Finite Group Databasebeta

← #41013630

Subgroups — #41013630

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#27213186C211yesmaximal, characteristic, complemented
3C2×#20269905C211yesmaximal, characteristic
4#27213145C211yesmaximal, characteristic
5#27212915C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7#382×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269942—21no—
12#20269939—21no—
13C4×C22⋊D211—22no—
14#20269936C2211yescharacteristic
15#20269935—21no—
16C4⋊C4×D211—21no—
17#20269910—21no—
18#20269908—21no—
19#20269905C414yescomplemented
20#20269900—21no—
21#20269897—21no—
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24C2×C42×C211C2211yescharacteristic
25#95×C211—22no—
26#16809962—21no—
27#16809962—21no—
28#16809962C2×C412yes—
29#16809962Q812yes—
30C2×Dic422D411yescharacteristic
31C2×D844D411yescharacteristic
32C2×C22⋊D211—21no—
33C2×C4×D211—22no—
34C2×C4×D211—21no—
35#16809946—21no—
36#16809946—21no—
37#16809946—21no—
38#16809946C2×C412yescomplemented
39C2×C211⋊C8—21no—
40C2×C211⋊C8—21no—
41C4⋊Dic211D411yescharacteristic
42C4⋊Dic211D411yescharacteristic
43Dic211⋊C4—22no—
44C22⋊Dic211—21no—
45(C2×D211)⋊C4—22no—
46C4×Dic211—22no—
47C4×Dic211—21no—
48C22×C4×C211—21no—
49C22×C4×C211—21no—
50C22×C4×C211D412yes—
51C22×C4×C211C2311yescharacteristic
52C4⋊C4×C211—21no—
53C4⋊C4×C211—21no—
54C4⋊C4×C211—21no—
55C4⋊C4×C211—21no—
56C42×C211—21no—
57C42×C211C2×C412yescomplemented
58C22×D211—21no—
59Dic422—21no—
60Dic422C4⋊C412yes—
61D844—21no—
62D844C4⋊C412yes—
63C22⋊D211—22no—
64C22⋊D211—22no—
65C2×Dic211—21no—
66C2×Dic211—21no—
67C2×Dic211—21no—
68C4×D211—22no—
69C4×D211—22no—
70C4×D211—82no—
71C211⋊C8—22no—
72C211⋊C8—41no—
73C23×C211C2×D411yescharacteristic
74C2×C4×C211—22no—
75C2×C4×C211—22no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—42no—
79C2×C4×C211—42no—
80C2×C4×C211—41no—
81C2×C4×C211C22×C411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×Q811yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C2×C4×C211#4111yescharacteristic
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×C211—42no—
97C4×C211—81no—
98C4×C211C2×C4⋊C411yescharacteristic
99C4×C211C4×D411yescharacteristic
100C4×C211C22⋊Q811yescharacteristic
101C4×C211#12311yescharacteristic
102Dic211—22no—
103Dic211—41no—
104Dic211—41no—
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#382D21111yescharacteristic, complemented
118#343—2111no—
119#339—2111no—
120C2×#41—2111no—
121C2×M16—2111no—
122C2×M16—2111no—
123C4×Q8—4221no—
124C4×D4—4221no—
125C4×D4—4222no—
126#117—4221no—
127#117—2111no—
128C2×C4⋊C4—4221no—
129C2×C4⋊C4C2×D21111yescharacteristic
130C2×C4⋊C4C2×D21111yescharacteristic
131C2×C42C2×D21111yescharacteristic
132#107—4221no—
133#106—4221no—
134#104—2114no—
135#103—4221no—
136#102—4221no—
137#95—22no—
138#41—4221no—
139#41—4221no—
140#41—2112no—
141#41—2112no—
142C2×Q8—2111no—
143C2×D4—4221no—
144C2×D4—2111no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—4222no—
148C22×C4—4221no—
149C22×C4C22⋊D21112yes—
150C22×C4C22×D21111yescharacteristic
151M16—4221no—
152M16—4221no—
153M16—4221no—
154M16—2112no—
155C2×C8—4221no—
156C2×C8—4221no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—4222no—
162C4⋊C4—2111no—
163C4⋊C4—2111no—
164C22⋊C4—4222no—
165C22⋊C4—4221no—
166C42—21no—
167C42—4222no—
168C42—4221no—
169C42C4×D21112yescomplemented
170C23—4221no—
171C23C2×C22⋊D21111yescharacteristic
172Q8—4221no—
173Q8—2112no—
174D4—4222no—
175D4—4222no—
176D4—4221no—
177D4—2112no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—16882no—
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#1680996211yescharacteristic
196C2×C4#1680996311yescharacteristic
197C8—8441no—
198C8—4222no—
199C22—21no—
200C22—21no—
201C22—8441no—
202C22—8441no—
203C22—4222no—
204C22C4×C22⋊D21111yescharacteristic
205C22#2026994511yescharacteristic
206C22#2026995311yescharacteristic
207C4—42no—
208C4—42no—
209C4—81no—
210C4—8441no—
211C4—8441no—
212C4—4222no—
213C4C4⋊C4×D21111yescharacteristic
214C4#2026993911yescharacteristic
215C4#2026995511yescharacteristic
216C4#2026996211yescharacteristic
217C2—21no—
218C2—21no—
219C2—8442no—
220C2#2721301711yescharacteristic
221C2#2721331811yescharacteristic
222C2#2721332311yescharacteristic
subgroupG/H
centerC22#20269945
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#382×C211C2
socleC22×C211#120
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#382D211
hypocenterC211#1860
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #27213186C2semidirect
5. #27212915C2semidirect
6. #27212906C2semidirect
7. #382×C211C2semidirect
19. #20269905C4semidirect
38. #16809946C2×C4semidirect
57. C42×C211C2×C4semidirect
111. C211#1860semidirect
117. #382D211semidirect
169. C42C4×D211semidirect

Maximal subgroups

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

subgroupindex
#272136272
#272131862
C2×#202699052
#272131452
#272129152
#272129062
#382×C2112
#1860211