Finite Group Databasebeta

← #37584100

Subgroups — #37584100

Order 24448.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259351C211yesmaximal, characteristic, complemented
2C2×#20259343C211yesmaximal, characteristic, complemented
3#27133333C211yesmaximal, characteristic, complemented
4#27133325C211yesmaximal, characteristic, complemented
5#27133050C211yesmaximal, characteristic, complemented
6#27132915C211yesmaximal, characteristic, complemented
7#27132891C211yesmaximal, characteristic, complemented
8#27132886C211yesmaximal, characteristic, complemented
9#27132826C211yesmaximal, characteristic, complemented
10#27132809C211yesmaximal, characteristic, complemented
11#27132807C211yesmaximal, characteristic, complemented
12#27132800C211yesmaximal, characteristic, complemented
13#27132768C211yesmaximal, characteristic, complemented
14#27132733C211yesmaximal, characteristic, complemented
15#460×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690286C2211yescharacteristic
18C2×#15690286C2211yescharacteristic
19C2×Q8×D191C2211yescharacteristic
20#20259351—24no—
21#20259343—24no—
22C2×Dic764C2211yescharacteristic
23C2×#15690277C2211yescharacteristic
24C2×#15690276C2211yescharacteristic
25C2×#15690275C2211yescharacteristic
26C2×#15690275C2211yescharacteristic
27C2×#15690272C2211yescharacteristic
28C2×#15690272C2211yescharacteristic
29C2×#15690270C2211yescharacteristic
30C2×C8×D191C2211yescharacteristic
31#20259318—21no—
32#20259310—21no—
33#20259310C2211yescharacteristic
34#20259302—21no—
35#20259302C2211yescharacteristic
36#20259297—21no—
37#20259297C2211yescharacteristic
38#20259294—21no—
39#20259294—21no—
40#20259293C2211yescharacteristic
41#20259293C2211yescharacteristic
42C4×Dic382C2211yescharacteristic
43C4×D764—21no—
44C4×D764C2211yescharacteristic
45D4×Dic191—21no—
46#20259253C2211yescharacteristic
47#20259247C2211yescharacteristic
48#20259246C2211yescharacteristic
49#20259245C2211yescharacteristic
50#20259244C2211yescharacteristic
51(C2×D191)⋊C8C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic
53#20259233C2211yescharacteristic
54C2×SD16×C191C2211yescharacteristic
55C2×SD16×C191C2211yescharacteristic
56#127×C191C2211yescharacteristic
57C4×Q8×C191C2211yescharacteristic
58C4⋊C8×C191C2211yescharacteristic
59#102×C191C2211yescharacteristic
60#102×C191C2211yescharacteristic
61#15690288—22no—
62#15690288—41no—
63#15690286—22no—
64#15690286—22no—
65#15690286—41no—
66#15690286—41no—
67C2×Dic382—21no—
68C2×Dic382C2311yescharacteristic
69C2×Dic382C2311yescharacteristic
70Q8×D191—22no—
71Q8×D191—41no—
72C2×D764—21no—
73C2×D764—21no—
74C2×D764D411yescharacteristic
75C2×D764C2311yescharacteristic
76C2×C22⋊D191—21no—
77C2×C22⋊D191—21no—
78C2×C22⋊D191—21no—
79C2×C22⋊D191—21no—
80C22×Dic191—21no—
81C22×Dic191—21no—
82C2×C4×D191—21no—
83C2×C4×D191—21no—
84C2×C4×D191C2311yescharacteristic
85C2×C4×D191C2311yescharacteristic
86Dic764—22no—
87#15690277D414yescomplemented
88#15690276—22no—
89#15690275—22no—
90#15690275—41no—
91#15690272—22no—
92#15690272—41no—
93#15690270—22no—
94C2×C191⋊C8C2311yescharacteristic
95C2×C191⋊C8C2311yescharacteristic
96C8×D191—22no—
97C4⋊Dic191—21no—
98C4⋊Dic191D411yescharacteristic
99C4⋊Dic191C2311yescharacteristic
100Dic191⋊C4—21no—
101C22⋊Dic191—21no—
102C22⋊Dic191—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4D411yescharacteristic
108(C2×D191)⋊C4D411yescharacteristic
109C4×Dic191—21no—
110C2×Q8×C191C2311yescharacteristic
111C2×D4×C191—21no—
112C2×D4×C191—21no—
113C2×D4×C191C2311yescharacteristic
114C2×D4×C191C2311yescharacteristic
115SD16×C191—22no—
116SD16×C191—22no—
117C2×C8×C191C2311yescharacteristic
118C2×C8×C191C2311yescharacteristic
119C4⋊C4×C191—21no—
120C4⋊C4×C191C2311yescharacteristic
121C42×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191C2×D411yescharacteristic
125C22×D191C2×D411yescharacteristic
126Dic382—21no—
127Dic382—21no—
128Dic382—21no—
129Dic382—42no—
130Dic382C2×D412yes—
131D764—21no—
132D764—21no—
133D764—41no—
134D764—42no—
135D764—42no—
136C22⋊D191—42no—
137C22⋊D191—42no—
138C22⋊D191—81no—
139C22⋊D191—81no—
140C2×Dic191—21no—
141C2×Dic191—21no—
142C2×Dic191—21no—
143C2×Dic191—42no—
144C2×Dic191—42no—
145C2×Dic191C2×D411yescharacteristic
146C2×Dic191C2×D411yescharacteristic
147C4×D191—21no—
148C4×D191—21no—
149C4×D191—21no—
150C4×D191—21no—
151C4×D191—42no—
152C4×D191—42no—
153C191⋊C8—21no—
154C191⋊C8C2×D412yes—
155C23×C191—21no—
156C23×C191—21no—
157Q8×C191—21no—
158Q8×C191C2×D412yes—
159D4×C191—21no—
160D4×C191—21no—
161D4×C191—21no—
162D4×C191—21no—
163D4×C191—42no—
164D4×C191—81no—
165C2×C4×C191—21no—
166C2×C4×C191—21no—
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2×D411yescharacteristic
169C2×C4×C191C2411yescharacteristic
170C8×C191—21no—
171C8×C191—21no—
172C22×C191—41no—
173C22×C191—41no—
174C22×C191—41no—
175C22×C191—41no—
176C22×C191—41no—
177C22×C191—41no—
178C22×C191C22×D411yescharacteristic
179C2×D191—21no—
180C2×D191—21no—
181C2×D191—21no—
182C2×D191—21no—
183C2×D191—21no—
184C2×D191—21no—
185C2×D191—41no—
186C2×D191—41no—
187C2×D191—41no—
188C4×C191—22no—
189C4×C191—21no—
190C4×C191—21no—
191C4×C191—41no—
192C4×C191C22×D411yescharacteristic
193C4×C191H(5,𝔽2)11yescharacteristic
194Dic191—21no—
195Dic191—21no—
196Dic191—22no—
197Dic191—41no—
198Dic191—41no—
199C2×C191—81no—
200C2×C191—81no—
201C2×C191D4211yescharacteristic
202C2×C191#57511yescharacteristic
203C2×C191#57711yescharacteristic
204D191—41no—
205D191—41no—
206D191—81no—
207C191#327111yescharacteristic, complemented, Sylow
208#3271—1911nomaximal, Sylow
209C2×#137—1911no—
210C2×#135—1911no—
211#550—1911no—
212#548—1911no—
213#496—1911no—
214#469—1911no—
215#465—1911no—
216#464—1911no—
217#460D19111yescharacteristic, complemented
218#452—1911no—
219#449—1911no—
220#449—1911no—
221#448—1911no—
222#441—1911no—
223#435—1911no—
224C2×#41—1911no—
225C2×#41—1911no—
226C2×#41—1911no—
227C22×Q8—1911no—
228#137—3824no—
229#135—3824no—
230C2×Q16—1911no—
231C2×Q16—1911no—
232C2×SD16—1911no—
233C2×SD16—1911no—
234C2×SD16—1911no—
235C2×SD16C2×D19111yescharacteristic
236C2×SD16C2×D19111yescharacteristic
237C2×D8—1911no—
238C2×D8—1911no—
239C2×M16—1911no—
240C22×C8—1911no—
241#127—3821no—
242#127C2×D19111yescharacteristic
243#124—3821no—
244#124—1911no—
245C22⋊Q8—3821no—
246C22⋊Q8—1911no—
247#121—3821no—
248#121—3821no—
249#121—3821no—
250#121—1911no—
251#121—1911no—
252#121—1911no—
253C4×Q8—1911no—
254C4×Q8C2×D19111yescharacteristic
255C4×D4—3821no—
256C4×D4—3821no—
257C4×D4—1911no—
258#106—1911no—
259C4⋊C8C2×D19111yescharacteristic
260#103—1911no—
261#103—1911no—
262#103—1911no—
263#102—1911no—
264#102C2×D19111yescharacteristic
265#102C2×D19111yescharacteristic
266C22⋊C8—1911no—
267C22⋊C8—1911no—
268#97—1911no—
269#41—3822no—
270#41—3822no—
271#41—3822no—
272#41—7641no—
273#41—7641no—
274#41—7641no—
275C2×Q8—3821no—
276C2×Q8—3822no—
277C2×Q8—1911no—
278C2×Q8—1911no—
279C2×Q8—7641no—
280C2×Q8C22×D19111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—3821no—
284C2×D4—3821no—
285C2×D4—3821no—
286C2×D4—3821no—
287C2×D4—3821no—
288C2×D4—3821no—
289C2×D4—1911no—
290C2×D4—1911no—
291C2×D4C22×D19111yescharacteristic
292C2×D4C22×D19111yescharacteristic
293C22×C4—3821no—
294C22×C4—3821no—
295C22×C4—3821no—
296C22×C4—3821no—
297C22×C4—1911no—
298C22×C4—1911no—
299Q16—3822no—
300Q16—1914no—
301SD16—22no—
302SD16—22no—
303SD16—3822no—
304SD16—3822no—
305SD16—7641no—
306D8—3822no—
307D8—7641no—
308M16—3822no—
309C2×C8—3822no—
310C2×C8—1911no—
311C2×C8—1911no—
312C2×C8C22×D19111yescharacteristic
313C2×C8C22×D19111yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—3821no—
316C4⋊C4—3821no—
317C4⋊C4—1911no—
318C4⋊C4—1911no—
319C4⋊C4C22×D19111yescharacteristic
320C22⋊C4—3821no—
321C22⋊C4—3821no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4—3821no—
325C22⋊C4—3821no—
326C22⋊C4—1911no—
327C22⋊C4—1911no—
328C42—21no—
329C42—3821no—
330C42C22×D19111yescharacteristic
331C23—21no—
332C23—21no—
333C23—3821no—
334C23—1911no—
335C23—1911no—
336Q8—21no—
337Q8—3821no—
338Q8—3821no—
339Q8—3821no—
340Q8—1912no—
341Q8—7642no—
342Q8D4×D19112yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—42no—
348D4—81no—
349D4—3821no—
350D4—3821no—
351D4—15281no—
352D4—15281no—
353D4—7641no—
354D4—7642no—
355D4—7642no—
356D4—7642no—
357D4—7642no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—3821no—
361C2×C4—3821no—
362C2×C4—3821no—
363C2×C4—3821no—
364C2×C4—3821no—
365C2×C4—3821no—
366C2×C4—3821no—
367C2×C4—1911no—
368C2×C4—1911no—
369C2×C4—7642no—
370C2×C4—7642no—
371C2×C4—7642no—
372C2×C4—7642no—
373C2×C4D4×D19111yescharacteristic
374C2×C4D4×D19111yescharacteristic
375C2×C4C23×D19111yescharacteristic
376C8—21no—
377C8—21no—
378C8—3821no—
379C8—1912no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—3821no—
387C22—3821no—
388C22—3821no—
389C22—3821no—
390C22—3821no—
391C22—3821no—
392C22—7641no—
393C22—7641no—
394C22—7641no—
395C22C2×D4×D19111yescharacteristic
396C4—21no—
397C4—21no—
398C4—22no—
399C4—41no—
400C4—3821no—
401C4—3821no—
402C4—3822no—
403C4—7641no—
404C4—7641no—
405C4C2×D4×D19111yescharacteristic
406C4#2025936811yescharacteristic
407C2—81no—
408C2—81no—
409C2—15281no—
410C2—7641no—
411C2—7641no—
412C2#2713329711yescharacteristic
413C2#2713345911yescharacteristic
414C2#2713346911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#460×C191C2
socleC22×C191C22×D4
hypercenter#460D191
hypocenterC191#3271
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#20259351C2semidirect
2. C2×#20259343C2semidirect
3. #27133333C2semidirect
4. #27133325C2semidirect
5. #27133050C2semidirect
6. #27132915C2semidirect
7. #27132891C2semidirect
8. #27132886C2semidirect
9. #27132826C2semidirect
10. #27132809C2semidirect
11. #27132807C2semidirect
12. #27132800C2semidirect
13. #27132768C2semidirect
14. #27132733C2semidirect
15. #460×C191C2semidirect
87. #15690277D4semidirect
207. C191#3271semidirect
217. #460D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593512
C2×#202593432
#271333332
#271333252
#271330502
#271329152
#271328912
#271328862
#271328262
#271328092
#271328072
#271328002
#271327682
#271327332
#460×C1912
#3271191