Finite Group Databasebeta

← #37582990

Subgroups — #37582990

Order 24448.

564 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20259353C211yesmaximal, characteristic, complemented
2C2×#20259348C211yesmaximal, characteristic, complemented
3C2×#20259344C211yesmaximal, characteristic, complemented
4#27133237C211yesmaximal, characteristic, complemented
5#27133230C211yesmaximal, characteristic, complemented
6#27133054C212yesmaximal, complemented
7#27133049C212yesmaximal, complemented
8#27133044C212yesmaximal, complemented
9#27133034C212yesmaximal, complemented
10#27132618C211yesmaximal, characteristic, complemented
11C22×SD16×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C2×#15690287C2211yescharacteristic
14#20259353—24no—
15#20259348—24no—
16#20259344—24no—
17C2×Dic764C2211yescharacteristic, complemented
18C2×#15690277C2212yescomplemented
19C2×#15690276C2211yescharacteristic, complemented
20C2×#15690276C2211yescharacteristic, complemented
21C2×#15690275C2212yes—
22C2×#15690274C2212yescomplemented
23C2×D1528C2211yescharacteristic, complemented
24C2×#15690272C2212yes—
25C2×#15690271C2211yescharacteristic
26#20259320C2212yescomplemented
27#20259318C2212yes—
28#20259311C2212yescomplemented
29#20259309C2212yes—
30#20259305—22no—
31#20259298—22no—
32#20259293—22no—
33#20259291—22no—
34#20259281C2211yescharacteristic
35#20259232C2212yescomplemented
36C8×Dic191C2212yescomplemented
37C22×Q8×C191C2211yescharacteristic
38C22×D4×C191C2211yescharacteristic
39C2×SD16×C191—24no—
40C2×SD16×C191C2212yes—
41C2×SD16×C191C2212yes—
42C22×C8×C191C2211yescharacteristic
43#15690287—22no—
44#15690287—22no—
45#15690287—22no—
46#15690287—22no—
47C2×Dic382C2311yescharacteristic
48C2×Dic382C2311yescharacteristic
49C2×D764C2311yescharacteristic
50C2×D764C2311yescharacteristic
51C2×C22⋊D191—24no—
52C2×C22⋊D191—21no—
53C2×C22⋊D191—21no—
54C2×C4×D191—21no—
55C2×C4×D191—21no—
56Dic764—22no—
57#15690277—24no—
58#15690276—22no—
59#15690276—22no—
60#15690275—24no—
61#15690274—24no—
62D1528—22no—
63#15690272—24no—
64#15690271D414yescomplemented
65C2×C191⋊C8C2312yes—
66C4⋊Dic191D412yes—
67Dic191⋊C4—24no—
68C22⋊Dic191—24no—
69C22⋊Dic191D412yes—
70(C2×D191)⋊C4—24no—
71C4×Dic191C2312yes—
72C24×C191—21no—
73C2×Q8×C191—21no—
74C2×Q8×C191—22no—
75C2×Q8×C191—22no—
76C2×Q8×C191C2312yes—
77C2×D4×C191—22no—
78C2×D4×C191—22no—
79C2×D4×C191—21no—
80C2×D4×C191C2312yes—
81C22×C4×C191—21no—
82C22×C4×C191C2311yescharacteristic
83SD16×C191—24no—
84SD16×C191—24no—
85C2×C8×C191D414yescomplemented
86C2×C8×C191C2311yescharacteristic
87C2×C8×C191C2311yescharacteristic
88C22×D191—21no—
89C22×D191—21no—
90Dic382—21no—
91Dic382—21no—
92Dic382—21no—
93Dic382—21no—
94D764—21no—
95D764—21no—
96D764—21no—
97D764—21no—
98C22⋊D191—42no—
99C22⋊D191—42no—
100C22⋊D191—84no—
101C2×Dic191—21no—
102C2×Dic191—21no—
103C2×Dic191C2×D414yes—
104C4×D191—42no—
105C4×D191—42no—
106C191⋊C8C2×D414yes—
107C23×C191—22no—
108C23×C191—42no—
109C23×C191—82no—
110C23×C191C2×D411yescharacteristic
111Q8×C191—22no—
112Q8×C191—22no—
113Q8×C191—42no—
114D4×C191—22no—
115D4×C191—22no—
116D4×C191—42no—
117C2×C4×C191—22no—
118C2×C4×C191—42no—
119C2×C4×C191C2×D411yescharacteristic
120C2×C4×C191C2×D412yes—
121C2×C4×C191C2×D412yes—
122C2×C4×C191C2411yescharacteristic
123C8×C191C2×D412yes—
124C8×C191C2×D412yes—
125C22×C191—22no—
126C22×C191—42no—
127C22×C191—44no—
128C22×C191—84no—
129C22×C191#12712yes—
130C22×C191C22×D411yescharacteristic
131C2×D191—41no—
132C2×D191—41no—
133C2×D191—42no—
134C2×D191—42no—
135C4×C191—42no—
136C4×C191#12712yes—
137C4×C191C22×D411yescharacteristic
138C4×C191C22×D411yescharacteristic
139Dic191—24no—
140Dic191—41no—
141Dic191—41no—
142C2×C191—22no—
143C2×C191—82no—
144C2×C191C2×#12711yescharacteristic
145C2×C191#57712yes—
146D191—81no—
147D191—81no—
148C191#315411yescharacteristic, complemented, Sylow
149#3154—1911nomaximal, Sylow
150C2×#137—1911no—
151C2×AGL(1,ℤ/8)—1911no—
152C2×#135—1911no—
153C22×SD16D19111yescharacteristic, complemented
154#536—1911no—
155#535—1911no—
156#497—1912no—
157#496—1912no—
158#495—1912no—
159#492—1912no—
160#405—1911no—
161C2×#41—1911no—
162C2×#41—1911no—
163C22×Q8C2×D19111yescharacteristic
164C22×D4C2×D19111yescharacteristic
165#137—3824no—
166AGL(1,ℤ/8)—3824no—
167#135—3824no—
168C2×Q16—1912no—
169C2×Q16—1911no—
170C2×SD16—24no—
171C2×SD16—1912no—
172C2×SD16—1912no—
173C2×SD16—1911no—
174C2×SD16—1911no—
175C2×SD16C2×D19112yes—
176C2×SD16C2×D19112yes—
177C2×D8—1912no—
178C2×D8—1911no—
179C2×M16—1911no—
180C22×C8C2×D19111yescharacteristic
181C4⋊Q8—1912no—
182#127—1912no—
183#124—1912no—
184#124—1912no—
185#123—3822no—
186C22⋊Q8—3822no—
187#121—3822no—
188#120—3822no—
189#117—1911no—
190#97—1912no—
191C4×C8—1912no—
192C24—21no—
193#41—3822no—
194#41—3822no—
195#41—3822no—
196#41—3822no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—1911no—
201C2×Q8—1911no—
202C2×Q8C22×D19112yes—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—3821no—
207C2×D4—3821no—
208C2×D4—3824no—
209C2×D4—1911no—
210C2×D4—1911no—
211C2×D4C22×D19112yes—
212C22×C4—21no—
213C22×C4—3821no—
214C22×C4—3821no—
215C22×C4C22×D19111yescharacteristic
216Q16—3824no—
217Q16—3822no—
218SD16—24no—
219SD16—24no—
220SD16—3824no—
221SD16—3824no—
222SD16—3822no—
223SD16—3822no—
224D8—3824no—
225D8—3822no—
226M16—1914no—
227C2×C8—1912no—
228C2×C8C22⋊D19114yescomplemented
229C2×C8C22×D19111yescharacteristic
230C2×C8C22×D19111yescharacteristic
231C4⋊C4—3824no—
232C4⋊C4—1912no—
233C22⋊C4—3824no—
234C22⋊C4—3824no—
235C22⋊C4—1912no—
236C42—1912no—
237C23—22no—
238C23—42no—
239C23—82no—
240C23—3821no—
241C23—3821no—
242C23D4×D19111yescharacteristic
243Q8—22no—
244Q8—22no—
245Q8—42no—
246Q8—3821no—
247Q8—3821no—
248Q8—3821no—
249Q8—3821no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—3821no—
254D4—3821no—
255D4—3821no—
256D4—3821no—
257D4—15284no—
258D4—7642no—
259D4—7642no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—3821no—
263C2×C4—3821no—
264C2×C4—1914no—
265C2×C4—7642no—
266C2×C4—7642no—
267C2×C4D4×D19112yes—
268C2×C4D4×D19111yescharacteristic
269C2×C4C2×C22⋊D19112yes—
270C2×C4C23×D19111yescharacteristic
271C8—1914no—
272C8C2×C22⋊D19112yes—
273C8C2×C22⋊D19112yes—
274C22—22no—
275C22—42no—
276C22—44no—
277C22—84no—
278C22—7641no—
279C22—7641no—
280C22—7642no—
281C22—7642no—
282C22#2025931812yes—
283C22C2×D4×D19111yescharacteristic
284C4—42no—
285C4—3824no—
286C4—7641no—
287C4—7641no—
288C4#2025931812yes—
289C4C2×D4×D19111yescharacteristic
290C4C22×C22⋊D19111yescharacteristic
291C2—22no—
292C2—82no—
293C2—15281no—
294C2—15281no—
295C2C2×#2025931811yescharacteristic
296C2#2713346912yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC22×SD16×C191C2
socleC22×C191C22×D4
hypercenterC22×SD16D191
hypocenterC191#3154
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#20259353C2semidirect
2. C2×#20259348C2semidirect
3. C2×#20259344C2semidirect
4. #27133237C2semidirect
5. #27133230C2semidirect
6. #27133054C2semidirect
7. #27133049C2semidirect
8. #27133044C2semidirect
9. #27133034C2semidirect
10. #27132618C2semidirect
11. C22×SD16×C191C2semidirect
17. C2×Dic764C22semidirect
18. C2×#15690277C22semidirect
19. C2×#15690276C22semidirect
20. C2×#15690276C22semidirect
22. C2×#15690274C22semidirect
23. C2×D1528C22semidirect
26. #20259320C22semidirect
28. #20259311C22semidirect
35. #20259232C22semidirect
36. C8×Dic191C22semidirect
64. #15690271D4semidirect
85. C2×C8×C191D4semidirect
148. C191#3154semidirect
153. C22×SD16D191semidirect
228. C2×C8C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593532
C2×#202593482
C2×#202593442
#271332372
#271332302
#271330542
#271330492
#271330442
#271330342
#271326182
C22×SD16×C1912
#3154191