Finite Group Databasebeta

← #41013854

Subgroups — #41013854

Order 27008.

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

normal subgroups only

This group has 378 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×#20269960C211yesmaximal, characteristic
2C2×#20269936C211yesmaximal, characteristic, complemented
3C2×#20269891C212yesmaximal
4#27212912C212yesmaximal, complemented
5#381×C211C211yesmaximal, characteristic, complemented
6C22×C22⋊D211C2211yescharacteristic
7#20269960—22no—
8#20269960—21no—
9#20269960—21no—
10#20269936—24no—
11C2×C4⋊Dic211—21no—
12C2×Dic211⋊C4C412yescomplemented
13C2×C22⋊Dic211C2211yescharacteristic
14C2×C22⋊Dic211C2212yes—
15C2×(C2×D211)⋊C4C412yescomplemented
16C2×C4×Dic211—21no—
17#20269891—24no—
18C23×C4×C211C2211yescharacteristic
19C2×C22⋊C4×C211C2212yes—
20#95×C211—22no—
21C23×D211C2×C411yescharacteristic
22C2×C22⋊D211—22no—
23C2×C22⋊D211—21no—
24C2×C22⋊D211—21no—
25C2×C22⋊D211—22no—
26C22×Dic211—22no—
27C22×Dic211C2×C411yescharacteristic
28C22×Dic211C2×C412yes—
29C4⋊Dic211—24no—
30Dic211⋊C4—22no—
31Dic211⋊C4—22no—
32C22⋊Dic211—22no—
33C22⋊Dic211—21no—
34C22⋊Dic211—21no—
35C22⋊Dic211D414yes—
36(C2×D211)⋊C4—22no—
37(C2×D211)⋊C4—22no—
38C4×Dic211—24no—
39C24×C211C2311yescharacteristic
40C22×C4×C211—22no—
41C22×C4×C211—22no—
42C22×C4×C211—22no—
43C22×C4×C211C2×C412yes—
44C22×C4×C211D412yes—
45C22×C4×C211D411yescharacteristic
46C22×C4×C211D411yescharacteristic
47C22⋊C4×C211—22no—
48C22⋊C4×C211—22no—
49C22×D211—21no—
50C22×D211—22no—
51C22×D211—42no—
52C22⋊D211—41no—
53C22⋊D211—42no—
54C22⋊D211—41no—
55C2×Dic211—22no—
56C2×Dic211—21no—
57C2×Dic211—22no—
58C2×Dic211—24no—
59C2×Dic211—44no—
60C2×Dic211—42no—
61C23×C211—42no—
62C23×C211C22×C411yescharacteristic
63C23×C211C2×D412yes—
64C23×C211C2×D411yescharacteristic
65C23×C211C2×D411yescharacteristic
66C23×C211#4112yes—
67C2×C4×C211—24no—
68C2×C4×C211—21no—
69C2×C4×C211—21no—
70C2×C4×C211—24no—
71C2×C4×C211—21no—
72C2×C4×C211—21no—
73C2×C4×C211—44no—
74C2×C4×C211—42no—
75C2×C4×C211C22⋊C412yes—
76C2×C4×C211C22⋊C412yes—
77C22×C211—22no—
78C22×C211—22no—
79C22×C211—22no—
80C22×C211—42no—
81C22×C211—42no—
82C22×C211C2×C22⋊C411yescharacteristic
83C22×C211C4×D412yes—
84C22×C211#12011yescharacteristic
85C22×C211#12112yes—
86C22×C211#12311yescharacteristic
87C2×D211—41no—
88C2×D211—42no—
89C2×D211—82no—
90C4×C211—22no—
91C4×C211—22no—
92C4×C211—82no—
93Dic211—44no—
94Dic211—41no—
95Dic211—42no—
96C2×C211—21no—
97C2×C211—22no—
98C2×C211—22no—
99C2×C211—21no—
100C2×C211#38611yescharacteristic
101C2×C211#41012yes—
102D211—81no—
103C211#189911yescharacteristic, complemented, Sylow
104#1899—2111nomaximal, Sylow
105C2×#123—2111no—
106C2×#117—2111no—
107C2×#99—2112no—
108#381D21111yescharacteristic, complemented
109#342—2112no—
110C22×D4—2111no—
111C23×C4C2×D21111yescharacteristic
112#123—4222no—
113#123—4221no—
114#123—4221no—
115#117—4224no—
116C2×C4⋊C4—4221no—
117C2×C4⋊C4—2112no—
118C2×C22⋊C4—2111no—
119C2×C22⋊C4—2112no—
120C2×C22⋊C4—2112no—
121C2×C22⋊C4C2×D21112yes—
122C2×C42—4221no—
123#99—4224no—
124#95—22no—
125C24—2111no—
126C24C22×D21111yescharacteristic
127C2×D4—4222no—
128C2×D4—4221no—
129C2×D4—4221no—
130C2×D4—4222no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—4222no—
135C22×C4—2112no—
136C22×C4—2111no—
137C22×C4C4×D21112yes—
138C22×C4C22⋊D21112yes—
139C22×C4D84411yescharacteristic
140C22×C4D84411yescharacteristic
141C4⋊C4—4224no—
142C4⋊C4—4222no—
143C4⋊C4—4222no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—4222no—
147C22⋊C4—4221no—
148C22⋊C4—4221no—
149C22⋊C4—4222no—
150C22⋊C4—4222no—
151C22⋊C4—2114no—
152C42—4224no—
153C23—42no—
154C23—8442no—
155C23—4221no—
156C23—4222no—
157C23C2×C4×D21111yescharacteristic
158C23D4×D21112yes—
159C23C2×C22⋊D21111yescharacteristic
160C23C2×D84411yescharacteristic
161C23#1680996112yes—
162D4—8441no—
163D4—8442no—
164D4—8441no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—8444no—
174C2×C4—8442no—
175C2×C4—4222no—
176C2×C4—4221no—
177C2×C4—4222no—
178C2×C4—4224no—
179C2×C4(C2×D211)⋊C412yes—
180C2×C4(C2×D211)⋊C412yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—8442no—
187C22—8441no—
188C22—16882no—
189C22C2×(C2×D211)⋊C411yescharacteristic
190C22(C22⋊D211)⋊C412yes—
191C22#2026994711yescharacteristic
192C22#2026994912yes—
193C22#2026996111yescharacteristic
194C4—22no—
195C4—22no—
196C4—82no—
197C4—8444no—
198C4—8442no—
199C4—8441no—
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2—16881no—
205C2#2721302311yescharacteristic
206C2#2721311912yes—
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC23×C211C22×C4
FrattiniC24C22×D211
Fitting#381×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC22×C22⋊D211C22
cosocleC24×C211C23
hypercenter#381D211
hypocenterC211#1899
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#20269936C2semidirect
4. #27212912C2semidirect
5. #381×C211C2semidirect
12. C2×Dic211⋊C4C4semidirect
15. C2×(C2×D211)⋊C4C4semidirect
103. C211#1899semidirect
108. #381D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699602
C2×#202699362
C2×#202698912
#272129122
#381×C2112
#1899211