Finite Group Databasebeta

← #41013864

Subgroups — #41013864

Order 27008.

338 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20269956C211yesmaximal, characteristic
3C2×#20269895C212yesmaximal, complemented
4#27212914C212yesmaximal, complemented
5#407×C211C211yesmaximal, characteristic
6C22×Dic422C2211yescharacteristic
7C2×#16809946—24no—
8C2×#16809946—21no—
9C2×#16809946C2212yes—
10C22×C211⋊C8—21no—
11#20269956—21no—
12#20269956—21no—
13#20269956—21no—
14#20269956—21no—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×Dic211⋊C4C412yes—
17C2×C22⋊Dic211C412yes—
18#20269895—24no—
19C23×C4×C211C2211yescharacteristic
20C2×M16×C211C2212yes—
21C22⋊C8×C211—21no—
22C22⋊C8×C211—21no—
23C2×Dic422—22no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—22no—
27C22×Dic211C2×C412yes—
28C22×Dic211C2×C412yes—
29#16809946—24no—
30#16809946—22no—
31#16809946D414yes—
32C2×C211⋊C8—24no—
33C2×C211⋊C8—22no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36Dic211⋊C4—22no—
37Dic211⋊C4—22no—
38C22⋊Dic211—22no—
39C22⋊Dic211—22no—
40C24×C211C2×C411yescharacteristic
41C22×C4×C211—22no—
42C22×C4×C211—22no—
43C22×C4×C211C2×C411yescharacteristic
44C22×C4×C211D411yescharacteristic
45C22×C4×C211D411yescharacteristic
46C22×C4×C211D411yescharacteristic
47C22×C4×C211D411yescharacteristic
48C22×C4×C211C2311yescharacteristic
49M16×C211—22no—
50M16×C211—22no—
51C2×C8×C211—22no—
52Dic422—41no—
53Dic422—42no—
54Dic422—41no—
55C2×Dic211—22no—
56C2×Dic211—22no—
57C2×Dic211—24no—
58C2×Dic211—42no—
59C211⋊C8—24no—
60C23×C211—22no—
61C23×C211—22no—
62C23×C211—41no—
63C23×C211C22⋊C411yescharacteristic
64C23×C211C22⋊C411yescharacteristic
65C23×C211C22×C411yescharacteristic
66C2×C4×C211—22no—
67C2×C4×C211—21no—
68C2×C4×C211—21no—
69C2×C4×C211—22no—
70C2×C4×C211—22no—
71C2×C4×C211—21no—
72C2×C4×C211—21no—
73C2×C4×C211C22⋊C411yescharacteristic
74C2×C4×C211C22⋊C411yescharacteristic
75C2×C4×C211C2×D412yes—
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C2×C4×C211#4112yes—
79C8×C211—42no—
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#12011yescharacteristic
94C4×C211#12112yes—
95C4×C211#12311yescharacteristic
96Dic211—42no—
97Dic211—42no—
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⋊Q8—2111no—
108C2×#101—2112no—
109#407D21111yescharacteristic
110#343—2112no—
111C22×Q8—2111no—
112C23×C4C2×D21111yescharacteristic
113C2×M16—4221no—
114C2×M16—4224no—
115C2×M16—2112no—
116C2×M16C2×D21112yes—
117C22×C8—4221no—
118C22⋊Q8—4221no—
119C22⋊Q8—4221no—
120C22⋊Q8—4221no—
121C22⋊Q8—4221no—
122C2×C4⋊C4—2112no—
123C2×C4⋊C4—2111no—
124C2×C22⋊C4—2112no—
125#101—4224no—
126C22⋊C8—21no—
127C22⋊C8—21no—
128C24C4×D21111yescharacteristic
129C2×Q8—4222no—
130C2×Q8—4221no—
131C2×Q8—4221no—
132C2×Q8—4222no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—2112no—
136C22×C4—2112no—
137C22×C4C4×D21111yescharacteristic
138C22×C4C22⋊D21111yescharacteristic
139C22×C4C22⋊D21111yescharacteristic
140C22×C4D84411yescharacteristic
141C22×C4D84411yescharacteristic
142C22×C4C22×D21111yescharacteristic
143M16—22no—
144M16—22no—
145M16—4224no—
146M16—4222no—
147M16—2114no—
148C2×C8—22no—
149C2×C8—4224no—
150C2×C8—4222no—
151C4⋊C4—4222no—
152C4⋊C4—4222no—
153C4⋊C4—4221no—
154C4⋊C4—4221no—
155C22⋊C4—4222no—
156C22⋊C4—4222no—
157C23—22no—
158C23—22no—
159C23—41no—
160C23(C2×D211)⋊C411yescharacteristic
161C23(C2×D211)⋊C411yescharacteristic
162C23C2×C4×D21111yescharacteristic
163Q8—8441no—
164Q8—8442no—
165Q8—8441no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—8442no—
174C2×C4—4222no—
175C2×C4—4222no—
176C2×C4—4224no—
177C2×C4(C2×D211)⋊C411yescharacteristic
178C2×C4(C2×D211)⋊C411yescharacteristic
179C2×C4D4×D21112yes—
180C2×C4C2×C22⋊D21111yescharacteristic
181C2×C4C2×D84411yescharacteristic
182C2×C4#1680996112yes—
183C8—42no—
184C8—4224no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—44no—
193C22#2026989512yescomplemented
194C22C2×(C2×D211)⋊C411yescharacteristic
195C22(C22⋊D211)⋊C412yes—
196C4—21no—
197C4—21no—
198C4—8442no—
199C4—8442no—
200C4#2026994711yescharacteristic
201C4#2026994912yes—
202C4#2026996111yescharacteristic
203C2—22no—
204C2—22no—
205C2—41no—
206C2#2721302311yescharacteristic
207C2C2×#2026989511yescharacteristic
208C2#2721312811yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#407×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC24×C211C2×C4
cosocleC22×C4×C211C23
hypercenter#407D211
hypocenterC211#1901
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. C2×#20269895C2semidirect
4. #27212914C2semidirect
88. C22×C211#101semidirect
104. C211#1901semidirect
193. C22#20269895semidirect

Maximal subgroups

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

subgroupindex
C22×#168099462
C2×#202699562
C2×#202698952
#272129142
#407×C2112
#1901211