Finite Group Databasebeta

← #41014527

Subgroups — #41014527

Order 27008.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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×#20269958C212yesmaximal, complemented
2C2×#20269892C212yesmaximal, complemented
3#27213007C211yesmaximal, characteristic, complemented
4#27212913C211yesmaximal, characteristic, complemented
5#534×C211C211yesmaximal, characteristic
6C23×Dic211C2211yescharacteristic
7#20269958—24no—
8#20269958—24no—
9C2×Dic211⋊C4—22no—
10C2×C22⋊Dic211—22no—
11C2×C22⋊Dic211C2211yescharacteristic
12C2×C22⋊Dic211C2212yes—
13#20269892—24no—
14C211⋊#95—22no—
15C22×D4×C211C2212yes—
16#123×C211—22no—
17#121×C211—22no—
18#120×C211—22no—
19C2×C22⋊C4×C211C2211yescharacteristic
20C22×Dic211—21no—
21C22×Dic211—22no—
22C22×Dic211—22no—
23C22×Dic211—22no—
24C22×Dic211—21no—
25C22×Dic211—21no—
26C22×Dic211D412yescomplemented
27Dic211⋊C4—28no—
28C22⋊Dic211—28no—
29C22⋊Dic211—22no—
30C22⋊Dic211—22no—
31C22⋊Dic211D414yescomplemented
32C24×C211D412yes—
33C24×C211C2311yescharacteristic
34C2×D4×C211—22no—
35C2×D4×C211—22no—
36C2×D4×C211—22no—
37C2×D4×C211—22no—
38C2×D4×C211—22no—
39C2×D4×C211—22no—
40C2×D4×C211—22no—
41C22×C4×C211D412yes—
42C22×C4×C211D412yes—
43C4⋊C4×C211—22no—
44C22⋊C4×C211—24no—
45C22⋊C4×C211—22no—
46C2×Dic211—24no—
47C2×Dic211—24no—
48C2×Dic211—22no—
49C2×Dic211—44no—
50C2×Dic211—41no—
51C2×Dic211—42no—
52C2×Dic211—41no—
53C2×Dic211—42no—
54C2×Dic211—41no—
55C23×C211—22no—
56C23×C211—22no—
57C23×C211—22no—
58C23×C211—42no—
59C23×C211—44no—
60C23×C211C2×D411yescharacteristic
61C23×C211C2×D412yes—
62C23×C211C2×D412yes—
63C23×C211C2×D411yescharacteristic
64C23×C211#4111yescharacteristic
65D4×C211—42no—
66D4×C211—42no—
67D4×C211—42no—
68D4×C211—42no—
69D4×C211—82no—
70C2×C4×C211—22no—
71C2×C4×C211—22no—
72C2×C4×C211—22no—
73C2×C4×C211—22no—
74C2×C4×C211—42no—
75C22×C211—22no—
76C22×C211—24no—
77C22×C211—41no—
78C22×C211—42no—
79C22×C211—41no—
80C22×C211—44no—
81C22×C211—42no—
82C22×C211—44no—
83C22×C211—82no—
84C22×C211#12012yes—
85C22×C211#12011yescharacteristic
86C22×C211#12111yescharacteristic
87C22×C211#12112yes—
88C22×C211#12711yescharacteristic
89C4×C211—42no—
90C4×C211—42no—
91Dic211—44no—
92Dic211—42no—
93Dic211—81no—
94C2×C211—22no—
95C2×C211—21no—
96C2×C211—22no—
97C2×C211—21no—
98C2×C211—82no—
99C2×C211#39211yescharacteristic
100C2×C211#45812yes—
101C211#202211yescharacteristic, complemented, Sylow
102#2022—2111nomaximal, Sylow
103#534D21111yescharacteristic
104C2×#123—2112no—
105C2×#99—2112no—
106#381—2111no—
107#342—2111no—
108C22×D4C2×D21112yes—
109C23×C4—2111no—
110#123—22no—
111#123—4224no—
112#123—4224no—
113#121—22no—
114#120—22no—
115C2×C4⋊C4—4222no—
116C2×C22⋊C4—4222no—
117C2×C22⋊C4—2111no—
118C2×C22⋊C4—2112no—
119C2×C22⋊C4C2×D21111yescharacteristic
120#99—4224no—
121#95—4222no—
122C24C22⋊D21112yes—
123C24C22×D21111yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C22×C4—4221no—
132C22×C4—4222no—
133C22×C4—4221no—
134C22×C4—4222no—
135C22×C4—4222no—
136C22×C4—4221no—
137C22×C4—2112no—
138C22×C4C22⋊D21112yes—
139C22×C4C22⋊D21112yes—
140C4⋊C4—22no—
141C4⋊C4—4228no—
142C22⋊C4—24no—
143C22⋊C4—22no—
144C22⋊C4—4222no—
145C22⋊C4—4228no—
146C22⋊C4—4222no—
147C22⋊C4—2114no—
148C23—22no—
149C23—22no—
150C23—22no—
151C23—42no—
152C23—44no—
153C23D4×D21111yescharacteristic
154C23D4×D21112yes—
155C23C2×C22⋊D21112yes—
156C23C2×C22⋊D21111yescharacteristic
157C23#1680996111yescharacteristic
158D4—42no—
159D4—42no—
160D4—42no—
161D4—42no—
162D4—82no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—8444no—
169C2×C4—8441no—
170C2×C4—8442no—
171C2×C4—8441no—
172C2×C4—8442no—
173C2×C4—8441no—
174C2×C4—4224no—
175C2×C4—4224no—
176C2×C4—4222no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—44no—
183C22—44no—
184C22—42no—
185C22—82no—
186C22#2026994512yes—
187C22#2026994611yescharacteristic
188C22#2026994811yescharacteristic
189C22#2026994912yes—
190C22#2026997311yescharacteristic
191C4—42no—
192C4—42no—
193C4—8444no—
194C4—8442no—
195C4—16881no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—82no—
201C2#2721305411yescharacteristic
202C2#2721333612yes—
subgroupG/H
centerC22#20269946
commutator, cosocleC24×C211C23
FrattiniC24C22×D211
Fitting, prime radical#534×C211C2
socleC22×C211#120
hypercenter#534D211
hypocenterC211#2022
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#20269958C2semidirect
2. C2×#20269892C2semidirect
3. #27213007C2semidirect
4. #27212913C2semidirect
26. C22×Dic211D4semidirect
31. C22⋊Dic211D4semidirect
101. C211#2022semidirect

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
#534×C2112
#2022211