Finite Group Databasebeta

← #41013129

Subgroups — #41013129

Order 27008.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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×#20269933C212yesmaximal
2C2×#20269933C211yesmaximal, characteristic
3#27212913C211yesmaximal, characteristic
4#27212912C212yesmaximal
5#342×C211C211yesmaximal, characteristic
6C23×Dic211C2211yescharacteristic
7#20269933—24no—
8#20269933—22no—
9#20269933—22no—
10#20269933C418yescomplemented
11C2×Dic211⋊C4—22no—
12C2×Dic211⋊C4C412yescomplemented
13C2×C22⋊Dic211C2211yescharacteristic
14C2×C22⋊Dic211C2212yes—
15C2×C4×Dic211—22no—
16C2×C4×Dic211C412yescomplemented
17C2×C22⋊C4×C211C2211yescharacteristic
18C2×C22⋊C4×C211C2212yes—
19C22×Dic211—22no—
20C22×Dic211C2×C412yes—
21C22×Dic211C2×C411yescharacteristic
22C22×Dic211C2×C411yescharacteristic
23C22×Dic211C2×C414yes—
24C22×Dic211D412yes—
25C22×Dic211D412yes—
26C22×Dic211D412yes—
27C22×Dic211Q812yes—
28Dic211⋊C4—28no—
29Dic211⋊C4—24no—
30C22⋊Dic211—22no—
31C22⋊Dic211—22no—
32C22⋊Dic211C2×C414yes—
33C4×Dic211—28no—
34C4×Dic211—22no—
35C4×Dic211—22no—
36C24×C211C2311yescharacteristic
37C22×C4×C211—22no—
38C22×C4×C211C2×C412yes—
39C22⋊C4×C211—22no—
40C22⋊C4×C211—22no—
41C22⋊C4×C211C2×C414yes—
42C2×Dic211—22no—
43C2×Dic211—24no—
44C2×Dic211—44no—
45C2×Dic211—42no—
46C2×Dic211C4212yes—
47C2×Dic211C4212yes—
48C2×Dic211C22⋊C414yes—
49C2×Dic211C22⋊C411yescharacteristic
50C2×Dic211C22⋊C411yescharacteristic
51C2×Dic211C22⋊C414yes—
52C2×Dic211C22⋊C412yes—
53C2×Dic211C4⋊C412yes—
54C2×Dic211C4⋊C412yes—
55C2×Dic211C4⋊C414yes—
56C2×Dic211C4⋊C414yes—
57C23×C211—42no—
58C23×C211C22×C411yescharacteristic
59C23×C211C22×C412yes—
60C23×C211C2×D412yes—
61C23×C211C2×D411yescharacteristic
62C23×C211C2×Q811yescharacteristic
63C2×C4×C211—22no—
64C2×C4×C211—24no—
65C2×C4×C211—44no—
66C2×C4×C211—42no—
67C22×C211—22no—
68C22×C211—24no—
69C22×C211—42no—
70C22×C211—41no—
71C22×C211—41no—
72C22×C211C2×C4211yescharacteristic
73C22×C211C2×C22⋊C412yes—
74C22×C211C2×C22⋊C411yescharacteristic
75C22×C211C2×C4⋊C411yescharacteristic
76C22×C211C2×C4⋊C412yes—
77C4×C211—44no—
78C4×C211—42no—
79Dic211—44no—
80Dic211—42no—
81Dic211#9512yes—
82Dic211#9514yes—
83Dic211#9512yes—
84C2×C211—22no—
85C2×C211—21no—
86C2×C211—21no—
87C2×C211—22no—
88C2×C211C2×#9511yescharacteristic
89C2×C211#41012yes—
90C211#174011yescharacteristic, complemented, Sylow
91#1740—2111nomaximal, Sylow
92C2×#117—2112no—
93C2×#117—2111no—
94#342—2112no—
95#342—2111no—
96#342D21111yescharacteristic
97C23×C4—2111no—
98#117—4224no—
99#117—4222no—
100#117—4222no—
101#117—2118no—
102C2×C4⋊C4—4222no—
103C2×C4⋊C4—2112no—
104C2×C22⋊C4—2111no—
105C2×C22⋊C4—2112no—
106C2×C22⋊C4C2×D21111yescharacteristic
107C2×C22⋊C4C2×D21112yes—
108C2×C42—4222no—
109C2×C42—2112no—
110C24C22×D21111yescharacteristic
111C22×C4—22no—
112C22×C4—4222no—
113C22×C4—2112no—
114C22×C4—2111no—
115C22×C4—2111no—
116C22×C4—2114no—
117C22×C4—2112no—
118C22×C4—2112no—
119C22×C4—2112no—
120C22×C4—2112no—
121C22×C4C4×D21112yes—
122C4⋊C4—4228no—
123C4⋊C4—4224no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—4222no—
127C22⋊C4—4222no—
128C22⋊C4—2114no—
129C22⋊C4C4×D21114yes—
130C42—4228no—
131C42—4222no—
132C42—4222no—
133C23—42no—
134C23C2×C4×D21111yescharacteristic
135C23C2×C4×D21112yes—
136C23D4×D21112yes—
137C23D4×D21111yescharacteristic
138C23Q8×D21111yescharacteristic
139C2×C4—24no—
140C2×C4—22no—
141C2×C4—44no—
142C2×C4—42no—
143C2×C4—8444no—
144C2×C4—8442no—
145C2×C4—4224no—
146C2×C4—4222no—
147C2×C4—2112no—
148C2×C4—2112no—
149C2×C4—2114no—
150C2×C4—2114no—
151C2×C4—2111no—
152C2×C4—2111no—
153C2×C4—2112no—
154C2×C4—2112no—
155C2×C4—2112no—
156C2×C4—2114no—
157C2×C4—2114no—
158C22—22no—
159C22—24no—
160C22—42no—
161C22—41no—
162C22—41no—
163C22C42×D21111yescharacteristic
164C22C22⋊C4×D21112yes—
165C22C22⋊C4×D21111yescharacteristic
166C22C4⋊C4×D21111yescharacteristic
167C22C4⋊C4×D21112yes—
168C4—44no—
169C4—42no—
170C4—8444no—
171C4—8442no—
172C4—2112no—
173C4—2114no—
174C4—2112no—
175C2—22no—
176C2—21no—
177C2—21no—
178C2—22no—
179C2#95×D21111yescharacteristic
180C2#2721311812yes—
subgroupG/H
centerC22C22⋊C4×D211
commutatorC22×C211C2×C42
FrattiniC24C22×D211
Fitting#342×C211C2
socleC22×C211C2×C22⋊C4
prime radical, cosocleC24×C211C23
hypercenter#342D211
hypocenterC211#1740
LCSG ▷ C22×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
10. #20269933C4semidirect
12. C2×Dic211⋊C4C4semidirect
16. C2×C4×Dic211C4semidirect
90. C211#1740semidirect

Maximal subgroups

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

subgroupindex
C2×#202699332
#272129132
#272129122
#342×C2112
#1740211