Finite Group Databasebeta

← #38734739

Subgroups — #38734739

Order 24896.

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×#20261966C211yesmaximal, characteristic, complemented
2C22×C22⋊D389C211yesmaximal, characteristic, complemented
3#27159370C211yesmaximal, characteristic, complemented
4#27159367C211yesmaximal, characteristic, complemented
5#27159366C211yesmaximal, characteristic, complemented
6#27159361C211yesmaximal, characteristic, complemented
7#27159358C211yesmaximal, characteristic, complemented
8#27159357C211yesmaximal, characteristic, complemented
9#27159357C211yesmaximal, characteristic, complemented
10#27159354C211yesmaximal, characteristic, complemented
11#27159353C211yesmaximal, characteristic, complemented
12C4×C22⋊D389C211yesmaximal, characteristic, complemented
13#27159344C211yesmaximal, characteristic, complemented
14C22⋊C4×D389C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C389C211yesmaximal, characteristic, complemented
16C23×D389C2211yescharacteristic
17#20261966—24no—
18C2×Dic778C2211yescharacteristic, complemented
19C2×D1556C2211yescharacteristic, complemented
20C2×D1556C2211yescharacteristic, complemented
21C2×C22⋊D389—24no—
22C2×C22⋊D389C2211yescharacteristic
23C2×C22⋊D389C2211yescharacteristic
24C2×C22⋊D389C2211yescharacteristic
25C2×C22⋊D389C2211yescharacteristic
26C2×C22⋊D389C2211yescharacteristic
27C2×C22⋊D389C2211yescharacteristic, complemented
28C2×C22⋊D389C2211yescharacteristic, complemented
29C22×Dic389C2211yescharacteristic
30C2×C4×D389C2211yescharacteristic, complemented
31C2×C4×D389C2211yescharacteristic, complemented
32C2×C4×D389C2211yescharacteristic, complemented
33C4⋊Dic389C2211yescharacteristic, complemented
34#20261945C2211yescharacteristic, complemented
35#20261945C2211yescharacteristic, complemented
36#20261945C2211yescharacteristic, complemented
37C22⋊Dic389C2211yescharacteristic
38C22⋊Dic389C2211yescharacteristic, complemented
39C22⋊Dic389C2211yescharacteristic, complemented
40#20261943C2211yescharacteristic, complemented
41#20261943C2211yescharacteristic, complemented
42#20261943C2211yescharacteristic, complemented
43#20261943C2211yescharacteristic, complemented
44#20261943C2211yescharacteristic, complemented
45C4×Dic389C2211yescharacteristic, complemented
46C24×C389C2211yescharacteristic
47C22×C4×C389C2211yescharacteristic, complemented
48C22×C4×C389C2211yescharacteristic, complemented
49C22⋊C4×C389C2211yescharacteristic
50C22⋊C4×C389C2211yescharacteristic
51C22⋊C4×C389C2211yescharacteristic, complemented
52C22⋊C4×C389C2211yescharacteristic, complemented
53C22×D389—22no—
54C22×D389—22no—
55C22×D389—41no—
56C22×D389C2311yescharacteristic
57C22×D389C2311yescharacteristic
58C22×D389C2311yescharacteristic
59Dic778—22no—
60D1556—22no—
61D1556—41no—
62C22⋊D389—22no—
63C22⋊D389—22no—
64C22⋊D389—22no—
65C22⋊D389—22no—
66C22⋊D389—22no—
67C22⋊D389—41no—
68C22⋊D389D414yescomplemented
69C2×Dic389—22no—
70C2×Dic389C2311yescharacteristic
71C2×Dic389C2311yescharacteristic
72C2×Dic389C2311yescharacteristic
73C2×Dic389C2311yescharacteristic
74C2×Dic389C2311yescharacteristic
75C4×D389—22no—
76C4×D389—22no—
77C4×D389—41no—
78C23×C389—22no—
79C23×C389—22no—
80C23×C389—41no—
81C23×C389C2311yescharacteristic
82C23×C389C2311yescharacteristic
83C23×C389C2311yescharacteristic
84C2×C4×C389—22no—
85C2×C4×C389—41no—
86C2×C4×C389C2311yescharacteristic
87C2×C4×C389C2311yescharacteristic
88C2×C4×C389C2311yescharacteristic
89C2×C4×C389C2311yescharacteristic
90C22×C389—21no—
91C22×C389—21no—
92C22×C389—21no—
93C22×C389—21no—
94C22×C389—21no—
95C22×C389—21no—
96C22×C389—21no—
97C22×C389—44no—
98C22×C389C2×D412yes—
99C22×C389#4112yescomplemented
100C22×C389C2411yescharacteristic
101C2×D389—21no—
102C2×D389—21no—
103C2×D389—21no—
104C2×D389—21no—
105C2×D389—21no—
106C2×D389—21no—
107C2×D389—21no—
108C2×D389—21no—
109C2×D389—44no—
110C2×D389C2×D412yes—
111C4×C389—21no—
112C4×C389—21no—
113C4×C389—21no—
114C4×C389—21no—
115Dic389—21no—
116Dic389—21no—
117Dic389—21no—
118Dic389—21no—
119Dic389C2×D412yes—
120C2×C389—22no—
121C2×C389—22no—
122C2×C389—41no—
123C2×C389C22×D411yescharacteristic
124C2×C389C2×#4111yescharacteristic
125C2×C389H(5,𝔽2)11yescharacteristic
126D389—22no—
127D389—41no—
128D389—41no—
129C389#54611yescharacteristic, complemented, Sylow
130#546—3891nomaximal, Sylow
131C2×#41—3891no—
132C22×D4—3891no—
133#124—3891no—
134#123—3891no—
135#123—3891no—
136C22⋊Q8—3891no—
137#121—3891no—
138#121—3891no—
139#121—3891no—
140#120—3891no—
141#120—3891no—
142C4×D4—3891no—
143C4×D4—3891no—
144C2×C22⋊C4—3891no—
145C2×C22⋊C4D38911yescharacteristic, complemented
146C24—3891no—
147C24C2×D38911yescharacteristic
148#41—7784no—
149C2×Q8—3891no—
150C2×D4—7784no—
151C2×D4—3891no—
152C2×D4—3891no—
153C2×D4—3891no—
154C2×D4—3891no—
155C2×D4—3891no—
156C2×D4—3891no—
157C2×D4—3891no—
158C2×D4—3891no—
159C2×D4—3891no—
160C22×C4—3891no—
161C22×C4—3891no—
162C22×C4—3891no—
163C22×C4—3891no—
164C22×C4C2×D38911yescharacteristic, complemented
165C22×C4C2×D38911yescharacteristic, complemented
166C4⋊C4—3891no—
167C4⋊C4—3891no—
168C4⋊C4—3891no—
169C4⋊C4—3891no—
170C22⋊C4—3891no—
171C22⋊C4—3891no—
172C22⋊C4—3891no—
173C22⋊C4—3891no—
174C22⋊C4—3891no—
175C22⋊C4—3891no—
176C22⋊C4—3891no—
177C22⋊C4—3891no—
178C22⋊C4C2×D38911yescharacteristic
179C22⋊C4C2×D38911yescharacteristic
180C22⋊C4C2×D38911yescharacteristic, complemented
181C22⋊C4C2×D38911yescharacteristic, complemented
182C42—3891no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—7782no—
187C23—7782no—
188C23—15561no—
189C23—3891no—
190C23—3891no—
191C23—3891no—
192C23C22×D38911yescharacteristic
193C23C22×D38911yescharacteristic
194C23C22×D38911yescharacteristic
195Q8—7782no—
196D4—7782no—
197D4—7782no—
198D4—7782no—
199D4—7782no—
200D4—7782no—
201D4—7782no—
202D4—15561no—
203D4—15561no—
204D4—3894no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—7782no—
208C2×C4—7782no—
209C2×C4—7782no—
210C2×C4—15561no—
211C2×C4—3891no—
212C2×C4—3891no—
213C2×C4—3891no—
214C2×C4—3891no—
215C2×C4—3891no—
216C2×C4C22×D38911yescharacteristic
217C2×C4C22×D38911yescharacteristic
218C2×C4C22×D38911yescharacteristic
219C2×C4C22×D38911yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—44no—
228C22—7781no—
229C22—7781no—
230C22—7781no—
231C22—7781no—
232C22—7781no—
233C22—7781no—
234C22—7781no—
235C22—7781no—
236C22—15564no—
237C22—3892no—
238C22D4×D38912yes—
239C22#2026196612yescomplemented
240C22C23×D38911yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—7781no—
246C4—7781no—
247C4—7781no—
248C4—7781no—
249C4—3892no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—7782no—
254C2—15561no—
255C2—15561no—
256C2C2×D4×D38911yescharacteristic
257C2C2×#2026196611yescharacteristic
258C2#2715942911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D389
commutator, socle, cosocleC22×C389C24
FittingC2×C22⋊C4×C389C2
hypercenterC2×C22⋊C4D389
hypocenterC389#546
LCSG ▷ C22×C389 ▷ C389
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#20261966C2semidirect
2. C22×C22⋊D389C2semidirect
3. #27159370C2semidirect
4. #27159367C2semidirect
5. #27159366C2semidirect
6. #27159361C2semidirect
7. #27159358C2semidirect
8. #27159357C2semidirect
9. #27159357C2semidirect
10. #27159354C2semidirect
11. #27159353C2semidirect
12. C4×C22⋊D389C2semidirect
13. #27159344C2semidirect
14. C22⋊C4×D389C2semidirect
15. C2×C22⋊C4×C389C2semidirect
18. C2×Dic778C22semidirect
19. C2×D1556C22semidirect
20. C2×D1556C22semidirect
27. C2×C22⋊D389C22semidirect
28. C2×C22⋊D389C22semidirect
30. C2×C4×D389C22semidirect
31. C2×C4×D389C22semidirect
32. C2×C4×D389C22semidirect
33. C4⋊Dic389C22semidirect
34. #20261945C22semidirect
35. #20261945C22semidirect
36. #20261945C22semidirect
38. C22⋊Dic389C22semidirect
39. C22⋊Dic389C22semidirect
40. #20261943C22semidirect
41. #20261943C22semidirect
42. #20261943C22semidirect
43. #20261943C22semidirect
44. #20261943C22semidirect
45. C4×Dic389C22semidirect
47. C22×C4×C389C22semidirect
48. C22×C4×C389C22semidirect
51. C22⋊C4×C389C22semidirect
52. C22⋊C4×C389C22semidirect
68. C22⋊D389D4semidirect
99. C22×C389#41semidirect
129. C389#546semidirect
145. C2×C22⋊C4D389semidirect
164. C22×C4C2×D389semidirect
165. C22×C4C2×D389semidirect
180. C22⋊C4C2×D389semidirect
181. C22⋊C4C2×D389semidirect
239. C22#20261966semidirect

Maximal subgroups

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

subgroupindex
C2×#202619662
C22×C22⋊D3892
#271593702
#271593672
#271593662
#271593612
#271593582
#271593572
#271593542
#271593532
C4×C22⋊D3892
#271593442
C22⋊C4×D3892
C2×C22⋊C4×C3892
#546389