Finite Group Databasebeta

← #41014529

Subgroups — #41014529

Order 27008.

430 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 73 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
1#27213708C211yesmaximal, characteristic
2C2×#20269960C212yesmaximal, complemented
3C2×#20269891C212yesmaximal, complemented
4#27212913C211yesmaximal, characteristic, complemented
5#381×C211C211yesmaximal, characteristic, complemented
6C22×C22⋊D211C2212yes—
7#20269961—22no—
8#20269960—24no—
9#20269960—24no—
10#20269951—22no—
11#20269947—22no—
12C2×Dic211⋊C4—22no—
13C2×C22⋊Dic211C2212yes—
14C2×(C2×D211)⋊C4—22no—
15#20269891—24no—
16C23×C4×C211C2211yescharacteristic
17C2×C22⋊C4×C211C2211yescharacteristic
18C2×C22⋊C4×C211C2211yescharacteristic
19#95×C211—22no—
20C23×D211D412yes—
21C2×D844—22no—
22C2×C22⋊D211—22no—
23C2×C22⋊D211—22no—
24C2×C22⋊D211—22no—
25C2×C22⋊D211—22no—
26C2×C22⋊D211—22no—
27C2×C22⋊D211—22no—
28C22×Dic211—22no—
29C22×Dic211D412yes—
30C4⋊Dic211—22no—
31Dic211⋊C4—28no—
32C22⋊Dic211—22no—
33C22⋊Dic211D414yescomplemented
34(C2×D211)⋊C4—28no—
35(C2×D211)⋊C4—24no—
36C24×C211C2311yescharacteristic
37C22×C4×C211—21no—
38C22×C4×C211—22no—
39C22×C4×C211—22no—
40C22×C4×C211—21no—
41C22×C4×C211—21no—
42C22×C4×C211D412yes—
43C22×C4×C211D412yescomplemented
44C22⋊C4×C211—22no—
45C22⋊C4×C211—22no—
46C22×D211—22no—
47C22×D211—22no—
48C22×D211—22no—
49C22×D211—44no—
50D844—82no—
51C22⋊D211—42no—
52C22⋊D211—42no—
53C22⋊D211—42no—
54C22⋊D211—42no—
55C2×Dic211—24no—
56C2×Dic211—22no—
57C2×Dic211—22no—
58C2×Dic211—22no—
59C2×Dic211—44no—
60C23×C211—42no—
61C23×C211C2×D412yes—
62C23×C211C2×D412yes—
63C23×C211C2×D411yescharacteristic
64C23×C211C2×D411yescharacteristic
65C23×C211#4111yescharacteristic
66C2×C4×C211—22no—
67C2×C4×C211—24no—
68C2×C4×C211—22no—
69C2×C4×C211—41no—
70C2×C4×C211—42no—
71C2×C4×C211—41no—
72C2×C4×C211—42no—
73C2×C4×C211—42no—
74C2×C4×C211—41no—
75C22×C211—22no—
76C22×C211—24no—
77C22×C211—42no—
78C22×C211—41no—
79C22×C211—41no—
80C22×C211#12011yescharacteristic
81C22×C211#12012yes—
82C22×C211#12112yes—
83C22×C211#12111yescharacteristic
84C22×C211#12711yescharacteristic
85C2×D211—44no—
86C2×D211—42no—
87C2×D211—44no—
88C2×D211—82no—
89C4×C211—42no—
90C4×C211—42no—
91C4×C211—81no—
92Dic211—44no—
93Dic211—42no—
94C2×C211—22no—
95C2×C211—21no—
96C2×C211—22no—
97C2×C211—21no—
98C2×C211#39211yescharacteristic
99C2×C211#45812yes—
100D211—82no—
101C211#202211yescharacteristic, complemented, Sylow
102#2022—2111nomaximal, Sylow
103#534—2111no—
104C2×#123—2112no—
105C2×#99—2112no—
106#381D21111yescharacteristic, complemented
107#342—2111no—
108C22×D4—2112no—
109C23×C4C2×D21111yescharacteristic
110#123—4222no—
111#123—4224no—
112#123—4224no—
113#121—4222no—
114#120—4222no—
115C2×C4⋊C4—4222no—
116C2×C22⋊C4—4222no—
117C2×C22⋊C4—2112no—
118C2×C22⋊C4C2×D21111yescharacteristic
119C2×C22⋊C4C2×D21111yescharacteristic
120#99—4224no—
121#95—22no—
122C24—2112no—
123C24C22×D21111yescharacteristic
124C2×D4—4222no—
125C2×D4—4222no—
126C2×D4—4222no—
127C2×D4—4222no—
128C2×D4—4222no—
129C2×D4—4222no—
130C2×D4—4222no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—4222no—
137C22×C4—2112no—
138C22×C4C22⋊D21112yes—
139C22×C4D84412yescomplemented
140C4⋊C4—4228no—
141C4⋊C4—4222no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—4228no—
145C22⋊C4—4224no—
146C22⋊C4—4222no—
147C22⋊C4—2114no—
148C23—42no—
149C23—8444no—
150C23—4222no—
151C23—4222no—
152C23—4222no—
153C23D4×D21112yes—
154C23D4×D21112yes—
155C23C2×C22⋊D21111yescharacteristic
156C23C2×D84411yescharacteristic
157C23#1680996211yescharacteristic
158D4—8442no—
159D4—8442no—
160D4—8442no—
161D4—8442no—
162D4—16882no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—8444no—
173C2×C4—4222no—
174C2×C4—4224no—
175C2×C4—4222no—
176C2×C4—4222no—
177C22—22no—
178C22—24no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—8444no—
183C22—8444no—
184C22—8442no—
185C22—16882no—
186C22#2026994511yescharacteristic
187C22#2026994712yes—
188C22#2026995012yes—
189C22#2026995111yescharacteristic
190C22#2026997311yescharacteristic
191C4—42no—
192C4—42no—
193C4—81no—
194C4—8444no—
195C4—8442no—
196C2—21no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—16882no—
201C2#2721305511yescharacteristic
202C2#2721333512yes—
subgroupG/H
centerC22#20269945
commutator, cosocleC24×C211C23
FrattiniC24C22×D211
Fitting#381×C211C2
socleC22×C211#120
prime radical#27213708C2
hypercenter#381D211
hypocenterC211#2022
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#20269960C2semidirect
3. C2×#20269891C2semidirect
4. #27212913C2semidirect
5. #381×C211C2semidirect
33. C22⋊Dic211D4semidirect
43. C22×C4×C211D4semidirect
101. C211#2022semidirect
106. #381D211semidirect
139. C22×C4D844semidirect

Maximal subgroups

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

subgroupindex
#272137082
C2×#202699602
C2×#202698912
#272129132
#381×C2112
#2022211