Finite Group Databasebeta

← #41011736

Subgroups — #41011736

Order 27008.

316 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 89 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#27213110C212yesmaximal, complemented
2#27212872C212yesmaximal, complemented
3#27212870C212yesmaximal, complemented
4C2×C4×D4×C211C211yesmaximal, characteristic
5C2×#16809946—22no—
6C22⋊(C211⋊C8)—22no—
7C22⋊(C211⋊C8)C2214yes—
8C22×D4×C211C411yescharacteristic
9C23×C4×C211C411yescharacteristic
10C23×C4×C211C2211yescharacteristic
11C4×D4×C211—22no—
12C4×D4×C211—22no—
13C2×C4⋊C4×C211C2211yescharacteristic
14C2×C22⋊C4×C211C411yescharacteristic
15C2×C22⋊C4×C211C2211yescharacteristic
16C2×C42×C211C411yescharacteristic
17#16809946—44no—
18C2×C211⋊C8—24no—
19C24×C211C2×C411yescharacteristic
20C24×C211C2×C411yescharacteristic
21C2×D4×C211—22no—
22C2×D4×C211—22no—
23C2×D4×C211—22no—
24C22×C4×C211—21no—
25C22×C4×C211—21no—
26C22×C4×C211—42no—
27C22×C4×C211—42no—
28C22×C4×C211C2×C411yescharacteristic
29C22×C4×C211C2×C411yescharacteristic
30C22×C4×C211C2×C411yescharacteristic
31C22×C4×C211C2×C411yescharacteristic
32C22×C4×C211D412yes—
33C22×C4×C211D412yes—
34C22×C4×C211C2311yescharacteristic
35C4⋊C4×C211—22no—
36C22⋊C4×C211—22no—
37C22⋊C4×C211—22no—
38C42×C211—21no—
39C42×C211—21no—
40C211⋊C8—44no—
41C23×C211—21no—
42C23×C211—21no—
43C23×C211—21no—
44C23×C211—21no—
45C23×C211—41no—
46C23×C211—41no—
47C23×C211—41no—
48C23×C211—41no—
49C23×C211C22⋊C412yes—
50C23×C211M1612yescomplemented
51C23×C211C22×C411yescharacteristic
52D4×C211—28no—
53C2×C4×C211—21no—
54C2×C4×C211—21no—
55C2×C4×C211—22no—
56C2×C4×C211—24no—
57C2×C4×C211—21no—
58C2×C4×C211—21no—
59C2×C4×C211—42no—
60C2×C4×C211—41no—
61C2×C4×C211—42no—
62C2×C4×C211—42no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211C22⋊C411yescharacteristic
66C2×C4×C211C22⋊C411yescharacteristic
67C2×C4×C211M1612yescomplemented
68C2×C4×C211C2×D412yes—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211—22no—
72C22×C211—22no—
73C22×C211—22no—
74C22×C211—22no—
75C22×C211—42no—
76C22×C211—42no—
77C22×C211—42no—
78C22×C211—42no—
79C22×C211—41no—
80C22×C211—41no—
81C22×C211C2×C22⋊C411yescharacteristic
82C22×C211C2×M1611yescharacteristic
83C22×C211C2×M1611yescharacteristic
84C4×C211—22no—
85C4×C211—21no—
86C4×C211—22no—
87C4×C211—41no—
88C4×C211—41no—
89C4×C211#9912yes—
90C2×C211—21no—
91C2×C211—21no—
92C2×C211—42no—
93C2×C211—42no—
94C2×C211#40711yescharacteristic
95C2×C211C2×#9911yescharacteristic
96C2×C211#41311yescharacteristic
97C211#146511yescharacteristic, complemented, Sylow
98#1465—2111nomaximal, Sylow
99C2×C4×D4D21111yescharacteristic
100#407—2112no—
101#324—2112no—
102#323—2112no—
103C22×D4Dic21111yescharacteristic
104C23×C4Dic21111yescharacteristic
105C23×C4C2×D21111yescharacteristic
106C2×M16—4222no—
107C4×D4—22no—
108C4×D4—22no—
109C2×C4⋊C4C2×D21111yescharacteristic
110C2×C22⋊C4Dic21111yescharacteristic
111C2×C22⋊C4C2×D21111yescharacteristic
112C2×C42Dic21111yescharacteristic
113C22⋊C8—4222no—
114C22⋊C8—2114no—
115C24C2×Dic21111yescharacteristic
116C24C2×Dic21111yescharacteristic
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×Dic21111yescharacteristic
125C22×C4C2×Dic21111yescharacteristic
126C22×C4C2×Dic21111yescharacteristic
127C22×C4C2×Dic21111yescharacteristic
128C22×C4C22⋊D21112yes—
129C22×C4C22⋊D21112yes—
130C22×C4C22×D21111yescharacteristic
131M16—8444no—
132C2×C8—4224no—
133C4⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C42—21no—
137C42—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23C22⋊Dic21112yes—
147C23#1680994612yescomplemented
148C23C22×Dic21111yescharacteristic
149D4—28no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—24no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4C22⋊Dic21111yescharacteristic
163C2×C4C22⋊Dic21111yescharacteristic
164C2×C4#1680994612yescomplemented
165C2×C4C2×C22⋊D21112yes—
166C8—8444no—
167C22—21no—
168C22—21no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—42no—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22C2×C22⋊Dic21111yescharacteristic
180C22C2×#1680994611yescharacteristic
181C22C2×#1680994611yescharacteristic
182C4—22no—
183C4—22no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4#2026989212yes—
188C2—21no—
189C2—21no—
190C2—42no—
191C2—42no—
192C2#2721311011yescharacteristic
193C2C2×#2026989211yescharacteristic
194C2#2721313011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×C4×D4×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC22×D4×C211C4
cosocleC22×C4×C211C23
hypercenterC2×C4×D4D211
hypocenterC211#1465
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #27213110C2semidirect
2. #27212872C2semidirect
3. #27212870C2semidirect
50. C23×C211M16semidirect
67. C2×C4×C211M16semidirect
97. C211#1465semidirect
147. C23#16809946semidirect
164. C2×C4#16809946semidirect

Maximal subgroups

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

subgroupindex
#272131102
#272128722
#272128702
C2×C4×D4×C2112
#1465211