Finite Group Databasebeta

← #33242803

Subgroups — #33242803

Order 18688.

654 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22622276C211yesmaximal, characteristic, complemented
2D8×C22⋊D73C211yesmaximal, characteristic, complemented
3#22612153C211yesmaximal, characteristic, complemented
4#22612111C211yesmaximal, characteristic, complemented
5#22612110C211yesmaximal, characteristic, complemented
6#22612066C211yesmaximal, characteristic, complemented
7#22612017C211yesmaximal, characteristic, complemented
8#22612016C211yesmaximal, characteristic, complemented
9#22611965C211yesmaximal, characteristic, complemented
10#22611844C211yesmaximal, characteristic, complemented
11#22611843C211yesmaximal, characteristic, complemented
12#22611775C211yesmaximal, characteristic, complemented
13C16×C22⋊D73C211yesmaximal, characteristic, complemented
14#357×D73C211yesmaximal, characteristic, complemented
15C2×#357×C73C211yesmaximal, characteristic, complemented
16C2×D8×D73C2211yescharacteristic
17#17989238—21no—
18D4×C22⋊D73—21no—
19C2×D1168C2211yescharacteristic, complemented
20C2×#14556532C2211yescharacteristic
21C2×#14556532C2211yescharacteristic, complemented
22C2×C16×D73C2211yescharacteristic, complemented
23#17988991C2211yescharacteristic, complemented
24#17988962C2211yescharacteristic
25#17988902—21no—
26#17988900—21no—
27#17988866—21no—
28#17988824C2211yescharacteristic, complemented
29#17988823C2211yescharacteristic
30#17988823C2211yescharacteristic, complemented
31#17988820C2211yescharacteristic
32#17988780—21no—
33#17988721—21no—
34#17988719—21no—
35D8×Dic73C2211yescharacteristic
36#17988669C2211yescharacteristic, complemented
37C8×C22⋊D73C2211yescharacteristic
38#107×D73C2211yescharacteristic, complemented
39#17988413C2211yescharacteristic, complemented
40#17988412C2211yescharacteristic, complemented
41#17988412C2211yescharacteristic, complemented
42#17988405C2211yescharacteristic, complemented
43#17988389C2211yescharacteristic, complemented
44#17988389C2211yescharacteristic, complemented
45#17988388C2211yescharacteristic
46#17988388C2211yescharacteristic, complemented
47#17988387C2211yescharacteristic, complemented
48#17988387C2211yescharacteristic, complemented
49C22⋊#11924003C2211yescharacteristic, complemented
50#17988366C2211yescharacteristic, complemented
51C16×Dic73C2211yescharacteristic, complemented
52C22×D8×C73C2211yescharacteristic
53C22×C16×C73C2211yescharacteristic, complemented
54C2×#107×C73C2211yescharacteristic, complemented
55#357×C73C2211yescharacteristic
56#357×C73C2211yescharacteristic
57#357×C73C2211yescharacteristic, complemented
58#357×C73C2211yescharacteristic, complemented
59C2×#11924028—41no—
60C22×C22⋊D73—41no—
61C2×D4×D73—21no—
62C2×D584C2311yescharacteristic
63C2×D584C2311yescharacteristic
64C2×#11924013—21no—
65C2×#11924013—21no—
66D8×D73—22no—
67D8×D73—41no—
68C2×C8×D73C2311yescharacteristic
69#14556590—21no—
70#14556587—21no—
71#14556576—41no—
72#14556574—41no—
73#14556569—41no—
74#14556566—21no—
75#14556565—21no—
76#14556565—21no—
77#14556563—41no—
78#14556562—21no—
79#14556560—41no—
80#14556559—41no—
81D4×Dic73—21no—
82#14556553—21no—
83C4×C22⋊D73D411yescharacteristic
84C4⋊C4×D73—21no—
85D1168—41no—
86#14556532—22no—
87#14556532—41no—
88C2×#11924003C2311yescharacteristic
89C2×#11924003C2311yescharacteristic
90C16×D73—41no—
91#14556525C2311yescharacteristic
92#14556525C2311yescharacteristic
93#14556524—21no—
94#14556524—21no—
95#14556520D411yescharacteristic
96#14556513—21no—
97#14556513—21no—
98#14556512—21no—
99#14556512—21no—
100#14556511—21no—
101#14556511—21no—
102C22⋊#10596831D411yescharacteristic
103#14556503D411yescharacteristic
104C8×Dic73C2311yescharacteristic
105C22×D4×C73—21no—
106C2×D8×C73—22no—
107C2×D8×C73—21no—
108C2×D8×C73—41no—
109C2×D8×C73C2311yescharacteristic
110C2×D8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C2×C4⋊C4×C73—21no—
113C2×C16×C73—41no—
114C2×C16×C73C2311yescharacteristic
115C2×C16×C73C2311yescharacteristic
116#107×C73—21no—
117#107×C73C2311yescharacteristic
118#107×C73C2311yescharacteristic
119C23×D73—41no—
120#11924028—84no—
121C2×Dic146—41no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292—41no—
125C2×C22⋊D73—41no—
126C2×C22⋊D73—41no—
127C2×C22⋊D73—41no—
128C2×C22⋊D73—41no—
129C2×C22⋊D73—41no—
130C2×C22⋊D73—41no—
131C2×C22⋊D73—84no—
132C2×C22⋊D73D811yescharacteristic
133D4×D73—42no—
134D4×D73—81no—
135C22×Dic73—41no—
136C2×C4×D73—41no—
137C2×C4×D73—41no—
138C2×C4×D73C2×D411yescharacteristic
139D584—21no—
140D584—21no—
141D584—21no—
142D584—21no—
143#11924013—42no—
144#11924013—81no—
145C2×#10596831C2×D411yescharacteristic
146C2×#10596831C2×D411yescharacteristic
147C8×D73—21no—
148C8×D73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151#11924007—41no—
152#11924007—41no—
153#11924007D811yescharacteristic
154C22⋊Dic73—41no—
155C22⋊Dic73—41no—
156C22⋊Dic73D811yescharacteristic
157#11924005—41no—
158#11924005—41no—
159#11924005—41no—
160#11924005—41no—
161#11924005D811yescharacteristic
162C4×Dic73C2×D411yescharacteristic
163#11924003—21no—
164#11924003—21no—
165C24×C73—41no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—42no—
169C2×D4×C73—41no—
170C2×D4×C73—81no—
171C22×C4×C73—41no—
172C22×C4×C73C2×D411yescharacteristic
173D8×C73—21no—
174D8×C73—21no—
175D8×C73—21no—
176D8×C73—42no—
177D8×C73#4112yescomplemented
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73C2×D411yescharacteristic
181C2×C8×C73C2411yescharacteristic
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—41no—
185C16×C73—21no—
186C16×C73—21no—
187C23×C73—41no—
188C23×C73—41no—
189C23×C73—82no—
190C23×C73—82no—
191C23×C73—161no—
192C23×C73C2×D811yescharacteristic
193C22×D73—41no—
194C22×D73—41no—
195C22×D73—82no—
196C22×D73—82no—
197C22×D73—161no—
198C22×D73C2×D811yescharacteristic
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×C73—82no—
209C2×C4×C73C2×D811yescharacteristic
210C2×C4×C73C22×D411yescharacteristic
211C22⋊D73—82no—
212C22⋊D73—82no—
213C22⋊D73—82no—
214C22⋊D73—82no—
215C22⋊D73—82no—
216C22⋊D73—161no—
217C22⋊D73D1614yescomplemented
218C2×Dic73—41no—
219C2×Dic73—41no—
220C2×Dic73—82no—
221C2×Dic73C2×D811yescharacteristic
222C2×Dic73C2×D811yescharacteristic
223C2×Dic73C2×D811yescharacteristic
224D292—41no—
225D292—41no—
226D292—41no—
227D292—41no—
228D292—82no—
229C4×D73—21no—
230C4×D73—21no—
231C4×D73—82no—
232C4×D73—82no—
233Dic146—82no—
234C8×C73—21no—
235C8×C73C2×#4111yescharacteristic
236C8×C73H(5,𝔽2)11yescharacteristic
237#10596831—21no—
238#10596831—21no—
239C22×C73—21no—
240C22×C73—21no—
241C22×C73—42no—
242C22×C73—81no—
243C22×C73—81no—
244C22×C73—81no—
245C22×C73—81no—
246C22×C73—81no—
247C22×C73—164no—
248C22×C73C2×D1612yes—
249C22×C73C22×D811yescharacteristic
250C2×D73—21no—
251C2×D73—21no—
252C2×D73—81no—
253C2×D73—81no—
254C2×D73—81no—
255C2×D73—81no—
256C2×D73—81no—
257C2×D73—81no—
258C2×D73—164no—
259C2×D73C2×D1612yes—
260C4×C73—21no—
261C4×C73—81no—
262C4×C73—81no—
263C4×C73#54611yescharacteristic
264C4×C73#57711yescharacteristic
265Dic73—21no—
266Dic73—21no—
267Dic73—81no—
268Dic73—81no—
269Dic73C2×D1612yes—
270#33861—731nomaximal, Sylow
271C2×C73—22no—
272C2×C73—82no—
273C2×C73—161no—
274C2×C73#329411yescharacteristic
275C2×C73C22×D1611yescharacteristic
276C2×C73#341611yescharacteristic
277D73—22no—
278D73—161no—
279D73—161no—
280#3334—731no—
281D4×D8—731no—
282#2240—731no—
283#2232—731no—
284#2232—731no—
285#2224—731no—
286#2215—731no—
287#2215—731no—
288#2206—731no—
289#2184—731no—
290#2184—731no—
291C4×D16—731no—
292D4×C16—731no—
293C2×#357—731no—
294C2×#357D7311yescharacteristic, complemented
295C73#3386111yescharacteristic, complemented, Sylow
296C22×D8—731no—
297C22×D8C2×D7311yescharacteristic
298#547—1461no—
299D42—1461no—
300C2×D16—731no—
301C2×D16—731no—
302C2×D16—731no—
303C22×C16—731no—
304C22×C16C2×D7311yescharacteristic, complemented
305#500—731no—
306#493—731no—
307#480—1461no—
308#480—1461no—
309#474—1461no—
310#466—731no—
311#466—731no—
312#466—731no—
313#466—731no—
314#459—1461no—
315#447—1461no—
316#447—1461no—
317C4×D8—731no—
318C4×D8—731no—
319C8×D4—731no—
320C2×#107—731no—
321C2×#107C2×D7311yescharacteristic, complemented
322#366—731no—
323#366—731no—
324#366—731no—
325C4⋊C16—731no—
326#357—731no—
327#357—731no—
328#357—731no—
329#357—731no—
330#357—731no—
331#357—731no—
332#357C2×D7311yescharacteristic
333#357C2×D7311yescharacteristic
334#357C2×D7311yescharacteristic, complemented
335#357C2×D7311yescharacteristic, complemented
336C22⋊C16—731no—
337C22⋊C16—731no—
338C4×C16—731no—
339C2×#41—2921no—
340C22×D4—21no—
341C22×D4—2921no—
342C22×D4—1461no—
343C2×D8—22no—
344C2×D8—21no—
345C2×D8—41no—
346C2×D8—2921no—
347C2×D8—731no—
348C2×D8—731no—
349C2×D8—1461no—
350C2×D8—1461no—
351C2×D8—1462no—
352C2×D8C22×D7311yescharacteristic
353C2×D8C22×D7311yescharacteristic
354C22×C8—731no—
355C22×C8C22×D7311yescharacteristic
356C4⋊Q8—1461no—
357#127—1461no—
358#123—2921no—
359#123—2921no—
360C22⋊Q8—2921no—
361#121—2921no—
362#121—1461no—
363#121—1461no—
364#121—1461no—
365#121—1461no—
366#120—2921no—
367#120—2921no—
368C4×D4—731no—
369C4×D4—1461no—
370C4×D4—1461no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—1461no—
373D16—2921no—
374D16—2921no—
375D16—1462no—
376C2×C16—41no—
377C2×C16—2921no—
378C2×C16—731no—
379C2×C16—731no—
380C2×C16C22×D7311yescharacteristic
381C2×C16C22×D7311yescharacteristic
382#107—21no—
383#107—731no—
384#107—731no—
385#107—1461no—
386#107—1461no—
387#107C22×D7311yescharacteristic
388#107C22×D7311yescharacteristic
389C4⋊C8—731no—
390#102—1461no—
391#102—1461no—
392#102—1461no—
393#102—1461no—
394#102—1461no—
395#102—1461no—
396C22⋊C8—731no—
397C22⋊C8—731no—
398C4×C8—731no—
399C24—41no—
400C24—2921no—
401#41—5844no—
402C2×Q8—2921no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—42no—
406C2×D4—41no—
407C2×D4—81no—
408C2×D4—2921no—
409C2×D4—2921no—
410C2×D4—2921no—
411C2×D4—2921no—
412C2×D4—2921no—
413C2×D4—2921no—
414C2×D4—2921no—
415C2×D4—2922no—
416C2×D4—5844no—
417C2×D4—5841no—
418C2×D4—731no—
419C2×D4—1461no—
420C2×D4—1461no—
421C22×C4—41no—
422C22×C4—2921no—
423C22×C4—2921no—
424C22×C4—2921no—
425C22×C4—731no—
426C22×C4D4×D7311yescharacteristic
427D8—21no—
428D8—21no—
429D8—21no—
430D8—42no—
431D8—2922no—
432D8—5841no—
433D8—1461no—
434D8—1461no—
435D8—1461no—
436D8—1461no—
437D8#1192402812yescomplemented
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—731no—
441C2×C8—731no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8D4×D7311yescharacteristic
445C2×C8C23×D7311yescharacteristic
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—41no—
449C4⋊C4—2921no—
450C4⋊C4—2921no—
451C4⋊C4—731no—
452C4⋊C4—1461no—
453C4⋊C4—1461no—
454C22⋊C4—2921no—
455C22⋊C4—2921no—
456C22⋊C4—2921no—
457C22⋊C4—2921no—
458C22⋊C4—2921no—
459C22⋊C4—2921no—
460C22⋊C4—731no—
461C22⋊C4—731no—
462C42—731no—
463C16—21no—
464C16—21no—
465C16—1461no—
466C16—1461no—
467C23—41no—
468C23—41no—
469C23—82no—
470C23—82no—
471C23—161no—
472C23—2921no—
473C23—2921no—
474C23—5842no—
475C23—5842no—
476C23—731no—
477C23—11681no—
478C23D8×D7311yescharacteristic
479Q8—5842no—
480D4—22no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—82no—
485D4—2921no—
486D4—2921no—
487D4—2921no—
488D4—2921no—
489D4—5842no—
490D4—5842no—
491D4—5842no—
492D4—5842no—
493D4—5842no—
494D4—5842no—
495D4—734no—
496D4—11681no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—82no—
502C2×C4—2921no—
503C2×C4—2921no—
504C2×C4—5842no—
505C2×C4—5842no—
506C2×C4—5842no—
507C2×C4—731no—
508C2×C4—731no—
509C2×C4—731no—
510C2×C4—1461no—
511C2×C4—1461no—
512C2×C4D8×D7311yescharacteristic
513C2×C4C2×D4×D7311yescharacteristic
514C8—21no—
515C8—1461no—
516C8—1461no—
517C8C2×#1192402811yescharacteristic
518C8#1455663611yescharacteristic
519C22—21no—
520C22—21no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—164no—
528C22—5841no—
529C22—5841no—
530C22—5841no—
531C22—5841no—
532C22—5841no—
533C22—5841no—
534C22—732no—
535C22—11684no—
536C22—1461no—
537C22—1461no—
538C22D16×D7312yes—
539C22C2×D8×D7311yescharacteristic
540C4—21no—
541C4—81no—
542C4—81no—
543C4—5841no—
544C4—5841no—
545C4—732no—
546C4—1461no—
547C4—1461no—
548C4#1798922711yescharacteristic
549C4#1798939311yescharacteristic
550C2—22no—
551C2—82no—
552C2—161no—
553C2—11681no—
554C2—11681no—
555C2—1462no—
556C2#2262174011yescharacteristic
557C2C2×D16×D7311yescharacteristic
558C2#2262307311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC2×#357×C73C2
socleC22×C73C22×D8
hypercenterC2×#357D73
hypocenterC73#33861
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
1. #22622276C2semidirect
2. D8×C22⋊D73C2semidirect
3. #22612153C2semidirect
4. #22612111C2semidirect
5. #22612110C2semidirect
6. #22612066C2semidirect
7. #22612017C2semidirect
8. #22612016C2semidirect
9. #22611965C2semidirect
10. #22611844C2semidirect
11. #22611843C2semidirect
12. #22611775C2semidirect
13. C16×C22⋊D73C2semidirect
14. #357×D73C2semidirect
15. C2×#357×C73C2semidirect
19. C2×D1168C22semidirect
21. C2×#14556532C22semidirect
22. C2×C16×D73C22semidirect
23. #17988991C22semidirect
28. #17988824C22semidirect
30. #17988823C22semidirect
36. #17988669C22semidirect
38. #107×D73C22semidirect
39. #17988413C22semidirect
40. #17988412C22semidirect
41. #17988412C22semidirect
42. #17988405C22semidirect
43. #17988389C22semidirect
44. #17988389C22semidirect
46. #17988388C22semidirect
47. #17988387C22semidirect
48. #17988387C22semidirect
49. C22⋊#11924003C22semidirect
50. #17988366C22semidirect
51. C16×Dic73C22semidirect
53. C22×C16×C73C22semidirect
54. C2×#107×C73C22semidirect
57. #357×C73C22semidirect
58. #357×C73C22semidirect
177. D8×C73#41semidirect
217. C22⋊D73D16semidirect
294. C2×#357D73semidirect
295. C73#33861semidirect
304. C22×C16C2×D73semidirect
321. C2×#107C2×D73semidirect
334. #357C2×D73semidirect
335. #357C2×D73semidirect
437. D8#11924028semidirect

Maximal subgroups

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

subgroupindex
#226222762
D8×C22⋊D732
#226121532
#226121112
#226121102
#226120662
#226120172
#226120162
#226119652
#226118442
#226118432
#226117752
C16×C22⋊D732
#357×D732
C2×#357×C732
#3386173