Finite Group Databasebeta

← #41013819

Subgroups — #41013819

Order 27008.

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

normal subgroups only

This group has 380 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
1C23×C211⋊C8C211yesmaximal, characteristic
2C2×#20269956C211yesmaximal, characteristic
3C2×#20269900C212yesmaximal, complemented
4#27212908C212yesmaximal, complemented
5#380×C211C211yesmaximal, characteristic
6C22×Dic422C2211yescharacteristic
7C22×C211⋊C8—24no—
8C22×C211⋊C8—22no—
9C22×C211⋊C8C2212yes—
10#20269956—22no—
11#20269956C414yescomplemented
12C2×C4⋊Dic211C2211yescharacteristic
13C2×Dic211⋊C4—21no—
14C2×C22⋊Dic211—21no—
15#20269900—24no—
16C23×C4×C211C2211yescharacteristic
17C2×C4⋊C4×C211C2212yes—
18C2×C22⋊C4×C211—21no—
19#95×C211—21no—
20C2×Dic422—21no—
21C2×Dic422—22no—
22C2×Dic422—22no—
23C2×Dic422C2×C412yes—
24C22×Dic211—21no—
25C22×Dic211—21no—
26C2×C211⋊C8—28no—
27C2×C211⋊C8—22no—
28C2×C211⋊C8—22no—
29C2×C211⋊C8D414yes—
30C4⋊Dic211—21no—
31C4⋊Dic211C2×C412yes—
32Dic211⋊C4—24no—
33C22⋊Dic211—24no—
34C24×C211D411yescharacteristic
35C22×C4×C211—22no—
36C22×C4×C211—22no—
37C22×C4×C211—22no—
38C22×C4×C211C2×C412yes—
39C22×C4×C211D411yescharacteristic
40C22×C4×C211D412yes—
41C22×C4×C211C2311yescharacteristic
42C4⋊C4×C211—22no—
43C4⋊C4×C211—22no—
44C22⋊C4×C211—42no—
45Dic422—24no—
46Dic422—41no—
47Dic422—41no—
48C2×Dic211—22no—
49C2×Dic211—22no—
50C2×Dic211—42no—
51C2×Dic211—42no—
52C211⋊C8—24no—
53C23×C211—22no—
54C23×C211—22no—
55C23×C211C22⋊C412yes—
56C23×C211SD1612yes—
57C23×C211Q1612yes—
58C23×C211C2×D411yescharacteristic
59C2×C4×C211—22no—
60C2×C4×C211—24no—
61C2×C4×C211—22no—
62C2×C4×C211—22no—
63C2×C4×C211—42no—
64C2×C4×C211—42no—
65C2×C4×C211—42no—
66C2×C4×C211C22⋊C412yes—
67C2×C4×C211C22×C411yescharacteristic
68C2×C4×C211C2×D412yes—
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211#4112yes—
71C22×C211—22no—
72C22×C211—22no—
73C22×C211—24no—
74C22×C211—24no—
75C22×C211#10314yescomplemented
76C22×C211C2×C22⋊C411yescharacteristic
77C22×C211#12112yes—
78C22×C211C2×SD1611yescharacteristic
79C22×C211C2×Q1611yescharacteristic
80C22×C211#13512yes—
81C4×C211—22no—
82C4×C211—82no—
83C4×C211C4×D412yes—
84C4×C211#12011yescharacteristic
85C4×C211#12311yescharacteristic
86Dic211—42no—
87Dic211—42no—
88C2×C211—24no—
89C2×C211#38611yescharacteristic
90C2×C211C2×#10311yescharacteristic
91C2×C211#41611yescharacteristic
92C2×C211#46512yes—
93C2×C211#46712yes—
94C211#189211yescharacteristic, complemented, Sylow
95#1892—2111nomaximal, Sylow
96C23×C8—2111no—
97C2×C22⋊Q8—2111no—
98C2×#103—2112no—
99#380D21111yescharacteristic
100#340—2112no—
101C22×Q8—2111no—
102C23×C4C2×D21111yescharacteristic
103C22×C8—4222no—
104C22×C8—4224no—
105C22×C8—2112no—
106C22⋊Q8—4222no—
107C22⋊Q8—2114no—
108C2×C4⋊C4—4221no—
109C2×C4⋊C4—2111no—
110C2×C4⋊C4C2×D21112yes—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—4221no—
113#103—4224no—
114#95—21no—
115C24C22⋊D21111yescharacteristic
116C2×Q8—4222no—
117C2×Q8—4222no—
118C2×Q8—4221no—
119C2×Q8—2112no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—4221no—
124C22×C4—4221no—
125C22×C4C4×D21112yes—
126C22×C4C22⋊D21111yescharacteristic
127C22×C4D84412yes—
128C22×C4C22×D21111yescharacteristic
129C2×C8—4228no—
130C2×C8—4222no—
131C2×C8—4222no—
132C2×C8—2114no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—4224no—
136C4⋊C4—4221no—
137C4⋊C4—2112no—
138C22⋊C4—42no—
139C22⋊C4—4224no—
140C23—22no—
141C23—22no—
142C23(C2×D211)⋊C412yes—
143C23#1680995012yes—
144C23#1680995212yes—
145C23C2×C22⋊D21111yescharacteristic
146Q8—8441no—
147Q8—8441no—
148Q8—4224no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—8442no—
157C2×C4—8442no—
158C2×C4—4222no—
159C2×C4—4222no—
160C2×C4(C2×D211)⋊C412yes—
161C2×C4C2×C4×D21111yescharacteristic
162C2×C4D4×D21112yes—
163C2×C4C2×D84411yescharacteristic
164C2×C4#1680996112yes—
165C8—4224no—
166C22—24no—
167C22—24no—
168C22—22no—
169C22—22no—
170C22#2026990014yescomplemented
171C22C2×(C2×D211)⋊C411yescharacteristic
172C22#2026994912yes—
173C22C2×#1680995011yescharacteristic
174C22C2×#1680995211yescharacteristic
175C22#2026999712yes—
176C4—22no—
177C4—82no—
178C4—8442no—
179C4—8442no—
180C4(C22⋊D211)⋊C412yes—
181C4#2026994711yescharacteristic
182C4#2026996111yescharacteristic
183C2—24no—
184C2#2721302311yescharacteristic
185C2C2×#2026990011yescharacteristic
186C2#2721314011yescharacteristic
187C2#2721337312yes—
188C2#2721338512yes—
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#380×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenter#380D211
hypocenterC211#1892
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#20269900C2semidirect
4. #27212908C2semidirect
11. #20269956C4semidirect
75. C22×C211#103semidirect
94. C211#1892semidirect
170. C22#20269900semidirect

Maximal subgroups

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

subgroupindex
C23×C211⋊C82
C2×#202699562
C2×#202699002
#272129082
#380×C2112
#1892211