Finite Group Databasebeta

← #37560061

Subgroups — #37560061

Order 23872.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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
2C2×D4×D373C211yesmaximal, characteristic, complemented
3#26045778C211yesmaximal, characteristic, complemented
4#26045777C211yesmaximal, characteristic, complemented
5#26045773C211yesmaximal, characteristic, complemented
6#26045770C211yesmaximal, characteristic, complemented
7#26045765C211yesmaximal, characteristic, complemented
8#26045764C211yesmaximal, characteristic, complemented
9#26045760C211yesmaximal, characteristic, complemented
10#26045760C211yesmaximal, characteristic, complemented
11C4×C22⋊D373C211yesmaximal, characteristic, complemented
12#26045752C211yesmaximal, characteristic, complemented
13C2×#20238587C211yesmaximal, characteristic, complemented
14C22⋊C4×D373C211yesmaximal, characteristic, complemented
15#121×C373C211yesmaximal, characteristic, complemented
16C23×D373C2211yescharacteristic
17C23×D373C2211yescharacteristic
18#20238609—24no—
19C2×Dic746C2211yescharacteristic, complemented
20C2×D1492C2211yescharacteristic
21C2×C22⋊D373C2211yescharacteristic
22C2×C22⋊D373C2211yescharacteristic
23C2×C22⋊D373C2211yescharacteristic
24C2×C22⋊D373C2211yescharacteristic, complemented
25C2×C22⋊D373C2211yescharacteristic, complemented
26D4×D373—24no—
27C22×Dic373C2211yescharacteristic, complemented
28C22×Dic373C2211yescharacteristic, complemented
29C2×C4×D373C2211yescharacteristic
30C2×C4×D373C2211yescharacteristic, complemented
31C2×C4×D373C2211yescharacteristic, complemented
32#20238589C2211yescharacteristic, complemented
33#20238589C2211yescharacteristic, complemented
34#20238589C2211yescharacteristic, complemented
35C22⋊Dic373C2211yescharacteristic, complemented
36C22⋊Dic373C2211yescharacteristic, complemented
37C22⋊Dic373C2211yescharacteristic, complemented
38#20238587C2211yescharacteristic
39#20238587C2211yescharacteristic
40#20238587C2211yescharacteristic
41#20238587C2211yescharacteristic, complemented
42#20238587C2211yescharacteristic, complemented
43#20238587C2211yescharacteristic, complemented
44#20238587C2211yescharacteristic, complemented
45C4×Dic373C2211yescharacteristic, complemented
46C2×D4×C373C2211yescharacteristic
47C2×D4×C373C2211yescharacteristic, complemented
48C2×D4×C373C2211yescharacteristic, complemented
49C22×C4×C373C2211yescharacteristic, complemented
50C4⋊C4×C373C2211yescharacteristic, complemented
51C22⋊C4×C373C2211yescharacteristic, complemented
52C22⋊C4×C373C2211yescharacteristic, complemented
53C22×D373—22no—
54C22×D373—22no—
55C22×D373—22no—
56C22×D373—22no—
57C22×D373—41no—
58C22×D373—41no—
59C22×D373C2311yescharacteristic
60C22×D373C2311yescharacteristic
61C22×D373C2311yescharacteristic
62Dic746—22no—
63D1492—22no—
64C22⋊D373—22no—
65C22⋊D373—22no—
66C22⋊D373—22no—
67C22⋊D373—41no—
68C22⋊D373D414yescomplemented
69C2×Dic373—22no—
70C2×Dic373—22no—
71C2×Dic373C2311yescharacteristic
72C2×Dic373C2311yescharacteristic
73C2×Dic373C2311yescharacteristic
74C2×Dic373C2311yescharacteristic
75C2×Dic373C2311yescharacteristic
76C4×D373—22no—
77C4×D373—22no—
78C4×D373—41no—
79C23×C373C2311yescharacteristic
80C23×C373C2311yescharacteristic
81C23×C373C2311yescharacteristic
82D4×C373—22no—
83D4×C373—22no—
84D4×C373—41no—
85C2×C4×C373—41no—
86C2×C4×C373C2311yescharacteristic
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×C373C2×D412yes—
99C22×C373C2411yescharacteristic
100C2×D373—21no—
101C2×D373—21no—
102C2×D373—21no—
103C2×D373—21no—
104C2×D373—21no—
105C2×D373—21no—
106C2×D373—21no—
107C2×D373—44no—
108C2×D373—44no—
109C2×D373C2×D412yes—
110C2×D373#4112yescomplemented
111C4×C373—21no—
112C4×C373—21no—
113C4×C373—21no—
114C4×C373—21no—
115Dic373—21no—
116Dic373—21no—
117Dic373—21no—
118Dic373—21no—
119Dic373C2×D412yes—
120C2×C373—22no—
121C2×C373—41no—
122C2×C373—41no—
123C2×C373C22×D411yescharacteristic
124C2×C373C2×#4111yescharacteristic
125C2×C373H(5,𝔽2)11yescharacteristic
126D373—22no—
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#121D37311yescharacteristic, complemented
140#120—3731no—
141#120—3731no—
142C4×D4—3731no—
143C4×D4—3731no—
144C2×C22⋊C4—3731no—
145C2×C22⋊C4—3731no—
146C24—3731no—
147C24—3731no—
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×C4C2×D37311yescharacteristic, complemented
166C4⋊C4—3731no—
167C4⋊C4—3731no—
168C4⋊C4—3731no—
169C4⋊C4C2×D37311yescharacteristic, complemented
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⋊C4—3731no—
180C22⋊C4C2×D37311yescharacteristic, complemented
181C22⋊C4C2×D37311yescharacteristic, complemented
182C42—3731no—
183C23—3731no—
184C23—3731no—
185C23—3731no—
186C23—14921no—
187C23—14921no—
188C23—7462no—
189C23—7462no—
190C23—7462no—
191C23—7462no—
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—41no—
206C2×C4—3731no—
207C2×C4—3731no—
208C2×C4—3731no—
209C2×C4—3731no—
210C2×C4—3731no—
211C2×C4—14921no—
212C2×C4—7462no—
213C2×C4—7462no—
214C2×C4—7462no—
215C2×C4—7462no—
216C2×C4C22×D37311yescharacteristic
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—3732no—
229C22—3732no—
230C22—14924no—
231C22—14924no—
232C22—7461no—
233C22—7461no—
234C22—7461no—
235C22—7461no—
236C22—7461no—
237C22—7461no—
238C22—7461no—
239C22D4×D37312yes—
240C22C23×D37311yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—3732no—
246C4—7461no—
247C4—7461no—
248C4—7461no—
249C4—7461no—
250C2—22no—
251C2—41no—
252C2—41no—
253C2—14921no—
254C2—7462no—
255C2—7462no—
256C2C2×D4×D37311yescharacteristic
257C2#41×D37311yescharacteristic
258C2#2604583711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#121×C373C2
hypercenter#121D373
hypocenterC373#546
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20238609C2semidirect
2. C2×D4×D373C2semidirect
3. #26045778C2semidirect
4. #26045777C2semidirect
5. #26045773C2semidirect
6. #26045770C2semidirect
7. #26045765C2semidirect
8. #26045764C2semidirect
9. #26045760C2semidirect
10. #26045760C2semidirect
11. C4×C22⋊D373C2semidirect
12. #26045752C2semidirect
13. C2×#20238587C2semidirect
14. C22⋊C4×D373C2semidirect
15. #121×C373C2semidirect
19. C2×Dic746C22semidirect
24. C2×C22⋊D373C22semidirect
25. C2×C22⋊D373C22semidirect
27. C22×Dic373C22semidirect
28. C22×Dic373C22semidirect
30. C2×C4×D373C22semidirect
31. C2×C4×D373C22semidirect
32. #20238589C22semidirect
33. #20238589C22semidirect
34. #20238589C22semidirect
35. C22⋊Dic373C22semidirect
36. C22⋊Dic373C22semidirect
37. C22⋊Dic373C22semidirect
41. #20238587C22semidirect
42. #20238587C22semidirect
43. #20238587C22semidirect
44. #20238587C22semidirect
45. C4×Dic373C22semidirect
47. C2×D4×C373C22semidirect
48. C2×D4×C373C22semidirect
49. C22×C4×C373C22semidirect
50. C4⋊C4×C373C22semidirect
51. C22⋊C4×C373C22semidirect
52. C22⋊C4×C373C22semidirect
68. C22⋊D373D4semidirect
110. C2×D373#41semidirect
129. C373#546semidirect
139. #121D373semidirect
158. C2×D4C2×D373semidirect
159. C2×D4C2×D373semidirect
165. C22×C4C2×D373semidirect
169. C4⋊C4C2×D373semidirect
180. C22⋊C4C2×D373semidirect
181. C22⋊C4C2×D373semidirect

Maximal subgroups

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

subgroupindex
C2×#202386092
C2×D4×D3732
#260457782
#260457772
#260457732
#260457702
#260457652
#260457642
#260457602
C4×C22⋊D3732
#260457522
C2×#202385872
C22⋊C4×D3732
#121×C3732
#546373