Finite Group Databasebeta

← #14580251

Subgroups — #14580251

Order 2624.

238 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C22×C22⋊D41C211yesmaximal, characteristic
2C2×#10620279C211yesmaximal, characteristic, complemented
3C2×C22⋊Dic41C211yesmaximal, characteristic, complemented
4C2×#10620281C211yesmaximal, characteristic, complemented
5C2×#10620281C211yesmaximal, characteristic, complemented
6#13050071C211yesmaximal, characteristic, complemented
7C2×C22⋊C4×C41C211yesmaximal, characteristic, complemented
8C24×C41C2211yescharacteristic
9C23×D41C2211yescharacteristic
10C22×C4×C41C2211yescharacteristic
11C22×C4×C41C2211yescharacteristic
12C2×C22⋊D41—22no—
13C2×C22⋊D41—22no—
14C2×C22⋊D41—22no—
15C22×Dic41C2211yescharacteristic
16C22×Dic41C2211yescharacteristic
17C22×Dic41C2211yescharacteristic
18C22⋊C4×C41—22no—
19C22⋊Dic41—22no—
20#10620281—22no—
21#10620281—22no—
22#10620279—22no—
23C23×C41—21no—
24C23×C41—21no—
25C23×C41—21no—
26C23×C41—21no—
27C23×C41—21no—
28C23×C41—21no—
29C23×C41D412yes—
30C23×C41C2311yescharacteristic
31C22×D41—21no—
32C22×D41—21no—
33C22×D41—21no—
34C22×D41—21no—
35C22×D41—21no—
36C22×D41—21no—
37C22×D41D412yes—
38C2×C4×C41—21no—
39C2×C4×C41—21no—
40C2×C4×C41—21no—
41C2×C4×C41—21no—
42C2×C4×C41—21no—
43C2×C4×C41—21no—
44C22⋊D41—28no—
45C2×Dic41—21no—
46C2×Dic41—21no—
47C2×Dic41—21no—
48C2×Dic41—21no—
49C2×Dic41—21no—
50C2×Dic41—21no—
51C2×Dic41—21no—
52C2×Dic41D412yes—
53C2×Dic41D412yescomplemented
54C22×C41—22no—
55C22×C41—22no—
56C22×C41—22no—
57C22×C41—41no—
58C22×C41—41no—
59C22×C41—41no—
60C22×C41—41no—
61C22×C41C2×D411yescharacteristic
62C22×C41C2×D411yescharacteristic
63C22×C41C2×D411yescharacteristic
64C22×C41C2×D411yescharacteristic
65C22×C41#4111yescharacteristic
66C22×C41#4111yescharacteristic
67C22×C41#4111yescharacteristic
68C2×D41—22no—
69C2×D41—22no—
70C2×D41—22no—
71C2×D41—41no—
72C2×D41—41no—
73C2×D41—41no—
74C2×D41—41no—
75C4×C41—41no—
76C4×C41—41no—
77Dic41—22no—
78Dic41—22no—
79Dic41—41no—
80C2×C41—42no—
81C2×C41#12011yescharacteristic
82C2×C41#12111yescharacteristic
83C2×C41#12111yescharacteristic
84C2×C41#12111yescharacteristic
85C2×C41#12311yescharacteristic
86C2×C41#12311yescharacteristic
87C2×C41#12411yescharacteristic
88D41—42no—
89#394—411nomaximal, Sylow
90C41#39411yescharacteristic, complemented, Sylow
91C22×D4—411no—
92C2×C4⋊C4—411no—
93C2×C22⋊C4—411no—
94C2×C22⋊C4—411no—
95C2×C22⋊C4—411no—
96C2×C22⋊C4D4111yescharacteristic, complemented
97#95—411no—
98C24—411no—
99C24C2×D4111yescharacteristic
100C2×D4—822no—
101C2×D4—822no—
102C2×D4—822no—
103C22×C4—411no—
104C22×C4—411no—
105C22×C4—411no—
106C22×C4C2×D4111yescharacteristic
107C22×C4C2×D4111yescharacteristic
108C4⋊C4—822no—
109C22⋊C4—22no—
110C22⋊C4—822no—
111C22⋊C4—822no—
112C22⋊C4—822no—
113C23—21no—
114C23—21no—
115C23—21no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—412no—
120C23—821no—
121C23—821no—
122C23—821no—
123C23—821no—
124C23—821no—
125C23—821no—
126C23C22⋊D4112yes—
127C23C22×D4111yescharacteristic
128D4—828no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—412no—
136C2×C4—412no—
137C2×C4—821no—
138C2×C4—821no—
139C2×C4—821no—
140C2×C4—821no—
141C2×C4—821no—
142C2×C4—821no—
143C2×C4—821no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—41no—
148C22—41no—
149C22—41no—
150C22—41no—
151C22—822no—
152C22—822no—
153C22—822no—
154C22—1641no—
155C22—1641no—
156C22—1641no—
157C22—1641no—
158C22#1062030611yescharacteristic
159C22#1062030611yescharacteristic
160C22D4×D4111yescharacteristic
161C22D4×D4111yescharacteristic
162C22D4×D4111yescharacteristic
163C22#1062030811yescharacteristic
164C22C2×C22⋊D4111yescharacteristic
165C4—41no—
166C4—41no—
167C4—822no—
168C4—822no—
169C4—1641no—
170C2—42no—
171C2—1642no—
172C2#1305013211yescharacteristic
173C2#1305013611yescharacteristic
174C2#1305013711yescharacteristic
175C2#1305013711yescharacteristic
176C2#1305014611yescharacteristic
177C2#1305014711yescharacteristic
178C2#1305015011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D41
commutator, socle, cosocleC23×C41C23
FittingC2×C22⋊C4×C41C2
prime radicalC22×C22⋊D41C2
hypercenterC2×C22⋊C4D41
hypocenterC41#394
LCSG ▷ C23×C41 ▷ C41
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#10620279C2semidirect
3. C2×C22⋊Dic41C2semidirect
4. C2×#10620281C2semidirect
5. C2×#10620281C2semidirect
6. #13050071C2semidirect
7. C2×C22⋊C4×C41C2semidirect
53. C2×Dic41D4semidirect
90. C41#394semidirect
96. C2×C22⋊C4D41semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D412
C2×#106202792
C2×C22⋊Dic412
C2×#106202812
#130500712
C2×C22⋊C4×C412
#39441