Finite Group Databasebeta

← #37575830

Subgroups — #37575830

Order 24448.

1564 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊D191C211yesmaximal, characteristic, complemented
2C2×#20259291C212yesmaximal, complemented
3C2×#20259290C212yesmaximal, complemented
4#27132571C214yesmaximal, complemented
5#27132543C212yesmaximal, complemented
6#27132542C212yesmaximal, complemented
7#27132519C211yesmaximal, characteristic, complemented
8C23×D4×C191C211yesmaximal, characteristic, complemented
9C24×D191C2211yescharacteristic, complemented
10C22×C22⋊D191—28no—
11C22×C22⋊D191—24no—
12C22×C22⋊D191C2212yes—
13C22×C22⋊D191C2212yescomplemented
14C22×C22⋊D191C2214yescomplemented
15C23×Dic191C2211yescharacteristic, complemented
16#20259291—28no—
17#20259290—28no—
18C2×C22⋊Dic191C2212yescomplemented
19C2×C22⋊Dic191C2212yescomplemented
20C2×C22⋊Dic191C2214yescomplemented
21C2×C22⋊Dic191C2212yescomplemented
22C2×(C2×D191)⋊C4C2214yescomplemented
23C191⋊#95C2214yescomplemented
24C25×C191C2211yescharacteristic
25C25×C191C2211yescharacteristic, complemented
26C22×D4×C191—24no—
27C22×D4×C191—28no—
28C22×D4×C191C2212yes—
29C22×D4×C191C2212yescomplemented
30C23×C4×C191C2211yescharacteristic, complemented
31C23×D191—24no—
32C23×D191—22no—
33C23×D191—24no—
34C23×D191—42no—
35C23×D191C2312yes—
36C2×C22⋊D191—232no—
37C2×C22⋊D191—24no—
38C2×C22⋊D191—28no—
39C2×C22⋊D191—24no—
40C2×C22⋊D191—24no—
41C2×C22⋊D191—24no—
42C2×C22⋊D191—24no—
43C2×C22⋊D191—28no—
44C2×C22⋊D191—44no—
45C2×C22⋊D191D418yescomplemented
46C22×Dic191—24no—
47C22×Dic191—22no—
48C22×Dic191C2314yes—
49C22×Dic191C2312yes—
50C22⋊Dic191—28no—
51C22⋊Dic191—24no—
52C22⋊Dic191—24no—
53C22⋊Dic191—42no—
54(C2×D191)⋊C4—28no—
55C24×C191—24no—
56C24×C191—24no—
57C24×C191—22no—
58C24×C191—24no—
59C24×C191—24no—
60C24×C191—42no—
61C24×C191—42no—
62C24×C191D414yescomplemented
63C24×C191C2311yescharacteristic
64C24×C191C2312yes—
65C24×C191C2311yescharacteristic
66C24×C191C2311yescharacteristic
67C2×D4×C191—232no—
68C2×D4×C191—24no—
69C2×D4×C191—24no—
70C2×D4×C191—24no—
71C2×D4×C191—24no—
72C2×D4×C191—24no—
73C2×D4×C191D418yescomplemented
74C22×C4×C191—22no—
75C22×C4×C191—24no—
76C22×C4×C191C2312yes—
77C22×D191—28no—
78C22×D191—24no—
79C22×D191—22no—
80C22×D191—22no—
81C22×D191—24no—
82C22×D191—416no—
83C22×D191—44no—
84C22×D191—44no—
85C22×D191C2×D414yescomplemented
86C22⋊D191—216no—
87C22⋊D191—216no—
88C22⋊D191—416no—
89C2×Dic191—28no—
90C2×Dic191—22no—
91C2×Dic191—22no—
92C2×Dic191—24no—
93C2×Dic191—24no—
94C2×Dic191—24no—
95C2×Dic191C2×D414yescomplemented
96C23×C191—22no—
97C23×C191—22no—
98C23×C191—22no—
99C23×C191—22no—
100C23×C191—28no—
101C23×C191—28no—
102C23×C191—28no—
103C23×C191—22no—
104C23×C191—22no—
105C23×C191—22no—
106C23×C191—24no—
107C23×C191—22no—
108C23×C191—22no—
109C23×C191—24no—
110C23×C191—44no—
111C23×C191—416no—
112C23×C191—416no—
113C23×C191—44no—
114C23×C191—44no—
115C23×C191C2×D414yes—
116C23×C191C2×D412yes—
117C23×C191C2×D412yes—
118C23×C191C2×D412yes—
119C23×C191C2×D414yes—
120C23×C191C2×D414yescomplemented
121C23×C191C2411yescharacteristic
122D4×C191—216no—
123D4×C191—216no—
124C2×C4×C191—28no—
125C2×C4×C191—22no—
126C2×C4×C191—22no—
127C2×C4×C191C2×D414yescomplemented
128C22×C191—22no—
129C22×C191—22no—
130C22×C191—22no—
131C22×C191—22no—
132C22×C191—28no—
133C22×C191—28no—
134C22×C191—22no—
135C22×C191—24no—
136C22×C191—22no—
137C22×C191—22no—
138C22×C191—28no—
139C22×C191—416no—
140C22×C191—416no—
141C22×C191—42no—
142C22×C191—42no—
143C22×C191—44no—
144C22×C191—44no—
145C22×C191#12014yescomplemented
146C22×C191#12014yescomplemented
147C22×C191C22×D412yes—
148C22×C191C22×D411yescharacteristic
149C22×C191C22×D412yes—
150C22×C191H(5,𝔽2)12yes—
151C2×D191—24no—
152C2×D191—28no—
153C2×D191—416no—
154C2×D191—42no—
155C2×D191—42no—
156C2×D191—44no—
157C4×C191—24no—
158Dic191—24no—
159Dic191—44no—
160C2×C191—24no—
161C2×C191—24no—
162C2×C191—42no—
163C2×C191—44no—
164C2×C191C2×#12011yescharacteristic
165C2×C191C2×#12011yescharacteristic
166C2×C191#53411yescharacteristic
167C2×C191D4214yes—
168D191—44no—
169C191#240311yescharacteristic, complemented, Sylow
170#2403—1911nomaximal, Sylow
171C23×D4—1911no—
172C23×D4D19111yescharacteristic, complemented
173C2×#120—1912no—
174C2×#120—1912no—
175#392—1914no—
176#386—1912no—
177#386—1912no—
178#379—1911no—
179C25—1911no—
180C25C2×D19111yescharacteristic
181C25C2×D19111yescharacteristic, complemented
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—3824no—
185C22×D4—3828no—
186C22×D4—1912no—
187C22×D4—1912no—
188C22×D4—1914no—
189C22×D4C2×D19112yes—
190C22×D4C2×D19112yescomplemented
191C23×C4—1911no—
192C23×C4C2×D19111yescharacteristic, complemented
193#120—3828no—
194#120—3828no—
195C2×C22⋊C4—1912no—
196C2×C22⋊C4—1912no—
197C2×C22⋊C4—1914no—
198C2×C22⋊C4—1914no—
199C2×C22⋊C4—1912no—
200#95—1914no—
201C24—24no—
202C24—24no—
203C24—22no—
204C24—24no—
205C24—24no—
206C24—42no—
207C24—42no—
208C24—3824no—
209C24—3822no—
210C24—3824no—
211C24—1912no—
212C24—7642no—
213C24C22⋊D19114yescomplemented
214C24C22×D19111yescharacteristic
215C24C22×D19112yes—
216C24C22×D19111yescharacteristic
217C24C22×D19111yescharacteristic
218C2×D4—232no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—38232no—
225C2×D4—3824no—
226C2×D4—3824no—
227C2×D4—3824no—
228C2×D4—3824no—
229C2×D4—3828no—
230C2×D4—3828no—
231C2×D4—3824no—
232C2×D4—1918no—
233C2×D4—7644no—
234C2×D4C22⋊D19118yescomplemented
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—3822no—
238C22×C4—3824no—
239C22×C4—1914no—
240C22×C4—1912no—
241C22×C4C22×D19112yes—
242C22⋊C4—3824no—
243C22⋊C4—3824no—
244C22⋊C4—3828no—
245C22⋊C4—3828no—
246C22⋊C4—7642no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—28no—
252C23—24no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—24no—
259C23—28no—
260C23—28no—
261C23—44no—
262C23—416no—
263C23—416no—
264C23—44no—
265C23—44no—
266C23—3828no—
267C23—3822no—
268C23—3824no—
269C23—3822no—
270C23—3824no—
271C23—1914no—
272C23—76416no—
273C23—7644no—
274C23—7644no—
275C23D4×D19114yes—
276C23C2×C22⋊D19112yes—
277C23C2×C22⋊D19112yes—
278C23C2×C22⋊D19114yes—
279C23C2×C22⋊D19112yes—
280C23C2×C22⋊D19114yescomplemented
281C23C23×D19111yescharacteristic
282D4—216no—
283D4—216no—
284D4—38216no—
285D4—38216no—
286D4—76416no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—3828no—
291C2×C4—3822no—
292C2×C4—3822no—
293C2×C4—3824no—
294C2×C4—3824no—
295C2×C4—3824no—
296C2×C4—1914no—
297C2×C4C2×C22⋊D19114yescomplemented
298C22—22no—
299C22—22no—
300C22—28no—
301C22—28no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—28no—
306C22—22no—
307C22—22no—
308C22—24no—
309C22—416no—
310C22—416no—
311C22—42no—
312C22—44no—
313C22—44no—
314C22—42no—
315C22—3828no—
316C22—3824no—
317C22—76416no—
318C22—7642no—
319C22—7644no—
320C22—7642no—
321C22#2025929014yescomplemented
322C22#2025929114yescomplemented
323C22C2×D4×D19112yes—
324C22C22×C22⋊D19111yescharacteristic
325C22C22×C22⋊D19112yes—
326C22#2025936712yes—
327C4—24no—
328C4—3824no—
329C4—7644no—
330C2—24no—
331C2—24no—
332C2—42no—
333C2—44no—
334C2—7644no—
335C2C2×#2025929011yescharacteristic
336C2C2×#2025929111yescharacteristic
337C2#2713322211yescharacteristic
338C2D4×C22⋊D19114yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
FittingC23×D4×C191C2
hypercenterC23×D4D191
hypocenterC191#2403
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D191C2semidirect
2. C2×#20259291C2semidirect
3. C2×#20259290C2semidirect
4. #27132571C2semidirect
5. #27132543C2semidirect
6. #27132542C2semidirect
7. #27132519C2semidirect
8. C23×D4×C191C2semidirect
9. C24×D191C22semidirect
13. C22×C22⋊D191C22semidirect
14. C22×C22⋊D191C22semidirect
15. C23×Dic191C22semidirect
18. C2×C22⋊Dic191C22semidirect
19. C2×C22⋊Dic191C22semidirect
20. C2×C22⋊Dic191C22semidirect
21. C2×C22⋊Dic191C22semidirect
22. C2×(C2×D191)⋊C4C22semidirect
23. C191⋊#95C22semidirect
25. C25×C191C22semidirect
29. C22×D4×C191C22semidirect
30. C23×C4×C191C22semidirect
45. C2×C22⋊D191D4semidirect
62. C24×C191D4semidirect
73. C2×D4×C191D4semidirect
85. C22×D191C2×D4semidirect
95. C2×Dic191C2×D4semidirect
120. C23×C191C2×D4semidirect
127. C2×C4×C191C2×D4semidirect
145. C22×C191#120semidirect
146. C22×C191#120semidirect
169. C191#2403semidirect
172. C23×D4D191semidirect
181. C25C2×D191semidirect
190. C22×D4C2×D191semidirect
192. C23×C4C2×D191semidirect
213. C24C22⋊D191semidirect
234. C2×D4C22⋊D191semidirect
280. C23C2×C22⋊D191semidirect
297. C2×C4C2×C22⋊D191semidirect
321. C22#20259290semidirect
322. C22#20259291semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1912
C2×#202592912
C2×#202592902
#271325712
#271325432
#271325422
#271325192
C23×D4×C1912
#2403191