Finite Group Databasebeta

← #41014384

Subgroups — #41014384

Order 27008.

324 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 101 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#27213006C211yesmaximal, characteristic
2C4×C22⋊Dic211C211yesmaximal, characteristic
3#27212913C214yesmaximal
4C2×C22⋊Q8×C211C211yesmaximal, characteristic
5C2×C4⋊Dic211—21no—
6C2×C22⋊Dic211C2212yes—
7C2×C22⋊Dic211C2212yes—
8C2×C4×Dic211—21no—
9C211⋊#95—21no—
10C211⋊#95—21no—
11C22×Q8×C211C411yescharacteristic, complemented
12C23×C4×C211C2211yescharacteristic
13C22⋊Q8×C211—21no—
14C22⋊Q8×C211—21no—
15C22⋊Q8×C211—21no—
16C22⋊Q8×C211—21no—
17C2×C4⋊C4×C211C411yescharacteristic, complemented
18C2×C4⋊C4×C211C412yescomplemented
19C2×C22⋊C4×C211C2212yes—
20C22×Dic211—22no—
21C22×Dic211—22no—
22C4⋊Dic211—42no—
23C22⋊Dic211—22no—
24C22⋊Dic211—24no—
25C22⋊Dic211Q814yescomplemented
26C4×Dic211—42no—
27C24×C211C2311yescharacteristic
28C2×Q8×C211—21no—
29C2×Q8×C211—21no—
30C2×Q8×C211—42no—
31C22×C4×C211—42no—
32C22×C4×C211C2×C411yescharacteristic
33C22×C4×C211C2×C412yes—
34C22×C4×C211C2×C412yes—
35C22×C4×C211C2×C411yescharacteristic
36C22×C4×C211D411yescharacteristic
37C22×C4×C211D411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D411yescharacteristic
40C4⋊C4×C211—21no—
41C4⋊C4×C211—21no—
42C4⋊C4×C211—22no—
43C4⋊C4×C211—22no—
44C22⋊C4×C211—22no—
45C22⋊C4×C211—22no—
46C2×Dic211—24no—
47C2×Dic211—42no—
48C2×Dic211—42no—
49C2×Dic211—42no—
50C2×Dic211—42no—
51C2×Dic211—42no—
52C23×C211—41no—
53C23×C211—41no—
54C23×C211C22×C411yescharacteristic
55C23×C211C2×D411yescharacteristic
56C23×C211C2×D411yescharacteristic
57C23×C211C2×Q812yes—
58C23×C211#4112yes—
59Q8×C211—42no—
60Q8×C211—42no—
61C2×C4×C211—22no—
62C2×C4×C211—22no—
63C2×C4×C211—22no—
64C2×C4×C211—22no—
65C2×C4×C211—22no—
66C2×C4×C211—41no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211—42no—
70C2×C4×C211—42no—
71C2×C4×C211C22⋊C411yescharacteristic
72C2×C4×C211C22⋊C411yescharacteristic
73C2×C4×C211C22⋊C411yescharacteristic
74C2×C4×C211C22⋊C411yescharacteristic
75C22×C211—21no—
76C22×C211—21no—
77C22×C211—21no—
78C22×C211—21no—
79C22×C211—21no—
80C22×C211—21no—
81C22×C211—42no—
82C22×C211—42no—
83C22×C211C2×C22⋊C411yescharacteristic
84C22×C211C4×Q812yes—
85C22×C211C22⋊Q812yes—
86C22×C211#12411yescharacteristic
87C22×C211C4⋊Q811yescharacteristic
88C4×C211—21no—
89C4×C211—21no—
90C4×C211—21no—
91C4×C211—42no—
92C4×C211—42no—
93C4×C211#9912yes—
94Dic211—44no—
95Dic211—82no—
96C2×C211—22no—
97C2×C211—21no—
98C2×C211—22no—
99C2×C211—21no—
100C2×C211#39111yescharacteristic
101C2×C211C2×#9911yescharacteristic
102C2×C211#41011yescharacteristic
103C211#199611yescharacteristic, complemented, Sylow
104#1996—2111nomaximal, Sylow
105C2×C22⋊Q8D21111yescharacteristic
106#380—2111no—
107C4×C22⋊C4—2111no—
108#342—2114no—
109C22×Q8Dic21111yescharacteristic, complemented
110C23×C4C2×D21111yescharacteristic
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8—21no—
115C2×C4⋊C4—4221no—
116C2×C4⋊C4Dic21111yescharacteristic, complemented
117C2×C4⋊C4Dic21112yescomplemented
118C2×C22⋊C4—2112no—
119C2×C22⋊C4—2112no—
120C2×C22⋊C4C2×D21112yes—
121C2×C42—4221no—
122#95—4221no—
123#95—4221no—
124C24C22×D21111yescharacteristic
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—42no—
128C22×C4—42no—
129C22×C4—4222no—
130C22×C4—4222no—
131C22×C4C2×Dic21111yescharacteristic
132C22×C4C2×Dic21112yes—
133C22×C4C2×Dic21112yes—
134C22×C4C2×Dic21111yescharacteristic
135C22×C4C22⋊D21111yescharacteristic
136C22×C4C22⋊D21111yescharacteristic
137C22×C4C22⋊D21111yescharacteristic
138C22×C4C22⋊D21111yescharacteristic
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—8442no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—4224no—
147C22⋊C4—4222no—
148C22⋊C4—2114no—
149C42—8442no—
150C23—41no—
151C23—41no—
152C23C22×Dic21111yescharacteristic
153C23C2×C22⋊D21111yescharacteristic
154C23C2×C22⋊D21111yescharacteristic
155C23Q8×D21112yes—
156C23#1680996312yes—
157Q8—42no—
158Q8—42no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—8442no—
170C2×C4—8442no—
171C2×C4—8442no—
172C2×C4—8442no—
173C2×C4—8442no—
174C2×C4—4224no—
175C2×C4C22⋊Dic21111yescharacteristic
176C2×C4C22⋊Dic21111yescharacteristic
177C2×C4C22⋊Dic21111yescharacteristic
178C2×C4C22⋊Dic21111yescharacteristic
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22C2×C22⋊Dic21111yescharacteristic
188C22Q8×Dic21112yes—
189C22#2026995312yes—
190C22#2026996611yescharacteristic
191C22#2026997511yescharacteristic
192C4—21no—
193C4—21no—
194C4—21no—
195C4—42no—
196C4—42no—
197C4—8444no—
198C4—16882no—
199C4#2026989212yes—
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2#2721305311yescharacteristic
205C2C2×#2026989211yescharacteristic
206C2#2721312011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC23×C211C22×C4
FrattiniC24C22×D211
FittingC2×C22⋊Q8×C211C2
socleC22×C211C2×C22⋊C4
prime radical, cosocleC24×C211C23
hypercenterC2×C22⋊Q8D211
hypocenterC211#1996
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
11. C22×Q8×C211C4semidirect
17. C2×C4⋊C4×C211C4semidirect
18. C2×C4⋊C4×C211C4semidirect
25. C22⋊Dic211Q8semidirect
103. C211#1996semidirect
109. C22×Q8Dic211semidirect
116. C2×C4⋊C4Dic211semidirect
117. C2×C4⋊C4Dic211semidirect

Maximal subgroups

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

subgroupindex
#272130062
C4×C22⋊Dic2112
#272129132
C2×C22⋊Q8×C2112
#1996211