Finite Group Databasebeta

← #37560059

Subgroups — #37560059

Order 23872.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#20238609C211yesmaximal, characteristic, complemented
2C22×C22⋊D373C211yesmaximal, characteristic, complemented
3#26045778C211yesmaximal, characteristic, complemented
4#26045776C211yesmaximal, characteristic, complemented
5#26045774C211yesmaximal, characteristic, complemented
6#26045769C211yesmaximal, characteristic, complemented
7#26045764C211yesmaximal, characteristic, complemented
8#26045764C211yesmaximal, characteristic, complemented
9#26045763C211yesmaximal, characteristic, complemented
10#26045760C211yesmaximal, characteristic, complemented
11D4×Dic373C211yesmaximal, characteristic, complemented
12#26045752C211yesmaximal, characteristic, complemented
13C2×C22⋊Dic373C211yesmaximal, characteristic, complemented
14C22⋊C4×D373C211yesmaximal, characteristic, complemented
15#120×C373C211yesmaximal, characteristic, complemented
16C23×D373C2211yescharacteristic
17#20238609—24no—
18C2×Dic746C2211yescharacteristic, complemented
19C2×C22⋊D373—24no—
20C2×C22⋊D373C2211yescharacteristic
21C2×C22⋊D373C2211yescharacteristic
22C2×C22⋊D373C2211yescharacteristic
23C2×C22⋊D373C2211yescharacteristic
24C2×C22⋊D373C2211yescharacteristic, complemented
25C2×C22⋊D373C2211yescharacteristic, complemented
26C22×Dic373C2211yescharacteristic
27C22×Dic373C2211yescharacteristic, complemented
28C22×Dic373C2211yescharacteristic, complemented
29C22×Dic373C2211yescharacteristic, complemented
30C2×C4×D373C2211yescharacteristic, complemented
31C2×C4×D373C2211yescharacteristic, complemented
32C4⋊Dic373C2211yescharacteristic, complemented
33C4⋊Dic373C2211yescharacteristic, complemented
34#20238589C2211yescharacteristic, complemented
35#20238589C2211yescharacteristic, complemented
36C22⋊Dic373C2211yescharacteristic
37C22⋊Dic373C2211yescharacteristic
38C22⋊Dic373C2211yescharacteristic, complemented
39C22⋊Dic373C2211yescharacteristic, complemented
40C22⋊Dic373C2211yescharacteristic, complemented
41#20238587C2211yescharacteristic, complemented
42#20238587C2211yescharacteristic, complemented
43#20238587C2211yescharacteristic, complemented
44#20238587C2211yescharacteristic, complemented
45C4×Dic373C2211yescharacteristic, complemented
46C24×C373C2211yescharacteristic
47C2×D4×C373C2211yescharacteristic
48C2×D4×C373C2211yescharacteristic, complemented
49C2×D4×C373C2211yescharacteristic, complemented
50C22⋊C4×C373C2211yescharacteristic
51C22⋊C4×C373C2211yescharacteristic, complemented
52C22⋊C4×C373C2211yescharacteristic, complemented
53C22×D373—22no—
54C22×D373—22no—
55C22×D373—41no—
56C22×D373C2311yescharacteristic
57C22×D373C2311yescharacteristic
58Dic746—22no—
59C22⋊D373—22no—
60C22⋊D373—22no—
61C22⋊D373—22no—
62C22⋊D373—22no—
63C22⋊D373—41no—
64C22⋊D373D414yescomplemented
65C2×Dic373—22no—
66C2×Dic373—22no—
67C2×Dic373—22no—
68C2×Dic373—41no—
69C2×Dic373C2311yescharacteristic
70C2×Dic373C2311yescharacteristic
71C2×Dic373C2311yescharacteristic
72C2×Dic373C2311yescharacteristic
73C2×Dic373C2311yescharacteristic
74C2×Dic373C2311yescharacteristic
75C4×D373—22no—
76C4×D373—41no—
77C23×C373—22no—
78C23×C373—22no—
79C23×C373—41no—
80C23×C373C2311yescharacteristic
81C23×C373C2311yescharacteristic
82C23×C373C2311yescharacteristic
83C23×C373C2311yescharacteristic
84D4×C373—22no—
85D4×C373—22no—
86D4×C373—41no—
87C2×C4×C373C2311yescharacteristic
88C2×C4×C373C2311yescharacteristic
89C2×C4×C373C2311yescharacteristic
90C22×C373—21no—
91C22×C373—21no—
92C22×C373—21no—
93C22×C373—21no—
94C22×C373—21no—
95C22×C373—21no—
96C22×C373—21no—
97C22×C373—21no—
98C22×C373—21no—
99C22×C373—21no—
100C22×C373—44no—
101C22×C373C2×D412yes—
102C22×C373#4112yescomplemented
103C22×C373C2411yescharacteristic
104C2×D373—21no—
105C2×D373—21no—
106C2×D373—21no—
107C2×D373—21no—
108C2×D373—21no—
109C2×D373—44no—
110C2×D373C2×D412yes—
111C4×C373—21no—
112C4×C373—21no—
113C4×C373—21no—
114Dic373—21no—
115Dic373—21no—
116Dic373—21no—
117Dic373—21no—
118Dic373—21no—
119Dic373C2×D412yes—
120C2×C373—22no—
121C2×C373—22no—
122C2×C373—41no—
123C2×C373—41no—
124C2×C373C22×D411yescharacteristic
125C2×C373C2×#4111yescharacteristic
126C2×C373H(5,𝔽2)11yescharacteristic
127D373—22no—
128D373—41no—
129C373#54611yescharacteristic, complemented, Sylow
130#546—3731nomaximal, Sylow
131C2×#41—3731no—
132C22×D4—3731no—
133#124—3731no—
134#123—3731no—
135#123—3731no—
136C22⋊Q8—3731no—
137#121—3731no—
138#121—3731no—
139#121—3731no—
140#120—3731no—
141#120D37311yescharacteristic, complemented
142C4×D4—3731no—
143C4×D4—3731no—
144C2×C22⋊C4—3731no—
145C2×C22⋊C4—3731no—
146C24—3731no—
147C24C2×D37311yescharacteristic
148#41—7464no—
149C2×Q8—3731no—
150C2×D4—3731no—
151C2×D4—3731no—
152C2×D4—3731no—
153C2×D4—3731no—
154C2×D4—3731no—
155C2×D4—3731no—
156C2×D4—7464no—
157C2×D4C2×D37311yescharacteristic
158C2×D4C2×D37311yescharacteristic, complemented
159C2×D4C2×D37311yescharacteristic, complemented
160C22×C4—3731no—
161C22×C4—3731no—
162C22×C4—3731no—
163C22×C4—3731no—
164C22×C4—3731no—
165C22×C4—3731no—
166C4⋊C4—3731no—
167C4⋊C4—3731no—
168C4⋊C4—3731no—
169C4⋊C4—3731no—
170C22⋊C4—3731no—
171C22⋊C4—3731no—
172C22⋊C4—3731no—
173C22⋊C4—3731no—
174C22⋊C4—3731no—
175C22⋊C4—3731no—
176C22⋊C4—3731no—
177C22⋊C4—3731no—
178C22⋊C4—3731no—
179C22⋊C4C2×D37311yescharacteristic
180C22⋊C4C2×D37311yescharacteristic, complemented
181C22⋊C4C2×D37311yescharacteristic, complemented
182C42—3731no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—3731no—
187C23—3731no—
188C23—14921no—
189C23—7462no—
190C23—7462no—
191C23C22×D37311yescharacteristic
192C23C22×D37311yescharacteristic
193C23C22×D37311yescharacteristic
194C23C22×D37311yescharacteristic
195Q8—7462no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—3734no—
200D4—14921no—
201D4—7462no—
202D4—7462no—
203D4—7462no—
204D4—7462no—
205C2×C4—3731no—
206C2×C4—3731no—
207C2×C4—3731no—
208C2×C4—3731no—
209C2×C4—3731no—
210C2×C4—3731no—
211C2×C4—14921no—
212C2×C4—14921no—
213C2×C4—7462no—
214C2×C4—7462no—
215C2×C4—7462no—
216C2×C4—7462no—
217C2×C4C22×D37311yescharacteristic
218C2×C4C22×D37311yescharacteristic
219C2×C4C22×D37311yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—44no—
231C22—3732no—
232C22—14924no—
233C22—7461no—
234C22—7461no—
235C22—7461no—
236C22—7461no—
237C22—7461no—
238C22D4×D37312yes—
239C22#2023860912yescomplemented
240C22C23×D37311yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—3732no—
245C4—7461no—
246C4—7461no—
247C4—7461no—
248C4—7461no—
249C4—7461no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—41no—
254C2—14921no—
255C2—7462no—
256C2C2×D4×D37311yescharacteristic
257C2C2×#2023860911yescharacteristic
258C2#2604583711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#120×C373C2
hypercenter#120D373
hypocenterC373#546
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#20238609C2semidirect
2. C22×C22⋊D373C2semidirect
3. #26045778C2semidirect
4. #26045776C2semidirect
5. #26045774C2semidirect
6. #26045769C2semidirect
7. #26045764C2semidirect
8. #26045764C2semidirect
9. #26045763C2semidirect
10. #26045760C2semidirect
11. D4×Dic373C2semidirect
12. #26045752C2semidirect
13. C2×C22⋊Dic373C2semidirect
14. C22⋊C4×D373C2semidirect
15. #120×C373C2semidirect
18. C2×Dic746C22semidirect
24. C2×C22⋊D373C22semidirect
25. C2×C22⋊D373C22semidirect
27. C22×Dic373C22semidirect
28. C22×Dic373C22semidirect
29. C22×Dic373C22semidirect
30. C2×C4×D373C22semidirect
31. C2×C4×D373C22semidirect
32. C4⋊Dic373C22semidirect
33. C4⋊Dic373C22semidirect
34. #20238589C22semidirect
35. #20238589C22semidirect
38. C22⋊Dic373C22semidirect
39. C22⋊Dic373C22semidirect
40. C22⋊Dic373C22semidirect
41. #20238587C22semidirect
42. #20238587C22semidirect
43. #20238587C22semidirect
44. #20238587C22semidirect
45. C4×Dic373C22semidirect
48. C2×D4×C373C22semidirect
49. C2×D4×C373C22semidirect
51. C22⋊C4×C373C22semidirect
52. C22⋊C4×C373C22semidirect
64. C22⋊D373D4semidirect
102. C22×C373#41semidirect
129. C373#546semidirect
141. #120D373semidirect
158. C2×D4C2×D373semidirect
159. C2×D4C2×D373semidirect
180. C22⋊C4C2×D373semidirect
181. C22⋊C4C2×D373semidirect
239. C22#20238609semidirect

Maximal subgroups

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

subgroupindex
C2×#202386092
C22×C22⋊D3732
#260457782
#260457762
#260457742
#260457692
#260457642
#260457632
#260457602
D4×Dic3732
#260457522
C2×C22⋊Dic3732
C22⋊C4×D3732
#120×C3732
#546373