Finite Group Databasebeta

← #33242801

Subgroups — #33242801

Order 18688.

654 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D584C211yesmaximal, characteristic, complemented
2#22612156C211yesmaximal, characteristic, complemented
3#22612111C212yesmaximal, complemented
4#22612065C211yesmaximal, characteristic, complemented
5#22612015C212yesmaximal, complemented
6#22611966C211yesmaximal, characteristic, complemented
7#22611842C212yesmaximal, complemented
8C4×#14556532C211yesmaximal, characteristic, complemented
9D4×#11924003C211yesmaximal, characteristic, complemented
10C2×#17988387C212yesmaximal, complemented
11C73×#3334C211yesmaximal, characteristic, complemented
12C22×D584C2212yes—
13D4×D292—21no—
14C2×#14556532C2211yescharacteristic
15C2×#14556532C2212yescomplemented
16C22×#11924003C2212yescomplemented
17#17988964C2211yescharacteristic
18#17988823C2212yes—
19#17988783—21no—
20#17988723—22no—
21C4×D584C2211yescharacteristic
22#17988412C2212yescomplemented
23#17988412C2211yescharacteristic, complemented
24C4⋊#11924003C2211yescharacteristic, complemented
25#17988388C2211yescharacteristic, complemented
26#17988387C2211yescharacteristic
27#17988387C2212yes—
28#17988387C2212yescomplemented
29#17988387C2212yescomplemented
30#17988387C2212yescomplemented
31C22⋊#11924003C2212yescomplemented
32C4×#11924003C2211yescharacteristic, complemented
33#547×C73—21no—
34#500×C73C2211yescharacteristic, complemented
35#480×C73—22no—
36#474×C73—21no—
37#466×C73C2212yescomplemented
38C4×D8×C73C2211yescharacteristic, complemented
39C8×D4×C73C2211yescharacteristic
40C2×#107×C73C2212yescomplemented
41C22×D292—22no—
42C2×D4×D73—41no—
43C2×D584—24no—
44C2×D584—22no—
45C2×D584—21no—
46C2×D584—42no—
47C2×D584C2312yes—
48C2×D584C2311yescharacteristic
49#14556588—21no—
50#14556566—41no—
51#14556565—22no—
52#14556561—42no—
53C4×D292—21no—
54#14556532—22no—
55#14556532—42no—
56C2×#11924003—42no—
57C2×#11924003C2311yescharacteristic
58C2×#11924003C2311yescharacteristic
59C2×#11924003C2312yes—
60#14556525C2311yescharacteristic
61#14556513—21no—
62#14556513—22no—
63C2×#41×C73—41no—
64C2×D8×C73C2311yescharacteristic
65C22×C8×C73C2312yes—
66C4⋊Q8×C73—21no—
67#123×C73—42no—
68C22⋊Q8×C73—41no—
69#121×C73—22no—
70C4×D4×C73—21no—
71C4×D4×C73D411yescharacteristic
72C2×C4⋊C4×C73—22no—
73#107×C73—22no—
74#107×C73—21no—
75#107×C73C2311yescharacteristic
76#107×C73C2312yes—
77C4⋊C8×C73D411yescharacteristic
78#102×C73—21no—
79#102×C73—22no—
80C22⋊C8×C73D412yes—
81C4×C8×C73C2311yescharacteristic
82C23×D73—42no—
83C2×D292—21no—
84C2×D292—22no—
85C2×D292—44no—
86C2×D292—42no—
87C2×D292—41no—
88C2×D292—41no—
89C2×D292—82no—
90C2×C22⋊D73—42no—
91D4×D73—84no—
92C2×C4×D73—41no—
93D584—22no—
94D584—21no—
95D584—22no—
96D584—44no—
97D584—81no—
98D584#4112yescomplemented
99C4⋊Dic73—21no—
100#11924005—41no—
101#11924005—42no—
102#11924003—21no—
103#11924003—21no—
104#11924003—22no—
105#41×C73—84no—
106C2×Q8×C73—41no—
107C2×D4×C73—21no—
108C2×D4×C73—42no—
109C2×D4×C73D811yescharacteristic
110C22×C4×C73—42no—
111C22×C4×C73—41no—
112C22×C4×C73C2×D412yes—
113D8×C73—21no—
114D8×C73—21no—
115C2×C8×C73—22no—
116C2×C8×C73—22no—
117C2×C8×C73C2×D411yescharacteristic
118C2×C8×C73C2×D412yes—
119C2×C8×C73C2411yescharacteristic
120C4⋊C4×C73—21no—
121C4⋊C4×C73—22no—
122C4⋊C4×C73—42no—
123C4⋊C4×C73—41no—
124C4⋊C4×C73D811yescharacteristic
125C22⋊C4×C73—41no—
126C22⋊C4×C73—42no—
127C22⋊C4×C73D812yes—
128C42×C73C2×D411yescharacteristic
129C23×C73—41no—
130C23×C73C2×D812yes—
131C22×D73—41no—
132C22×D73—42no—
133C22×D73—84no—
134C22×D73—84no—
135C22×D73—162no—
136Q8×C73—82no—
137D4×C73—41no—
138D4×C73—41no—
139D4×C73—84no—
140D4×C73D1614yescomplemented
141C2×C4×C73—22no—
142C2×C4×C73—22no—
143C2×C4×C73—41no—
144C2×C4×C73—42no—
145C2×C4×C73—84no—
146C2×C4×C73—82no—
147C2×C4×C73C2×D811yescharacteristic
148C2×C4×C73C2×D812yes—
149C2×C4×C73C2×D811yescharacteristic
150C2×C4×C73C22×D411yescharacteristic
151C22⋊D73—84no—
152C2×Dic73—41no—
153D292—22no—
154D292—42no—
155D292—41no—
156D292—42no—
157D292—84no—
158D292—82no—
159D292—161no—
160C4×D73—82no—
161C8×C73—22no—
162C8×C73—21no—
163C8×C73C2×#4111yescharacteristic
164C8×C73H(5,𝔽2)11yescharacteristic
165C22×C73—22no—
166C22×C73—22no—
167C22×C73—81no—
168C22×C73—81no—
169C22×C73—81no—
170C22×C73C2×D1614yes—
171C22×C73C22×D811yescharacteristic
172C2×D73—42no—
173C2×D73—81no—
174C2×D73—81no—
175C2×D73—82no—
176C2×D73—82no—
177C2×D73—82no—
178C2×D73—168no—
179C4×C73—21no—
180C4×C73—22no—
181C4×C73—81no—
182C4×C73—82no—
183C4×C73C2×D1612yes—
184C4×C73#54611yescharacteristic
185C4×C73#57711yescharacteristic
186Dic73—81no—
187#33861—731nomaximal, Sylow
188C2×C73—24no—
189C2×C73—161no—
190C2×C73#329411yescharacteristic
191C2×C73C22×D1611yescharacteristic
192C2×C73#341611yescharacteristic
193D73—82no—
194D73—162no—
195#3334D7311yescharacteristic, complemented
196D4×D8—731no—
197#2240—731no—
198#2232—732no—
199#2224—731no—
200#2215—732no—
201#2206—731no—
202#2184—732no—
203C4×D16—731no—
204D4×C16—731no—
205C2×#357—732no—
206C73#3386111yescharacteristic, complemented, Sylow
207C22×D8—732no—
208#547—21no—
209D42—1461no—
210C2×D16—732no—
211C2×D16—731no—
212C22×C16—732no—
213#500C2×D7311yescharacteristic, complemented
214#493—731no—
215#480—22no—
216#474—21no—
217#466—732no—
218#466C2×D7312yescomplemented
219#459—1461no—
220#447—1462no—
221C4×D8—731no—
222C4×D8C2×D7311yescharacteristic, complemented
223C8×D4C2×D7311yescharacteristic
224C2×#107C2×D7312yescomplemented
225#366—731no—
226#366—732no—
227C4⋊C16—731no—
228#357—731no—
229#357—731no—
230#357—732no—
231#357—732no—
232#357—732no—
233#357—732no—
234C22⋊C16—732no—
235C4×C16—731no—
236C2×#41—41no—
237C22×D4—2921no—
238C22×D4—1462no—
239C2×D8—2922no—
240C2×D8—731no—
241C2×D8—732no—
242C2×D8—1464no—
243C2×D8—1462no—
244C2×D8—1461no—
245C2×D8C22×D7311yescharacteristic
246C22×C8C22×D7312yes—
247C4⋊Q8—21no—
248#127—1461no—
249#123—42no—
250C22⋊Q8—41no—
251#121—22no—
252#121—2921no—
253#121—1462no—
254#120—2922no—
255C4×D4—21no—
256C4×D4—1461no—
257C4×D4C22⋊D7311yescharacteristic
258C2×C4⋊C4—22no—
259D16—2922no—
260D16—1462no—
261C2×C16—2922no—
262C2×C16—731no—
263C2×C16—731no—
264C2×C16—732no—
265#107—22no—
266#107—21no—
267#107—731no—
268#107C22×D7311yescharacteristic
269#107C22×D7312yes—
270C4⋊C8C22⋊D7311yescharacteristic
271#102—21no—
272#102—22no—
273#102—1461no—
274#102—1462no—
275C22⋊C8C22⋊D7312yes—
276C4×C8C22×D7311yescharacteristic
277C24—2922no—
278#41—84no—
279C2×Q8—41no—
280C2×D4—21no—
281C2×D4—42no—
282C2×D4—2924no—
283C2×D4—2922no—
284C2×D4—2922no—
285C2×D4—2921no—
286C2×D4—2921no—
287C2×D4—5842no—
288C2×D4—5844no—
289C2×D4—1462no—
290C2×D4—1461no—
291C2×D4#1192401311yescharacteristic
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—2921no—
295C22×C4C2×C22⋊D7312yes—
296D8—21no—
297D8—21no—
298D8—2924no—
299D8—5841no—
300D8—732no—
301D8—1462no—
302D8—1462no—
303D8—1461no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8C2×C22⋊D7311yescharacteristic
307C2×C8C2×C22⋊D7312yes—
308C2×C8C23×D7311yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—1461no—
314C4⋊C4#1192401311yescharacteristic
315C22⋊C4—41no—
316C22⋊C4—42no—
317C22⋊C4—2921no—
318C22⋊C4—2922no—
319C22⋊C4#1192401312yes—
320C42C2×C22⋊D7311yescharacteristic
321C16—1461no—
322C16—1461no—
323C16—1462no—
324C23—41no—
325C23—2921no—
326C23—2922no—
327C23—5844no—
328C23—5844no—
329C23—11682no—
330C23C2×#1192401312yes—
331Q8—82no—
332D4—41no—
333D4—41no—
334D4—84no—
335D4—2921no—
336D4—2922no—
337D4—2922no—
338D4—5844no—
339D4—5844no—
340D4—5842no—
341D4—11681no—
342D4—1462no—
343D4#1455653214yescomplemented
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—84no—
349C2×C4—82no—
350C2×C4—2921no—
351C2×C4—5842no—
352C2×C4C2×#1192401311yescharacteristic
353C2×C4C2×#1192401312yes—
354C2×C4C2×#1192401311yescharacteristic
355C2×C4C22×C22⋊D7311yescharacteristic
356C8—21no—
357C8—22no—
358C8#41×D7311yescharacteristic
359C8#1455663711yescharacteristic
360C22—22no—
361C22—22no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—2922no—
366C22—5841no—
367C22—5841no—
368C22—5842no—
369C22—5842no—
370C22—5842no—
371C22—11688no—
372C22C2×#1455653214yes—
373C22C22×#1192401311yescharacteristic
374C4—22no—
375C4—21no—
376C4—81no—
377C4—82no—
378C4—5841no—
379C4C2×#1455653212yes—
380C4#1798922511yescharacteristic
381C4#1798939511yescharacteristic
382C2—24no—
383C2—161no—
384C2—5842no—
385C2—11682no—
386C2#2262173811yescharacteristic
387C2C22×#1455653211yescharacteristic
388C2#2262307511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3334C2
socleC22×C73C22×D8
hypercenter#3334D73
hypocenterC73#33861
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. D4×D584C2semidirect
2. #22612156C2semidirect
3. #22612111C2semidirect
4. #22612065C2semidirect
5. #22612015C2semidirect
6. #22611966C2semidirect
7. #22611842C2semidirect
8. C4×#14556532C2semidirect
9. D4×#11924003C2semidirect
10. C2×#17988387C2semidirect
11. C73×#3334C2semidirect
15. C2×#14556532C22semidirect
16. C22×#11924003C22semidirect
22. #17988412C22semidirect
23. #17988412C22semidirect
24. C4⋊#11924003C22semidirect
25. #17988388C22semidirect
28. #17988387C22semidirect
29. #17988387C22semidirect
30. #17988387C22semidirect
31. C22⋊#11924003C22semidirect
32. C4×#11924003C22semidirect
34. #500×C73C22semidirect
37. #466×C73C22semidirect
38. C4×D8×C73C22semidirect
40. C2×#107×C73C22semidirect
98. D584#41semidirect
140. D4×C73D16semidirect
195. #3334D73semidirect
206. C73#33861semidirect
213. #500C2×D73semidirect
218. #466C2×D73semidirect
222. C4×D8C2×D73semidirect
224. C2×#107C2×D73semidirect
343. D4#14556532semidirect

Maximal subgroups

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

subgroupindex
D4×D5842
#226121562
#226121112
#226120652
#226120152
#226119662
#226118422
C4×#145565322
D4×#119240032
C2×#179883872
C73×#33342
#3386173