Finite Group Databasebeta

← #41013634

Subgroups — #41013634

Order 27008.

286 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 83 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#27213627C211yesmaximal, characteristic
2#27213188C211yesmaximal, characteristic, complemented
3C2×#20269903C211yesmaximal, characteristic
4#27213148C211yesmaximal, characteristic
5#27213011C211yesmaximal, characteristic, complemented
6#27212907C211yesmaximal, characteristic, complemented
7#343×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20269942—21no—
10#20269939—21no—
11C4×C22⋊D211—22no—
12#20269936C2211yescharacteristic
13#20269935—21no—
14C2×C4⋊Dic211C2211yescharacteristic
15C4⋊C4×D211—21no—
16C2×C4×Dic211C2211yescharacteristic
17#20269911—21no—
18#20269909—21no—
19#20269903C414yescomplemented
20#20269902—21no—
21#20269899—21no—
22C211⋊#95—22no—
23C2×M16×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25C2×C4⋊C4×C211C2211yescharacteristic
26#16809962—21no—
27#16809962—21no—
28#16809962C2×C412yes—
29#16809962Q812yes—
30C2×Dic422D411yescharacteristic
31C2×D844D411yescharacteristic
32C2×C22⋊D211—21no—
33C22×Dic211—21no—
34C22×Dic211—21no—
35C2×C4×D211—22no—
36C2×C4×D211—21no—
37C4⋊Dic211—21no—
38C4⋊Dic211—21no—
39C4⋊Dic211D411yescharacteristic
40C4⋊Dic211D411yescharacteristic
41Dic211⋊C4—22no—
42C22⋊Dic211—21no—
43(C2×D211)⋊C4—22no—
44C4×Dic211—22no—
45C4×Dic211—21no—
46C4×Dic211—21no—
47C4×Dic211C2×C412yescomplemented
48C22×C4×C211D412yes—
49C22×C4×C211C2311yescharacteristic
50M16×C211—21no—
51M16×C211—21no—
52M16×C211—21no—
53M16×C211C2×C412yescomplemented
54C2×C8×C211—21no—
55C2×C8×C211—21no—
56C4⋊C4×C211—21no—
57C4⋊C4×C211—21no—
58C22×D211—21no—
59Dic422—21no—
60Dic422C4⋊C412yes—
61D844—21no—
62D844C4⋊C412yes—
63C22⋊D211—22no—
64C22⋊D211—22no—
65C2×Dic211—21no—
66C2×Dic211—22no—
67C2×Dic211—21no—
68C2×Dic211—21no—
69C2×Dic211—41no—
70C2×Dic211—41no—
71C2×Dic211—42no—
72C2×Dic211—41no—
73C4×D211—22no—
74C4×D211—22no—
75C4×D211—82no—
76C23×C211C2×D411yescharacteristic
77C2×C4×C211—22no—
78C2×C4×C211—42no—
79C2×C4×C211C22×C411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×Q811yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—22no—
86C8×C211—41no—
87C22×C211—21no—
88C22×C211—21no—
89C22×C211C4×D411yescharacteristic
90C22×C211#12011yescharacteristic
91C22×C211C22⋊Q811yescharacteristic
92C2×D211—22no—
93C2×D211—41no—
94C2×D211—41no—
95C4×C211—42no—
96C4×C211C2×C4⋊C411yescharacteristic
97C4×C211C4×D411yescharacteristic
98C4×C211C22⋊Q811yescharacteristic
99C4×C211#12311yescharacteristic
100Dic211—22no—
101Dic211—42no—
102Dic211—41no—
103Dic211—41no—
104Dic211—81no—
105C2×C211—21no—
106C2×C211—21no—
107C2×C211#38511yescharacteristic
108C2×C211#45411yescharacteristic
109C2×C211#45511yescharacteristic
110D211—42no—
111C211#186011yescharacteristic, complemented, Sylow
112#1860—2111nomaximal, Sylow
113#520—2111no—
114#428—2111no—
115C2×#104—2111no—
116#417—2111no—
117#382—2111no—
118#343D21111yescharacteristic, complemented
119#339—2111no—
120C2×#41—2111no—
121C2×M16C2×D21111yescharacteristic
122C2×M16C2×D21111yescharacteristic
123C4×Q8—4221no—
124C4×D4—4221no—
125C4×D4—4222no—
126#117—4221no—
127#117—2111no—
128C2×C4⋊C4—4221no—
129C2×C4⋊C4—2111no—
130C2×C4⋊C4C2×D21111yescharacteristic
131C2×C42—2111no—
132#107—4221no—
133#106—4221no—
134#104—2114no—
135#103—4221no—
136#102—4221no—
137#95—4222no—
138#41—4221no—
139#41—4221no—
140#41—2112no—
141#41—2112no—
142C2×Q8—2111no—
143C2×D4—4221no—
144C2×D4—2111no—
145C22×C4—4222no—
146C22×C4—4221no—
147C22×C4—4221no—
148C22×C4—4221no—
149C22×C4D84412yes—
150C22×C4C22×D21111yescharacteristic
151M16—21no—
152M16—21no—
153M16—21no—
154M16C4×D21112yescomplemented
155C2×C8—21no—
156C2×C8—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—4222no—
160C4⋊C4—4221no—
161C4⋊C4—4221no—
162C4⋊C4—2111no—
163C4⋊C4—2111no—
164C22⋊C4—4222no—
165C22⋊C4—4221no—
166C42—4222no—
167C42—4221no—
168C42—4221no—
169C42—2112no—
170C23—4221no—
171C23C2×D84411yescharacteristic
172Q8—4221no—
173Q8—2112no—
174D4—4222no—
175D4—4222no—
176D4—4221no—
177D4—2112no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—8441no—
181C2×C4—8441no—
182C2×C4—8442no—
183C2×C4—8441no—
184C2×C4—16882no—
185C2×C4—4222no—
186C2×C4—4221no—
187C2×C4—4221no—
188C2×C4—4221no—
189C2×C4—4222no—
190C2×C4—4222no—
191C2×C4C2×C4×D21111yescharacteristic
192C2×C4D4×D21111yescharacteristic
193C2×C4D4×D21111yescharacteristic
194C2×C4Q8×D21111yescharacteristic
195C2×C4#1680996111yescharacteristic
196C2×C4#1680996211yescharacteristic
197C8—22no—
198C8—41no—
199C22—21no—
200C22—21no—
201C22—8441no—
202C22—8441no—
203C22—4222no—
204C22C4×D84411yescharacteristic
205C22#2026994711yescharacteristic
206C22#2026995711yescharacteristic
207C4—42no—
208C4—8442no—
209C4—8441no—
210C4—8441no—
211C4—16881no—
212C4—4222no—
213C4C4⋊C4×D21111yescharacteristic
214C4(C22⋊D211)⋊C411yescharacteristic
215C4#2026995511yescharacteristic
216C4#2026995911yescharacteristic
217C2—21no—
218C2—21no—
219C2—8442no—
220C2#2721302011yescharacteristic
221C2#2721332111yescharacteristic
222C2#2721332611yescharacteristic
subgroupG/H
centerC22#20269947
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#120
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#343D211
hypocenterC211#1860
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. #27213188C2semidirect
5. #27213011C2semidirect
6. #27212907C2semidirect
7. #343×C211C2semidirect
19. #20269903C4semidirect
47. C4×Dic211C2×C4semidirect
53. M16×C211C2×C4semidirect
111. C211#1860semidirect
118. #343D211semidirect
154. M16C4×D211semidirect

Maximal subgroups

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

subgroupindex
#272136272
#272131882
C2×#202699032
#272131482
#272130112
#272129072
#343×C2112
#1860211