Finite Group Databasebeta

← #41011932

Subgroups — #41011932

Order 27008.

274 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 73 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×#20269956C211yesmaximal, characteristic
2#27213009C211yesmaximal, characteristic
3#27212884C214yesmaximal, complemented
4#407×C211C211yesmaximal, characteristic
5C22×Dic422C411yescharacteristic, complemented
6#20269956—21no—
7#20269956—21no—
8#20269956C412yescomplemented
9#20269956C2212yes—
10C2×C4⋊Dic211C411yescharacteristic, complemented
11C2×Dic211⋊C4—21no—
12C2×C22⋊Dic211—21no—
13C2×C22⋊Dic211—21no—
14C211⋊#95—21no—
15C211⋊#95C2212yes—
16C23×C4×C211C2211yescharacteristic
17C2×M16×C211—21no—
18C22⋊C8×C211—21no—
19C22⋊C8×C211C2212yes—
20C2×Dic422—21no—
21C2×Dic422—42no—
22C2×Dic422C2×C412yes—
23C22×Dic211—21no—
24C22×Dic211—21no—
25C22×Dic211—22no—
26C4⋊Dic211—21no—
27C4⋊Dic211C2×C412yes—
28Dic211⋊C4—22no—
29Dic211⋊C4—22no—
30C22⋊Dic211—22no—
31C22⋊Dic211—22no—
32C22⋊Dic211—42no—
33C24×C211D411yescharacteristic
34C22×C4×C211—42no—
35C22×C4×C211C2×C411yescharacteristic
36C22×C4×C211C2×C411yescharacteristic
37C22×C4×C211D411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D411yescharacteristic
40C22×C4×C211C2311yescharacteristic
41M16×C211—42no—
42C2×C8×C211—22no—
43Dic422—42no—
44Dic422—42no—
45C2×Dic211—22no—
46C2×Dic211—22no—
47C2×Dic211—42no—
48C2×Dic211—42no—
49C2×Dic211—42no—
50C2×Dic211—42no—
51C2×Dic211—42no—
52C23×C211—22no—
53C23×C211—22no—
54C23×C211—41no—
55C23×C211C22⋊C411yescharacteristic
56C23×C211C22⋊C411yescharacteristic
57C23×C211C2×D411yescharacteristic
58C2×C4×C211—22no—
59C2×C4×C211—22no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211C22⋊C411yescharacteristic
65C2×C4×C211C22⋊C411yescharacteristic
66C2×C4×C211C22×C411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C8×C211—42no—
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—
84Dic211—42no—
85Dic211—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⋊Q8—2111no—
96#407D21111yescharacteristic
97#381—2111no—
98#328—2114no—
99C22×Q8—2111no—
100C23×C4C2×D21111yescharacteristic
101C2×M16—21no—
102C22⋊Q8—4221no—
103C22⋊Q8—4221no—
104C22⋊Q8—2112no—
105C22⋊Q8—2112no—
106C2×C4⋊C4—4221no—
107C2×C4⋊C4—2111no—
108C2×C22⋊C4—4221no—
109C2×C22⋊C4—4221no—
110C22⋊C8—21no—
111C22⋊C8C2×D21112yes—
112#95—4221no—
113#95—2112no—
114C24D84411yescharacteristic
115C2×Q8—8442no—
116C2×Q8—4221no—
117C2×Q8—2112no—
118C22×C4—42no—
119C22×C4—4222no—
120C22×C4—4221no—
121C22×C4—4221no—
122C22×C4C4×D21111yescharacteristic
123C22×C4C4×D21111yescharacteristic
124C22×C4C22⋊D21111yescharacteristic
125C22×C4C22⋊D21111yescharacteristic
126C22×C4D84411yescharacteristic
127C22×C4C22×D21111yescharacteristic
128M16—42no—
129C2×C8—22no—
130C4⋊C4—4222no—
131C4⋊C4—4221no—
132C4⋊C4—4222no—
133C4⋊C4—2112no—
134C22⋊C4—8442no—
135C22⋊C4—4222no—
136C22⋊C4—4222no—
137C23—22no—
138C23—22no—
139C23—41no—
140C23(C2×D211)⋊C411yescharacteristic
141C23(C2×D211)⋊C411yescharacteristic
142C23C2×D84411yescharacteristic
143Q8—8442no—
144Q8—8442no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—8442no—
152C2×C4—8442no—
153C2×C4—8442no—
154C2×C4—8442no—
155C2×C4—8442no—
156C2×C4—4222no—
157C2×C4—4222no—
158C2×C4(C2×D211)⋊C411yescharacteristic
159C2×C4(C2×D211)⋊C411yescharacteristic
160C2×C4C2×C4×D21111yescharacteristic
161C2×C4C2×C22⋊D21111yescharacteristic
162C8—42no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—44no—
171C22#2026989112yes—
172C22C2×(C2×D211)⋊C411yescharacteristic
173C22#2027001012yes—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—8442no—
179C4—8442no—
180C4—16882no—
181C2—22no—
182C2—22no—
183C2—41no—
184C2C2×#2026989111yescharacteristic
185C2#2721315611yescharacteristic
186C2#2721316211yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#407×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenter#407D211
hypocenterC211#1511
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. #27212884C2semidirect
5. C22×Dic422C4semidirect
8. #20269956C4semidirect
10. C2×C4⋊Dic211C4semidirect
93. C211#1511semidirect

Maximal subgroups

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

subgroupindex
C2×#202699562
#272130092
#272128842
#407×C2112
#1511211