Finite Group Databasebeta

← #38734776

Subgroups — #38734776

Order 24896.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#27159379C211yesmaximal, characteristic, complemented
2#27159379C211yesmaximal, characteristic, complemented
3#27159373C211yesmaximal, characteristic, complemented
4#27159369C211yesmaximal, characteristic, complemented
5#27159369C211yesmaximal, characteristic, complemented
6#27159368C211yesmaximal, characteristic, complemented
7#27159360C211yesmaximal, characteristic, complemented
8#27159354C211yesmaximal, characteristic
9#27159346C211yesmaximal, characteristic, complemented
10C4×C22⋊D389C211yesmaximal, characteristic, complemented
11#27159342C211yesmaximal, characteristic, complemented
12#27159340C211yesmaximal, characteristic, complemented
13C2×#20261943C211yesmaximal, characteristic, complemented
14C22⋊C4×D389C211yesmaximal, characteristic, complemented
15C22⋊Q8×C389C211yesmaximal, characteristic, complemented
16C23×D389C2211yescharacteristic
17C2×D1556C2211yescharacteristic
18C2×D1556C2211yescharacteristic
19C2×C22⋊D389C2211yescharacteristic
20C22×Dic389C2211yescharacteristic
21C2×C4×D389C2211yescharacteristic
22C2×C4×D389C2211yescharacteristic
23C2×C4×D389C2211yescharacteristic, complemented
24C4⋊Dic389C2211yescharacteristic, complemented
25C4⋊Dic389C2211yescharacteristic, complemented
26#20261945C2211yescharacteristic, complemented
27#20261945C2211yescharacteristic, complemented
28#20261945C2211yescharacteristic, complemented
29C22⋊Dic389C2211yescharacteristic
30#20261943C2211yescharacteristic
31#20261943C2211yescharacteristic
32#20261943C2211yescharacteristic
33#20261943C2211yescharacteristic
34#20261943C2211yescharacteristic
35#20261943C2211yescharacteristic
36#20261943C2211yescharacteristic
37#20261943C2211yescharacteristic
38#20261943C2211yescharacteristic, complemented
39#20261943C2211yescharacteristic, complemented
40#20261943C2211yescharacteristic, complemented
41C4×Dic389C2211yescharacteristic, complemented
42C4×Dic389C2211yescharacteristic, complemented
43C4×Dic389C2211yescharacteristic, complemented
44C2×Q8×C389C2211yescharacteristic, complemented
45C22×C4×C389C2211yescharacteristic
46C4⋊C4×C389C2211yescharacteristic, complemented
47C4⋊C4×C389C2211yescharacteristic, complemented
48C4⋊C4×C389C2211yescharacteristic, complemented
49C22⋊C4×C389C2211yescharacteristic
50C22⋊C4×C389C2211yescharacteristic
51C22×D389—22no—
52C22×D389—22no—
53C22×D389—41no—
54C22×D389C2311yescharacteristic
55C22×D389C2311yescharacteristic
56C22×D389C2311yescharacteristic
57D1556—22no—
58D1556—41no—
59C22⋊D389—22no—
60C2×Dic389—22no—
61C2×Dic389C2311yescharacteristic
62C2×Dic389C2311yescharacteristic
63C2×Dic389C2311yescharacteristic
64C2×Dic389C2311yescharacteristic
65C2×Dic389C2311yescharacteristic
66C4×D389—22no—
67C4×D389—41no—
68C4×D389—41no—
69C23×C389C2311yescharacteristic
70Q8×C389—41no—
71C2×C4×C389—41no—
72C2×C4×C389C2311yescharacteristic
73C2×C4×C389C2311yescharacteristic
74C2×C4×C389C2311yescharacteristic
75C2×C4×C389C2311yescharacteristic
76C2×C4×C389C2311yescharacteristic
77C2×C4×C389C2311yescharacteristic
78C22×C389—21no—
79C22×C389—21no—
80C22×C389#4112yes—
81C22×C389C2411yescharacteristic
82C2×D389—21no—
83C2×D389—21no—
84C2×D389—21no—
85C2×D389—21no—
86C2×D389—21no—
87C2×D389—21no—
88C2×D389—21no—
89C2×D389—21no—
90C2×D389—44no—
91C2×D389#4112yes—
92C4×C389—21no—
93C4×C389—21no—
94C4×C389—21no—
95C4×C389—21no—
96C4×C389—21no—
97C4×C389—21no—
98Dic389—21no—
99Dic389—21no—
100Dic389—21no—
101Dic389—21no—
102Dic389—21no—
103C2×C389—22no—
104C2×C389C2×#4111yescharacteristic
105C2×C389C2×#4111yescharacteristic
106C2×C389H(5,𝔽2)11yescharacteristic
107D389—22no—
108D389—41no—
109D389—41no—
110C389#55111yescharacteristic, complemented, Sylow
111#551—3891nomaximal, Sylow
112#126—3891no—
113#126—3891no—
114#124—3891no—
115#123—3891no—
116#123—3891no—
117#123—3891no—
118C22⋊Q8—3891no—
119C22⋊Q8D38911yescharacteristic, complemented
120#120—3891no—
121C4×D4—3891no—
122C4×D4—3891no—
123#117—3891no—
124#117—3891no—
125C2×C22⋊C4—3891no—
126C2×C22⋊C4—3891no—
127C24—3891no—
128C2×Q8C2×D38911yescharacteristic, complemented
129C2×D4—3891no—
130C2×D4—3891no—
131C2×D4—3891no—
132C22×C4—3891no—
133C22×C4—3891no—
134C22×C4—3891no—
135C22×C4—3891no—
136C22×C4C2×D38911yescharacteristic
137C4⋊C4—3891no—
138C4⋊C4—3891no—
139C4⋊C4—3891no—
140C4⋊C4—3891no—
141C4⋊C4—3891no—
142C4⋊C4C2×D38911yescharacteristic, complemented
143C4⋊C4C2×D38911yescharacteristic, complemented
144C4⋊C4C2×D38911yescharacteristic, complemented
145C22⋊C4—3891no—
146C22⋊C4—3891no—
147C22⋊C4—3891no—
148C22⋊C4—3891no—
149C22⋊C4—3891no—
150C22⋊C4—3891no—
151C22⋊C4—3891no—
152C22⋊C4—3891no—
153C22⋊C4—3891no—
154C22⋊C4—3891no—
155C22⋊C4—3891no—
156C22⋊C4—3891no—
157C22⋊C4C2×D38911yescharacteristic
158C22⋊C4C2×D38911yescharacteristic
159C42—3891no—
160C42—3891no—
161C42—3891no—
162C23—7782no—
163C23—7782no—
164C23—15561no—
165C23—3891no—
166C23—3891no—
167C23—3891no—
168C23C22×D38911yescharacteristic
169Q8—41no—
170D4—7782no—
171D4—7782no—
172D4—15561no—
173C2×C4—41no—
174C2×C4—7782no—
175C2×C4—7782no—
176C2×C4—15561no—
177C2×C4—15561no—
178C2×C4—3891no—
179C2×C4—3891no—
180C2×C4—3891no—
181C2×C4—3891no—
182C2×C4—3891no—
183C2×C4C22×D38911yescharacteristic
184C2×C4C22×D38911yescharacteristic
185C2×C4C22×D38911yescharacteristic
186C2×C4C22×D38911yescharacteristic
187C2×C4C22×D38911yescharacteristic
188C2×C4C22×D38911yescharacteristic
189C22—21no—
190C22—21no—
191C22—7781no—
192C22—7781no—
193C22—7781no—
194C22—7781no—
195C22—7781no—
196C22—7781no—
197C22—7781no—
198C22—7781no—
199C22—15564no—
200C22—3892no—
201C22#2026196712yes—
202C22C23×D38911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—7781no—
210C4—7781no—
211C4—7781no—
212C4—7781no—
213C4—7781no—
214C2—22no—
215C2—7782no—
216C2—15561no—
217C2—15561no—
218C2#41×D38911yescharacteristic
219C2C2×#2026196711yescharacteristic
220C2#2715942911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D389
commutator, socle, cosocleC22×C389C24
FittingC22⋊Q8×C389C2
prime radical#27159354C2
hypercenterC22⋊Q8D389
hypocenterC389#551
LCSG ▷ C22×C389 ▷ C389
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #27159379C2semidirect
2. #27159379C2semidirect
3. #27159373C2semidirect
4. #27159369C2semidirect
5. #27159369C2semidirect
6. #27159368C2semidirect
7. #27159360C2semidirect
9. #27159346C2semidirect
10. C4×C22⋊D389C2semidirect
11. #27159342C2semidirect
12. #27159340C2semidirect
13. C2×#20261943C2semidirect
14. C22⋊C4×D389C2semidirect
15. C22⋊Q8×C389C2semidirect
23. C2×C4×D389C22semidirect
24. C4⋊Dic389C22semidirect
25. C4⋊Dic389C22semidirect
26. #20261945C22semidirect
27. #20261945C22semidirect
28. #20261945C22semidirect
38. #20261943C22semidirect
39. #20261943C22semidirect
40. #20261943C22semidirect
41. C4×Dic389C22semidirect
42. C4×Dic389C22semidirect
43. C4×Dic389C22semidirect
44. C2×Q8×C389C22semidirect
46. C4⋊C4×C389C22semidirect
47. C4⋊C4×C389C22semidirect
48. C4⋊C4×C389C22semidirect
110. C389#551semidirect
119. C22⋊Q8D389semidirect
128. C2×Q8C2×D389semidirect
142. C4⋊C4C2×D389semidirect
143. C4⋊C4C2×D389semidirect
144. C4⋊C4C2×D389semidirect

Maximal subgroups

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

subgroupindex
#271593792
#271593732
#271593692
#271593682
#271593602
#271593542
#271593462
C4×C22⋊D3892
#271593422
#271593402
C2×#202619432
C22⋊C4×D3892
C22⋊Q8×C3892
#551389