⋊D73—41no—
124C2×C22⋊D73—41no—
125C2×C22⋊D73—41no—
126C2×C22⋊D73—41no—
127C2×C22⋊D73—84no—
128C2×C22⋊D73D811yescharacteristic
129D4×D73—42no—
130D4×D73—81no—
131C22×Dic73—41no—
132C22×Dic73—41no—
133C2×C4×D73—41no—
134C2×C4×D73C2×D411yescharacteristic
135D584—21no—
136D584—21no—
137#11924013—42no—
138#11924013—81no—
139C2×#10596831C2×D411yescharacteristic
140C2×#10596831C2×D411yescharacteristic
141C8×D73—21no—
142C8×D73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—41no—
147C4⋊Dic73—41no—
148#11924007—41no—
149#11924007—41no—
150#11924007—41no—
151#11924007D811yescharacteristic
152C22⋊Dic73—41no—
153C22⋊Dic73—41no—
154C22⋊Dic73D811yescharacteristic
155#11924005—41no—
156#11924005—41no—
157#11924005D811yescharacteristic
158C4×Dic73—41no—
159C4×Dic73C2×D411yescharacteristic
160#11924003—21no—
161#11924003—21no—
162C24×C73—41no—
163C2×Q8×C73—21no—
164C2×D4×C73—21no—
165C2×D4×C73—21no—
166C2×D4×C73—42no—
167C2×D4×C73—41no—
168C2×D4×C73—81no—
169C22×C4×C73C2×D411yescharacteristic
170Q16×C73—21no—
171Q16×C73—21no—
172D8×C73—21no—
173D8×C73—21no—
174D8×C73—21no—
175D8×C73—42no—
176D8×C73#4112yes—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73C2×D411yescharacteristic
180C2×C8×C73C2411yescharacteristic
181C4⋊C4×C73—21no—
182C4⋊C4×C73—41no—
183C22⋊C4×C73—41no—
184C16×C73—21no—
185C16×C73—21no—
186C23×C73—41no—
187C23×C73—41no—
188C23×C73—82no—
189C23×C73—82no—
190C23×C73—161no—
191C23×C73C2×D811yescharacteristic
192C22×D73—41no—
193C22×D73—82no—
194C22×D73—82no—
195C22×D73—161no—
196C22×D73C2×D811yescharacteristic
197Q8×C73—41no—
198Q8×C73—41no—
199D4×C73—22no—
200D4×C73—41no—
201D4×C73—41no—
202D4×C73—41no—
203D4×C73—82no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—41no—
207C2×C4×C73—41no—
208C2×C4×C73C2×D811yescharacteristic
209C2×C4×C73C22×D411yescharacteristic
210C22⋊D73—82no—
211C22⋊D73—82no—
212C22⋊D73—82no—
213C22⋊D73—82no—
214C22⋊D73—161no—
215C22⋊D73SD3214yescomplemented
216C2×Dic73—41no—
217C2×Dic73—41no—
218C2×Dic73—41no—
219C2×Dic73—82no—
220C2×Dic73—161no—
221C2×Dic73C2×D811yescharacteristic
222C2×Dic73C2×D811yescharacteristic
223C2×Dic73C2×D811yescharacteristic
224D292—41no—
225D292—41no—
226C4×D73—21no—
227C4×D73—21no—
228C4×D73—161no—
229Dic146—161no—
230C8×C73—21no—
231C8×C73C2×#4111yescharacteristic
232C8×C73H(5,𝔽2)11yescharacteristic
233#10596831—21no—
234#10596831—21no—
235C22×C73—21no—
236C22×C73—21no—
237C22×C73—42no—
238C22×C73—81no—
239C22×C73—81no—
240C22×C73—81no—
241C22×C73—81no—
242C22×C73—81no—
243C22×C73—164no—
244C22×C73C2×SD3212yes—
245C22×C73C22×D811yescharacteristic
246C2×D73—21no—
247C2×D73—21no—
248C2×D73—81no—
249C2×D73—81no—
250C2×D73—81no—
251C2×D73—164no—
252C2×D73C2×SD3212yes—
253C4×C73—21no—
254C4×C73—81no—
255C4×C73—81no—
256C4×C73#54611yescharacteristic
257C4×C73#57711yescharacteristic
258Dic73—21no—
259Dic73—21no—
260Dic73—81no—
261Dic73—81no—
262Dic73—81no—
263Dic73C2×SD3212yes—
264#33862—731nomaximal, Sylow
265C2×C73—22no—
266C2×C73—82no—
267C2×C73—161no—
268C2×C73#329411yescharacteristic
269C2×C73C22×SD3211yescharacteristic
270C2×C73#341511yescharacteristic
271D73—22no—
272D73—161no—
273#3338—731no—
274D4×D8—731no—
275#2240—731no—
276#2233—731no—
277#2233—731no—
278#2226—731no—
279#2217—731no—
280#2217—731no—
281#2207—731no—
282#2187—731no—
283#2187D7311yescharacteristic, complemented
284C4×SD32—731no—
285D4×C16—731no—
286C2×#357—731no—
287C2×#357—731no—
288C73#3386211yescharacteristic, complemented, Sylow
289C22×D8—731no—
290C22×D8C2×D7311yescharacteristic
291#554—1461no—
292D42—1461no—
293C2×SD32—731no—
294C2×SD32C2×D7311yescharacteristic
295C2×SD32C2×D7311yescharacteristic, complemented
296C22×C16—731no—
297C22×C16—731no—
298#500—731no—
299#493—731no—
300#484—1461no—
301#484—1461no—
302#477—1461no—
303#467—731no—
304#467C2×D7311yescharacteristic, complemented
305#466—731no—
306#466—731no—
307#459—1461no—
308#447—1461no—
309#447—1461no—
310C4×Q16—731no—
311C4×D8—731no—
312C8×D4—731no—
313C2×#107—731no—
314C2×#107—731no—
315#367—731no—
316#367—731no—
317#367—731no—
318C4⋊C16—731no—
319#358—731no—
320#357—731no—
321#357—731no—
322#357—731no—
323#357—731no—
324#357—731no—
325#357—731no—
326#357—731no—
327#357C2×D7311yescharacteristic
328#357C2×D7311yescharacteristic, complemented
329C22⋊C16—731no—
330C22⋊C16C2×D7311yescharacteristic, complemented
331C4×C16—731no—
332C22×D4—21no—
333C22×D4—2921no—
334C22×D4—1461no—
335C2×Q16C22×D7311yescharacteristic
336C2×D8—22no—
337C2×D8—21no—
338C2×D8—41no—
339C2×D8—2921no—
340C2×D8—731no—
341C2×D8—1461no—
342C2×D8—1461no—
343C2×D8—1462no—
344C2×D8C22×D7311yescharacteristic
345C2×D8C22×D7311yescharacteristic
346C22×C8—731no—
347C22×C8C22×D7311yescharacteristic
348C4⋊Q8—1461no—
349#127—1461no—
350#125—2921no—
351C22⋊Q8—21no—
352C22⋊Q8—2921no—
353C22⋊Q8—2921no—
354C22⋊Q8—1461no—
355#121—2921no—
356#121—1461no—
357#121—1461no—
358#120—2921no—
359#120—2921no—
360C4×Q8—1461no—
361C4×D4—2921no—
362C4×D4—731no—
363C4×D4—1461no—
364C2×C4⋊C4—1461no—
365C2×C4⋊C4—1461no—
366SD32—22no—
367SD32—41no—
368SD32—2921no—
369C2×C16—2921no—
370C2×C16—2921no—
371C2×C16—731no—
372C2×C16—731no—
373C2×C16C22×D7311yescharacteristic
374C2×C16C22×D7311yescharacteristic
375#107—731no—
376#107—731no—
377#107—731no—
378#107—1461no—
379#107—1461no—
380#107—1461no—
381#107C22×D7311yescharacteristic
382C4⋊C8—731no—
383#103—21no—
384#103—1461no—
385#103—1461no—
386#102—1461no—
387#102—1461no—
388#102—1461no—
389C22⋊C8—731no—
390C22⋊C8—731no—
391C4×C8—731no—
392C24—41no—
393C24—2921no—
394C2×Q8—21no—
395C2×Q8—2921no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—42no—
399C2×D4—41no—
400C2×D4—81no—
401C2×D4—2921no—
402C2×D4—2921no—
403C2×D4—2921no—
404C2×D4—2921no—
405C2×D4—2921no—
406C2×D4—2922no—
407C2×D4—5844no—
408C2×D4—5841no—
409C2×D4—731no—
410C2×D4—1461no—
411C22×C4—2921no—
412C22×C4—2921no—
413C22×C4—2921no—
414C22×C4—731no—
415C22×C4D4×D7311yescharacteristic
416Q16—21no—
417Q16—21no—
418D8—21no—
419D8—21no—
420D8—21no—
421D8—42no—
422D8—2922no—
423D8—5841no—
424D8—1461no—
425D8—1461no—
426D8#1192402712yes—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—731no—
430C2×C8—731no—
431C2×C8—1461no—
432C2×C8—1461no—
433C2×C8D4×D7311yescharacteristic
434C2×C8C23×D7311yescharacteristic
435C4⋊C4—21no—
436C4⋊C4—41no—
437C4⋊C4—2921no—
438C4⋊C4—2921no—
439C4⋊C4—2921no—
440C4⋊C4—2921no—
441C4⋊C4—2921no—
442C4⋊C4—731no—
443C4⋊C4—1461no—
444C4⋊C4—1461no—
445C4⋊C4—1461no—
446C22⋊C4—41no—
447C22⋊C4—2921no—
448C22⋊C4—2921no—
449C22⋊C4—2921no—
450C22⋊C4—2921no—
451C22⋊C4—731no—
452C22⋊C4—731no—
453C42—2921no—
454C42—731no—
455C16—21no—
456C16—21no—
457C16—1461no—
458C16—1461no—
459C23—41no—
460C23—41no—
461C23—82no—
462C23—82no—
463C23—161no—
464C23—2921no—
465C23—5842no—
466C23—5842no—
467C23—731no—
468C23—11681no—
469C23D8×D7311yescharacteristic
470Q8—41no—
471Q8—41no—
472Q8—11681no—
473D4—22no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—82no—
478D4—2921no—
479D4—2921no—
480D4—5842no—
481D4—5842no—
482D4—5842no—
483D4—5842no—
484D4—734no—
485D4—11681no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—2921no—
491C2×C4—2921no—
492C2×C4—2921no—
493C2×C4—5842no—
494C2×C4—731no—
495C2×C4—731no—
496C2×C4—731no—
497C2×C4—11681no—
498C2×C4—11681no—
499C2×C4—1461no—
500C2×C4—1461no—
501C2×C4D8×D7311yescharacteristic
502C2×C4C2×D4×D7311yescharacteristic
503C8—21no—
504C8—1461no—
505C8—1461no—
506C8C2×#1192402711yescharacteristic
507C8#1455663611yescharacteristic
508C22—21no—
509C22—21no—
510C22—42no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—164no—
517C22—5841no—
518C22—5841no—
519C22—5841no—
520C22—732no—
521C22—11684no—
522C22—1461no—
523C22—1461no—
524C22SD32×D7312yes—
525C22C2×D8×D7311yescharacteristic
526C4—21no—
527C4—81no—
528C4—81no—
529C4—5841no—
530C4—5841no—
531C4—5841no—
532C4—732no—
533C4—1461no—
534C4—1461no—
535C4#1798922411yescharacteristic
536C4#1798939711yescharacteristic
537C2—22no—
538C2—82no—
539C2—161no—
540C2—11681no—
541C2—1462no—
542C2#2262173711yescharacteristic
543C2C2×SD32×D7311yescharacteristic
544C2#2262307111yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2187C2
socleC22×C73C22×D8
prime radicalD8×C22⋊D73C2
hypercenter#2187D73
hypocenterC73#33862
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. #22622320C2semidirect
3. #22612153C2semidirect
4. #22612117C2semidirect
5. #22612115C2semidirect
6. #22612076C2semidirect
7. #22612029C2semidirect
8. #22612024C2semidirect
9. #22611969C2semidirect
10. #22611858C2semidirect
11. SD32×Dic73C2semidirect
12. #22611748C2semidirect
13. C2×#17988388C2semidirect
14. #357×D73C2semidirect
15. C73×#2187C2semidirect
19. C2×#14556534C22semidirect
20. C22×#11924003C22semidirect
21. C2×C16×D73C22semidirect
22. #17988991C22semidirect
27. #17988825C22semidirect
33. Q16×Dic73C22semidirect
36. C2×#14556525C22semidirect
37. #107×D73C22semidirect
38. #17988415C22semidirect
39. #17988415C22semidirect
40. #17988414C22semidirect
41. #17988405C22semidirect
42. #17988391C22semidirect
43. #17988389C22semidirect
44. #17988389C22semidirect
47. #17988388C22semidirect
48. #17988388C22semidirect
49. #17988387C22semidirect
50. #17988366C22semidirect
51. C16×Dic73C22semidirect
54. C2×SD32×C73C22semidirect
55. #467×C73C22semidirect
57. #357×C73C22semidirect
58. C22⋊C16×C73C22semidirect
215. C22⋊D73SD32semidirect
283. #2187D73semidirect
288. C73#33862semidirect
295. C2×SD32C2×D73semidirect
304. #467C2×D73semidirect
328. #357C2×D73semidirect
330. C22⋊C16C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226223202
D8×C22⋊D732
#226121532
#226121172
#226121152
#226120762
#226120292
#226120242
#226119692
#226118582
SD32×Dic732
#226117482
C2×#179883882
#357×D732
C73×#21872
#3386273