Finite Group Databasebeta

← #37584052

Subgroups — #37584052

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×#20259350C211yesmaximal, characteristic, complemented
2C2×#20259345C211yesmaximal, characteristic, complemented
3#27133315C211yesmaximal, characteristic, complemented
4#27133304C211yesmaximal, characteristic, complemented
5#27133056C211yesmaximal, characteristic, complemented
6#27132893C211yesmaximal, characteristic, complemented
7#27132881C211yesmaximal, characteristic, complemented
8#27132867C211yesmaximal, characteristic, complemented
9#27132828C211yesmaximal, characteristic, complemented
10#27132823C211yesmaximal, characteristic, complemented
11#27132813C211yesmaximal, characteristic, complemented
12#27132811C211yesmaximal, characteristic, complemented
13#27132774C211yesmaximal, characteristic, complemented
14#27132732C211yesmaximal, characteristic, complemented
15#470×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690288C2211yescharacteristic, complemented
18C2×#15690287C2211yescharacteristic
19C22×D764C2211yescharacteristic
20#20259350—24no—
21#20259345—24no—
22C2×#15690277C2211yescharacteristic, complemented
23C2×#15690276C2211yescharacteristic
24C2×#15690276C2211yescharacteristic
25C2×#15690274C2211yescharacteristic
26C2×#15690274C2211yescharacteristic
27C2×#15690274C2211yescharacteristic, complemented
28C2×D1528C2211yescharacteristic
29C2×D1528C2211yescharacteristic
30C2×C8×D191C2211yescharacteristic, complemented
31#20259320C2211yescharacteristic, complemented
32#20259311C2211yescharacteristic
33#20259307—21no—
34#20259307—21no—
35#20259304—21no—
36#20259302C2211yescharacteristic, complemented
37#20259297—21no—
38#20259297C2211yescharacteristic
39#20259295—21no—
40#20259292—21no—
41#20259284C2211yescharacteristic
42#20259284C2211yescharacteristic, complemented
43C4×C22⋊D191C2211yescharacteristic
44C4⋊C4×D191—21no—
45C22⋊C4×D191—21no—
46Dic191⋊C8C2211yescharacteristic, complemented
47#20259245C2211yescharacteristic, complemented
48#20259242C2211yescharacteristic
49#20259242C2211yescharacteristic
50#20259242C2211yescharacteristic, complemented
51C22⋊(C191⋊C8)C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53#20259232C2211yescharacteristic, complemented
54C2×Q16×C191C2211yescharacteristic, complemented
55C2×M16×C191C2211yescharacteristic
56C22⋊Q8×C191C2211yescharacteristic
57C22⋊Q8×C191C2211yescharacteristic
58#106×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60#103×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690288—22no—
63#15690288—22no—
64#15690288—41no—
65#15690288—41no—
66#15690287—22no—
67#15690287—41no—
68C2×Dic382C2311yescharacteristic
69C2×D764—22no—
70C2×D764—21no—
71C2×D764—21no—
72C2×D764—21no—
73C2×D764—21no—
74C2×D764—41no—
75C2×D764C2311yescharacteristic
76C2×D764C2311yescharacteristic
77C2×D764C2311yescharacteristic
78C2×C22⋊D191—21no—
79C2×C22⋊D191D411yescharacteristic
80C2×C4×D191—21no—
81C2×C4×D191—21no—
82C2×C4×D191—21no—
83C2×C4×D191—21no—
84C2×C4×D191—21no—
85C2×C4×D191C2311yescharacteristic
86#15690277—41no—
87#15690276—22no—
88#15690276—22no—
89#15690274—22no—
90#15690274—22no—
91#15690274—41no—
92D1528—22no—
93D1528D414yescomplemented
94C2×C191⋊C8C2311yescharacteristic
95C2×C191⋊C8C2311yescharacteristic
96C8×D191—22no—
97C4⋊Dic191—21no—
98Dic191⋊C4—21no—
99Dic191⋊C4—21no—
100Dic191⋊C4D411yescharacteristic
101C22⋊Dic191D411yescharacteristic
102(C2×D191)⋊C4—21no—
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)⋊C4D411yescharacteristic
109C4×Dic191C2311yescharacteristic
110C2×Q8×C191C2311yescharacteristic
111C2×Q8×C191C2311yescharacteristic
112C22×C4×C191C2311yescharacteristic
113Q16×C191—22no—
114M16×C191—22no—
115C2×C8×C191C2311yescharacteristic
116C2×C8×C191C2311yescharacteristic
117C4⋊C4×C191—21no—
118C4⋊C4×C191—21no—
119C4⋊C4×C191C2311yescharacteristic
120C4⋊C4×C191C2311yescharacteristic
121C22⋊C4×C191—21no—
122C22⋊C4×C191—21no—
123C22×D191—21no—
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—42no—
127C22×D191—42no—
128C22×D191—81no—
129C22×D191C2×D411yescharacteristic
130Dic382—21no—
131Dic382—21no—
132D764—21no—
133D764—21no—
134D764—21no—
135D764—21no—
136D764—42no—
137D764—42no—
138D764—42no—
139D764—81no—
140D764C2×D412yes—
141D764C2×D412yes—
142C22⋊D191—41no—
143C22⋊D191—42no—
144C2×Dic191—21no—
145C2×Dic191C2×D411yescharacteristic
146C2×Dic191C2×D411yescharacteristic
147C2×Dic191C2×D411yescharacteristic
148C4×D191—21no—
149C4×D191—21no—
150C4×D191—42no—
151C4×D191—42no—
152C4×D191—42no—
153C4×D191—42no—
154C4×D191—81no—
155C191⋊C8—21no—
156C191⋊C8—21no—
157C23×C191C2×D411yescharacteristic
158Q8×C191—21no—
159Q8×C191—21no—
160Q8×C191—21no—
161Q8×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—21no—
167C2×C4×C191—21no—
168C2×C4×C191C2×D411yescharacteristic
169C2×C4×C191C2411yescharacteristic
170C8×C191—21no—
171C8×C191C2×D412yes—
172C22×C191—21no—
173C22×C191—21no—
174C22×C191—21no—
175C22×C191C22×D411yescharacteristic
176C2×D191—21no—
177C2×D191—22no—
178C2×D191—22no—
179C2×D191—21no—
180C2×D191—21no—
181C2×D191—41no—
182C2×D191—41no—
183C2×D191—41no—
184C2×D191—41no—
185C2×D191—41no—
186C2×D191—41no—
187C2×D191—41no—
188C2×D191—84no—
189C4×C191—21no—
190C4×C191—41no—
191C4×C191—41no—
192C4×C191—41no—
193C4×C191—41no—
194C4×C191C22×D411yescharacteristic
195C4×C191H(5,𝔽2)11yescharacteristic
196Dic191—21no—
197Dic191—21no—
198Dic191—21no—
199Dic191—41no—
200C2×C191—41no—
201C2×C191D4211yescharacteristic
202C2×C191#57511yescharacteristic
203C2×C191#57711yescharacteristic
204D191—41no—
205D191—42no—
206D191—42no—
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#470D19111yescharacteristic, complemented
216#465—1911no—
217#463—1911no—
218#461—1911no—
219#452—1911no—
220#452—1911no—
221#450—1911no—
222#449—1911no—
223#442—1911no—
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×Q16C2×D19111yescharacteristic, complemented
233C2×SD16—1911no—
234C2×SD16—1911no—
235C2×SD16—1911no—
236C2×SD16—1911no—
237C2×SD16—1911no—
238C2×D8—1911no—
239C2×D8—1911no—
240C2×M16C2×D19111yescharacteristic
241C22×C8—1911no—
242C4⋊Q8—1911no—
243#124—1911no—
244#123—3821no—
245#123—3821no—
246#123—3821no—
247C22⋊Q8—1911no—
248C22⋊Q8C2×D19111yescharacteristic
249C22⋊Q8C2×D19111yescharacteristic
250#121—3821no—
251#121—3821no—
252#121—1911no—
253#120—3821no—
254C4×D4—1911no—
255C4×D4—1911no—
256C4×D4—1911no—
257C2×C4⋊C4—3821no—
258C2×C22⋊C4—3821no—
259#106C2×D19111yescharacteristic, complemented
260C4⋊C8—1911no—
261#103—1911no—
262#103C2×D19111yescharacteristic, complemented
263#103C2×D19111yescharacteristic, complemented
264#102—1911no—
265#102—1911no—
266#102—1911no—
267C22⋊C8—1911no—
268C22⋊C8—1911no—
269#97—1911no—
270C24—3821no—
271#41—3822no—
272#41—3822no—
273#41—3822no—
274#41—7641no—
275#41—7641no—
276#41—7641no—
277C2×Q8—1911no—
278C2×Q8C22×D19111yescharacteristic
279C2×Q8C22×D19111yescharacteristic
280C2×D4—3822no—
281C2×D4—3821no—
282C2×D4—3821no—
283C2×D4—3821no—
284C2×D4—3821no—
285C2×D4—3821no—
286C2×D4—1911no—
287C2×D4—1911no—
288C2×D4—1911no—
289C2×D4—1911no—
290C2×D4—7641no—
291C22×C4—3821no—
292C22×C4—3821no—
293C22×C4—3821no—
294C22×C4—3821no—
295C22×C4—3821no—
296C22×C4—1911no—
297C22×C4C22×D19111yescharacteristic
298Q16—22no—
299Q16—7641no—
300SD16—3822no—
301SD16—3822no—
302SD16—3822no—
303SD16—3822no—
304SD16—7641no—
305D8—3822no—
306D8—1914no—
307M16—22no—
308C2×C8—3822no—
309C2×C8—1911no—
310C2×C8—1911no—
311C2×C8C22×D19111yescharacteristic
312C2×C8C22×D19111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
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—
331C42—1911no—
332C23—3821no—
333C23—3821no—
334C23—3821no—
335C23—1911no—
336C23—15281no—
337C23—7642no—
338C23—7642no—
339C23D4×D19111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—3821no—
345Q8—3821no—
346D4—3821no—
347D4—3821no—
348D4—3821no—
349D4—3821no—
350D4—1912no—
351D4—1912no—
352D4—15281no—
353D4—7642no—
354D4—7642no—
355D4—7642no—
356D4—7641no—
357D4—7642no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—3821no—
365C2×C4—3821no—
366C2×C4—3821no—
367C2×C4—1911no—
368C2×C4—1911no—
369C2×C4—1911no—
370C2×C4—15281no—
371C2×C4—7642no—
372C2×C4—7642no—
373C2×C4—7642no—
374C2×C4—7642no—
375C2×C4D4×D19111yescharacteristic
376C2×C4C23×D19111yescharacteristic
377C8—21no—
378C8—3821no—
379C8—3821no—
380C8D4×D19112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—3821no—
385C22—3821no—
386C22—3821no—
387C22—3822no—
388C22—3822no—
389C22—15284no—
390C22—7641no—
391C22—7641no—
392C22—7641no—
393C22—7641no—
394C22—7641no—
395C22—7641no—
396C22—7641no—
397C22C2×D4×D19111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—3821no—
404C4—3821no—
405C4—3821no—
406C4—7641no—
407C4C2×D4×D19111yescharacteristic
408C4#2025936811yescharacteristic
409C2—41no—
410C2—15281no—
411C2—7641no—
412C2—7642no—
413C2—7642no—
414C2#2713329711yescharacteristic
415C2#2713346211yescharacteristic
416C2#2713347211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#470×C191C2
socleC22×C191C22×D4
hypercenter#470D191
hypocenterC191#3267
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#20259350C2semidirect
2. C2×#20259345C2semidirect
3. #27133315C2semidirect
4. #27133304C2semidirect
5. #27133056C2semidirect
6. #27132893C2semidirect
7. #27132881C2semidirect
8. #27132867C2semidirect
9. #27132828C2semidirect
10. #27132823C2semidirect
11. #27132813C2semidirect
12. #27132811C2semidirect
13. #27132774C2semidirect
14. #27132732C2semidirect
15. #470×C191C2semidirect
17. C2×#15690288C22semidirect
22. C2×#15690277C22semidirect
27. C2×#15690274C22semidirect
30. C2×C8×D191C22semidirect
31. #20259320C22semidirect
36. #20259302C22semidirect
42. #20259284C22semidirect
46. Dic191⋊C8C22semidirect
47. #20259245C22semidirect
50. #20259242C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. #20259232C22semidirect
54. C2×Q16×C191C22semidirect
58. #106×C191C22semidirect
59. #103×C191C22semidirect
60. #103×C191C22semidirect
93. D1528D4semidirect
208. C191#3267semidirect
215. #470D191semidirect
232. C2×Q16C2×D191semidirect
259. #106C2×D191semidirect
262. #103C2×D191semidirect
263. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593502
C2×#202593452
#271333152
#271333042
#271330562
#271328932
#271328812
#271328672
#271328282
#271328232
#271328132
#271328112
#271327742
#271327322
#470×C1912
#3267191