Finite Group Databasebeta

← #37583413

Subgroups — #37583413

Order 24448.

378 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133237C211yesmaximal, characteristic
2#27133236C211yesmaximal, characteristic
3#27133005C211yesmaximal, characteristic, complemented
4#27133003C211yesmaximal, characteristic, complemented
5#27133002C211yesmaximal, characteristic, complemented
6#27133001C211yesmaximal, characteristic, complemented
7#27132920C211yesmaximal, characteristic, complemented
8#27132918C211yesmaximal, characteristic, complemented
9#27132903C211yesmaximal, characteristic, complemented
10#27132901C211yesmaximal, characteristic, complemented
11#27132819C211yesmaximal, characteristic, complemented
12#27132817C211yesmaximal, characteristic, complemented
13#27132817C211yesmaximal, characteristic, complemented
14#27132628C211yesmaximal, characteristic
15#451×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18C2×#15690277C2211yescharacteristic
19C2×#15690277C2211yescharacteristic
20C2×#15690270C2211yescharacteristic
21C2×C8×D191C2211yescharacteristic
22#20259321—21no—
23#20259320—21no—
24#20259311—21no—
25#20259308—21no—
26#20259307—21no—
27#20259305—21no—
28#20259305—21no—
29#20259304—21no—
30#20259302C2211yescharacteristic
31#20259301C2211yescharacteristic
32#20259298C2211yescharacteristic
33#20259298C2211yescharacteristic
34#20259298C2211yescharacteristic
35#20259298C2211yescharacteristic
36#20259298C2211yescharacteristic
37#20259281C2211yescharacteristic
38#20259280C2211yescharacteristic
39#20259256C2211yescharacteristic
40#20259255C2211yescharacteristic
41#20259254C2211yescharacteristic
42#20259253C2211yescharacteristic
43#20259247C2211yescharacteristic
44#20259247C2211yescharacteristic
45#20259246C2211yescharacteristic
46#20259246C2211yescharacteristic
47#20259246C2211yescharacteristic
48#20259246C2211yescharacteristic
49C22⋊(C191⋊C8)C2211yescharacteristic
50(C2×D191)⋊C8C2211yescharacteristic
51(C2×D191)⋊C8C2211yescharacteristic
52C22×Q8×C191C2211yescharacteristic
53C2×Q16×C191C2211yescharacteristic
54C2×Q16×C191C2211yescharacteristic
55C22⋊Q8×C191C2211yescharacteristic
56#103×C191C2211yescharacteristic
57#103×C191C2211yescharacteristic
58C22⋊C8×C191C2211yescharacteristic
59#15690287—41no—
60#15690287—41no—
61C2×Dic382—21no—
62C2×Dic382D411yescharacteristic
63C2×Dic382C2311yescharacteristic
64Q8×D191—41no—
65Q8×D191—41no—
66C2×D764D411yescharacteristic
67C2×C22⋊D191D411yescharacteristic
68C2×C22⋊D191D411yescharacteristic
69C2×C4×D191—21no—
70C2×C4×D191C2311yescharacteristic
71C2×C4×D191C2311yescharacteristic
72#15690277—41no—
73#15690277—41no—
74#15690270—41no—
75C2×C191⋊C8C2311yescharacteristic
76C2×C191⋊C8C2311yescharacteristic
77C8×D191—41no—
78C4⋊Dic191—21no—
79C4⋊Dic191C2311yescharacteristic
80C4⋊Dic191C2311yescharacteristic
81C4⋊Dic191C2311yescharacteristic
82Dic191⋊C4—21no—
83Dic191⋊C4—21no—
84Dic191⋊C4—21no—
85Dic191⋊C4—21no—
86Dic191⋊C4—21no—
87Dic191⋊C4—21no—
88C22⋊Dic191—21no—
89C22⋊Dic191—21no—
90(C2×D191)⋊C4—21no—
91(C2×D191)⋊C4—21no—
92(C2×D191)⋊C4—21no—
93(C2×D191)⋊C4—21no—
94(C2×D191)⋊C4—21no—
95(C2×D191)⋊C4—21no—
96(C2×D191)⋊C4—21no—
97C4×Dic191—21no—
98C4×Dic191—21no—
99C2×Q8×C191—21no—
100C2×Q8×C191—41no—
101C2×Q8×C191—41no—
102C2×Q8×C191C2311yescharacteristic
103C2×Q8×C191C2311yescharacteristic
104C2×Q8×C191C2311yescharacteristic
105C22×C4×C191—21no—
106C22×C4×C191C2311yescharacteristic
107Q16×C191—41no—
108Q16×C191—41no—
109C2×C8×C191C2311yescharacteristic
110C2×C8×C191C2311yescharacteristic
111C4⋊C4×C191—21no—
112C4⋊C4×C191C2311yescharacteristic
113C22⋊C4×C191—21no—
114C22×D191C2×D411yescharacteristic
115C22×D191C2×D411yescharacteristic
116Dic382—21no—
117Dic382—21no—
118Dic382—41no—
119Dic382—42no—
120D764—41no—
121C22⋊D191—41no—
122C22⋊D191—41no—
123C2×Dic191—21no—
124C2×Dic191—21no—
125C2×Dic191—21no—
126C2×Dic191—21no—
127C2×Dic191C2×D411yescharacteristic
128C2×Dic191C2×D411yescharacteristic
129C4×D191—21no—
130C4×D191—21no—
131C4×D191—21no—
132C4×D191—21no—
133C4×D191—42no—
134C191⋊C8—21no—
135C191⋊C8—21no—
136C23×C191C2×D411yescharacteristic
137Q8×C191—21no—
138Q8×C191—21no—
139Q8×C191—21no—
140Q8×C191—21no—
141Q8×C191—21no—
142Q8×C191—21no—
143Q8×C191—42no—
144C2×C4×C191—21no—
145C2×C4×C191—21no—
146C2×C4×C191—21no—
147C2×C4×C191—21no—
148C2×C4×C191—21no—
149C2×C4×C191—21no—
150C2×C4×C191—42no—
151C2×C4×C191C2×D411yescharacteristic
152C2×C4×C191C2411yescharacteristic
153C8×C191—21no—
154C8×C191—21no—
155C22×C191—21no—
156C22×C191—21no—
157C22×C191—21no—
158C22×C191C22×D411yescharacteristic
159C2×D191—21no—
160C2×D191—21no—
161C2×D191—21no—
162C2×D191—21no—
163C2×D191—21no—
164C2×D191—21no—
165C4×C191—21no—
166C4×C191—41no—
167C4×C191—41no—
168C4×C191—41no—
169C4×C191—41no—
170C4×C191H(5,𝔽2)11yescharacteristic
171C4×C191H(5,𝔽2)11yescharacteristic
172Dic191—21no—
173Dic191—21no—
174Dic191—41no—
175Dic191—41no—
176Dic191—41no—
177Dic191—41no—
178C2×C191—41no—
179C2×C191#53411yescharacteristic
180C2×C191#57811yescharacteristic
181C2×C191#57811yescharacteristic
182D191—41no—
183D191—41no—
184C191#320211yescharacteristic, complemented, Sylow
185#3202—1911nomaximal, Sylow
186#536—1911no—
187#536—1911no—
188#485—1911no—
189#485—1911no—
190#485—1911no—
191#485—1911no—
192#470—1911no—
193#470—1911no—
194#467—1911no—
195#467—1911no—
196#451—1911no—
197#451—1911no—
198#451—1911no—
199#451D19111yescharacteristic, complemented
200#408—1911no—
201C2×#41—1911no—
202C22×Q8—1911no—
203C22×Q8C2×D19111yescharacteristic
204C2×Q16—1911no—
205C2×Q16—1911no—
206C2×Q16C2×D19111yescharacteristic
207C2×Q16C2×D19111yescharacteristic
208C2×M16—1911no—
209C22×C8—1911no—
210C4⋊Q8—3821no—
211C4⋊Q8—3821no—
212#124—3821no—
213#124—3821no—
214#123—3821no—
215#123—3821no—
216#123—3821no—
217#123—3821no—
218C22⋊Q8—1911no—
219C22⋊Q8—1911no—
220C22⋊Q8—1911no—
221C22⋊Q8—1911no—
222C22⋊Q8—1911no—
223C22⋊Q8—1911no—
224C22⋊Q8—1911no—
225C22⋊Q8C2×D19111yescharacteristic
226#117—1911no—
227#117—1911no—
228#107—1911no—
229#107—1911no—
230#106—1911no—
231#106—1911no—
232#103—1911no—
233#103—1911no—
234#103—1911no—
235#103—1911no—
236#103—1911no—
237#103—1911no—
238#103C2×D19111yescharacteristic
239#103C2×D19111yescharacteristic
240C22⋊C8—1911no—
241C22⋊C8—1911no—
242C22⋊C8—1911no—
243C22⋊C8C2×D19111yescharacteristic
244#41—7641no—
245#41—7641no—
246C2×Q8—21no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—3821no—
250C2×Q8—1911no—
251C2×Q8—1911no—
252C2×Q8—7641no—
253C2×Q8—7641no—
254C2×Q8C22×D19111yescharacteristic
255C2×Q8C22×D19111yescharacteristic
256C2×Q8C22×D19111yescharacteristic
257C2×D4—1911no—
258C2×D4—1911no—
259C2×D4—1911no—
260C22×C4—21no—
261C22×C4—3821no—
262C22×C4—1911no—
263C22×C4—1911no—
264C22×C4C22×D19111yescharacteristic
265Q16—41no—
266Q16—41no—
267Q16—7641no—
268Q16—7641no—
269M16—7641no—
270C2×C8—1911no—
271C2×C8—1911no—
272C2×C8—7641no—
273C2×C8C22×D19111yescharacteristic
274C2×C8C22×D19111yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—3821no—
277C4⋊C4—3821no—
278C4⋊C4—3821no—
279C4⋊C4—3821no—
280C4⋊C4—3821no—
281C4⋊C4—3821no—
282C4⋊C4—3821no—
283C4⋊C4—1911no—
284C4⋊C4—1911no—
285C4⋊C4—1911no—
286C4⋊C4C22×D19111yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—3821no—
289C22⋊C4—3821no—
290C22⋊C4—3821no—
291C22⋊C4—3821no—
292C22⋊C4—3821no—
293C22⋊C4—3821no—
294C22⋊C4—3821no—
295C22⋊C4—3821no—
296C22⋊C4—3821no—
297C42—3821no—
298C42—3821no—
299C23—1911no—
300C23—1911no—
301C23D4×D19111yescharacteristic
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—42no—
309Q8—3821no—
310Q8—3821no—
311Q8—7641no—
312Q8—7642no—
313D4—7641no—
314D4—7641no—
315D4—7641no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—3821no—
324C2×C4—3821no—
325C2×C4—3821no—
326C2×C4—3821no—
327C2×C4—3821no—
328C2×C4—3821no—
329C2×C4—3821no—
330C2×C4—3821no—
331C2×C4—1911no—
332C2×C4—1911no—
333C2×C4—7642no—
334C2×C4D4×D19111yescharacteristic
335C2×C4C23×D19111yescharacteristic
336C8—21no—
337C8—21no—
338C8—3821no—
339C8—3821no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—3821no—
344C22—3821no—
345C22—3821no—
346C22—3821no—
347C22—3821no—
348C22—3821no—
349C22C2×D4×D19111yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3821no—
356C4—3821no—
357C4—7641no—
358C4—7641no—
359C4—7641no—
360C4—7641no—
361C4#2025936711yescharacteristic
362C4#2025936711yescharacteristic
363C2—41no—
364C2—7641no—
365C2—7641no—
366C2#2713322311yescharacteristic
367C2#2713347611yescharacteristic
368C2#2713347911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#451×C191C2
socleC22×C191C22×D4
prime radicalC2×#15690287C22
hypercenter#451D191
hypocenterC191#3202
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #27133005C2semidirect
4. #27133003C2semidirect
5. #27133002C2semidirect
6. #27133001C2semidirect
7. #27132920C2semidirect
8. #27132918C2semidirect
9. #27132903C2semidirect
10. #27132901C2semidirect
11. #27132819C2semidirect
12. #27132817C2semidirect
13. #27132817C2semidirect
15. #451×C191C2semidirect
184. C191#3202semidirect
199. #451D191semidirect

Maximal subgroups

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

subgroupindex
#271332372
#271332362
#271330052
#271330032
#271330022
#271330012
#271329202
#271329182
#271329032
#271329012
#271328192
#271328172
#271326282
#451×C1912
#3202191