Finite Group Databasebeta

← #33242816

Subgroups — #33242816

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#22622321C211yesmaximal, characteristic, complemented
2#22621605C211yesmaximal, characteristic
3#22612153C211yesmaximal, characteristic, complemented
4#22612118C211yesmaximal, characteristic, complemented
5#22612077C211yesmaximal, characteristic, complemented
6#22612028C211yesmaximal, characteristic, complemented
7#22612025C211yesmaximal, characteristic, complemented
8#22611970C211yesmaximal, characteristic, complemented
9#22611862C211yesmaximal, characteristic, complemented
10#22611861C211yesmaximal, characteristic, complemented
11#22611782C211yesmaximal, characteristic, complemented
12#22611748C211yesmaximal, characteristic, complemented
13C2×#17988387C211yesmaximal, characteristic, complemented
14#357×D73C211yesmaximal, characteristic, complemented
15C73×#2233C211yesmaximal, characteristic, complemented
16C22×D584C2211yescharacteristic
17C2×D8×D73C2211yescharacteristic
18#17989292—21no—
19#17989221—21no—
20C2×#14556536C2211yescharacteristic
21C2×#14556536C2211yescharacteristic, complemented
22C2×#14556535C2211yescharacteristic, complemented
23C22×#11924003C2211yescharacteristic, complemented
24C2×C16×D73C2211yescharacteristic, complemented
25#17988991C2211yescharacteristic, complemented
26#17988962C2211yescharacteristic
27#17988924—21no—
28#17988922—21no—
29#17988881—21no—
30#17988829C2211yescharacteristic, complemented
31#17988828C2211yescharacteristic, complemented
32#17988824C2211yescharacteristic
33#17988781—21no—
34#17988722—21no—
35#17988720—21no—
36#17988681C2211yescharacteristic, complemented
37#17988669C2211yescharacteristic
38C8×C22⋊D73C2211yescharacteristic
39#107×D73C2211yescharacteristic, complemented
40#17988414C2211yescharacteristic, complemented
41#17988405C2211yescharacteristic, complemented
42#17988390C2211yescharacteristic, complemented
43#17988389C2211yescharacteristic
44#17988389C2211yescharacteristic, complemented
45#17988388C2211yescharacteristic, complemented
46#17988387C2211yescharacteristic
47#17988387C2211yescharacteristic
48#17988387C2211yescharacteristic, complemented
49#17988387C2211yescharacteristic, complemented
50#17988366C2211yescharacteristic, complemented
51C16×Dic73C2211yescharacteristic, complemented
52#466×C73C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic, complemented
54#367×C73C2211yescharacteristic, complemented
55#367×C73C2211yescharacteristic, complemented
56#357×C73C2211yescharacteristic, complemented
57#357×C73C2211yescharacteristic, complemented
58C22⋊C16×C73C2211yescharacteristic, complemented
59C22×D292—21no—
60C2×D4×D73—21no—
61C2×D4×D73—41no—
62C2×Dic292C2311yescharacteristic
63C2×D584—22no—
64C2×D584—21no—
65C2×D584—41no—
66C2×D584C2311yescharacteristic
67C2×D584C2311yescharacteristic
68C2×#11924013—21no—
69C2×#11924013—21no—
70D8×D73—22no—
71D8×D73—41no—
72C2×C8×D73C2311yescharacteristic
73#14556590—21no—
74#14556587—21no—
75#14556581—41no—
76#14556571—21no—
77#14556570—21no—
78#14556570—41no—
79#14556569—41no—
80#14556566—21no—
81#14556564—41no—
82#14556561—41no—
83#14556559—41no—
84#14556556—21no—
85#14556553—21no—
86C4×C22⋊D73—41no—
87C4×C22⋊D73D411yescharacteristic
88C4⋊C4×D73—21no—
89#14556536—22no—
90#14556536—41no—
91#14556535—41no—
92C2×#11924003—41no—
93C2×#11924003C2311yescharacteristic
94C2×#11924003C2311yescharacteristic
95C16×D73—41no—
96#14556525C2311yescharacteristic
97#14556524—21no—
98#14556524—21no—
99#14556520D411yescharacteristic
100#14556516—21no—
101#14556514—21no—
102#14556514—21no—
103#14556511—21no—
104#14556511—21no—
105C22⋊#10596831D411yescharacteristic
106#14556503D411yescharacteristic
107C8×Dic73C2311yescharacteristic
108C2×D8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110#121×C73—21no—
111C2×C4⋊C4×C73—21no—
112C2×C16×C73C2311yescharacteristic
113C2×C16×C73C2311yescharacteristic
114#107×C73—21no—
115#107×C73C2311yescharacteristic
116#107×C73C2311yescharacteristic
117#107×C73C2311yescharacteristic
118#102×C73—21no—
119C23×D73—41no—
120C23×D73—41no—
121C2×Dic146—21no—
122C2×Dic146—41no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292—42no—
126C2×D292—41no—
127C2×D292—41no—
128C2×D292—81no—
129C2×C22⋊D73—41no—
130C2×C22⋊D73—41no—
131C2×C22⋊D73—41no—
132C2×C22⋊D73D811yescharacteristic
133D4×D73—42no—
134D4×D73—81no—
135D4×D73—84no—
136C2×C4×D73—41no—
137C2×C4×D73—41no—
138C2×C4×D73C2×D411yescharacteristic
139Dic292—21no—
140Dic292—21no—
141D584—21no—
142D584—21no—
143D584—21no—
144D584—42no—
145D584#4112yes—
146#11924013—42no—
147#11924013—81no—
148C2×#10596831C2×D411yescharacteristic
149C2×#10596831C2×D411yescharacteristic
150C8×D73—21no—
151C8×D73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—41no—
154#11924007—41no—
155#11924007—41no—
156#11924007—41no—
157#11924007—41no—
158#11924007D811yescharacteristic
159C22⋊Dic73—41no—
160C22⋊Dic73D811yescharacteristic
161#11924005—41no—
162#11924005—41no—
163#11924005—41no—
164#11924005D811yescharacteristic
165C4×Dic73—41no—
166C4×Dic73C2×D411yescharacteristic
167#11924003—21no—
168#11924003—21no—
169C2×D4×C73—21no—
170C2×D4×C73—41no—
171C22×C4×C73—41no—
172C22×C4×C73C2×D411yescharacteristic
173D8×C73—21no—
174D8×C73—21no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73C2×D411yescharacteristic
178C2×C8×C73C2411yescharacteristic
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—41no—
183C22⋊C4×C73—41no—
184C16×C73—21no—
185C16×C73—21no—
186C23×C73—41no—
187C23×C73C2×D811yescharacteristic
188C22×D73—41no—
189C22×D73—41no—
190C22×D73—82no—
191C22×D73—82no—
192C22×D73—82no—
193C22×D73—82no—
194C22×D73—161no—
195C22×D73—161no—
196C22×D73C2×D811yescharacteristic
197D4×C73—41no—
198D4×C73—41no—
199D4×C73—82no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—41no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—161no—
206C2×C4×C73C2×D811yescharacteristic
207C2×C4×C73C22×D411yescharacteristic
208C22⋊D73—82no—
209C22⋊D73—82no—
210C22⋊D73—161no—
211C22⋊D73SD3214yescomplemented
212C2×Dic73—41no—
213C2×Dic73—41no—
214C2×Dic73C2×D811yescharacteristic
215C2×Dic73C2×D811yescharacteristic
216C2×Dic73C2×D811yescharacteristic
217D292—22no—
218D292—41no—
219D292—41no—
220D292—41no—
221D292—82no—
222D292—82no—
223C4×D73—21no—
224C4×D73—21no—
225C4×D73—82no—
226C4×D73—161no—
227Dic146—41no—
228Dic146—41no—
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—81no—
238C22×C73—81no—
239C22×C73—81no—
240C22×C73C2×SD3212yes—
241C22×C73C22×D811yescharacteristic
242C2×D73—21no—
243C2×D73—21no—
244C2×D73—42no—
245C2×D73—81no—
246C2×D73—81no—
247C2×D73—81no—
248C2×D73—81no—
249C2×D73—81no—
250C2×D73—164no—
251C2×D73—164no—
252C2×D73C2×SD3212yes—
253C4×C73—21no—
254C4×C73—81no—
255C4×C73—81no—
256C4×C73—81no—
257C4×C73#54611yescharacteristic
258C4×C73#57711yescharacteristic
259Dic73—21no—
260Dic73—21no—
261Dic73—81no—
262Dic73—81no—
263Dic73C2×SD3212yes—
264#33862—731nomaximal, Sylow
265C2×C73—22no—
266C2×C73—161no—
267C2×C73#329411yescharacteristic
268C2×C73C22×SD3211yescharacteristic
269C2×C73#341511yescharacteristic
270D73—22no—
271D73—82no—
272D73—161no—
273#3338—731no—
274D4×D8—731no—
275#2240—731no—
276#2233—731no—
277#2233D7311yescharacteristic, complemented
278#2226—731no—
279#2217—731no—
280#2217—731no—
281#2207—731no—
282#2187—731no—
283#2187—731no—
284C4×SD32—731no—
285D4×C16—731no—
286C2×#357—731no—
287C2×#357—731no—
288C73#3386211yescharacteristic, complemented, Sylow
289C22×D8—731no—
290C22×D8—731no—
291#554—1461no—
292D42—1461no—
293C2×SD32—731no—
294C2×SD32—731no—
295C2×SD32—731no—
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#467—731no—
305#466—731no—
306#466C2×D7311yescharacteristic
307#459—1461no—
308#447—1461no—
309#447—1461no—
310C4×Q16—731no—
311C4×D8—731no—
312C8×D4—731no—
313C2×#107—731no—
314C2×#107C2×D7311yescharacteristic, complemented
315#367—731no—
316#367C2×D7311yescharacteristic, complemented
317#367C2×D7311yescharacteristic, complemented
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, complemented
328#357C2×D7311yescharacteristic, complemented
329C22⋊C16—731no—
330C22⋊C16C2×D7311yescharacteristic, complemented
331C4×C16—731no—
332C22×D4—2921no—
333C22×D4—1461no—
334C22×D4—1461no—
335C2×Q16—731no—
336C2×D8—2921no—
337C2×D8—2921no—
338C2×D8—731no—
339C2×D8—731no—
340C2×D8—1462no—
341C2×D8—1461no—
342C2×D8—1461no—
343C2×D8—1462no—
344C2×D8—1461no—
345C2×D8C22×D7311yescharacteristic
346C22×C8—731no—
347C22×C8C22×D7311yescharacteristic
348C4⋊Q8—1461no—
349#127—1461no—
350#125—2921no—
351C22⋊Q8—2921no—
352C22⋊Q8—2921no—
353C22⋊Q8—1461no—
354C22⋊Q8—1461no—
355#121—21no—
356#121—2921no—
357#121—1461no—
358#120—2921no—
359#120—2921no—
360C4×Q8—1461no—
361C4×D4—2921no—
362C4×D4—731no—
363C4×D4—1461no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—1461no—
366SD32—2921no—
367SD32—2921no—
368SD32—1462no—
369C2×C16—2921no—
370C2×C16—2921no—
371C2×C16—731no—
372C2×C16—731no—
373C2×C16C22×D7311yescharacteristic
374C2×C16C22×D7311yescharacteristic
375#107—21no—
376#107—731no—
377#107—1461no—
378#107—1461no—
379#107C22×D7311yescharacteristic
380#107C22×D7311yescharacteristic
381#107C22×D7311yescharacteristic
382C4⋊C8—731no—
383#103—1461no—
384#103—1461no—
385#103—1461no—
386#102—21no—
387#102—1461no—
388#102—1461no—
389C22⋊C8—731no—
390C22⋊C8—731no—
391C4×C8—731no—
392C24—2921no—
393C24—2921no—
394C2×Q8—2921no—
395C2×Q8—1461no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—2922no—
399C2×D4—2921no—
400C2×D4—2921no—
401C2×D4—2921no—
402C2×D4—2921no—
403C2×D4—2922no—
404C2×D4—2921no—
405C2×D4—5841no—
406C2×D4—5844no—
407C2×D4—5841no—
408C2×D4—731no—
409C2×D4—1461no—
410C2×D4—1461no—
411C22×C4—41no—
412C22×C4—2921no—
413C22×C4—2921no—
414C22×C4—731no—
415C22×C4D4×D7311yescharacteristic
416Q16—1461no—
417Q16—1461no—
418D8—21no—
419D8—21no—
420D8—2922no—
421D8—2922no—
422D8—5841no—
423D8—732no—
424D8—1461no—
425D8—1461no—
426D8—1461no—
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—21no—
437C4⋊C4—21no—
438C4⋊C4—41no—
439C4⋊C4—2921no—
440C4⋊C4—2921no—
441C4⋊C4—2921no—
442C4⋊C4—2921no—
443C4⋊C4—2921no—
444C4⋊C4—731no—
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—2921no—
461C23—2921no—
462C23—5842no—
463C23—5842no—
464C23—5842no—
465C23—5842no—
466C23—731no—
467C23—11681no—
468C23—11681no—
469C23D8×D7311yescharacteristic
470Q8—2921no—
471Q8—2921no—
472Q8—11681no—
473D4—41no—
474D4—41no—
475D4—82no—
476D4—2921no—
477D4—2921no—
478D4—2921no—
479D4—5842no—
480D4—5842no—
481D4—5842no—
482D4—5842no—
483D4—734no—
484D4—11681no—
485D4—1462no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—161no—
492C2×C4—2921no—
493C2×C4—2921no—
494C2×C4—5842no—
495C2×C4—731no—
496C2×C4—731no—
497C2×C4—731no—
498C2×C4—11681no—
499C2×C4—1461no—
500C2×C4—1461no—
501C2×C4D8×D7311yescharacteristic
502C2×C4C2×D4×D7311yescharacteristic
503C8—21no—
504C8—1461no—
505C8—1461no—
506C8#41×D7311yescharacteristic
507C8#1455663711yescharacteristic
508C22—21no—
509C22—21no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—2922no—
514C22—5841no—
515C22—5841no—
516C22—5841no—
517C22—5841no—
518C22—5841no—
519C22—732no—
520C22—11684no—
521C22—11684no—
522C22—1461no—
523C22—1461no—
524C22SD32×D7312yes—
525C22C2×D8×D7311yescharacteristic
526C4—21no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—5841no—
531C4—5841no—
532C4—732no—
533C4—1461no—
534C4—1461no—
535C4#1798922911yescharacteristic
536C4#1798939611yescharacteristic
537C2—22no—
538C2—161no—
539C2—5842no—
540C2—11681no—
541C2—1462no—
542C2#2262174211yescharacteristic
543C2C2×SD32×D7311yescharacteristic
544C2#2262307111yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2233C2
socleC22×C73C22×D8
prime radical#22621605C2
hypercenter#2233D73
hypocenterC73#33862
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2233

Complements

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

NH ≅ G/Ntype
1. #22622321C2semidirect
3. #22612153C2semidirect
4. #22612118C2semidirect
5. #22612077C2semidirect
6. #22612028C2semidirect
7. #22612025C2semidirect
8. #22611970C2semidirect
9. #22611862C2semidirect
10. #22611861C2semidirect
11. #22611782C2semidirect
12. #22611748C2semidirect
13. C2×#17988387C2semidirect
14. #357×D73C2semidirect
15. C73×#2233C2semidirect
21. C2×#14556536C22semidirect
22. C2×#14556535C22semidirect
23. C22×#11924003C22semidirect
24. C2×C16×D73C22semidirect
25. #17988991C22semidirect
30. #17988829C22semidirect
31. #17988828C22semidirect
36. #17988681C22semidirect
39. #107×D73C22semidirect
40. #17988414C22semidirect
41. #17988405C22semidirect
42. #17988390C22semidirect
44. #17988389C22semidirect
45. #17988388C22semidirect
48. #17988387C22semidirect
49. #17988387C22semidirect
50. #17988366C22semidirect
51. C16×Dic73C22semidirect
53. C2×#107×C73C22semidirect
54. #367×C73C22semidirect
55. #367×C73C22semidirect
56. #357×C73C22semidirect
57. #357×C73C22semidirect
58. C22⋊C16×C73C22semidirect
211. C22⋊D73SD32semidirect
277. #2233D73semidirect
288. C73#33862semidirect
314. C2×#107C2×D73semidirect
316. #367C2×D73semidirect
317. #367C2×D73semidirect
327. #357C2×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
#226223212
#226216052
#226121532
#226121182
#226120772
#226120282
#226120252
#226119702
#226118622
#226118612
#226117822
#226117482
C2×#179883872
#357×D732
C73×#22332
#3386273