Finite Group Databasebeta

← #41013811

Subgroups — #41013811

Order 27008.

386 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20269958C211yesmaximal, characteristic
2C2×#20269892C211yesmaximal, characteristic
3#27213007C212yesmaximal, complemented
4#27212913C212yesmaximal, complemented
5#518×C211C211yesmaximal, characteristic
6C23×Dic211C2211yescharacteristic
7#20269958—22no—
8#20269958C414yescomplemented
9C2×Dic211⋊C4—21no—
10C2×C22⋊Dic211—21no—
11C2×C22⋊Dic211C2212yes—
12C2×C22⋊Dic211C2211yescharacteristic
13#20269892—22no—
14C211⋊#95—24no—
15C22×D4×C211C2211yescharacteristic
16C4×D4×C211—24no—
17#117×C211—21no—
18C2×C22⋊C4×C211—21no—
19C2×C22⋊C4×C211C2212yes—
20C22×Dic211—22no—
21C22×Dic211—21no—
22C22×Dic211—21no—
23C22×Dic211—21no—
24C22×Dic211—22no—
25C22×Dic211—22no—
26C22×Dic211C2×C412yescomplemented
27Dic211⋊C4—24no—
28C22⋊Dic211—24no—
29C22⋊Dic211—21no—
30C22⋊Dic211—22no—
31C22⋊Dic211—22no—
32C22⋊Dic211C2×C412yescomplemented
33C24×C211D411yescharacteristic
34C24×C211C2311yescharacteristic
35C2×D4×C211—22no—
36C2×D4×C211—22no—
37C2×D4×C211C2×C412yes—
38C2×D4×C211D412yes—
39C22×C4×C211—22no—
40C22×C4×C211D411yescharacteristic
41C22×C4×C211D414yes—
42C4⋊C4×C211—22no—
43C22⋊C4×C211—22no—
44C22⋊C4×C211—22no—
45C22⋊C4×C211—22no—
46C42×C211—22no—
47C2×Dic211—22no—
48C2×Dic211—22no—
49C2×Dic211—24no—
50C2×Dic211—42no—
51C2×Dic211—41no—
52C2×Dic211—42no—
53C2×Dic211—41no—
54C2×Dic211—44no—
55C2×Dic211—42no—
56C23×C211—21no—
57C23×C211—21no—
58C23×C211—42no—
59C23×C211—42no—
60C23×C211C22⋊C412yes—
61C23×C211C22×C411yescharacteristic
62C23×C211C2×D411yescharacteristic
63C23×C211C2×D412yes—
64C23×C211C2×D411yescharacteristic
65C23×C211#4112yes—
66D4×C211—24no—
67D4×C211—24no—
68C2×C4×C211—21no—
69C2×C4×C211—21no—
70C2×C4×C211—24no—
71C2×C4×C211—44no—
72C2×C4×C211—82no—
73C2×C4×C211C22⋊C412yes—
74C22×C211—24no—
75C22×C211—22no—
76C22×C211—22no—
77C22×C211—41no—
78C22×C211—41no—
79C22×C211—42no—
80C22×C211—42no—
81C22×C211—42no—
82C22×C211—81no—
83C22×C211C2×C22⋊C411yescharacteristic
84C22×C211C4×D412yes—
85C22×C211#12011yescharacteristic
86C22×C211#12112yes—
87C22×C211#12311yescharacteristic
88C4×C211—22no—
89C4×C211—44no—
90Dic211—42no—
91Dic211—42no—
92Dic211—82no—
93C2×C211—21no—
94C2×C211—22no—
95C2×C211—21no—
96C2×C211—22no—
97C2×C211—42no—
98C2×C211#38611yescharacteristic
99C2×C211#45812yes—
100C211#189011yescharacteristic, complemented, Sylow
101#1890—2111nomaximal, Sylow
102C2×#123—2111no—
103#518D21111yescharacteristic
104C2×#99—2111no—
105#381—2112no—
106#342—2112no—
107C22×D4C2×D21111yescharacteristic
108C23×C4—2111no—
109#123—4222no—
110#123—2114no—
111C4×D4—24no—
112#117—21no—
113C2×C4⋊C4—4221no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—4221no—
116C2×C22⋊C4—2112no—
117C2×C22⋊C4—2111no—
118C2×C22⋊C4C2×D21112yes—
119#99—4222no—
120#95—4224no—
121C24C22⋊D21111yescharacteristic
122C24C22×D21111yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C4×D21112yes—
126C2×D4D84412yes—
127C22×C4—22no—
128C22×C4—4222no—
129C22×C4—4222no—
130C22×C4—4221no—
131C22×C4—4221no—
132C22×C4—4221no—
133C22×C4—4222no—
134C22×C4—2112no—
135C22×C4C22⋊D21111yescharacteristic
136C22×C4C22⋊D21114yes—
137C4⋊C4—22no—
138C4⋊C4—4224no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—4222no—
143C22⋊C4—4222no—
144C22⋊C4—4224no—
145C22⋊C4—4221no—
146C22⋊C4—2112no—
147C42—22no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—42no—
152C23(C2×D211)⋊C412yes—
153C23C2×C4×D21111yescharacteristic
154C23C2×C22⋊D21111yescharacteristic
155C23C2×C22⋊D21112yes—
156C23C2×D84411yescharacteristic
157C23#1680996212yes—
158D4—24no—
159D4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—24no—
163C2×C4—44no—
164C2×C4—82no—
165C2×C4—8442no—
166C2×C4—8441no—
167C2×C4—8442no—
168C2×C4—8441no—
169C2×C4—8444no—
170C2×C4—8442no—
171C2×C4—4222no—
172C2×C4—4222no—
173C2×C4—4224no—
174C2×C4(C2×D211)⋊C412yes—
175C22—24no—
176C22—22no—
177C22—22no—
178C22—41no—
179C22—41no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22C2×(C2×D211)⋊C411yescharacteristic
185C22C4×C22⋊D21112yes—
186C22#2026994611yescharacteristic
187C22#2026995112yes—
188C22#2026996011yescharacteristic
189C4—22no—
190C4—44no—
191C4—8442no—
192C4—8442no—
193C4—16882no—
194C2—21no—
195C2—22no—
196C2—21no—
197C2—22no—
198C2—42no—
199C2#2721302511yescharacteristic
200C2#2721333612yes—
subgroupG/H
centerC22#20269946
commutatorC23×C211C22×C4
FrattiniC24C22×D211
Fitting#518×C211C2
socleC22×C211#120
prime radicalC22×D4×C211C22
cosocleC24×C211C23
hypercenter#518D211
hypocenterC211#1890
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
3. #27213007C2semidirect
4. #27212913C2semidirect
8. #20269958C4semidirect
26. C22×Dic211C2×C4semidirect
32. C22⋊Dic211C2×C4semidirect
100. C211#1890semidirect

Maximal subgroups

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

subgroupindex
C2×#202699582
C2×#202698922
#272130072
#272129132
#518×C2112
#1890211