Finite Group Databasebeta

← #41011969

Subgroups — #41011969

Order 27008.

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

normal subgroups only

This group has 288 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
1C2×#20269956C211yesmaximal, characteristic
2C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
3#27212882C214yesmaximal, complemented
4#380×C211C211yesmaximal, characteristic
5C22×Dic422C411yescharacteristic, complemented
6C22×C211⋊C8—21no—
7#20269956—21no—
8#20269956—21no—
9#20269956C412yescomplemented
10#20269956C2212yes—
11C2×C4⋊Dic211C411yescharacteristic, complemented
12C2×Dic211⋊C4—21no—
13C2×C22⋊Dic211—21no—
14C22⋊(C211⋊C8)—21no—
15C22⋊(C211⋊C8)C2212yes—
16C23×C4×C211C2211yescharacteristic
17C2×C4⋊C4×C211—21no—
18C2×C22⋊C4×C211—21no—
19#95×C211C2212yes—
20C2×Dic422—21no—
21C2×Dic422—42no—
22C2×Dic422C2×C412yes—
23C22×Dic211—21no—
24C22×Dic211—21no—
25C2×C211⋊C8—22no—
26C2×C211⋊C8—42no—
27C4⋊Dic211—21no—
28C4⋊Dic211C2×C412yes—
29Dic211⋊C4—22no—
30Dic211⋊C4—22no—
31C22⋊Dic211—22no—
32C22⋊Dic211—22no—
33C24×C211D411yescharacteristic
34C22×C4×C211—22no—
35C22×C4×C211—42no—
36C22×C4×C211C2×C411yescharacteristic
37C22×C4×C211C2×C411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D411yescharacteristic
40C22×C4×C211D411yescharacteristic
41C22×C4×C211C2311yescharacteristic
42C4⋊C4×C211—42no—
43C22⋊C4×C211—42no—
44Dic422—42no—
45Dic422—42no—
46C2×Dic211—22no—
47C2×Dic211—22no—
48C2×Dic211—42no—
49C2×Dic211—42no—
50C211⋊C8—42no—
51C23×C211—41no—
52C23×C211—41no—
53C23×C211C22⋊C411yescharacteristic
54C23×C211C22⋊C411yescharacteristic
55C23×C211SD1612yes—
56C23×C211Q1612yes—
57C23×C211C2×D411yescharacteristic
58C2×C4×C211—22no—
59C2×C4×C211—22no—
60C2×C4×C211—22no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—42no—
65C2×C4×C211—42no—
66C2×C4×C211—42no—
67C2×C4×C211C22⋊C411yescharacteristic
68C2×C4×C211C22⋊C411yescharacteristic
69C2×C4×C211C22×C411yescharacteristic
70C2×C4×C211C2×D411yescharacteristic
71C22×C211—21no—
72C22×C211—21no—
73C22×C211—21no—
74C22×C211—21no—
75C22×C211—21no—
76C22×C211—21no—
77C22×C211—42no—
78C22×C211—42no—
79C22×C211#10312yes—
80C22×C211#10312yes—
81C22×C211C2×C22⋊C411yescharacteristic
82C22×C211C2×SD1611yescharacteristic
83C22×C211C2×Q1611yescharacteristic
84C4×C211—21no—
85C4×C211—21no—
86C4×C211—21no—
87C4×C211—82no—
88C4×C211#9912yes—
89Dic211—42no—
90Dic211—42no—
91C2×C211—21no—
92C2×C211—22no—
93C2×C211—22no—
94C2×C211—21no—
95C2×C211C2×#9911yescharacteristic
96C2×C211C2×#10311yescharacteristic
97C2×C211#42111yescharacteristic
98C211#152011yescharacteristic, complemented, Sylow
99#1520—2111nomaximal, Sylow
100C2×C22⋊Q8—2111no—
101C2×C22⋊C8—2111no—
102#380D21111yescharacteristic
103#328—2114no—
104C22×Q8—2111no—
105C23×C4C2×D21111yescharacteristic
106C22×C8—4221no—
107C22⋊Q8—4221no—
108C22⋊Q8—4221no—
109C22⋊Q8—2112no—
110C22⋊Q8—2112no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—4221no—
113C2×C4⋊C4—2111no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—4221no—
116C22⋊C8—4221no—
117C22⋊C8—2112no—
118#95C2×D21112yes—
119C24C22⋊D21111yescharacteristic
120C2×Q8—8442no—
121C2×Q8—4221no—
122C2×Q8—2112no—
123C22×C4—22no—
124C22×C4—42no—
125C22×C4—4221no—
126C22×C4—4221no—
127C22×C4C4×D21111yescharacteristic
128C22×C4C4×D21111yescharacteristic
129C22×C4C22⋊D21111yescharacteristic
130C22×C4D84411yescharacteristic
131C22×C4D84411yescharacteristic
132C22×C4C22×D21111yescharacteristic
133C2×C8—8442no—
134C2×C8—4222no—
135C4⋊C4—42no—
136C4⋊C4—4222no—
137C4⋊C4—4221no—
138C4⋊C4—4222no—
139C4⋊C4—2112no—
140C22⋊C4—42no—
141C22⋊C4—4222no—
142C22⋊C4—4222no—
143C23—41no—
144C23—41no—
145C23(C2×D211)⋊C411yescharacteristic
146C23(C2×D211)⋊C411yescharacteristic
147C23#1680995012yes—
148C23#1680995212yes—
149C23C2×C22⋊D21111yescharacteristic
150Q8—8442no—
151Q8—8442no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—8442no—
162C2×C4—8442no—
163C2×C4—4222no—
164C2×C4—4222no—
165C2×C4(C2×D211)⋊C411yescharacteristic
166C2×C4(C2×D211)⋊C411yescharacteristic
167C2×C4C2×C4×D21111yescharacteristic
168C2×C4C2×D84411yescharacteristic
169C8—8442no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22#2026990012yes—
179C22#2026990012yes—
180C22C2×(C2×D211)⋊C411yescharacteristic
181C22C2×#1680995011yescharacteristic
182C22C2×#1680995211yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—82no—
187C4—8442no—
188C4—8442no—
189C4#2026989112yes—
190C2—21no—
191C2—22no—
192C2—22no—
193C2—21no—
194C2C2×#2026989111yescharacteristic
195C2C2×#2026990011yescharacteristic
196C2#2721316411yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#380×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenter#380D211
hypocenterC211#1520
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #27212882C2semidirect
5. C22×Dic422C4semidirect
9. #20269956C4semidirect
11. C2×C4⋊Dic211C4semidirect
98. C211#1520semidirect

Maximal subgroups

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

subgroupindex
C2×#202699562
C2×C22⋊(C211⋊C8)2
#272128822
#380×C2112
#1520211