Finite Group Databasebeta

← #41013867

Subgroups — #41013867

Order 27008.

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

normal subgroups only

This group has 338 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
1C22×#16809946C211yesmaximal, characteristic
2C2×#20269896C212yesmaximal, complemented
3#27213110C211yesmaximal, characteristic
4#27212915C212yesmaximal, complemented
5C2×C22⋊Q8×C211C211yesmaximal, characteristic
6C2×#16809946—24no—
7C2×#16809946—21no—
8C2×#16809946C2212yes—
9C2×#16809946C2212yes—
10C22×C211⋊C8—21no—
11#20269896—24no—
12C22⋊(C211⋊C8)—21no—
13C22⋊(C211⋊C8)—21no—
14C22×Q8×C211C2211yescharacteristic
15C23×C4×C211C2211yescharacteristic
16C22⋊Q8×C211—21no—
17C22⋊Q8×C211—21no—
18C22⋊Q8×C211—21no—
19C22⋊Q8×C211—21no—
20C2×C4⋊C4×C211C412yes—
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C22⋊C4×C211C412yes—
23#16809946—24no—
24#16809946—22no—
25#16809946—22no—
26#16809946—22no—
27#16809946D414yes—
28C2×C211⋊C8—24no—
29C2×C211⋊C8—22no—
30C2×C211⋊C8—22no—
31C24×C211C2×C411yescharacteristic
32C2×Q8×C211—21no—
33C2×Q8×C211—21no—
34C2×Q8×C211—22no—
35C2×Q8×C211—22no—
36C22×C4×C211—22no—
37C22×C4×C211—22no—
38C22×C4×C211C2×C412yes—
39C22×C4×C211C2×C411yescharacteristic
40C22×C4×C211C2×C412yes—
41C22×C4×C211D411yescharacteristic
42C22×C4×C211D411yescharacteristic
43C22×C4×C211D411yescharacteristic
44C22×C4×C211D411yescharacteristic
45C22×C4×C211C2311yescharacteristic
46C4⋊C4×C211—22no—
47C4⋊C4×C211—22no—
48C4⋊C4×C211—21no—
49C4⋊C4×C211—21no—
50C22⋊C4×C211—22no—
51C22⋊C4×C211—22no—
52C211⋊C8—24no—
53C211⋊C8—42no—
54C23×C211—22no—
55C23×C211—22no—
56C23×C211—41no—
57C23×C211C22⋊C411yescharacteristic
58C23×C211C22⋊C411yescharacteristic
59C23×C211C22×C411yescharacteristic
60Q8×C211—41no—
61Q8×C211—42no—
62Q8×C211—41no—
63C2×C4×C211—22no—
64C2×C4×C211—21no—
65C2×C4×C211—21no—
66C2×C4×C211—22no—
67C2×C4×C211—22no—
68C2×C4×C211—21no—
69C2×C4×C211—21no—
70C2×C4×C211—22no—
71C2×C4×C211—22no—
72C2×C4×C211—24no—
73C2×C4×C211—42no—
74C2×C4×C211C22⋊C411yescharacteristic
75C2×C4×C211C22⋊C411yescharacteristic
76C2×C4×C211C2×D412yes—
77C2×C4×C211C2×D411yescharacteristic
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211#4112yes—
80C22×C211—21no—
81C22×C211—21no—
82C22×C211—21no—
83C22×C211—21no—
84C22×C211—21no—
85C22×C211—21no—
86C22×C211—21no—
87C22×C211—44no—
88C22×C211#10112yescomplemented
89C22×C211C2×C22⋊C411yescharacteristic
90C22×C211C4×D412yes—
91C4×C211—21no—
92C4×C211—21no—
93C4×C211—42no—
94C4×C211—42no—
95C4×C211#12011yescharacteristic
96C4×C211#12112yes—
97C4×C211#12311yescharacteristic
98C2×C211—22no—
99C2×C211—22no—
100C2×C211—41no—
101C2×C211#38611yescharacteristic
102C2×C211C2×#10111yescharacteristic
103C2×C211#41311yescharacteristic
104C211#190111yescharacteristic, complemented, Sylow
105#1901—2111nomaximal, Sylow
106C22×M16—2111no—
107C2×C22⋊Q8D21111yescharacteristic
108C2×#101—2112no—
109#407—2111no—
110#343—2112no—
111C22×Q8C2×D21111yescharacteristic
112C23×C4C2×D21111yescharacteristic
113C2×M16—4221no—
114C2×M16—4224no—
115C2×M16—2112no—
116C2×M16—2112no—
117C22×C8—4221no—
118C22⋊Q8—21no—
119C22⋊Q8—21no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122C2×C4⋊C4Dic21112yes—
123C2×C4⋊C4C2×D21111yescharacteristic
124C2×C22⋊C4Dic21112yes—
125#101—4224no—
126C22⋊C8—4221no—
127C22⋊C8—4221no—
128C24C2×Dic21111yescharacteristic
129C2×Q8—22no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4C2×Dic21112yes—
136C22×C4C2×Dic21111yescharacteristic
137C22×C4C2×Dic21112yes—
138C22×C4C22⋊D21111yescharacteristic
139C22×C4C22⋊D21111yescharacteristic
140C22×C4C22⋊D21111yescharacteristic
141C22×C4C22⋊D21111yescharacteristic
142C22×C4C22×D21111yescharacteristic
143M16—4222no—
144M16—4222no—
145M16—4224no—
146M16—4222no—
147M16—2114no—
148C2×C8—4222no—
149C2×C8—4224no—
150C2×C8—4222no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C22⋊C4—22no—
156C22⋊C4—22no—
157C23—22no—
158C23—22no—
159C23—41no—
160C23C22⋊Dic21111yescharacteristic
161C23C22⋊Dic21111yescharacteristic
162C23C22×Dic21111yescharacteristic
163Q8—41no—
164Q8—42no—
165Q8—41no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—42no—
177C2×C4C22⋊Dic21111yescharacteristic
178C2×C4C22⋊Dic21111yescharacteristic
179C2×C4D4×D21112yes—
180C2×C4C2×C22⋊D21111yescharacteristic
181C2×C4C2×C22⋊D21111yescharacteristic
182C2×C4#1680996112yes—
183C8—8442no—
184C8—4224no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—44no—
193C22#2026989612yescomplemented
194C22C2×C22⋊Dic21111yescharacteristic
195C22D4×Dic21112yes—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—42no—
200C4#2026994511yescharacteristic
201C4#2026994912yes—
202C4#2026995811yescharacteristic
203C2—22no—
204C2—22no—
205C2—41no—
206C2#2721302611yescharacteristic
207C2C2×#2026989611yescharacteristic
208C2#2721313011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×C22⋊Q8×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC24×C211C2×C4
cosocleC22×C4×C211C23
hypercenterC2×C22⋊Q8D211
hypocenterC211#1901
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
2. C2×#20269896C2semidirect
4. #27212915C2semidirect
88. C22×C211#101semidirect
104. C211#1901semidirect
193. C22#20269896semidirect

Maximal subgroups

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

subgroupindex
C22×#168099462
C2×#202698962
#272131102
#272129152
C2×C22⋊Q8×C2112
#1901211