Finite Group Databasebeta

← #38734775

Subgroups — #38734775

Order 24896.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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#27159377C211yesmaximal, characteristic, complemented
2#27159377C211yesmaximal, characteristic, complemented
3#27159370C211yesmaximal, characteristic, complemented
4#27159368C211yesmaximal, characteristic, complemented
5#27159367C211yesmaximal, characteristic, complemented
6#27159366C211yesmaximal, characteristic, complemented
7#27159363C211yesmaximal, characteristic, complemented
8#27159361C211yesmaximal, characteristic, complemented
9#27159353C211yesmaximal, characteristic
10C4×C22⋊D389C211yesmaximal, characteristic, complemented
11#27159344C211yesmaximal, characteristic, complemented
12#27159343C211yesmaximal, characteristic, complemented
13#27159340C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic389C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C389C211yesmaximal, characteristic, complemented
16C2×Dic778C2211yescharacteristic, complemented
17C2×C22⋊D389C2211yescharacteristic
18C2×C22⋊D389C2211yescharacteristic
19C2×C22⋊D389C2211yescharacteristic
20C22×Dic389C2211yescharacteristic
21C22×Dic389C2211yescharacteristic
22C2×C4×D389C2211yescharacteristic, complemented
23C4⋊Dic389C2211yescharacteristic, complemented
24C4⋊Dic389C2211yescharacteristic, complemented
25C4⋊Dic389C2211yescharacteristic, complemented
26#20261945C2211yescharacteristic, complemented
27#20261945C2211yescharacteristic, complemented
28#20261945C2211yescharacteristic, complemented
29#20261945C2211yescharacteristic, complemented
30#20261945C2211yescharacteristic, complemented
31C22⋊Dic389C2211yescharacteristic
32C22⋊Dic389C2211yescharacteristic
33C22⋊Dic389C2211yescharacteristic
34C22⋊Dic389C2211yescharacteristic
35C22⋊Dic389C2211yescharacteristic
36C22⋊Dic389C2211yescharacteristic
37C22⋊Dic389C2211yescharacteristic
38#20261943C2211yescharacteristic, complemented
39#20261943C2211yescharacteristic, complemented
40#20261943C2211yescharacteristic, complemented
41C4×Dic389C2211yescharacteristic, complemented
42C4×Dic389C2211yescharacteristic, complemented
43C4×Dic389C2211yescharacteristic, complemented
44C24×C389C2211yescharacteristic
45C22×C4×C389C2211yescharacteristic
46C22×C4×C389C2211yescharacteristic
47C22⋊C4×C389C2211yescharacteristic
48C22⋊C4×C389C2211yescharacteristic
49C22⋊C4×C389C2211yescharacteristic
50C22⋊C4×C389C2211yescharacteristic
51C22×D389C2311yescharacteristic
52Dic778—41no—
53C22⋊D389—22no—
54C22⋊D389—22no—
55C22⋊D389—41no—
56C2×Dic389—22no—
57C2×Dic389—41no—
58C2×Dic389C2311yescharacteristic
59C2×Dic389C2311yescharacteristic
60C2×Dic389C2311yescharacteristic
61C2×Dic389C2311yescharacteristic
62C2×Dic389C2311yescharacteristic
63C2×Dic389C2311yescharacteristic
64C2×Dic389C2311yescharacteristic
65C4×D389—41no—
66C23×C389—22no—
67C23×C389—22no—
68C23×C389—41no—
69C23×C389C2311yescharacteristic
70C23×C389C2311yescharacteristic
71C23×C389C2311yescharacteristic
72C2×C4×C389—22no—
73C2×C4×C389—41no—
74C2×C4×C389C2311yescharacteristic
75C2×C4×C389C2311yescharacteristic
76C2×C4×C389C2311yescharacteristic
77C2×C4×C389C2311yescharacteristic
78C22×C389—21no—
79C22×C389—21no—
80C22×C389—21no—
81C22×C389—21no—
82C22×C389—21no—
83C22×C389—21no—
84C22×C389—21no—
85C22×C389—44no—
86C22×C389#4112yes—
87C22×C389#4112yes—
88C22×C389C2411yescharacteristic
89C2×D389—21no—
90C2×D389—21no—
91C2×D389—21no—
92C4×C389—21no—
93C4×C389—21no—
94C4×C389—21no—
95C4×C389—21no—
96Dic389—21no—
97Dic389—21no—
98Dic389—21no—
99Dic389—21no—
100Dic389—21no—
101Dic389—21no—
102Dic389—21no—
103C2×C389—22no—
104C2×C389—22no—
105C2×C389—41no—
106C2×C389C2×#4111yescharacteristic
107C2×C389C2×#4111yescharacteristic
108C2×C389H(5,𝔽2)11yescharacteristic
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⋊Q8—3891no—
120#120—3891no—
121C4×D4—3891no—
122C4×D4—3891no—
123#117—3891no—
124#117—3891no—
125C2×C22⋊C4—3891no—
126C2×C22⋊C4D38911yescharacteristic, complemented
127C24C2×D38911yescharacteristic
128C2×Q8—3891no—
129C2×D4—3891no—
130C2×D4—3891no—
131C2×D4—3891no—
132C22×C4—3891no—
133C22×C4—3891no—
134C22×C4—3891no—
135C22×C4C2×D38911yescharacteristic
136C22×C4C2×D38911yescharacteristic
137C4⋊C4—3891no—
138C4⋊C4—3891no—
139C4⋊C4—3891no—
140C4⋊C4—3891no—
141C4⋊C4—3891no—
142C4⋊C4—3891no—
143C4⋊C4—3891no—
144C4⋊C4—3891no—
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⋊C4C2×D38911yescharacteristic
156C22⋊C4C2×D38911yescharacteristic
157C22⋊C4C2×D38911yescharacteristic
158C22⋊C4C2×D38911yescharacteristic
159C42—3891no—
160C42—3891no—
161C42—3891no—
162C23—22no—
163C23—22no—
164C23—41no—
165C23—3891no—
166C23C22×D38911yescharacteristic
167C23C22×D38911yescharacteristic
168C23C22×D38911yescharacteristic
169Q8—15561no—
170D4—7782no—
171D4—7782no—
172D4—15561no—
173C2×C4—22no—
174C2×C4—41no—
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×C4—3891no—
184C2×C4—3891no—
185C2×C4C22×D38911yescharacteristic
186C2×C4C22×D38911yescharacteristic
187C2×C4C22×D38911yescharacteristic
188C2×C4C22×D38911yescharacteristic
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—44no—
197C22—7781no—
198C22—7781no—
199C22—7781no—
200C22#2026196512yes—
201C22#2026196612yes—
202C22C23×D38911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—7781no—
208C4—7781no—
209C4—7781no—
210C4—7781no—
211C4—7781no—
212C4—7781no—
213C4—7781no—
214C2—22no—
215C2—22no—
216C2—41no—
217C2—15561no—
218C2C2×#2026196511yescharacteristic
219C2C2×#2026196611yescharacteristic
220C2#2715942911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D389
commutator, socle, cosocleC22×C389C24
FittingC2×C22⋊C4×C389C2
prime radical#27159353C2
hypercenterC2×C22⋊C4D389
hypocenterC389#551
LCSG ▷ C22×C389 ▷ C389
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #27159377C2semidirect
2. #27159377C2semidirect
3. #27159370C2semidirect
4. #27159368C2semidirect
5. #27159367C2semidirect
6. #27159366C2semidirect
7. #27159363C2semidirect
8. #27159361C2semidirect
10. C4×C22⋊D389C2semidirect
11. #27159344C2semidirect
12. #27159343C2semidirect
13. #27159340C2semidirect
14. C2×C22⋊Dic389C2semidirect
15. C2×C22⋊C4×C389C2semidirect
16. C2×Dic778C22semidirect
22. C2×C4×D389C22semidirect
23. C4⋊Dic389C22semidirect
24. C4⋊Dic389C22semidirect
25. C4⋊Dic389C22semidirect
26. #20261945C22semidirect
27. #20261945C22semidirect
28. #20261945C22semidirect
29. #20261945C22semidirect
30. #20261945C22semidirect
38. #20261943C22semidirect
39. #20261943C22semidirect
40. #20261943C22semidirect
41. C4×Dic389C22semidirect
42. C4×Dic389C22semidirect
43. C4×Dic389C22semidirect
110. C389#551semidirect
126. C2×C22⋊C4D389semidirect

Maximal subgroups

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

subgroupindex
#271593772
#271593702
#271593682
#271593672
#271593662
#271593632
#271593612
#271593532
C4×C22⋊D3892
#271593442
#271593432
#271593402
C2×C22⋊Dic3892
C2×C22⋊C4×C3892
#551389