Finite Group Databasebeta

← #41014235

Subgroups — #41014235

Order 27008.

408 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 101 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×#20269892C214yesmaximal, complemented
2#27213006C211yesmaximal, characteristic, complemented
3C4×C22⋊Dic211C211yesmaximal, characteristic, complemented
4C2×#121×C211C211yesmaximal, characteristic
5C2×C4⋊Dic211—21no—
6C2×C22⋊Dic211C2212yes—
7C2×C22⋊Dic211C2212yes—
8C2×C4×Dic211—21no—
9#20269892—28no—
10C211⋊#95—21no—
11C211⋊#95—21no—
12C22×D4×C211C411yescharacteristic, complemented
13C22×D4×C211C2212yes—
14C23×C4×C211C2211yescharacteristic
15#121×C211—21no—
16#121×C211—21no—
17#121×C211—21no—
18#121×C211—21no—
19C2×C4⋊C4×C211C411yescharacteristic, complemented
20C2×C22⋊C4×C211C412yescomplemented
21C22×Dic211—22no—
22C22×Dic211—22no—
23C4⋊Dic211—42no—
24C22⋊Dic211—24no—
25C22⋊Dic211—22no—
26C22⋊Dic211D414yescomplemented
27C4×Dic211—42no—
28C24×C211C2×C412yes—
29C24×C211C2311yescharacteristic
30C2×D4×C211—22no—
31C2×D4×C211—22no—
32C2×D4×C211—24no—
33C2×D4×C211—24no—
34C2×D4×C211—21no—
35C2×D4×C211—21no—
36C2×D4×C211—42no—
37C22×C4×C211—42no—
38C22×C4×C211C2×C412yes—
39C22×C4×C211C2×C411yescharacteristic
40C22×C4×C211C2×C411yescharacteristic
41C22×C4×C211D411yescharacteristic
42C22×C4×C211D411yescharacteristic
43C22×C4×C211D411yescharacteristic
44C22×C4×C211D411yescharacteristic
45C4⋊C4×C211—21no—
46C4⋊C4×C211—21no—
47C22⋊C4×C211—22no—
48C22⋊C4×C211—22no—
49C2×Dic211—24no—
50C2×Dic211—42no—
51C2×Dic211—42no—
52C2×Dic211—42no—
53C2×Dic211—42no—
54C2×Dic211—42no—
55C23×C211—22no—
56C23×C211—24no—
57C23×C211—41no—
58C23×C211—41no—
59C23×C211—42no—
60C23×C211—42no—
61C23×C211C22×C411yescharacteristic
62C23×C211C2×D412yes—
63C23×C211C2×D411yescharacteristic
64C23×C211C2×D411yescharacteristic
65C23×C211#4112yes—
66D4×C211—42no—
67D4×C211—44no—
68D4×C211—42no—
69D4×C211—42no—
70D4×C211—42no—
71C2×C4×C211—22no—
72C2×C4×C211—22no—
73C2×C4×C211—22no—
74C2×C4×C211—22no—
75C2×C4×C211—24no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211C22⋊C411yescharacteristic
80C2×C4×C211C22⋊C411yescharacteristic
81C2×C4×C211C22⋊C411yescharacteristic
82C2×C4×C211C22⋊C411yescharacteristic
83C22×C211—21no—
84C22×C211—21no—
85C22×C211—21no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211—21no—
89C22×C211—42no—
90C22×C211—42no—
91C22×C211—42no—
92C22×C211—42no—
93C22×C211—42no—
94C22×C211—84no—
95C22×C211C2×C22⋊C411yescharacteristic
96C22×C211C4×D412yes—
97C22×C211#12112yes—
98C22×C211#12411yescharacteristic
99C22×C211#12711yescharacteristic
100C4×C211—21no—
101C4×C211—21no—
102C4×C211—21no—
103C4×C211—42no—
104C4×C211#9912yescomplemented
105Dic211—44no—
106Dic211—82no—
107C2×C211—22no—
108C2×C211—21no—
109C2×C211—22no—
110C2×C211—21no—
111C2×C211—82no—
112C2×C211#39011yescharacteristic
113C2×C211C2×#9911yescharacteristic
114C2×C211#41011yescharacteristic
115C211#196411yescharacteristic, complemented, Sylow
116#1964—2111nomaximal, Sylow
117C2×#121D21111yescharacteristic
118C2×#99—2114no—
119#380—2111no—
120C4×C22⋊C4—2111no—
121C22×D4Dic21111yescharacteristic, complemented
122C22×D4C2×D21112yes—
123C23×C4C2×D21111yescharacteristic
124#121—21no—
125#121—21no—
126#121—21no—
127#121—21no—
128C2×C4⋊C4—4221no—
129C2×C4⋊C4Dic21111yescharacteristic, complemented
130C2×C22⋊C4—2112no—
131C2×C22⋊C4—2112no—
132C2×C22⋊C4Dic21112yescomplemented
133C2×C42—4221no—
134#99—4228no—
135#95—4221no—
136#95—4221no—
137C24C2×Dic21112yes—
138C24C22×D21111yescharacteristic
139C2×D4—24no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—24no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—42no—
146C22×C4—42no—
147C22×C4—4222no—
148C22×C4—4222no—
149C22×C4C2×Dic21112yes—
150C22×C4C2×Dic21111yescharacteristic
151C22×C4C2×Dic21111yescharacteristic
152C22×C4C22⋊D21111yescharacteristic
153C22×C4C22⋊D21111yescharacteristic
154C22×C4C22⋊D21111yescharacteristic
155C22×C4C22⋊D21111yescharacteristic
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—8442no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—4224no—
162C22⋊C4—4222no—
163C22⋊C4—2114no—
164C42—8442no—
165C23—22no—
166C23—24no—
167C23—41no—
168C23—41no—
169C23—42no—
170C23—42no—
171C23C22×Dic21111yescharacteristic
172C23D4×D21112yes—
173C23C2×C22⋊D21111yescharacteristic
174C23C2×C22⋊D21111yescharacteristic
175C23#1680996112yes—
176D4—42no—
177D4—44no—
178D4—42no—
179D4—42no—
180D4—42no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—8442no—
190C2×C4—8442no—
191C2×C4—8442no—
192C2×C4—8442no—
193C2×C4—8442no—
194C2×C4—4224no—
195C2×C4C22⋊Dic21111yescharacteristic
196C2×C4C22⋊Dic21111yescharacteristic
197C2×C4C22⋊Dic21111yescharacteristic
198C2×C4C22⋊Dic21111yescharacteristic
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—84no—
211C22C2×C22⋊Dic21111yescharacteristic
212C22D4×Dic21112yes—
213C22#2026994812yes—
214C22#2026996411yescharacteristic
215C22#2026997311yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—8444no—
221C4—16882no—
222C4#2026989212yescomplemented
223C2—22no—
224C2—21no—
225C2—22no—
226C2—21no—
227C2—82no—
228C2#2721304911yescharacteristic
229C2C2×#2026989211yescharacteristic
230C2#2721312011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC23×C211C22×C4
FrattiniC24C22×D211
Fitting, prime radicalC2×#121×C211C2
socleC22×C211C2×C22⋊C4
cosocleC24×C211C23
hypercenterC2×#121D211
hypocenterC211#1964
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#20269892C2semidirect
2. #27213006C2semidirect
3. C4×C22⋊Dic211C2semidirect
12. C22×D4×C211C4semidirect
19. C2×C4⋊C4×C211C4semidirect
20. C2×C22⋊C4×C211C4semidirect
26. C22⋊Dic211D4semidirect
104. C4×C211#99semidirect
115. C211#1964semidirect
121. C22×D4Dic211semidirect
129. C2×C4⋊C4Dic211semidirect
132. C2×C22⋊C4Dic211semidirect
222. C4#20269892semidirect

Maximal subgroups

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

subgroupindex
C2×#202698922
#272130062
C4×C22⋊Dic2112
C2×#121×C2112
#1964211