Finite Group Databasebeta

← #41013788

Subgroups — #41013788

Order 27008.

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

normal subgroups only

This group has 324 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
1C2×#20269975C211yesmaximal, characteristic
2#27213623C211yesmaximal, characteristic
3#27213155C211yesmaximal, characteristic
4C2×#20269895C211yesmaximal, characteristic
5#27212907C211yesmaximal, characteristic
6#27212905C211yesmaximal, characteristic
7#419×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269975—22no—
11#20269975C414yescomplemented
12C4×Dic422—22no—
13#20269942—22no—
14#20269936C2211yescharacteristic
15#20269933—22no—
16C2×Dic211⋊C4—22no—
17C2×C4×Dic211C2211yescharacteristic
18#20269901—22no—
19#20269895—22no—
20C22×Q8×C211C2211yescharacteristic
21C2×M16×C211C2211yescharacteristic
22#117×C211C2211yescharacteristic
23#103×C211—22no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422C2×C412yes—
29C2×Dic422D412yes—
30C22×Dic211—21no—
31C22×Dic211D412yes—
32#16809946—21no—
33#16809946—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211D412yes—
36Dic211⋊C4—24no—
37Dic211⋊C4—28no—
38C22⋊Dic211—21no—
39C4×Dic211—24no—
40C4×Dic211—21no—
41C4×Dic211—21no—
42C4×Dic211C2×C412yes—
43C2×Q8×C211—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211C2×C412yes—
49C22×C4×C211—21no—
50C22×C4×C211C2311yescharacteristic
51M16×C211—21no—
52M16×C211—21no—
53C2×C8×C211—21no—
54C4⋊C4×C211D412yes—
55C22⋊C4×C211—21no—
56C42×C211—21no—
57Dic422—22no—
58Dic422—22no—
59Dic422—22no—
60Dic422—22no—
61C2×Dic211—24no—
62C2×Dic211—22no—
63C2×Dic211—22no—
64C2×Dic211—22no—
65C2×Dic211—22no—
66C2×Dic211C22⋊C414yes—
67C211⋊C8—41no—
68C23×C211C2×D411yescharacteristic
69Q8×C211—22no—
70Q8×C211—22no—
71Q8×C211—41no—
72Q8×C211—41no—
73C2×C4×C211—22no—
74C2×C4×C211—22no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211C22×C411yescharacteristic
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211#4111yescharacteristic
82C2×C4×C211#4111yescharacteristic
83C8×C211—41no—
84C22×C211—22no—
85C22×C211C2×C22⋊C411yescharacteristic
86C22×C211#12011yescharacteristic
87C22×C211#12311yescharacteristic
88C4×C211—42no—
89C4×C211—42no—
90C4×C211C4×D411yescharacteristic
91C4×C211C4×D411yescharacteristic
92C4×C211#12111yescharacteristic
93C4×C211#12111yescharacteristic
94Dic211—24no—
95Dic211—24no—
96Dic211—42no—
97C2×C211—21no—
98C2×C211—21no—
99C2×C211#38611yescharacteristic
100C2×C211#45612yes—
101C211#188611yescharacteristic, complemented, Sylow
102#1886—2111nomaximal, Sylow
103C2×C4⋊Q8—2111no—
104#519—2111no—
105#419—2111no—
106#419D21111yescharacteristic
107C2×#101—2111no—
108#339—2111no—
109#339—2111no—
110C22×Q8—2111no—
111C22×Q8C2×D21111yescharacteristic
112C2×M16—2111no—
113C2×M16C2×D21111yescharacteristic
114C4⋊Q8—4222no—
115C4⋊Q8—2114no—
116C4×Q8—4222no—
117C4×Q8—4222no—
118#117—4222no—
119#117—2111no—
120#117C2×D21111yescharacteristic
121C2×C4⋊C4—4222no—
122C2×C42—2111no—
123#103—22no—
124#103—4222no—
125#101—4222no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—4221no—
132C2×Q8—4221no—
133C2×Q8—4221no—
134C2×Q8—4221no—
135C2×Q8—2112no—
136C2×Q8—2112no—
137C2×Q8C4×D21112yes—
138C22×C4—21no—
139C22×C4—4221no—
140C22×C4—2112no—
141C22×C4C22×D21111yescharacteristic
142M16—21no—
143M16—21no—
144M16—4221no—
145M16—4221no—
146C2×C8—21no—
147C2×C8—4221no—
148C4⋊C4—4224no—
149C4⋊C4—4228no—
150C4⋊C4—2112no—
151C4⋊C4C22⋊D21112yes—
152C22⋊C4—21no—
153C22⋊C4—4221no—
154C42—21no—
155C42—4224no—
156C42—4221no—
157C42—4221no—
158C42—2112no—
159C23D4×D21111yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164Q8—4222no—
165Q8—4222no—
166Q8—4222no—
167Q8—4222no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—4222no—
173C2×C4—4224no—
174C2×C4—4222no—
175C2×C4—4222no—
176C2×C4—4222no—
177C2×C4—2114no—
178C2×C4C2×C4×D21111yescharacteristic
179C2×C4D4×D21111yescharacteristic
180C2×C4D4×D21111yescharacteristic
181C2×C4C2×C22⋊D21111yescharacteristic
182C2×C4#1680996111yescharacteristic
183C2×C4#1680996211yescharacteristic
184C8—41no—
185C8—8441no—
186C22—22no—
187C22C22⋊C4×D21111yescharacteristic
188C22#2026994511yescharacteristic
189C22#2026995911yescharacteristic
190C4—42no—
191C4—42no—
192C4—8442no—
193C4—4224no—
194C4—4224no—
195C4(C22⋊D211)⋊C411yescharacteristic
196C4C4×C22⋊D21111yescharacteristic
197C4#2026994811yescharacteristic
198C4#2026995011yescharacteristic
199C2—21no—
200C2—21no—
201C2#2721302211yescharacteristic
202C2#2721332712yes—
subgroupG/H
centerC22#20269945
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#419×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenter#419D211
hypocenterC211#1886
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
11. #20269975C4semidirect
101. C211#1886semidirect

Maximal subgroups

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

subgroupindex
C2×#202699752
#272136232
#272131552
C2×#202698952
#272129072
#272129052
#419×C2112
#1886211