Finite Group Databasebeta

← #37584047

Subgroups — #37584047

Order 24448.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259346C211yesmaximal, characteristic, complemented
2C2×#20259341C211yesmaximal, characteristic, complemented
3#27133317C211yesmaximal, characteristic, complemented
4#27133308C211yesmaximal, characteristic, complemented
5#27133057C211yesmaximal, characteristic, complemented
6#27132922C211yesmaximal, characteristic, complemented
7#27132893C211yesmaximal, characteristic, complemented
8#27132878C211yesmaximal, characteristic, complemented
9#27132868C211yesmaximal, characteristic, complemented
10#27132826C211yesmaximal, characteristic, complemented
11#27132826C211yesmaximal, characteristic, complemented
12#27132815C211yesmaximal, characteristic, complemented
13#27132809C211yesmaximal, characteristic, complemented
14#27132736C211yesmaximal, characteristic, complemented
15#442×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×#15690286C2211yescharacteristic
18C2×#15690286C2211yescharacteristic, complemented
19C2×D4×D191C2211yescharacteristic
20#20259346—24no—
21#20259341—24no—
22C2×Dic764C2211yescharacteristic, complemented
23C2×Dic764C2211yescharacteristic, complemented
24C2×#15690276C2211yescharacteristic
25C2×#15690276C2211yescharacteristic
26C2×#15690275C2211yescharacteristic
27C2×#15690275C2211yescharacteristic
28C2×#15690275C2211yescharacteristic, complemented
29C2×#15690272C2211yescharacteristic
30C2×#15690270C2211yescharacteristic
31C2×C8×D191C2211yescharacteristic, complemented
32#20259322C2211yescharacteristic, complemented
33#20259310C2211yescharacteristic
34#20259306—21no—
35#20259306—21no—
36#20259306—21no—
37#20259302C2211yescharacteristic
38#20259302C2211yescharacteristic
39#20259302C2211yescharacteristic, complemented
40#20259297C2211yescharacteristic
41#20259296—21no—
42#20259296—21no—
43#20259292—21no—
44C4×D764C2211yescharacteristic
45C2×C4⋊Dic191—21no—
46C2×(C2×D191)⋊C4—21no—
47C4⋊(C191⋊C8)C2211yescharacteristic, complemented
48#20259245C2211yescharacteristic, complemented
49#20259245C2211yescharacteristic, complemented
50#20259245C2211yescharacteristic, complemented
51#20259242C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic
53(C2×D191)⋊C8C2211yescharacteristic, complemented
54C2×D8×C191C2211yescharacteristic
55C4×D4×C191C2211yescharacteristic
56C4×D4×C191C2211yescharacteristic, complemented
57#106×C191C2211yescharacteristic, complemented
58#102×C191C2211yescharacteristic
59#102×C191C2211yescharacteristic, complemented
60#97×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690286—22no—
63#15690286—22no—
64#15690286—22no—
65#15690286—41no—
66#15690286—41no—
67#15690286—41no—
68C2×Dic382C2311yescharacteristic
69C2×Dic382C2311yescharacteristic
70C2×Dic382C2311yescharacteristic
71C2×D764—21no—
72C2×D764D411yescharacteristic
73C2×D764C2311yescharacteristic
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77C2×C22⋊D191—21no—
78D4×D191—22no—
79D4×D191—41no—
80C22×Dic191—21no—
81C22×Dic191—21no—
82C22×Dic191—21no—
83C2×C4×D191C2311yescharacteristic
84C2×C4×D191C2311yescharacteristic
85Dic764—22no—
86Dic764—41no—
87#15690276—22no—
88#15690276—41no—
89#15690275—22no—
90#15690275—22no—
91#15690275—22no—
92#15690272—22no—
93#15690270—22no—
94C2×C191⋊C8C2311yescharacteristic
95C2×C191⋊C8C2311yescharacteristic
96C8×D191—22no—
97C4⋊Dic191—21no—
98C4⋊Dic191—21no—
99C4⋊Dic191—21no—
100C4⋊Dic191—21no—
101C4⋊Dic191—21no—
102C4⋊Dic191D411yescharacteristic
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4—21no—
108(C2×D191)⋊C4—21no—
109(C2×D191)⋊C4D411yescharacteristic
110(C2×D191)⋊C4D411yescharacteristic
111C2×D4×C191C2311yescharacteristic
112C2×D4×C191C2311yescharacteristic
113C22×C4×C191—21no—
114C22×C4×C191—21no—
115D8×C191D414yescomplemented
116C2×C8×C191C2311yescharacteristic
117C2×C8×C191C2311yescharacteristic
118C4⋊C4×C191C2311yescharacteristic
119C4⋊C4×C191C2311yescharacteristic
120C22⋊C4×C191—21no—
121C22⋊C4×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191—42no—
125C22×D191—42no—
126C22×D191—81no—
127C22×D191C2×D411yescharacteristic
128C22×D191C2×D411yescharacteristic
129Dic382—21no—
130Dic382—21no—
131Dic382—21no—
132Dic382—21no—
133Dic382—21no—
134Dic382—21no—
135D764—21no—
136D764—21no—
137D764—41no—
138D764—81no—
139C22⋊D191—42no—
140C22⋊D191—42no—
141C22⋊D191—42no—
142C22⋊D191—42no—
143C2×Dic191—21no—
144C2×Dic191—21no—
145C2×Dic191—21no—
146C2×Dic191—42no—
147C2×Dic191—42no—
148C2×Dic191—42no—
149C2×Dic191C2×D411yescharacteristic
150C2×Dic191C2×D411yescharacteristic
151C4×D191—21no—
152C4×D191—21no—
153C4×D191—21no—
154C4×D191—21no—
155C191⋊C8—21no—
156C191⋊C8—21no—
157C23×C191—21no—
158C23×C191—21no—
159D4×C191—21no—
160D4×C191—21no—
161D4×C191C2×D412yes—
162D4×C191C2×D412yes—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—42no—
166C2×C4×C191—81no—
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2×D411yescharacteristic
169C2×C4×C191C2411yescharacteristic
170C8×C191—21no—
171C8×C191C2×D412yes—
172C22×C191—22no—
173C22×C191—22no—
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—
188C2×D191—84no—
189C4×C191—21no—
190C4×C191—21no—
191C4×C191—41no—
192C4×C191—41no—
193C4×C191C22×D411yescharacteristic
194C4×C191H(5,𝔽2)11yescharacteristic
195Dic191—21no—
196Dic191—21no—
197Dic191—41no—
198Dic191—41no—
199Dic191—41no—
200C2×C191—42no—
201C2×C191—42no—
202C2×C191D4211yescharacteristic
203C2×C191#57511yescharacteristic
204C2×C191#57711yescharacteristic
205D191—41no—
206D191—41no—
207D191—81no—
208C191#326711yescharacteristic, complemented, Sylow
209#3267—1911nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1911no—
211C2×#135—1911no—
212#547—1911no—
213#546—1911no—
214#497—1911no—
215#470—1911no—
216#465—1911no—
217#463—1911no—
218#461—1911no—
219#452—1911no—
220#452—1911no—
221#450—1911no—
222#449—1911no—
223#442D19111yescharacteristic, complemented
224#435—1911no—
225C2×#41—1911no—
226C2×#41—1911no—
227C2×#41—1911no—
228C22×D4—1911no—
229AGL(1,ℤ/8)—3824no—
230#135—3824no—
231C2×Q16—1911no—
232C2×Q16—1911no—
233C2×SD16—1911no—
234C2×SD16—1911no—
235C2×SD16—1911no—
236C2×SD16—1911no—
237C2×SD16—1911no—
238C2×D8—1911no—
239C2×D8C2×D19111yescharacteristic
240C2×M16—1911no—
241C22×C8—1911no—
242C4⋊Q8—1911no—
243#124—1911no—
244#123—3821no—
245#123—3821no—
246#123—3821no—
247C22⋊Q8—1911no—
248C22⋊Q8—1911no—
249C22⋊Q8—1911no—
250#121—3821no—
251#121—3821no—
252#121—1911no—
253#120—3821no—
254C4×D4—1911no—
255C4×D4C2×D19111yescharacteristic
256C4×D4C2×D19111yescharacteristic, complemented
257C2×C4⋊C4—3821no—
258C2×C22⋊C4—3821no—
259#106C2×D19111yescharacteristic, complemented
260C4⋊C8—1911no—
261#103—1911no—
262#103—1911no—
263#103—1911no—
264#102—1911no—
265#102C2×D19111yescharacteristic
266#102C2×D19111yescharacteristic, complemented
267C22⋊C8—1911no—
268C22⋊C8—1911no—
269#97C2×D19111yescharacteristic, complemented
270C24—3821no—
271#41—3822no—
272#41—3822no—
273#41—3822no—
274#41—7641no—
275#41—7641no—
276#41—7641no—
277C2×Q8—1911no—
278C2×Q8—1911no—
279C2×Q8—1911no—
280C2×D4—3821no—
281C2×D4—3822no—
282C2×D4—3821no—
283C2×D4—3821no—
284C2×D4—3821no—
285C2×D4—3821no—
286C2×D4—1911no—
287C2×D4—1911no—
288C2×D4—7641no—
289C2×D4C22×D19111yescharacteristic
290C2×D4C22×D19111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—3821no—
294C22×C4—3821no—
295C22×C4—3821no—
296C22×C4—1911no—
297C22×C4—1911no—
298Q16—3822no—
299Q16—7641no—
300SD16—3822no—
301SD16—3822no—
302SD16—3822no—
303SD16—3822no—
304SD16—7641no—
305D8—3822no—
306D8D76414yescomplemented
307M16—3822no—
308C2×C8—3822no—
309C2×C8—1911no—
310C2×C8—1911no—
311C2×C8C22×D19111yescharacteristic
312C2×C8C22×D19111yescharacteristic
313C4⋊C4—3821no—
314C4⋊C4—3821no—
315C4⋊C4—3821no—
316C4⋊C4—3821no—
317C4⋊C4—3821no—
318C4⋊C4—1911no—
319C4⋊C4C22×D19111yescharacteristic
320C4⋊C4C22×D19111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—3821no—
324C22⋊C4—3821no—
325C22⋊C4—3821no—
326C22⋊C4—3821no—
327C22⋊C4—3821no—
328C22⋊C4—3821no—
329C22⋊C4—1911no—
330C22⋊C4—1911no—
331C42C22×D19111yescharacteristic
332C23—21no—
333C23—21no—
334C23—3821no—
335C23—1911no—
336C23—1911no—
337C23—15281no—
338C23—7642no—
339C23—7642no—
340Q8—3821no—
341Q8—3821no—
342Q8—3821no—
343Q8—3821no—
344Q8—3821no—
345Q8—3821no—
346D4—21no—
347D4—21no—
348D4—3821no—
349D4—3821no—
350D4—15281no—
351D4—7641no—
352D4—7642no—
353D4—7642no—
354D4—7642no—
355D4—7642no—
356D4C2×D76412yes—
357D4C2×D76412yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—3821no—
363C2×C4—3821no—
364C2×C4—3821no—
365C2×C4—3821no—
366C2×C4—3821no—
367C2×C4—3821no—
368C2×C4—3821no—
369C2×C4—1911no—
370C2×C4—1911no—
371C2×C4—7642no—
372C2×C4—7642no—
373C2×C4—7642no—
374C2×C4D4×D19111yescharacteristic
375C2×C4D4×D19111yescharacteristic
376C2×C4C23×D19111yescharacteristic
377C8—21no—
378C8—3821no—
379C8—3821no—
380C8C2×D76412yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3821no—
388C22—3821no—
389C22—3821no—
390C22—3821no—
391C22—3821no—
392C22—3821no—
393C22—15284no—
394C22—7641no—
395C22—7641no—
396C22—7641no—
397C22C2×D4×D19111yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3821no—
403C4—3821no—
404C4—7641no—
405C4—7641no—
406C4—7641no—
407C4C22×D76411yescharacteristic
408C4#2025936811yescharacteristic
409C2—42no—
410C2—42no—
411C2—15281no—
412C2—7641no—
413C2—7641no—
414C2D4×D76411yescharacteristic
415C2#2713345911yescharacteristic
416C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#442×C191C2
socleC22×C191C22×D4
hypercenter#442D191
hypocenterC191#3267
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. C2×#20259341C2semidirect
3. #27133317C2semidirect
4. #27133308C2semidirect
5. #27133057C2semidirect
6. #27132922C2semidirect
7. #27132893C2semidirect
8. #27132878C2semidirect
9. #27132868C2semidirect
10. #27132826C2semidirect
11. #27132826C2semidirect
12. #27132815C2semidirect
13. #27132809C2semidirect
14. #27132736C2semidirect
15. #442×C191C2semidirect
18. C2×#15690286C22semidirect
22. C2×Dic764C22semidirect
23. C2×Dic764C22semidirect
28. C2×#15690275C22semidirect
31. C2×C8×D191C22semidirect
32. #20259322C22semidirect
39. #20259302C22semidirect
47. C4⋊(C191⋊C8)C22semidirect
48. #20259245C22semidirect
49. #20259245C22semidirect
50. #20259245C22semidirect
53. (C2×D191)⋊C8C22semidirect
56. C4×D4×C191C22semidirect
57. #106×C191C22semidirect
59. #102×C191C22semidirect
60. #97×C191C22semidirect
115. D8×C191D4semidirect
208. C191#3267semidirect
223. #442D191semidirect
256. C4×D4C2×D191semidirect
259. #106C2×D191semidirect
266. #102C2×D191semidirect
269. #97C2×D191semidirect
306. D8D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C2×#202593412
#271333172
#271333082
#271330572
#271329222
#271328932
#271328782
#271328682
#271328262
#271328152
#271328092
#271327362
#442×C1912
#3267191