Finite Group Databasebeta

← #41014530

Subgroups — #41014530

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
1#27213708C211yesmaximal, characteristic
2C2×#20269961C212yesmaximal, complemented
3C2×#20269891C212yesmaximal, complemented
4#27213007C211yesmaximal, characteristic, complemented
5#342×C211C211yesmaximal, characteristic, complemented
6C22×C22⋊D211C2212yes—
7C23×Dic211C2211yescharacteristic
8#20269961—22no—
9#20269961—24no—
10#20269961—24no—
11#20269951—22no—
12#20269947—22no—
13C2×C4⋊Dic211—22no—
14C2×C22⋊Dic211C2211yescharacteristic
15C2×(C2×D211)⋊C4—22no—
16#20269891—24no—
17C211⋊#95—22no—
18C2×C22⋊C4×C211C2211yescharacteristic
19C2×C22⋊C4×C211C2212yes—
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—21no—
29C22×Dic211—22no—
30C22×Dic211—22no—
31C22×Dic211—21no—
32C22×Dic211—21no—
33C22×Dic211D412yes—
34C22×Dic211D412yescomplemented
35C4⋊Dic211—28no—
36C4⋊Dic211—22no—
37C22⋊Dic211—22no—
38(C2×D211)⋊C4—28no—
39(C2×D211)⋊C4—24no—
40C24×C211C2311yescharacteristic
41C22×C4×C211—22no—
42C22×C4×C211D412yes—
43C22⋊C4×C211—22no—
44C22⋊C4×C211—22no—
45C22⋊C4×C211D414yescomplemented
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—22no—
60C2×Dic211—41no—
61C2×Dic211—42no—
62C2×Dic211—41no—
63C2×Dic211—42no—
64C2×Dic211—41no—
65C23×C211—42no—
66C23×C211C2×D412yes—
67C23×C211C2×D411yescharacteristic
68C23×C211C2×D411yescharacteristic
69C23×C211C2×D412yes—
70C23×C211#4111yescharacteristic
71C2×C4×C211—22no—
72C2×C4×C211—24no—
73C2×C4×C211—44no—
74C2×C4×C211—42no—
75C22×C211—22no—
76C22×C211—24no—
77C22×C211—41no—
78C22×C211—42no—
79C22×C211—41no—
80C22×C211#12012yes—
81C22×C211#12011yescharacteristic
82C22×C211#12111yescharacteristic
83C22×C211#12112yes—
84C22×C211#12711yescharacteristic
85C2×D211—44no—
86C2×D211—42no—
87C2×D211—44no—
88C2×D211—82no—
89C4×C211—44no—
90C4×C211—42no—
91Dic211—42no—
92Dic211—42no—
93Dic211—81no—
94C2×C211—21no—
95C2×C211—22no—
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#381—2111no—
107#342D21111yescharacteristic, complemented
108C22×D4—2112no—
109C23×C4—2111no—
110#123—4224no—
111#123—4222no—
112#123—4224no—
113#121—4222no—
114#120—4222no—
115C2×C4⋊C4—4222no—
116C2×C22⋊C4—4222no—
117C2×C22⋊C4—2111no—
118C2×C22⋊C4C2×D21111yescharacteristic
119C2×C22⋊C4C2×D21112yes—
120#99—4224no—
121#95—4222no—
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—4221no—
133C22×C4—4222no—
134C22×C4—4221no—
135C22×C4—4222no—
136C22×C4—4221no—
137C22×C4—2112no—
138C22×C4—2112no—
139C22×C4D84412yes—
140C4⋊C4—4228no—
141C4⋊C4—4222no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—4228no—
145C22⋊C4—4224no—
146C22⋊C4—4222no—
147C22⋊C4D84414yescomplemented
148C23—42no—
149C23—8444no—
150C23—4222no—
151C23—4222no—
152C23—4222no—
153C23D4×D21112yes—
154C23D4×D21111yescharacteristic
155C23C2×D84411yescharacteristic
156C23C2×D84412yes—
157C23#1680996311yescharacteristic
158D4—8442no—
159D4—8442no—
160D4—8442no—
161D4—8442no—
162D4—16882no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—8441no—
168C2×C4—8442no—
169C2×C4—8441no—
170C2×C4—8442no—
171C2×C4—8441no—
172C2×C4—4224no—
173C2×C4—4222no—
174C2×C4—4222no—
175C2×C4—4222no—
176C2×C4—4222no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—8444no—
183C22—8442no—
184C22—8444no—
185C22—16882no—
186C22#2026994712yes—
187C22#2026994711yescharacteristic
188C22#2026995212yes—
189C22#2026995211yescharacteristic
190C22#2026997411yescharacteristic
191C4—44no—
192C4—42no—
193C4—8442no—
194C4—8442no—
195C4—16881no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—16882no—
201C2#2721305611yescharacteristic
202C2#2721333712yes—
subgroupG/H
centerC22#20269947
commutator, cosocleC24×C211C23
FrattiniC24C22×D211
Fitting#342×C211C2
socleC22×C211#120
prime radical#27213708C2
hypercenter#342D211
hypocenterC211#2022
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#20269961C2semidirect
3. C2×#20269891C2semidirect
4. #27213007C2semidirect
5. #342×C211C2semidirect
34. C22×Dic211D4semidirect
45. C22⋊C4×C211D4semidirect
101. C211#2022semidirect
107. #342D211semidirect
147. C22⋊C4D844semidirect

Maximal subgroups

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

subgroupindex
#272137082
C2×#202699612
C2×#202698912
#272130072
#342×C2112
#2022211