Finite Group Databasebeta

← #41011735

Subgroups — #41011735

Order 27008.

316 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 69 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×#20269939C211yesmaximal, characteristic
2#27213108C212yesmaximal, complemented
3#27212872C211yesmaximal, characteristic, complemented
4#27212869C212yesmaximal, complemented
5#324×C211C211yesmaximal, characteristic, complemented
6C22×D844C411yescharacteristic
7C22×C4×D211C411yescharacteristic
8C22×C4×D211C2211yescharacteristic
9C2×#16809945—22no—
10#20269939—22no—
11#20269939—22no—
12C2×(C2×D211)⋊C4C411yescharacteristic
13C2×(C2×D211)⋊C4C2211yescharacteristic
14C2×C4×Dic211C411yescharacteristic
15C22⋊(C211⋊C8)C2212yes—
16(C2×D211)⋊C8—22no—
17C2×C4⋊C4×C211C2211yescharacteristic
18C22⋊C8×C211C2212yes—
19C23×D211C2×C411yescharacteristic
20C23×D211C2×C411yescharacteristic
21C2×D844—22no—
22C2×D844—22no—
23C2×D844—22no—
24C22×Dic211C2×C411yescharacteristic
25C22×Dic211C2×C411yescharacteristic
26C2×C4×D211—21no—
27C2×C4×D211—21no—
28C2×C4×D211—42no—
29C2×C4×D211—42no—
30C2×C4×D211D412yes—
31C2×C4×D211D412yes—
32#16809945—44no—
33C2×C211⋊C8—22no—
34(C2×D211)⋊C4—22no—
35(C2×D211)⋊C4—22no—
36C4×Dic211—21no—
37C4×Dic211—21no—
38C22×C4×C211C2×C411yescharacteristic
39C22×C4×C211C2×C411yescharacteristic
40C22×C4×C211C2311yescharacteristic
41C2×C8×C211—22no—
42C4⋊C4×C211—22no—
43C22×D211—21no—
44C22×D211—21no—
45C22×D211—21no—
46C22×D211—21no—
47C22×D211—41no—
48C22×D211—41no—
49C22×D211—41no—
50C22×D211—41no—
51C22×D211C22⋊C412yes—
52C22×D211M1612yescomplemented
53D844—28no—
54C2×Dic211—22no—
55C2×Dic211—21no—
56C2×Dic211—41no—
57C2×Dic211C22⋊C411yescharacteristic
58C2×Dic211C22⋊C411yescharacteristic
59C4×D211—24no—
60C4×D211—42no—
61C4×D211—42no—
62C4×D211—42no—
63C211⋊C8—42no—
64C23×C211C22×C411yescharacteristic
65C2×C4×C211—21no—
66C2×C4×C211—21no—
67C2×C4×C211—21no—
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211M1612yescomplemented
71C2×C4×C211C2×D412yes—
72C8×C211—42no—
73C22×C211—21no—
74C22×C211—21no—
75C22×C211C2×C22⋊C411yescharacteristic
76C22×C211C2×M1611yescharacteristic
77C22×C211C2×M1611yescharacteristic
78C2×D211—22no—
79C2×D211—22no—
80C2×D211—22no—
81C2×D211—22no—
82C2×D211—42no—
83C2×D211—42no—
84C2×D211—42no—
85C2×D211—41no—
86C2×D211—41no—
87C2×D211—42no—
88C4×C211—22no—
89C4×C211—22no—
90C4×C211—41no—
91Dic211—21no—
92Dic211—41no—
93Dic211#9912yes—
94C2×C211—21no—
95C2×C211—21no—
96C2×C211#40711yescharacteristic
97C2×C211C2×#9911yescharacteristic
98C2×C211#41311yescharacteristic
99D211—42no—
100D211—42no—
101C211#146511yescharacteristic, complemented, Sylow
102#1465—2111nomaximal, Sylow
103C2×C4×D4—2111no—
104#407—2112no—
105#324—2111no—
106#324D21111yescharacteristic, complemented
107#323—2112no—
108C22×D4—2111no—
109C23×C4—2111no—
110C23×C4—2111no—
111C2×M16—4222no—
112C4×D4—4222no—
113C4×D4—4222no—
114C2×C4⋊C4C2×D21111yescharacteristic
115C2×C22⋊C4—2111no—
116C2×C22⋊C4—2111no—
117C2×C42—2111no—
118C22⋊C8—4222no—
119C22⋊C8—2112no—
120C22⋊C8C2×D21112yes—
121C24—2111no—
122C24—2111no—
123C2×D4—4222no—
124C2×D4—4222no—
125C2×D4—4222no—
126C22×C4—8442no—
127C22×C4—8442no—
128C22×C4—4221no—
129C22×C4—4221no—
130C22×C4—2111no—
131C22×C4—2112no—
132C22×C4—2111no—
133C22×C4—2112no—
134C22×C4C4×D21111yescharacteristic
135C22×C4C4×D21111yescharacteristic
136C22×C4C22×D21111yescharacteristic
137M16—8444no—
138C2×C8—22no—
139C2×C8—4222no—
140C4⋊C4—22no—
141C22⋊C4—4222no—
142C22⋊C4—4222no—
143C42—4221no—
144C42—4221no—
145C23—8441no—
146C23—8441no—
147C23—8441no—
148C23—8441no—
149C23—4221no—
150C23—4221no—
151C23—4221no—
152C23—4221no—
153C23—2112no—
154C23—2112no—
155C23C2×C4×D21111yescharacteristic
156D4—4228no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—8442no—
163C2×C4—8442no—
164C2×C4—8442no—
165C2×C4—8441no—
166C2×C4—4224no—
167C2×C4—4221no—
168C2×C4—4222no—
169C2×C4—2111no—
170C2×C4—2111no—
171C2×C4#1680994512yescomplemented
172C2×C4D4×D21112yes—
173C8—42no—
174C8—8442no—
175C22—21no—
176C22—21no—
177C22—8442no—
178C22—8442no—
179C22—8442no—
180C22—8442no—
181C22—8441no—
182C22—8441no—
183C22—4222no—
184C22—4222no—
185C22—4222no—
186C22—4222no—
187C22C22⋊C4×D21111yescharacteristic
188C22M16×D21111yescharacteristic
189C22C2×#1680994511yescharacteristic
190C4—22no—
191C4—22no—
192C4—41no—
193C4—8441no—
194C4—4221no—
195C4—2112no—
196C2—21no—
197C2—21no—
198C2—8442no—
199C2—8442no—
200C2#2721310811yescharacteristic
201C2#99×D21111yescharacteristic
202C2#2721312911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#324×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC22×D844C4
cosocleC22×C4×C211C23
hypercenter#324D211
hypocenterC211#1465
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27213108C2semidirect
3. #27212872C2semidirect
4. #27212869C2semidirect
5. #324×C211C2semidirect
52. C22×D211M16semidirect
70. C2×C4×C211M16semidirect
101. C211#1465semidirect
106. #324D211semidirect
171. C2×C4#16809945semidirect

Maximal subgroups

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

subgroupindex
C2×#202699392
#272131082
#272128722
#272128692
#324×C2112
#1465211