Finite Group Databasebeta

← #37584101

Subgroups — #37584101

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×#20259352C211yesmaximal, characteristic, complemented
2C2×#20259342C211yesmaximal, characteristic, complemented
3#27133322C211yesmaximal, characteristic, complemented
4#27133050C211yesmaximal, characteristic, complemented
5#27132914C211yesmaximal, characteristic, complemented
6#27132890C211yesmaximal, characteristic, complemented
7#27132885C211yesmaximal, characteristic, complemented
8#27132860C211yesmaximal, characteristic, complemented
9#27132825C211yesmaximal, characteristic, complemented
10#27132808C211yesmaximal, characteristic, complemented
11#27132806C211yesmaximal, characteristic, complemented
12#27132801C211yesmaximal, characteristic, complemented
13#27132768C211yesmaximal, characteristic, complemented
14#27132735C211yesmaximal, characteristic, complemented
15#550×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C22×Dic382C2211yescharacteristic
18#20259352—24no—
19#20259342—24no—
20C2×#15690277C2211yescharacteristic
21C2×#15690277C2211yescharacteristic
22C2×#15690275C2211yescharacteristic
23C2×#15690275C2211yescharacteristic
24C2×#15690275C2211yescharacteristic
25C2×#15690275C2211yescharacteristic
26C2×#15690274C2211yescharacteristic
27C2×#15690272C2211yescharacteristic
28C2×#15690272C2211yescharacteristic
29C2×#15690271C2211yescharacteristic
30C22×C191⋊C8C2211yescharacteristic
31#20259310C2211yescharacteristic
32#20259308—21no—
33#20259301C2211yescharacteristic
34#20259300—21no—
35#20259296C2211yescharacteristic
36#20259294—21no—
37C4×Dic382C2211yescharacteristic
38(C22⋊D191)⋊C4—21no—
39#20259253C2211yescharacteristic
40C4⋊(C191⋊C8)C2211yescharacteristic
41#20259246C2211yescharacteristic
42#20259245C2211yescharacteristic
43#20259245C2211yescharacteristic
44#20259243C2211yescharacteristic
45#20259243C2211yescharacteristic
46#20259242C2211yescharacteristic
47C22⋊(C191⋊C8)C2211yescharacteristic
48C22⋊(C191⋊C8)C2211yescharacteristic
49#20259233C2211yescharacteristic
50C2×#41×C191C2211yescharacteristic
51C2×#41×C191C2211yescharacteristic
52#127×C191—21no—
53#127×C191C2211yescharacteristic
54#121×C191—21no—
55#121×C191—21no—
56#121×C191C2211yescharacteristic
57#121×C191C2211yescharacteristic
58C4×Q8×C191C2211yescharacteristic
59C4×D4×C191—21no—
60C4×D4×C191C2211yescharacteristic
61#15690287—22no—
62#15690287—41no—
63C2×Dic382—22no—
64C2×Dic382—21no—
65C2×Dic382—41no—
66C2×Dic382C2311yescharacteristic
67C2×Dic382C2311yescharacteristic
68C2×D764C2311yescharacteristic
69C2×C22⋊D191—21no—
70C2×C22⋊D191—21no—
71C22×Dic191—21no—
72C2×C4×D191—21no—
73#15690277—22no—
74#15690277D414yescomplemented
75#15690275—22no—
76#15690275—22no—
77#15690275—22no—
78#15690275—41no—
79#15690274—22no—
80#15690272—22no—
81#15690272—41no—
82#15690271—22no—
83C2×C191⋊C8—22no—
84C2×C191⋊C8C2311yescharacteristic
85C2×C191⋊C8C2311yescharacteristic
86C2×C191⋊C8C2311yescharacteristic
87C2×C191⋊C8C2311yescharacteristic
88C4⋊Dic191C2311yescharacteristic
89Dic191⋊C4—21no—
90Dic191⋊C4—21no—
91C22⋊Dic191—21no—
92(C2×D191)⋊C4—21no—
93(C2×D191)⋊C4—21no—
94(C2×D191)⋊C4—21no—
95C4×Dic191—21no—
96#41×C191—22no—
97#41×C191—22no—
98#41×C191—41no—
99#41×C191—41no—
100C2×Q8×C191C2311yescharacteristic
101C2×D4×C191—21no—
102C2×D4×C191—21no—
103C2×D4×C191—21no—
104C2×D4×C191—21no—
105C2×D4×C191—21no—
106C2×D4×C191—21no—
107C2×D4×C191D411yescharacteristic
108C2×D4×C191C2311yescharacteristic
109C2×D4×C191C2311yescharacteristic
110C22×C4×C191—21no—
111C22×C4×C191—21no—
112C22×C4×C191C2311yescharacteristic
113C22×C4×C191C2311yescharacteristic
114C4⋊C4×C191—21no—
115C4⋊C4×C191D411yescharacteristic
116C4⋊C4×C191C2311yescharacteristic
117C22⋊C4×C191—21no—
118C22⋊C4×C191—21no—
119C22⋊C4×C191D411yescharacteristic
120C22⋊C4×C191D411yescharacteristic
121C42×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124Dic382—21no—
125Dic382—21no—
126Dic382—21no—
127Dic382—42no—
128Dic382C2×D412yes—
129D764—21no—
130D764—21no—
131C22⋊D191—42no—
132C22⋊D191—42no—
133C2×Dic191—21no—
134C2×Dic191—21no—
135C2×Dic191—21no—
136C2×Dic191—42no—
137C4×D191—42no—
138C191⋊C8—21no—
139C191⋊C8—21no—
140C191⋊C8—21no—
141C191⋊C8C2×D412yes—
142C23×C191—21no—
143C23×C191—21no—
144C23×C191C2×D411yescharacteristic
145C23×C191C2×D411yescharacteristic
146Q8×C191—21no—
147Q8×C191C2×D412yes—
148D4×C191—21no—
149D4×C191—21no—
150D4×C191—21no—
151D4×C191—21no—
152D4×C191—41no—
153D4×C191—42no—
154D4×C191—42no—
155D4×C191—42no—
156D4×C191—81no—
157D4×C191—81no—
158D4×C191—81no—
159C2×C4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—42no—
166C2×C4×C191—42no—
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2×D411yescharacteristic
169C2×C4×C191C2×D411yescharacteristic
170C2×C4×C191C2×D411yescharacteristic
171C2×C4×C191C2411yescharacteristic
172C22×C191—21no—
173C22×C191—21no—
174C22×C191—21no—
175C22×C191—21no—
176C22×C191—21no—
177C22×C191—21no—
178C22×C191—41no—
179C22×C191—41no—
180C22×C191—41no—
181C22×C191—41no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191C22×D411yescharacteristic
185C2×D191—41no—
186C2×D191—41no—
187C2×D191—41no—
188C4×C191—21no—
189C4×C191—21no—
190C4×C191—22no—
191C4×C191—21no—
192C4×C191—21no—
193C4×C191—41no—
194C4×C191C22×D411yescharacteristic
195C4×C191H(5,𝔽2)11yescharacteristic
196Dic191—22no—
197Dic191—41no—
198Dic191—41no—
199C2×C191—41no—
200C2×C191—41no—
201C2×C191—81no—
202C2×C191—81no—
203C2×C191D4211yescharacteristic
204C2×C191#57511yescharacteristic
205C2×C191#57711yescharacteristic
206D191—81no—
207C191#327111yescharacteristic, complemented, Sylow
208#3271—1911nomaximal, Sylow
209C2×#137—1911no—
210C2×#135—1911no—
211#550D19111yescharacteristic, complemented
212#548—1911no—
213#496—1911no—
214#469—1911no—
215#465—1911no—
216#464—1911no—
217#460—1911no—
218#452—1911no—
219#449—1911no—
220#449—1911no—
221#448—1911no—
222#441—1911no—
223#435—1911no—
224C2×#41—1911no—
225C2×#41C2×D19111yescharacteristic
226C2×#41C2×D19111yescharacteristic
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×SD16—1911no—
236C2×SD16—1911no—
237C2×D8—1911no—
238C2×D8—1911no—
239C2×M16—1911no—
240C22×C8—1911no—
241#127—21no—
242#127C2×D19111yescharacteristic
243#124—3821no—
244#124—1911no—
245C22⋊Q8—3821no—
246C22⋊Q8—1911no—
247#121—21no—
248#121—21no—
249#121—3821no—
250#121—1911no—
251#121C2×D19111yescharacteristic
252#121C2×D19111yescharacteristic
253C4×Q8—1911no—
254C4×Q8C2×D19111yescharacteristic
255C4×D4—21no—
256C4×D4—3821no—
257C4×D4C2×D19111yescharacteristic
258#106—1911no—
259C4⋊C8—1911no—
260#103—1911no—
261#103—1911no—
262#103—1911no—
263#102—1911no—
264#102—1911no—
265#102—1911no—
266C22⋊C8—1911no—
267C22⋊C8—1911no—
268#97—1911no—
269#41—22no—
270#41—22no—
271#41—41no—
272#41—41no—
273#41—3822no—
274#41—7641no—
275C2×Q8—3822no—
276C2×Q8—3821no—
277C2×Q8—1911no—
278C2×Q8—1911no—
279C2×Q8—7641no—
280C2×Q8C22×D19111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—3821no—
288C2×D4—3821no—
289C2×D4—1911no—
290C2×D4C22⋊D19111yescharacteristic
291C2×D4C22×D19111yescharacteristic
292C2×D4C22×D19111yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—3821no—
296C22×C4—3821no—
297C22×C4C22×D19111yescharacteristic
298C22×C4C22×D19111yescharacteristic
299Q16—3822no—
300Q16—1914no—
301SD16—3822no—
302SD16—3822no—
303SD16—3822no—
304SD16—3822no—
305SD16—7641no—
306D8—3822no—
307D8—7641no—
308M16—3822no—
309C2×C8—3822no—
310C2×C8—1911no—
311C2×C8—1911no—
312C2×C8—1911no—
313C2×C8—1911no—
314C4⋊C4—21no—
315C4⋊C4—3821no—
316C4⋊C4—3821no—
317C4⋊C4—1911no—
318C4⋊C4C22⋊D19111yescharacteristic
319C4⋊C4C22×D19111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4—3821no—
325C22⋊C4—3821no—
326C22⋊C4C22⋊D19111yescharacteristic
327C22⋊C4C22⋊D19111yescharacteristic
328C42—21no—
329C42—3821no—
330C42C22×D19111yescharacteristic
331C23—21no—
332C23—21no—
333C23—3821no—
334C23C2×C22⋊D19111yescharacteristic
335C23C2×C22⋊D19111yescharacteristic
336Q8—21no—
337Q8—3821no—
338Q8—3821no—
339Q8—3821no—
340Q8—1912no—
341Q8—7642no—
342Q8D4×D19112yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—3821no—
355D4—3821no—
356D4—7642no—
357D4—7642no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—3821no—
367C2×C4—3821no—
368C2×C4—3821no—
369C2×C4—7642no—
370C2×C4—7642no—
371C2×C4C2×C22⋊D19111yescharacteristic
372C2×C4C2×C22⋊D19111yescharacteristic
373C2×C4C2×C22⋊D19111yescharacteristic
374C2×C4C2×C22⋊D19111yescharacteristic
375C2×C4C23×D19111yescharacteristic
376C8—3821no—
377C8—3821no—
378C8—3821no—
379C8—1912no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—7641no—
393C22—7641no—
394C22—7641no—
395C22C22×C22⋊D19111yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—22no—
401C4—41no—
402C4—3822no—
403C4—7641no—
404C4—7641no—
405C4C2×D4×D19111yescharacteristic
406C4#2025936711yescharacteristic
407C2—41no—
408C2—41no—
409C2—81no—
410C2—81no—
411C2—15281no—
412C2D4×C22⋊D19111yescharacteristic
413C2#2713346111yescharacteristic
414C2#2713347411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#550×C191C2
socleC22×C191C22×D4
hypercenter#550D191
hypocenterC191#3271
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#20259352C2semidirect
2. C2×#20259342C2semidirect
3. #27133322C2semidirect
4. #27133050C2semidirect
5. #27132914C2semidirect
6. #27132890C2semidirect
7. #27132885C2semidirect
8. #27132860C2semidirect
9. #27132825C2semidirect
10. #27132808C2semidirect
11. #27132806C2semidirect
12. #27132801C2semidirect
13. #27132768C2semidirect
14. #27132735C2semidirect
15. #550×C191C2semidirect
74. #15690277D4semidirect
207. C191#3271semidirect
211. #550D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593522
C2×#202593422
#271333222
#271330502
#271329142
#271328902
#271328852
#271328602
#271328252
#271328082
#271328062
#271328012
#271327682
#271327352
#550×C1912
#3271191