Finite Group Databasebeta

← #41011931

Subgroups — #41011931

Order 27008.

274 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 81 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#27213110C211yesmaximal, characteristic
2#27213009C211yesmaximal, characteristic
3#27212883C214yesmaximal, complemented
4C2×C22⋊Q8×C211C211yesmaximal, characteristic
5C2×#16809946—21no—
6C2×C22⋊Dic211—21no—
7C22⋊(C211⋊C8)—21no—
8C22⋊(C211⋊C8)C2212yes—
9C211⋊#95—21no—
10C211⋊#95C2212yes—
11C22×Q8×C211C411yescharacteristic, complemented
12C23×C4×C211C2211yescharacteristic
13C22⋊Q8×C211—21no—
14C22⋊Q8×C211—21no—
15C22⋊Q8×C211C412yescomplemented
16C22⋊Q8×C211C2212yes—
17C2×C4⋊C4×C211—21no—
18C2×C4⋊C4×C211C411yescharacteristic, complemented
19C2×C22⋊C4×C211—21no—
20C22×Dic211—22no—
21#16809946—42no—
22C2×C211⋊C8—22no—
23C22⋊Dic211—42no—
24C24×C211D411yescharacteristic
25C2×Q8×C211—21no—
26C2×Q8×C211—42no—
27C2×Q8×C211C2×C412yes—
28C22×C4×C211—21no—
29C22×C4×C211—21no—
30C22×C4×C211—42no—
31C22×C4×C211C2×C411yescharacteristic
32C22×C4×C211C2×C411yescharacteristic
33C22×C4×C211D411yescharacteristic
34C22×C4×C211D411yescharacteristic
35C22×C4×C211D411yescharacteristic
36C22×C4×C211C2311yescharacteristic
37C4⋊C4×C211—22no—
38C4⋊C4×C211—21no—
39C4⋊C4×C211—22no—
40C4⋊C4×C211C2×C412yes—
41C22⋊C4×C211—22no—
42C22⋊C4×C211—22no—
43C2×Dic211—42no—
44C2×Dic211—42no—
45C2×Dic211—42no—
46C211⋊C8—42no—
47C23×C211—22no—
48C23×C211—22no—
49C23×C211—41no—
50C23×C211C22⋊C411yescharacteristic
51C23×C211C22⋊C411yescharacteristic
52C23×C211C2×D411yescharacteristic
53Q8×C211—42no—
54Q8×C211—42no—
55C2×C4×C211—22no—
56C2×C4×C211—22no—
57C2×C4×C211—22no—
58C2×C4×C211—22no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—42no—
64C2×C4×C211—42no—
65C2×C4×C211C22⋊C411yescharacteristic
66C2×C4×C211C22⋊C411yescharacteristic
67C2×C4×C211C22×C411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C22×C211—21no—
70C22×C211—21no—
71C22×C211—21no—
72C22×C211—21no—
73C22×C211—21no—
74C22×C211—21no—
75C22×C211—21no—
76C22×C211—44no—
77C22×C211#9912yes—
78C22×C211C2×C22⋊C411yescharacteristic
79C22×C211#13712yes—
80C4×C211—21no—
81C4×C211—21no—
82C4×C211—21no—
83C4×C211—21no—
84C4×C211—42no—
85C4×C211—42no—
86Dic211—82no—
87C2×C211—22no—
88C2×C211—22no—
89C2×C211—41no—
90C2×C211C2×#9911yescharacteristic
91C2×C211#41911yescharacteristic
92C2×C211#42111yescharacteristic
93C211#151111yescharacteristic, complemented, Sylow
94#1511—2111nomaximal, Sylow
95C2×C22⋊Q8D21111yescharacteristic
96#407—2111no—
97#381—2111no—
98#328—2114no—
99C22×Q8Dic21111yescharacteristic, complemented
100C23×C4C2×D21111yescharacteristic
101C2×M16—4221no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104C22⋊Q8Dic21112yescomplemented
105C22⋊Q8C2×D21112yes—
106C2×C4⋊C4—21no—
107C2×C4⋊C4Dic21111yescharacteristic, complemented
108C2×C22⋊C4—21no—
109C2×C22⋊C4—4221no—
110C22⋊C8—4221no—
111C22⋊C8—2112no—
112#95—4221no—
113#95—2112no—
114C24C22⋊D21111yescharacteristic
115C2×Q8—21no—
116C2×Q8—42no—
117C2×Q8C2×Dic21112yes—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—42no—
121C22×C4—4222no—
122C22×C4C2×Dic21111yescharacteristic
123C22×C4C2×Dic21111yescharacteristic
124C22×C4C22⋊D21111yescharacteristic
125C22×C4C22⋊D21111yescharacteristic
126C22×C4C22⋊D21111yescharacteristic
127C22×C4C22×D21111yescharacteristic
128M16—8442no—
129C2×C8—4222no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4C2×Dic21112yes—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—8442no—
137C23—22no—
138C23—22no—
139C23—41no—
140C23C22⋊Dic21111yescharacteristic
141C23C22⋊Dic21111yescharacteristic
142C23C2×C22⋊D21111yescharacteristic
143Q8—42no—
144Q8—42no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—8442no—
156C2×C4—8442no—
157C2×C4—8442no—
158C2×C4C22⋊Dic21111yescharacteristic
159C2×C4C22⋊Dic21111yescharacteristic
160C2×C4C22×Dic21111yescharacteristic
161C2×C4C2×C22⋊D21111yescharacteristic
162C8—8442no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—44no—
171C22#2026989212yes—
172C22C2×C22⋊Dic21111yescharacteristic
173C22#2027000812yes—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—42no—
179C4—42no—
180C4—16882no—
181C2—22no—
182C2—22no—
183C2—41no—
184C2C2×#2026989211yescharacteristic
185C2#2721315511yescharacteristic
186C2#2721316311yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×C22⋊Q8×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenterC2×C22⋊Q8D211
hypocenterC211#1511
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #27212883C2semidirect
11. C22×Q8×C211C4semidirect
15. C22⋊Q8×C211C4semidirect
18. C2×C4⋊C4×C211C4semidirect
93. C211#1511semidirect
99. C22×Q8Dic211semidirect
104. C22⋊Q8Dic211semidirect
107. C2×C4⋊C4Dic211semidirect

Maximal subgroups

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

subgroupindex
#272131102
#272130092
#272128832
C2×C22⋊Q8×C2112
#1511211