Finite Group Databasebeta

← #37582151

Subgroups — #37582151

Order 24448.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#20259355C211yesmaximal, characteristic, complemented
2C2×M16×D191C211yesmaximal, characteristic, complemented
3#27133175C211yesmaximal, characteristic
4C2×#20259302C211yesmaximal, characteristic, complemented
5#27132922C212yesmaximal, complemented
6#27132910C212yesmaximal, complemented
7#27132905C212yesmaximal, complemented
8#27132893C212yesmaximal, complemented
9#27132671C211yesmaximal, characteristic, complemented
10#27132662C211yesmaximal, characteristic, complemented
11#428×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×Dic382C2211yescharacteristic
14C22×C4×D191C2211yescharacteristic
15#20259355—24no—
16C2×Dic764C2212yescomplemented
17C2×#15690276C2212yescomplemented
18C2×#15690271C2211yescharacteristic
19C2×#15690270C2212yes—
20M16×D191—24no—
21C2×C8×D191C2212yes—
22#20259306—21no—
23#20259302—21no—
24#20259302—21no—
25#20259302C2212yes—
26#20259302C2211yescharacteristic
27#20259302C2212yes—
28#20259302C2211yescharacteristic
29#20259297C2211yescharacteristic
30#20259297C2211yescharacteristic
31#20259292—21no—
32C4×D764—21no—
33C4×D764—21no—
34C2×C4⋊Dic191—21no—
35C2×(C2×D191)⋊C4—21no—
36#20259245C2212yescomplemented
37#20259245C2212yescomplemented
38#20259245C2212yescomplemented
39#20259242C2212yescomplemented
40C2×M16×C191C2211yescharacteristic
41#117×C191C2211yescharacteristic
42C2×C4⋊C4×C191C2211yescharacteristic
43#107×C191C2212yescomplemented
44#106×C191C2212yescomplemented
45C23×D191D411yescharacteristic
46#15690287—22no—
47#15690287—22no—
48C2×Dic382—21no—
49C2×Dic382—22no—
50C2×Dic382—22no—
51C2×Dic382C2312yes—
52C2×Dic382C2311yescharacteristic
53C2×D764—21no—
54C2×D764—21no—
55C2×D764C2311yescharacteristic
56C2×C22⋊D191—21no—
57C22×Dic191—21no—
58C22×Dic191D411yescharacteristic
59C2×C4×D191—22no—
60C2×C4×D191—21no—
61C2×C4×D191—22no—
62C2×C4×D191D411yescharacteristic
63C2×C4×D191D411yescharacteristic
64C2×C4×D191C2311yescharacteristic
65C2×C4×D191C2311yescharacteristic
66Dic764—24no—
67#15690276—24no—
68#15690271—22no—
69#15690270—24no—
70C2×C191⋊C8C2312yes—
71C8×D191—24no—
72C4⋊Dic191—22no—
73C4⋊Dic191—22no—
74C4⋊Dic191—21no—
75C4⋊Dic191—21no—
76(C2×D191)⋊C4—22no—
77(C2×D191)⋊C4—22no—
78(C2×D191)⋊C4—21no—
79(C2×D191)⋊C4—21no—
80(C2×D191)⋊C4—21no—
81(C2×D191)⋊C4—21no—
82C22×C4×C191—21no—
83C22×C4×C191C2311yescharacteristic
84M16×C191D414yescomplemented
85C2×C8×C191C2312yes—
86C4⋊C4×C191—21no—
87C4⋊C4×C191C2311yescharacteristic
88C4⋊C4×C191C2312yes—
89C4⋊C4×C191C2311yescharacteristic
90C22⋊C4×C191—21no—
91C42×C191—21no—
92C22×D191—22no—
93C22×D191—22no—
94C22×D191—21no—
95C22×D191—41no—
96C22×D191C2×D411yescharacteristic
97C22×D191C2×D411yescharacteristic
98Dic382—22no—
99Dic382—21no—
100Dic382—22no—
101Dic382—21no—
102Dic382—42no—
103D764—21no—
104D764—21no—
105D764—42no—
106D764—42no—
107C22⋊D191—42no—
108C2×Dic191—22no—
109C2×Dic191—21no—
110C2×Dic191—22no—
111C2×Dic191—42no—
112C2×Dic191C2×D411yescharacteristic
113C2×Dic191C2×D411yescharacteristic
114C4×D191—22no—
115C4×D191—22no—
116C4×D191—21no—
117C4×D191—21no—
118C4×D191—21no—
119C4×D191—21no—
120C4×D191—42no—
121C191⋊C8—22no—
122C23×C191C2×D411yescharacteristic
123C2×C4×C191—21no—
124C2×C4×C191—21no—
125C2×C4×C191—22no—
126C2×C4×C191—42no—
127C2×C4×C191C2×D411yescharacteristic
128C2×C4×C191C2×D412yes—
129C2×C4×C191#4112yes—
130C2×C4×C191C2411yescharacteristic
131C8×C191C2×D414yes—
132C22×C191—21no—
133C22×C191—21no—
134C22×C191#12112yes—
135C22×C191C22×D411yescharacteristic
136C2×D191—21no—
137C2×D191—21no—
138C2×D191—21no—
139C2×D191—21no—
140C2×D191—21no—
141C2×D191—44no—
142C2×D191—41no—
143C2×D191—41no—
144C2×D191—41no—
145C2×D191#13712yescomplemented
146C4×C191—41no—
147C4×C191—42no—
148C4×C191—41no—
149C4×C191#12112yes—
150C4×C191C22×D411yescharacteristic
151C4×C191C2×#4111yescharacteristic
152Dic191—21no—
153Dic191—21no—
154Dic191—41no—
155Dic191—42no—
156C2×C191—22no—
157C2×C191C2×#12111yescharacteristic
158C2×C191C2×#13711yescharacteristic
159C2×C191#57511yescharacteristic
160D191—22no—
161D191—41no—
162D191—81no—
163C191#305611yescharacteristic, complemented, Sylow
164#3056—1911nomaximal, Sylow
165C2×#137—1911no—
166C22×M16—1911no—
167#525—1911no—
168C2×C22⋊Q8—1911no—
169#470—1912no—
170#468—1912no—
171#467—1912no—
172#465—1912no—
173#428D19111yescharacteristic, complemented
174#419—1911no—
175#417—1911no—
176C2×#41—1911no—
177C22×Q8—1911no—
178C23×C4—1911no—
179#137—3824no—
180C2×Q16—1912no—
181C2×SD16—1912no—
182C2×M16—3824no—
183C2×M16—1911no—
184C2×M16—1912no—
185C2×M16C2×D19111yescharacteristic
186C22×C8—1912no—
187#123—3821no—
188C22⋊Q8—3821no—
189C22⋊Q8—3821no—
190C22⋊Q8—1912no—
191C22⋊Q8—1912no—
192C22⋊Q8—1911no—
193C22⋊Q8—1911no—
194#121—1911no—
195#121—1911no—
196#120—3821no—
197C4×D4—3821no—
198C4×D4—3821no—
199#117C2×D19111yescharacteristic
200C2×C4⋊C4—3821no—
201C2×C4⋊C4C2×D19111yescharacteristic
202C2×C22⋊C4—3821no—
203#107C2×D19112yescomplemented
204#106C2×D19112yescomplemented
205#103—1912no—
206#103—1912no—
207#103—1912no—
208#102—1912no—
209C24—1911no—
210#41—3822no—
211#41—3822no—
212C2×Q8—3821no—
213C2×Q8—3822no—
214C2×Q8—3822no—
215C2×Q8—1912no—
216C2×Q8—1911no—
217C2×D4—3821no—
218C2×D4—3821no—
219C2×D4—3821no—
220C2×D4—1911no—
221C22×C4—21no—
222C22×C4—3822no—
223C22×C4—3822no—
224C22×C4—3821no—
225C22×C4—3821no—
226C22×C4—1911no—
227C22×C4—1911no—
228C22×C4—1911no—
229C22×C4—1911no—
230C22×C4—1911no—
231C22×C4C22×D19111yescharacteristic
232Q16—3824no—
233SD16—3824no—
234M16—3822no—
235M16—3824no—
236M16D76414yescomplemented
237C2×C8—3824no—
238C2×C8—1912no—
239C2×C8C22×D19112yes—
240C4⋊C4—21no—
241C4⋊C4—3822no—
242C4⋊C4—3822no—
243C4⋊C4—3821no—
244C4⋊C4—3821no—
245C4⋊C4C22×D19112yes—
246C4⋊C4C22×D19111yescharacteristic
247C4⋊C4C22×D19111yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—3822no—
250C22⋊C4—3822no—
251C22⋊C4—3821no—
252C22⋊C4—3821no—
253C22⋊C4—3821no—
254C22⋊C4—3821no—
255C42—21no—
256C23—3822no—
257C23—3822no—
258C23—3821no—
259C23—1911no—
260C23—1911no—
261C23—7641no—
262C23D4×D19111yescharacteristic
263Q8—3822no—
264Q8—3821no—
265Q8—3822no—
266Q8—3821no—
267Q8—7642no—
268D4—3821no—
269D4—3821no—
270D4—7642no—
271D4—7642no—
272D4—7642no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—3822no—
278C2×C4—3821no—
279C2×C4—3821no—
280C2×C4—3821no—
281C2×C4—3821no—
282C2×C4—3822no—
283C2×C4—3822no—
284C2×C4—3821no—
285C2×C4—3822no—
286C2×C4—1911no—
287C2×C4—1911no—
288C2×C4—7642no—
289C2×C4—7642no—
290C2×C4D4×D19111yescharacteristic
291C2×C4C2×D76412yes—
292C2×C4#1569028812yes—
293C2×C4C23×D19111yescharacteristic
294C8—3822no—
295C8C2×D76414yes—
296C22—21no—
297C22—21no—
298C22—3821no—
299C22—3821no—
300C22—3821no—
301C22—3821no—
302C22—3821no—
303C22—1912no—
304C22—7644no—
305C22—7641no—
306C22—7641no—
307C22—7641no—
308C22#2025929712yes—
309C22C2×D4×D19111yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—3821no—
314C4—3821no—
315C4—7641no—
316C4—7642no—
317C4#2025929712yes—
318C4C22×D76411yescharacteristic
319C4C2×#1569028811yescharacteristic
320C2—22no—
321C2—3822no—
322C2—15281no—
323C2—7641no—
324C2C2×#2025929711yescharacteristic
325C2#137×D19111yescharacteristic
326C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#428×C191C2
socleC22×C191C22×D4
prime radical#27133175C2
hypercenter#428D191
hypocenterC191#3056
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#20259355C2semidirect
2. C2×M16×D191C2semidirect
4. C2×#20259302C2semidirect
5. #27132922C2semidirect
6. #27132910C2semidirect
7. #27132905C2semidirect
8. #27132893C2semidirect
9. #27132671C2semidirect
10. #27132662C2semidirect
11. #428×C191C2semidirect
16. C2×Dic764C22semidirect
17. C2×#15690276C22semidirect
36. #20259245C22semidirect
37. #20259245C22semidirect
38. #20259245C22semidirect
39. #20259242C22semidirect
43. #107×C191C22semidirect
44. #106×C191C22semidirect
84. M16×C191D4semidirect
145. C2×D191#137semidirect
163. C191#3056semidirect
173. #428D191semidirect
203. #107C2×D191semidirect
204. #106C2×D191semidirect
236. M16D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593552
C2×M16×D1912
#271331752
C2×#202593022
#271329222
#271329102
#271329052
#271328932
#271326712
#271326622
#428×C1912
#3056191