Finite Group Databasebeta

← #37584035

Subgroups — #37584035

Order 24448.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259355C211yesmaximal, characteristic, complemented
2C22×#15690276C211yesmaximal, characteristic, complemented
3D4×Dic382C211yesmaximal, characteristic, complemented
4#27133309C211yesmaximal, characteristic
5#27133057C211yesmaximal, characteristic, complemented
6#27132909C211yesmaximal, characteristic, complemented
7#27132904C211yesmaximal, characteristic, complemented
8#27132886C211yesmaximal, characteristic, complemented
9#27132869C211yesmaximal, characteristic, complemented
10#27132827C211yesmaximal, characteristic, complemented
11#27132821C211yesmaximal, characteristic, complemented
12#27132816C211yesmaximal, characteristic, complemented
13#27132804C211yesmaximal, characteristic, complemented
14#27132765C211yesmaximal, characteristic, complemented
15#435×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C22×Dic382C2211yescharacteristic
18C22×Dic382C2211yescharacteristic, complemented
19C22×D764C2211yescharacteristic
20#20259355—24no—
21C2×Dic764C2211yescharacteristic, complemented
22C2×Dic764C2211yescharacteristic, complemented
23C2×Dic764C2211yescharacteristic, complemented
24C2×#15690276—24no—
25C2×#15690276C2211yescharacteristic
26C2×#15690276C2211yescharacteristic
27C2×#15690276C2211yescharacteristic
28C2×#15690276C2211yescharacteristic
29C2×#15690276C2211yescharacteristic
30C2×#15690276C2211yescharacteristic
31#20259322C2211yescharacteristic, complemented
32#20259321—21no—
33#20259310C2211yescharacteristic
34#20259307—21no—
35#20259301C2211yescharacteristic
36#20259301C2211yescharacteristic
37#20259301C2211yescharacteristic, complemented
38#20259299—21no—
39#20259299—21no—
40#20259296C2211yescharacteristic
41#20259295—21no—
42#20259290—21no—
43C4×Dic382C2211yescharacteristic, complemented
44C4×D764C2211yescharacteristic
45D4×Dic191—21no—
46C22⋊C4×D191—21no—
47#20259256C2211yescharacteristic, complemented
48#20259247C2211yescharacteristic, complemented
49#20259247C2211yescharacteristic, complemented
50#20259247C2211yescharacteristic, complemented
51#20259247C2211yescharacteristic, complemented
52#20259244C2211yescharacteristic
53#20259244C2211yescharacteristic
54C2×M16×C191C2211yescharacteristic, complemented
55C22×C8×C191C2211yescharacteristic
56C4×D4×C191C2211yescharacteristic
57C4⋊C8×C191C2211yescharacteristic, complemented
58C22⋊C8×C191C2211yescharacteristic
59C22⋊C8×C191C2211yescharacteristic, complemented
60#97×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690287—22no—
63#15690287—41no—
64C2×Dic382—22no—
65C2×Dic382—22no—
66C2×Dic382—21no—
67C2×Dic382—21no—
68C2×Dic382—41no—
69C2×Dic382—41no—
70C2×Dic382C2311yescharacteristic
71C2×Dic382C2311yescharacteristic
72C2×Dic382C2311yescharacteristic
73C2×Dic382C2311yescharacteristic
74C2×D764—22no—
75C2×D764—21no—
76C2×D764—41no—
77C2×D764C2311yescharacteristic
78C2×D764C2311yescharacteristic
79C2×C22⋊D191—21no—
80C2×C22⋊D191—21no—
81C22×Dic191—21no—
82C22×Dic191—21no—
83C2×C4×D191—21no—
84C2×C4×D191—21no—
85Dic764—22no—
86Dic764—22no—
87Dic764—41no—
88#15690276—22no—
89#15690276—22no—
90#15690276—22no—
91#15690276—22no—
92#15690276—41no—
93#15690276D414yescomplemented
94C4⋊Dic191—21no—
95C4⋊Dic191—21no—
96C4⋊Dic191C2311yescharacteristic
97C4⋊Dic191C2311yescharacteristic
98Dic191⋊C4—21no—
99Dic191⋊C4—21no—
100Dic191⋊C4—21no—
101Dic191⋊C4—21no—
102C22⋊Dic191—21no—
103C22⋊Dic191—21no—
104C22⋊Dic191—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4—21no—
108(C2×D191)⋊C4—21no—
109C4×Dic191—21no—
110C2×D4×C191D411yescharacteristic
111C22×C4×C191C2311yescharacteristic
112C22×C4×C191C2311yescharacteristic
113M16×C191—22no—
114C2×C8×C191—22no—
115C2×C8×C191C2311yescharacteristic
116C2×C8×C191C2311yescharacteristic
117C2×C8×C191C2311yescharacteristic
118C2×C8×C191C2311yescharacteristic
119C4⋊C4×C191D411yescharacteristic
120C22⋊C4×C191D411yescharacteristic
121C22⋊C4×C191D411yescharacteristic
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191—21no—
125C22×D191—42no—
126C22×D191—42no—
127C22×D191—81no—
128Dic382—21no—
129Dic382—21no—
130Dic382—21no—
131Dic382—21no—
132Dic382—21no—
133Dic382—21no—
134Dic382—21no—
135Dic382—42no—
136Dic382—42no—
137Dic382C2×D412yes—
138D764—21no—
139D764—21no—
140D764—21no—
141D764—42no—
142D764C2×D412yes—
143C22⋊D191—42no—
144C22⋊D191—81no—
145C2×Dic191—21no—
146C2×Dic191—21no—
147C2×Dic191—21no—
148C2×Dic191—21no—
149C2×Dic191—21no—
150C2×Dic191—21no—
151C2×Dic191—42no—
152C2×Dic191—42no—
153C4×D191—42no—
154C4×D191—81no—
155C23×C191C2×D411yescharacteristic
156C23×C191C2×D411yescharacteristic
157D4×C191—22no—
158C2×C4×C191—21no—
159C2×C4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191C2×D411yescharacteristic
163C2×C4×C191C2×D411yescharacteristic
164C2×C4×C191C2×D411yescharacteristic
165C2×C4×C191C2×D411yescharacteristic
166C2×C4×C191C2411yescharacteristic
167C8×C191—21no—
168C8×C191—21no—
169C8×C191—21no—
170C8×C191C2×D412yes—
171C22×C191—21no—
172C22×C191—21no—
173C22×C191—21no—
174C22×C191—21no—
175C22×C191—21no—
176C22×C191#13712yescomplemented
177C22×C191C22×D411yescharacteristic
178C2×D191—22no—
179C2×D191—41no—
180C2×D191—41no—
181C2×D191—41no—
182C2×D191—41no—
183C2×D191—41no—
184C2×D191—84no—
185C4×C191—21no—
186C4×C191—21no—
187C4×C191—21no—
188C4×C191—21no—
189C4×C191C22×D411yescharacteristic
190C4×C191H(5,𝔽2)11yescharacteristic
191Dic191—22no—
192Dic191—41no—
193Dic191—41no—
194Dic191—41no—
195Dic191—41no—
196Dic191—41no—
197C2×C191—22no—
198C2×C191—41no—
199C2×C191D4211yescharacteristic
200C2×C191C2×#13711yescharacteristic
201C2×C191#57711yescharacteristic
202D191—42no—
203D191—81no—
204C191#326611yescharacteristic, complemented, Sylow
205#3266—1911nomaximal, Sylow
206C2×#137—1911no—
207C22×SD16—1911no—
208D4×Q8—1911no—
209#546—1911no—
210#497—1911no—
211#468—1911no—
212#467—1911no—
213#464—1911no—
214#461—1911no—
215#452—1911no—
216#451—1911no—
217#450—1911no—
218#448—1911no—
219#440—1911no—
220#435D19111yescharacteristic, complemented
221C2×#41—1911no—
222C22×Q8—1911no—
223C22×Q8—1911no—
224C22×D4—1911no—
225#137—3824no—
226C2×Q16—1911no—
227C2×Q16—1911no—
228C2×Q16—1911no—
229C2×SD16—3824no—
230C2×SD16—1911no—
231C2×SD16—1911no—
232C2×SD16—1911no—
233C2×SD16—1911no—
234C2×SD16—1911no—
235C2×SD16—1911no—
236C2×M16C2×D19111yescharacteristic, complemented
237C22×C8C2×D19111yescharacteristic
238C4⋊Q8—3821no—
239C4⋊Q8—1911no—
240#124—1911no—
241#123—3821no—
242C22⋊Q8—3821no—
243C22⋊Q8—3821no—
244C22⋊Q8—1911no—
245C22⋊Q8—1911no—
246C22⋊Q8—1911no—
247#121—3821no—
248#121—1911no—
249#120—3821no—
250C4×Q8—1911no—
251C4×D4—3821no—
252C4×D4—1911no—
253C4×D4C2×D19111yescharacteristic
254C2×C22⋊C4—3821no—
255#107—1911no—
256C4⋊C8C2×D19111yescharacteristic, complemented
257#103—1911no—
258#103—1911no—
259#103—1911no—
260#103—1911no—
261#102—1911no—
262#102—1911no—
263C22⋊C8C2×D19111yescharacteristic
264C22⋊C8C2×D19111yescharacteristic, complemented
265#97C2×D19111yescharacteristic, complemented
266C24—3821no—
267#41—3822no—
268#41—7641no—
269C2×Q8—3822no—
270C2×Q8—3822no—
271C2×Q8—3821no—
272C2×Q8—3821no—
273C2×Q8—1911no—
274C2×Q8—1911no—
275C2×Q8—1911no—
276C2×Q8—1911no—
277C2×Q8—7641no—
278C2×Q8—7641no—
279C2×D4—3822no—
280C2×D4—3821no—
281C2×D4—3821no—
282C2×D4—3821no—
283C2×D4—1911no—
284C2×D4—1911no—
285C2×D4—7641no—
286C2×D4D76411yescharacteristic
287C22×C4—3821no—
288C22×C4—3821no—
289C22×C4—3821no—
290C22×C4—3821no—
291C22×C4C22×D19111yescharacteristic
292C22×C4C22×D19111yescharacteristic
293Q16—3822no—
294Q16—3822no—
295Q16—7641no—
296SD16—3822no—
297SD16—3822no—
298SD16—3822no—
299SD16—3822no—
300SD16—1914no—
301SD16—7641no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D19111yescharacteristic
305C2×C8C22×D19111yescharacteristic
306C2×C8C22×D19111yescharacteristic
307C2×C8C22×D19111yescharacteristic
308C4⋊C4—3821no—
309C4⋊C4—3821no—
310C4⋊C4—3821no—
311C4⋊C4—3821no—
312C4⋊C4—3821no—
313C4⋊C4—3821no—
314C4⋊C4—1911no—
315C4⋊C4—1911no—
316C4⋊C4D76411yescharacteristic
317C22⋊C4—3821no—
318C22⋊C4—3821no—
319C22⋊C4—3821no—
320C22⋊C4—3821no—
321C22⋊C4—3821no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4D76411yescharacteristic
325C22⋊C4D76411yescharacteristic
326C42—3821no—
327C42C22×D19111yescharacteristic
328C23—3821no—
329C23—3821no—
330C23—15281no—
331C23—7642no—
332C23—7642no—
333C23C2×D76411yescharacteristic
334C23C2×D76411yescharacteristic
335Q8—3821no—
336Q8—3821no—
337Q8—3821no—
338Q8—3821no—
339Q8—3821no—
340Q8—3821no—
341Q8—3821no—
342Q8—1912no—
343Q8—7642no—
344Q8—7642no—
345D4—22no—
346D4—3821no—
347D4—3821no—
348D4—3821no—
349D4—1912no—
350D4—15281no—
351D4—7642no—
352D4—7642no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—3821no—
358C2×C4—3821no—
359C2×C4—3821no—
360C2×C4—3821no—
361C2×C4—3821no—
362C2×C4—3821no—
363C2×C4—15281no—
364C2×C4—7642no—
365C2×C4—7642no—
366C2×C4—7642no—
367C2×C4C2×D76411yescharacteristic
368C2×C4C2×D76411yescharacteristic
369C2×C4C2×D76411yescharacteristic
370C2×C4C2×D76411yescharacteristic
371C2×C4C23×D19111yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D19112yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—3822no—
382C22—15284no—
383C22—7641no—
384C22—7641no—
385C22—7641no—
386C22—7641no—
387C22—7641no—
388C22#2025935512yescomplemented
389C22C22×D76411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—3822no—
395C4—7641no—
396C4—7641no—
397C4—7641no—
398C4—7641no—
399C4—7641no—
400C4C2×D4×D19111yescharacteristic
401C4#2025936811yescharacteristic
402C2—22no—
403C2—41no—
404C2—15281no—
405C2—7642no—
406C2D4×D76411yescharacteristic
407C2C2×#2025935511yescharacteristic
408C2#2713347511yescharacteristic
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#435×C191C2
socleC22×C191C22×D4
prime radical#27133309C2
hypercenter#435D191
hypocenterC191#3266
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#20259355C2semidirect
2. C22×#15690276C2semidirect
3. D4×Dic382C2semidirect
5. #27133057C2semidirect
6. #27132909C2semidirect
7. #27132904C2semidirect
8. #27132886C2semidirect
9. #27132869C2semidirect
10. #27132827C2semidirect
11. #27132821C2semidirect
12. #27132816C2semidirect
13. #27132804C2semidirect
14. #27132765C2semidirect
15. #435×C191C2semidirect
18. C22×Dic382C22semidirect
21. C2×Dic764C22semidirect
22. C2×Dic764C22semidirect
23. C2×Dic764C22semidirect
31. #20259322C22semidirect
37. #20259301C22semidirect
43. C4×Dic382C22semidirect
47. #20259256C22semidirect
48. #20259247C22semidirect
49. #20259247C22semidirect
50. #20259247C22semidirect
51. #20259247C22semidirect
54. C2×M16×C191C22semidirect
57. C4⋊C8×C191C22semidirect
59. C22⋊C8×C191C22semidirect
60. #97×C191C22semidirect
93. #15690276D4semidirect
176. C22×C191#137semidirect
204. C191#3266semidirect
220. #435D191semidirect
236. C2×M16C2×D191semidirect
256. C4⋊C8C2×D191semidirect
264. C22⋊C8C2×D191semidirect
265. #97C2×D191semidirect
388. C22#20259355semidirect

Maximal subgroups

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

subgroupindex
C2×#202593552
C22×#156902762
D4×Dic3822
#271333092
#271330572
#271329092
#271329042
#271328862
#271328692
#271328272
#271328212
#271328162
#271328042
#271327652
#435×C1912
#3266191