Finite Group Databasebeta

← #41011908

Subgroups — #41011908

Order 27008.

228 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 63 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#27213698C211yesmaximal, characteristic
2#27213200C211yesmaximal, characteristic
3#27212991C211yesmaximal, characteristic
4#27212884C211yesmaximal, characteristic
5#27212883C211yesmaximal, characteristic
6#27212882C211yesmaximal, characteristic
7#328×C211C211yesmaximal, characteristic
8#20269976C411yescharacteristic, complemented
9#20269975C411yescharacteristic, complemented
10#20269967—21no—
11#20269956C2211yescharacteristic
12#20269954—21no—
13Q8×Dic211C411yescharacteristic, complemented
14#20269942—21no—
15#20269942C411yescharacteristic, complemented
16#20269933—21no—
17C2×C4×Dic211—21no—
18C2×C4×Dic211C2211yescharacteristic
19Dic211⋊C8—21no—
20C22⋊(C211⋊C8)C2211yescharacteristic
21#20269887—21no—
22C211⋊#95—21no—
23C211⋊#95C2211yescharacteristic
24C22⋊Q8×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C2×Dic422—21no—
28C2×Dic422C2×C411yescharacteristic
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211C2×C411yescharacteristic
35Dic211⋊C4—21no—
36Dic211⋊C4—21no—
37Dic211⋊C4—21no—
38Dic211⋊C4—21no—
39Dic211⋊C4—21no—
40C22⋊Dic211—21no—
41C4×Dic211—24no—
42C4×Dic211—21no—
43C4×Dic211—21no—
44C4×Dic211C2×C411yescharacteristic
45C4×Dic211C2×C411yescharacteristic
46C4×Dic211D411yescharacteristic
47C4×Dic211D411yescharacteristic
48C2×Q8×C211C2×C411yescharacteristic
49C22×C4×C211—21no—
50C22×C4×C211C2311yescharacteristic
51C2×C8×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211C2×C411yescharacteristic
54C22⋊C4×C211—21no—
55Dic422—42no—
56Dic422—41no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—22no—
60C2×Dic211—22no—
61C2×Dic211—21no—
62C2×Dic211—21no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211C22⋊C411yescharacteristic
66C2×Dic211C22⋊C411yescharacteristic
67C2×Dic211C22⋊C411yescharacteristic
68C2×Dic211C22⋊C411yescharacteristic
69C211⋊C8—41no—
70C23×C211C2×D411yescharacteristic
71Q8×C211—41no—
72C2×C4×C211—22no—
73C2×C4×C211—21no—
74C2×C4×C211—21no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211C22×C411yescharacteristic
79C2×C4×C211C2×D411yescharacteristic
80C8×C211—41no—
81C22×C211—21no—
82C22×C211—21no—
83C22×C211C2×C22⋊C411yescharacteristic
84C22×C211#13711yescharacteristic
85C22×C211#13711yescharacteristic
86C4×C211—21no—
87C4×C211—21no—
88C4×C211—42no—
89C4×C211—41no—
90C4×C211—41no—
91Dic211—21no—
92Dic211—21no—
93Dic211—21no—
94Dic211—42no—
95Dic211—41no—
96Dic211—41no—
97Dic211#10412yes—
98C2×C211—21no—
99C2×C211—21no—
100C2×C211#41011yescharacteristic
101C2×C211#41911yescharacteristic
102C2×C211C2×#10411yescharacteristic
103C211#150611yescharacteristic, complemented, Sylow
104#1506—2111nomaximal, Sylow
105#533—2111no—
106#432—2111no—
107#376—2111no—
108#328—2111no—
109#328—2111no—
110#328—2111no—
111#328D21111yescharacteristic
112C4⋊Q8—2111no—
113C4⋊Q8—2111no—
114#125—4221no—
115C22⋊Q8—4221no—
116C22⋊Q8—2111no—
117C22⋊Q8C2×D21111yescharacteristic
118C4×Q8—4221no—
119C4×Q8—2111no—
120C4×Q8—2111no—
121#117—4221no—
122C2×C42—4221no—
123C2×C42—2111no—
124C4⋊C8—4221no—
125C22⋊C8—2111no—
126C22⋊C8C2×D21111yescharacteristic
127#97—4221no—
128#95—4221no—
129#95—2111no—
130#95C2×D21111yescharacteristic
131C2×Q8—4221no—
132C2×Q8—2111no—
133C2×Q8C4×D21111yescharacteristic
134C22×C4—21no—
135C22×C4—4221no—
136C22×C4—2111no—
137C22×C4—2111no—
138C22×C4C22×D21111yescharacteristic
139C2×C8—21no—
140C2×C8—4221no—
141C4⋊C4—21no—
142C4⋊C4—4221no—
143C4⋊C4—4221no—
144C4⋊C4—4221no—
145C4⋊C4—4221no—
146C4⋊C4—4221no—
147C4⋊C4—4221no—
148C4⋊C4—2111no—
149C4⋊C4C4×D21111yescharacteristic
150C22⋊C4—21no—
151C22⋊C4—4221no—
152C42—4224no—
153C42—4221no—
154C42—4221no—
155C42—2111no—
156C42—2111no—
157C42—2111no—
158C42—2111no—
159C23D4×D21111yescharacteristic
160Q8—41no—
161Q8—8441no—
162Q8—8442no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—8441no—
170C2×C4—8441no—
171C2×C4—4222no—
172C2×C4—4221no—
173C2×C4—4221no—
174C2×C4—4221no—
175C2×C4—4221no—
176C2×C4—4222no—
177C2×C4—2111no—
178C2×C4—2111no—
179C2×C4—2111no—
180C2×C4—2111no—
181C2×C4C2×C4×D21111yescharacteristic
182C2×C4D4×D21111yescharacteristic
183C8—41no—
184C8—8441no—
185C22—21no—
186C22—21no—
187C22C22⋊C4×D21111yescharacteristic
188C22#2027000611yescharacteristic
189C22#2027000911yescharacteristic
190C4—21no—
191C4—21no—
192C4—42no—
193C4—41no—
194C4—41no—
195C4—8442no—
196C4—8441no—
197C4—8441no—
198C4—4221no—
199C4—4221no—
200C4—4221no—
201C4—2112no—
202C2—21no—
203C2—21no—
204C2#2721311811yescharacteristic
205C2#2721315311yescharacteristic
206C2#104×D21111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#328×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenter#328D211
hypocenterC211#1506
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
8. #20269976C4semidirect
9. #20269975C4semidirect
13. Q8×Dic211C4semidirect
15. #20269942C4semidirect
103. C211#1506semidirect

Maximal subgroups

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

subgroupindex
#272136982
#272132002
#272129912
#272128842
#272128832
#272128822
#328×C2112
#1506211