Finite Group Databasebeta

← #41013696

Subgroups — #41013696

Order 27008.

408 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 121 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, complemented
2C2×C22⋊(C211⋊C8)C212yesmaximal, complemented
3C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
4#27212901C212yesmaximal, complemented
5C2×C4×D4×C211C211yesmaximal, characteristic
6C2×#16809946—24no—
7C2×#16809946—22no—
8C22×C211⋊C8C2212yes—
9C22×C211⋊C8C2212yes—
10C22⋊(C211⋊C8)—24no—
11C22⋊(C211⋊C8)—22no—
12C22×D4×C211C411yescharacteristic
13C23×C4×C211C2211yescharacteristic
14C23×C4×C211C2211yescharacteristic
15C4×D4×C211—22no—
16C4×D4×C211—22no—
17C2×C4⋊C4×C211C411yescharacteristic
18C2×C22⋊C4×C211C411yescharacteristic
19C2×C22⋊C4×C211C411yescharacteristic
20C2×C42×C211C2211yescharacteristic
21#16809946—28no—
22C2×C211⋊C8—22no—
23C2×C211⋊C8—22no—
24C2×C211⋊C8—22no—
25C2×C211⋊C8—22no—
26C2×C211⋊C8—22no—
27C2×C211⋊C8D414yescomplemented
28C24×C211C2×C411yescharacteristic
29C24×C211C2×C411yescharacteristic
30C2×D4×C211—22no—
31C2×D4×C211—22no—
32C2×D4×C211—22no—
33C22×C4×C211—22no—
34C22×C4×C211—22no—
35C22×C4×C211—21no—
36C22×C4×C211—21no—
37C22×C4×C211—22no—
38C22×C4×C211—22no—
39C22×C4×C211C2×C411yescharacteristic
40C22×C4×C211C2×C411yescharacteristic
41C22×C4×C211C2×C411yescharacteristic
42C22×C4×C211C2×C411yescharacteristic
43C22×C4×C211D412yes—
44C22×C4×C211D412yes—
45C22×C4×C211C2311yescharacteristic
46C4⋊C4×C211—22no—
47C22⋊C4×C211—22no—
48C22⋊C4×C211—22no—
49C42×C211—21no—
50C42×C211—21no—
51C211⋊C8—24no—
52C211⋊C8—42no—
53C23×C211—21no—
54C23×C211—22no—
55C23×C211—22no—
56C23×C211—21no—
57C23×C211—21no—
58C23×C211—21no—
59C23×C211—22no—
60C23×C211—22no—
61C23×C211C22⋊C412yes—
62C23×C211M1612yescomplemented
63C23×C211C22×C411yescharacteristic
64D4×C211—28no—
65C2×C4×C211—21no—
66C2×C4×C211—22no—
67C2×C4×C211—21no—
68C2×C4×C211—22no—
69C2×C4×C211—24no—
70C2×C4×C211—22no—
71C2×C4×C211—21no—
72C2×C4×C211—21no—
73C2×C4×C211—22no—
74C2×C4×C211—22no—
75C2×C4×C211—41no—
76C2×C4×C211—42no—
77C2×C4×C211—41no—
78C2×C4×C211C22⋊C412yes—
79C2×C4×C211M1612yescomplemented
80C2×C4×C211C2×D412yes—
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211#4112yes—
84C22×C211—22no—
85C22×C211—22no—
86C22×C211—22no—
87C22×C211—22no—
88C22×C211—24no—
89C22×C211—41no—
90C22×C211—41no—
91C22×C211—42no—
92C22×C211—42no—
93C22×C211—42no—
94C22×C211C2×C22⋊C411yescharacteristic
95C22×C211C4×D412yes—
96C22×C211C2×M1611yescharacteristic
97C22×C211C2×M1611yescharacteristic
98C22×C211#13112yes—
99C4×C211—22no—
100C4×C211—22no—
101C4×C211—41no—
102C4×C211—41no—
103C4×C211#12011yescharacteristic
104C4×C211#12112yes—
105C4×C211#12311yescharacteristic
106C2×C211—42no—
107C2×C211—42no—
108C2×C211#38611yescharacteristic
109C2×C211#40711yescharacteristic
110C2×C211#40811yescharacteristic
111C2×C211#43512yes—
112C2×C211#43612yes—
113C211#187011yescharacteristic, complemented, Sylow
114#1870—2111nomaximal, Sylow
115C22×M16—2111no—
116C2×C4×D4D21111yescharacteristic
117C2×C22⋊C8—2111no—
118C2×C22⋊C8—2112no—
119#336—2112no—
120C22×D4Dic21111yescharacteristic
121C23×C4C2×D21111yescharacteristic
122C23×C4C2×D21111yescharacteristic
123C2×M16—4222no—
124C2×M16—4224no—
125C22×C8—2112no—
126C22×C8—2112no—
127C4×D4—22no—
128C4×D4—22no—
129C2×C4⋊C4Dic21111yescharacteristic
130C2×C22⋊C4Dic21111yescharacteristic
131C2×C22⋊C4Dic21111yescharacteristic
132C2×C42C2×D21111yescharacteristic
133C22⋊C8—4222no—
134C22⋊C8—4224no—
135C24C2×Dic21111yescharacteristic
136C24C2×Dic21111yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4C2×Dic21111yescharacteristic
147C22×C4C2×Dic21111yescharacteristic
148C22×C4C2×Dic21111yescharacteristic
149C22×C4C2×Dic21111yescharacteristic
150C22×C4C22⋊D21112yes—
151C22×C4C22⋊D21112yes—
152C22×C4C22×D21111yescharacteristic
153M16—4228no—
154C2×C8—4222no—
155C2×C8—4222no—
156C2×C8—4222no—
157C2×C8—4222no—
158C2×C8—4222no—
159C2×C8—2114no—
160C4⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C42—21no—
164C42—21no—
165C23—21no—
166C23—22no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—22no—
173C23C22⋊Dic21112yes—
174C23#1680994612yescomplemented
175C23C22×Dic21111yescharacteristic
176D4—28no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4C22⋊Dic21112yes—
191C2×C4#1680994612yescomplemented
192C2×C4D4×D21112yes—
193C2×C4C2×C22⋊D21111yescharacteristic
194C2×C4C2×C22⋊D21111yescharacteristic
195C2×C4#1680996112yes—
196C8—8442no—
197C8—4224no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—24no—
202C22—22no—
203C22—41no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—41no—
208C22C2×C22⋊Dic21111yescharacteristic
209C22D4×Dic21112yes—
210C22C2×#1680994611yescharacteristic
211C22C2×#1680994611yescharacteristic
212C22#2026998412yes—
213C4—22no—
214C4—22no—
215C4—41no—
216C4—41no—
217C4#2026994511yescharacteristic
218C4#2026994912yes—
219C4#2026995811yescharacteristic
220C2—42no—
221C2—42no—
222C2#2721302611yescharacteristic
223C2#2721311011yescharacteristic
224C2#2721311311yescharacteristic
225C2#2721321812yes—
226C2#2721322412yes—
subgroupG/H
center, FrattiniC22×C4C22×D211
commutator, socleC23×C211C22×C4
FittingC2×C4×D4×C211C2
prime radicalC22×D4×C211C4
cosocleC22×C4×C211C23
hypercenterC2×C4×D4D211
hypocenterC211#1870
LCSG ▷ C23×C211 ▷ C211
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#16809946C2semidirect
2. C2×C22⋊(C211⋊C8)C2semidirect
3. C2×C22⋊(C211⋊C8)C2semidirect
4. #27212901C2semidirect
27. C2×C211⋊C8D4semidirect
62. C23×C211M16semidirect
79. C2×C4×C211M16semidirect
113. C211#1870semidirect
174. C23#16809946semidirect
191. C2×C4#16809946semidirect

Maximal subgroups

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

subgroupindex
C22×#168099462
C2×C22⋊(C211⋊C8)2
#272129012
C2×C4×D4×C2112
#1870211