Finite Group Databasebeta

← #37583275

Subgroups — #37583275

Order 24448.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#20259298C211yesmaximal, characteristic, complemented
2#121×D191C211yesmaximal, characteristic
3#27132992C212yesmaximal, complemented
4#27132982C211yesmaximal, characteristic, complemented
5#27132979C211yesmaximal, characteristic, complemented
6#27132895C212yesmaximal, complemented
7#27132889C212yesmaximal, complemented
8#27132812C212yesmaximal, complemented
9#27132802C211yesmaximal, characteristic, complemented
10C22⋊C8×D191C211yesmaximal, characteristic, complemented
11#448×C191C211yesmaximal, characteristic, complemented
12C2×D4×D191—21no—
13C2×D4×D191C2211yescharacteristic
14C22×C4×D191C2211yescharacteristic
15C2×#15690274C2212yescomplemented
16C2×C8×D191C2212yescomplemented
17#20259298—21no—
18#20259298—21no—
19#20259298C2212yes—
20#20259298C2212yescomplemented
21#20259297C2211yescharacteristic
22#20259296C2211yescharacteristic
23#20259295—21no—
24#20259294—21no—
25#20259293C2211yescharacteristic
26C2×C4⋊Dic191C2211yescharacteristic, complemented
27C2×Dic191⋊C4—21no—
28C4⋊C4×D191C2211yescharacteristic
29C2×(C2×D191)⋊C4—21no—
30C22⋊C4×D191—21no—
31#20259254C2212yescomplemented
32#20259253C2212yescomplemented
33#20259246C2212yescomplemented
34#20259244C2212yescomplemented
35#20259243C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic, complemented
37(C2×D191)⋊C8C2212yes—
38C22×Q8×C191C2211yescharacteristic, complemented
39C2×SD16×C191C2212yescomplemented
40#121×C191C2211yescharacteristic
41#103×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic, complemented
43C23×D191—21no—
44C23×D191D411yescharacteristic
45C2×D764—21no—
46C2×D764C2311yescharacteristic
47C2×C22⋊D191—21no—
48C2×C22⋊D191—21no—
49C2×C22⋊D191—21no—
50D4×D191—42no—
51D4×D191—44no—
52C22×Dic191—21no—
53C22×Dic191D411yescharacteristic
54C2×C4×D191—21no—
55C2×C4×D191—42no—
56C2×C4×D191D411yescharacteristic
57C2×C4×D191D411yescharacteristic
58C2×C4×D191C2311yescharacteristic
59C2×C4×D191C2311yescharacteristic
60#15690274—42no—
61C2×C191⋊C8C2312yes—
62C8×D191—42no—
63C4⋊Dic191—21no—
64C4⋊Dic191C2312yes—
65C4⋊Dic191C2311yescharacteristic
66Dic191⋊C4—21no—
67Dic191⋊C4—22no—
68Dic191⋊C4—22no—
69C22⋊Dic191—21no—
70(C2×D191)⋊C4—22no—
71(C2×D191)⋊C4—22no—
72(C2×D191)⋊C4—21no—
73(C2×D191)⋊C4—21no—
74C2×Q8×C191—21no—
75C2×Q8×C191—42no—
76C2×Q8×C191C2312yes—
77C2×D4×C191—21no—
78C2×D4×C191C2311yescharacteristic
79C22×C4×C191—21no—
80C22×C4×C191C2311yescharacteristic
81SD16×C191—42no—
82C2×C8×C191C2312yes—
83C4⋊C4×C191C2311yescharacteristic
84C22⋊C4×C191—21no—
85C22×D191—21no—
86C22×D191—41no—
87C22×D191—41no—
88C22×D191—42no—
89C22×D191—42no—
90C22×D191—42no—
91C22×D191SD1614yescomplemented
92C22×D191C2×D411yescharacteristic
93C22×D191C2×D411yescharacteristic
94D764—22no—
95D764—42no—
96C22⋊D191—42no—
97C22⋊D191—42no—
98C22⋊D191—42no—
99C2×Dic191—22no—
100C2×Dic191—21no—
101C2×Dic191—41no—
102C2×Dic191—42no—
103C2×Dic191C2×D411yescharacteristic
104C2×Dic191C2×D411yescharacteristic
105C4×D191—22no—
106C4×D191—22no—
107C4×D191—41no—
108C4×D191—41no—
109C4×D191—42no—
110C191⋊C8—22no—
111C23×C191—21no—
112C23×C191C2×D411yescharacteristic
113Q8×C191—22no—
114Q8×C191—22no—
115Q8×C191—42no—
116D4×C191—22no—
117D4×C191—42no—
118C2×C4×C191—22no—
119C2×C4×C191—21no—
120C2×C4×C191—22no—
121C2×C4×C191—42no—
122C2×C4×C191C2×D411yescharacteristic
123C2×C4×C191C2411yescharacteristic
124C8×C191—22no—
125C22×C191—21no—
126C22×C191—21no—
127C22×C191—41no—
128C22×C191—41no—
129C22×C191—41no—
130C22×C191C2×SD1612yes—
131C22×C191C22×D411yescharacteristic
132C2×D191—21no—
133C2×D191—21no—
134C2×D191—21no—
135C2×D191—21no—
136C2×D191—44no—
137C2×D191—41no—
138C2×D191—41no—
139C2×D191—41no—
140C2×D191—84no—
141C2×D191C2×SD1612yes—
142C2×D191C2×SD1612yes—
143C4×C191—21no—
144C4×C191—42no—
145C4×C191—41no—
146C4×C191H(5,𝔽2)12yes—
147Dic191—21no—
148Dic191—21no—
149Dic191—41no—
150Dic191—42no—
151C2×C191—22no—
152C2×C191—81no—
153C2×C191#53411yescharacteristic
154C2×C191C22×SD1611yescharacteristic
155C2×C191#57511yescharacteristic
156D191—22no—
157D191—22no—
158D191—81no—
159C191#318711yescharacteristic, complemented, Sylow
160#3187—1911nomaximal, Sylow
161C2×C22⋊Q8—1911no—
162C2×#121—1911no—
163#483—1912no—
164#481—1911no—
165#481—1911no—
166#465—1912no—
167#465—1912no—
168#450—1912no—
169#448—1911no—
170#448D19111yescharacteristic, complemented
171C2×C22⋊C8—1911no—
172C22×Q8C2×D19111yescharacteristic, complemented
173C22×D4—3821no—
174C22×D4—1911no—
175C23×C4—1911no—
176C2×SD16—1912no—
177C2×SD16C2×D19112yescomplemented
178C22×C8—1912no—
179C22⋊Q8—3821no—
180C22⋊Q8—3821no—
181C22⋊Q8—1912no—
182C22⋊Q8—1912no—
183#121—3821no—
184#121—3821no—
185#121—1911no—
186#121—1911no—
187#121—1911no—
188#121C2×D19111yescharacteristic
189C2×C4⋊C4—3821no—
190C2×C4⋊C4—1911no—
191C2×C4⋊C4—1911no—
192C2×C22⋊C4—3821no—
193C2×C22⋊C4—3821no—
194#106—1912no—
195#106—1912no—
196#103—1912no—
197#103C2×D19112yescomplemented
198#102—1912no—
199#102—1912no—
200C22⋊C8—1911no—
201C22⋊C8—1912no—
202C22⋊C8C2×D19111yescharacteristic, complemented
203C24—3821no—
204C24—1911no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D19112yes—
208C2×D4—21no—
209C2×D4—3821no—
210C2×D4—3821no—
211C2×D4—3821no—
212C2×D4—3821no—
213C2×D4—1911no—
214C2×D4—7642no—
215C2×D4—7644no—
216C2×D4C22×D19111yescharacteristic
217C22×C4—21no—
218C22×C4—3821no—
219C22×C4—3821no—
220C22×C4—1911no—
221C22×C4—1911no—
222C22×C4—1911no—
223C22×C4—1911no—
224C22×C4—1911no—
225C22×C4—7642no—
226C22×C4C22×D19111yescharacteristic
227SD16—42no—
228SD16—7642no—
229C2×C8—1912no—
230C2×C8—7642no—
231C2×C8C22×D19112yes—
232C4⋊C4—3822no—
233C4⋊C4—3822no—
234C4⋊C4—3821no—
235C4⋊C4—3821no—
236C4⋊C4—1912no—
237C4⋊C4—1911no—
238C4⋊C4C22×D19111yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—3822no—
241C22⋊C4—3822no—
242C22⋊C4—3821no—
243C22⋊C4—3821no—
244C22⋊C4—3821no—
245C23—21no—
246C23—3821no—
247C23—1911no—
248C23—1911no—
249C23—1914no—
250C23—7641no—
251C23—7641no—
252C23—7642no—
253C23—7642no—
254C23—7642no—
255C23D4×D19111yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—3822no—
262D4—7642no—
263D4—7642no—
264D4—7642no—
265D4—7642no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—3822no—
271C2×C4—3822no—
272C2×C4—3822no—
273C2×C4—3821no—
274C2×C4—1911no—
275C2×C4—1911no—
276C2×C4—7641no—
277C2×C4—7641no—
278C2×C4—7642no—
279C2×C4—7641no—
280C2×C4—7642no—
281C2×C4D4×D19111yescharacteristic
282C2×C4C23×D19111yescharacteristic
283C8—22no—
284C8—3822no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—3821no—
291C22—3821no—
292C22—3821no—
293C22—3821no—
294C22—1912no—
295C22—1912no—
296C22—15284no—
297C22—7644no—
298C22—7641no—
299C22—7641no—
300C22—7641no—
301C22SD16×D19112yes—
302C22C2×D4×D19111yescharacteristic
303C4—21no—
304C4—42no—
305C4—41no—
306C4—3821no—
307C4—3821no—
308C4—7642no—
309C4—7641no—
310C4#2025936712yes—
311C2—22no—
312C2—81no—
313C2—3822no—
314C2—3822no—
315C2—15281no—
316C2#2713322311yescharacteristic
317C2C2×SD16×D19111yescharacteristic
318C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#448×C191C2
socleC22×C191C22×D4
prime radical#121×D191C2
hypercenter#448D191
hypocenterC191#3187
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#20259298C2semidirect
3. #27132992C2semidirect
4. #27132982C2semidirect
5. #27132979C2semidirect
6. #27132895C2semidirect
7. #27132889C2semidirect
8. #27132812C2semidirect
9. #27132802C2semidirect
10. C22⋊C8×D191C2semidirect
11. #448×C191C2semidirect
15. C2×#15690274C22semidirect
16. C2×C8×D191C22semidirect
20. #20259298C22semidirect
26. C2×C4⋊Dic191C22semidirect
31. #20259254C22semidirect
32. #20259253C22semidirect
33. #20259246C22semidirect
34. #20259244C22semidirect
35. #20259243C22semidirect
36. C22⋊(C191⋊C8)C22semidirect
38. C22×Q8×C191C22semidirect
39. C2×SD16×C191C22semidirect
41. #103×C191C22semidirect
42. C22⋊C8×C191C22semidirect
91. C22×D191SD16semidirect
159. C191#3187semidirect
170. #448D191semidirect
172. C22×Q8C2×D191semidirect
177. C2×SD16C2×D191semidirect
197. #103C2×D191semidirect
202. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592982
#121×D1912
#271329922
#271329822
#271329792
#271328952
#271328892
#271328122
#271328022
C22⋊C8×D1912
#448×C1912
#3187191