Finite Group Databasebeta

← #37575829

Subgroups — #37575829

Order 24448.

1564 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×D4×D191C212yesmaximal, complemented
2C2×#20259292C212yesmaximal, complemented
3C2×#20259290C212yesmaximal, complemented
4#27132572C211yesmaximal, characteristic, complemented
5#27132571C212yesmaximal, complemented
6#27132541C212yesmaximal, complemented
7#27132539C212yesmaximal, complemented
8#27132518C211yesmaximal, characteristic, complemented
9#392×C191C211yesmaximal, characteristic, complemented
10C24×D191C2211yescharacteristic
11C24×D191C2212yescomplemented
12C22×D764C2212yes—
13C22×D764C2211yescharacteristic, complemented
14C22×C22⋊D191C2212yes—
15C22×C22⋊D191C2212yescomplemented
16C22×C22⋊D191C2212yescomplemented
17C2×D4×D191—28no—
18C2×D4×D191—28no—
19C2×D4×D191—28no—
20C22×C4×D191C2212yescomplemented
21#20259292—28no—
22#20259290—28no—
23C2×C22⋊Dic191C2212yescomplemented
24C2×C22⋊Dic191C2211yescharacteristic, complemented
25C2×(C2×D191)⋊C4C2212yescomplemented
26C2×(C2×D191)⋊C4C2212yescomplemented
27C2×(C2×D191)⋊C4C2212yescomplemented
28C2×(C2×D191)⋊C4C2212yescomplemented
29C191⋊#95C2211yescharacteristic, complemented
30C191⋊#95C2212yescomplemented
31C22×D4×C191C2212yescomplemented
32C22×D4×C191C2211yescharacteristic, complemented
33C2×C22⋊C4×C191C2212yescomplemented
34C2×C22⋊C4×C191C2211yescharacteristic, complemented
35#95×C191C2211yescharacteristic, complemented
36C23×D191—24no—
37C23×D191—24no—
38C23×D191—24no—
39C23×D191—24no—
40C23×D191—24no—
41C23×D191—24no—
42C23×D191—24no—
43C23×D191—41no—
44C23×D191—41no—
45C23×D191—42no—
46C23×D191—42no—
47C23×D191D414yescomplemented
48C23×D191C2312yes—
49C23×D191C2312yes—
50C2×D764—24no—
51C2×D764—24no—
52C2×D764—24no—
53C2×D764—41no—
54C2×D764D418yescomplemented
55C2×C22⋊D191—24no—
56C2×C22⋊D191—24no—
57C2×C22⋊D191—24no—
58C2×C22⋊D191—24no—
59C2×C22⋊D191—24no—
60C2×C22⋊D191—24no—
61C2×C22⋊D191—24no—
62C2×C22⋊D191—42no—
63C2×C22⋊D191D418yescomplemented
64D4×D191—264no—
65C22×Dic191C2312yes—
66C22×Dic191C2312yes—
67C2×C4×D191—24no—
68C2×C4×D191—24no—
69C2×C4×D191—24no—
70C22⋊Dic191—24no—
71C22⋊Dic191—41no—
72(C2×D191)⋊C4—24no—
73(C2×D191)⋊C4—24no—
74(C2×D191)⋊C4—24no—
75(C2×D191)⋊C4—24no—
76C24×C191C2311yescharacteristic
77C24×C191C2312yes—
78C2×D4×C191—24no—
79C2×D4×C191—24no—
80C2×D4×C191—24no—
81C2×D4×C191—24no—
82C2×D4×C191—41no—
83C22×C4×C191C2311yescharacteristic
84C22×C4×C191C2311yescharacteristic
85C22×C4×C191C2312yes—
86C22⋊C4×C191—24no—
87C22⋊C4×C191—41no—
88C22×D191—22no—
89C22×D191—22no—
90C22×D191—22no—
91C22×D191—22no—
92C22×D191—216no—
93C22×D191—22no—
94C22×D191—22no—
95C22×D191—22no—
96C22×D191—22no—
97C22×D191—22no—
98C22×D191—22no—
99C22×D191—216no—
100C22×D191—44no—
101C22×D191—44no—
102C22×D191—48no—
103C22×D191—44no—
104C22×D191—48no—
105C22×D191—48no—
106C22×D191—48no—
107C22×D191—48no—
108C22×D191—48no—
109C22×D191—48no—
110C22×D191C2×D414yes—
111C22×D191C2×D414yes—
112C22×D191C2×D414yes—
113C22×D191C2×D414yescomplemented
114D764—216no—
115D764—44no—
116C22⋊D191—216no—
117C22⋊D191—216no—
118C22⋊D191—48no—
119C2×Dic191—22no—
120C2×Dic191—22no—
121C2×Dic191—22no—
122C2×Dic191—22no—
123C2×Dic191—22no—
124C2×Dic191C2×D414yescomplemented
125C4×D191—216no—
126C23×C191—22no—
127C23×C191—22no—
128C23×C191—22no—
129C23×C191—22no—
130C23×C191—22no—
131C23×C191—22no—
132C23×C191—21no—
133C23×C191—22no—
134C23×C191—21no—
135C23×C191—22no—
136C23×C191C2×D412yes—
137C23×C191C2×D414yescomplemented
138C23×C191C2411yescharacteristic
139D4×C191—216no—
140D4×C191—44no—
141C2×C4×C191—22no—
142C2×C4×C191—22no—
143C2×C4×C191—22no—
144C2×C4×C191—22no—
145C2×C4×C191—21no—
146C2×C4×C191—21no—
147C2×C4×C191C2×D414yescomplemented
148C22×C191—24no—
149C22×C191—24no—
150C22×C191—22no—
151C22×C191—24no—
152C22×C191—24no—
153C22×C191—42no—
154C22×C191—42no—
155C22×C191—41no—
156C22×C191—42no—
157C22×C191—42no—
158C22×C191—42no—
159C22×C191—41no—
160C22×C191C22×D411yescharacteristic
161C22×C191C22×D412yes—
162C22×C191C22×D412yes—
163C22×C191H(5,𝔽2)11yescharacteristic
164C22×C191H(5,𝔽2)11yescharacteristic
165C2×D191—24no—
166C2×D191—24no—
167C2×D191—24no—
168C2×D191—24no—
169C2×D191—24no—
170C2×D191—24no—
171C2×D191—24no—
172C2×D191—24no—
173C2×D191—24no—
174C2×D191—416no—
175C2×D191—432no—
176C2×D191—42no—
177C2×D191—42no—
178C2×D191—42no—
179C2×D191—42no—
180C2×D191#12018yescomplemented
181C4×C191—24no—
182C4×C191—41no—
183C4×C191—41no—
184Dic191—24no—
185Dic191—42no—
186C2×C191—42no—
187C2×C191—44no—
188C2×C191C2×#12012yes—
189C2×C191#53411yescharacteristic
190C2×C191D4211yescharacteristic
191C2×C191D4212yes—
192C2×C191D4211yescharacteristic
193D191—28no—
194D191—44no—
195C191#240311yescharacteristic, complemented, Sylow
196#2403—1911nomaximal, Sylow
197C23×D4—1912no—
198C2×#120—1912no—
199C2×#120—1912no—
200#392—1912no—
201#392—1911no—
202#392D19111yescharacteristic, complemented
203#386—1912no—
204#386—1912no—
205#379—1911no—
206C25—1912no—
207C25—1911no—
208C22×D4—3828no—
209C22×D4—3828no—
210C22×D4—3828no—
211C22×D4—1912no—
212C22×D4—1912no—
213C22×D4—1912no—
214C22×D4—1912no—
215C22×D4—1911no—
216C22×D4C2×D19112yescomplemented
217C22×D4C2×D19111yescharacteristic, complemented
218C23×C4—1912no—
219#120—3828no—
220#120—3828no—
221C2×C22⋊C4—1912no—
222C2×C22⋊C4—1912no—
223C2×C22⋊C4—1912no—
224C2×C22⋊C4—1912no—
225C2×C22⋊C4—1912no—
226C2×C22⋊C4—1911no—
227C2×C22⋊C4C2×D19112yescomplemented
228C2×C22⋊C4C2×D19111yescharacteristic, complemented
229#95—1912no—
230#95—1911no—
231#95C2×D19111yescharacteristic, complemented
232C24—3824no—
233C24—3824no—
234C24—3824no—
235C24—3824no—
236C24—3824no—
237C24—3824no—
238C24—3824no—
239C24—1912no—
240C24—1912no—
241C24—1914no—
242C24—7641no—
243C24—7642no—
244C24—7641no—
245C24—7642no—
246C24C22×D19111yescharacteristic
247C24C22×D19112yes—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—41no—
253C2×D4—3824no—
254C2×D4—3824no—
255C2×D4—3824no—
256C2×D4—38264no—
257C2×D4—3824no—
258C2×D4—3824no—
259C2×D4—3824no—
260C2×D4—3824no—
261C2×D4—3824no—
262C2×D4—3824no—
263C2×D4—3824no—
264C2×D4—1918no—
265C2×D4—1918no—
266C2×D4—7641no—
267C2×D4—7642no—
268C22×C4—3824no—
269C22×C4—3824no—
270C22×C4—3824no—
271C22×C4—1912no—
272C22×C4—1912no—
273C22×C4C22×D19111yescharacteristic
274C22×C4C22×D19111yescharacteristic
275C22×C4C22×D19112yes—
276C22⋊C4—24no—
277C22⋊C4—41no—
278C22⋊C4—3824no—
279C22⋊C4—3824no—
280C22⋊C4—3824no—
281C22⋊C4—3824no—
282C22⋊C4—3824no—
283C22⋊C4—7641no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—21no—
291C23—22no—
292C23—21no—
293C23—22no—
294C23—38216no—
295C23—38216no—
296C23—3822no—
297C23—3822no—
298C23—3822no—
299C23—3822no—
300C23—3822no—
301C23—3822no—
302C23—3822no—
303C23—3822no—
304C23—3822no—
305C23—3822no—
306C23—1914no—
307C23—1914no—
308C23—1914no—
309C23—1914no—
310C23—7644no—
311C23—7648no—
312C23—7648no—
313C23—7644no—
314C23—7648no—
315C23—7648no—
316C23—7644no—
317C23—7648no—
318C23—7648no—
319C23—7648no—
320C23D4×D19112yes—
321C23D4×D19114yescomplemented
322C23C23×D19111yescharacteristic
323D4—216no—
324D4—44no—
325D4—38216no—
326D4—38216no—
327D4—38216no—
328D4—7644no—
329D4—7648no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—38216no—
337C2×C4—3822no—
338C2×C4—3822no—
339C2×C4—3822no—
340C2×C4—3822no—
341C2×C4—3822no—
342C2×C4—1914no—
343C2×C4D4×D19114yescomplemented
344C22—22no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—24no—
349C22—42no—
350C22—41no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—41no—
355C22—42no—
356C22—3824no—
357C22—3824no—
358C22—3824no—
359C22—3824no—
360C22—3824no—
361C22—3824no—
362C22—3824no—
363C22—3824no—
364C22—3824no—
365C22—1918no—
366C22—76416no—
367C22—76432no—
368C22—7642no—
369C22—7642no—
370C22—7642no—
371C22—7642no—
372C22C2×D4×D19111yescharacteristic
373C22C2×D4×D19112yes—
374C22C2×D4×D19112yes—
375C22#2025936711yescharacteristic
376C22#2025936811yescharacteristic
377C4—24no—
378C4—41no—
379C4—41no—
380C4—3824no—
381C4—7642no—
382C2—42no—
383C2—44no—
384C2—3828no—
385C2—7644no—
386C2#120×D19112yes—
387C2#2713322111yescharacteristic
388C2#2713329511yescharacteristic
389C2#2713329712yes—
390C2#2713329811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#392×C191C2
hypercenter#392D191
hypocenterC191#2403
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D191C2semidirect
2. C2×#20259292C2semidirect
3. C2×#20259290C2semidirect
4. #27132572C2semidirect
5. #27132571C2semidirect
6. #27132541C2semidirect
7. #27132539C2semidirect
8. #27132518C2semidirect
9. #392×C191C2semidirect
11. C24×D191C22semidirect
13. C22×D764C22semidirect
15. C22×C22⋊D191C22semidirect
16. C22×C22⋊D191C22semidirect
20. C22×C4×D191C22semidirect
23. C2×C22⋊Dic191C22semidirect
24. C2×C22⋊Dic191C22semidirect
25. C2×(C2×D191)⋊C4C22semidirect
26. C2×(C2×D191)⋊C4C22semidirect
27. C2×(C2×D191)⋊C4C22semidirect
28. C2×(C2×D191)⋊C4C22semidirect
29. C191⋊#95C22semidirect
30. C191⋊#95C22semidirect
31. C22×D4×C191C22semidirect
32. C22×D4×C191C22semidirect
33. C2×C22⋊C4×C191C22semidirect
34. C2×C22⋊C4×C191C22semidirect
35. #95×C191C22semidirect
47. C23×D191D4semidirect
54. C2×D764D4semidirect
63. C2×C22⋊D191D4semidirect
113. C22×D191C2×D4semidirect
124. C2×Dic191C2×D4semidirect
137. C23×C191C2×D4semidirect
147. C2×C4×C191C2×D4semidirect
180. C2×D191#120semidirect
195. C191#2403semidirect
202. #392D191semidirect
216. C22×D4C2×D191semidirect
217. C22×D4C2×D191semidirect
227. C2×C22⋊C4C2×D191semidirect
228. C2×C22⋊C4C2×D191semidirect
231. #95C2×D191semidirect
321. C23D4×D191semidirect
343. C2×C4D4×D191semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1912
C2×#202592922
C2×#202592902
#271325722
#271325712
#271325412
#271325392
#271325182
#392×C1912
#2403191