Finite Group Databasebeta

← #27213766

Subgroups — #27213766

Order 13504.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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#20269975C211yesmaximal, characteristic, complemented
2#20269972C211yesmaximal, characteristic, complemented
3#20269970C211yesmaximal, characteristic, complemented
4#20269966C211yesmaximal, characteristic, complemented
5#20269963C211yesmaximal, characteristic, complemented
6#20269963C211yesmaximal, characteristic, complemented
7#20269962C211yesmaximal, characteristic, complemented
8#20269960C211yesmaximal, characteristic, complemented
9#20269955C211yesmaximal, characteristic, complemented
10#20269955C211yesmaximal, characteristic, complemented
11#20269950C211yesmaximal, characteristic
12#20269942C211yesmaximal, characteristic, complemented
13C4×C22⋊D211C211yesmaximal, characteristic, complemented
14#20269935C211yesmaximal, characteristic, complemented
15C22⋊Q8×C211C211yesmaximal, characteristic, complemented
16C2×Dic422C2211yescharacteristic
17C2×Dic422C2211yescharacteristic
18C2×D844C2211yescharacteristic
19C2×C22⋊D211C2211yescharacteristic
20C2×C22⋊D211C2211yescharacteristic
21C2×C4×D211C2211yescharacteristic
22C2×C4×D211C2211yescharacteristic
23C4⋊Dic211C2211yescharacteristic
24Dic211⋊C4C2211yescharacteristic
25Dic211⋊C4C2211yescharacteristic
26Dic211⋊C4C2211yescharacteristic
27Dic211⋊C4C2211yescharacteristic
28Dic211⋊C4C2211yescharacteristic
29Dic211⋊C4C2211yescharacteristic
30C22⋊Dic211C2211yescharacteristic
31C22⋊Dic211C2211yescharacteristic
32(C2×D211)⋊C4C2211yescharacteristic
33(C2×D211)⋊C4C2211yescharacteristic
34(C2×D211)⋊C4C2211yescharacteristic
35(C2×D211)⋊C4C2211yescharacteristic
36(C2×D211)⋊C4C2211yescharacteristic
37(C2×D211)⋊C4C2211yescharacteristic
38(C2×D211)⋊C4C2211yescharacteristic
39(C2×D211)⋊C4C2211yescharacteristic
40C4×Dic211C2211yescharacteristic
41C4×Dic211C2211yescharacteristic
42C4×Dic211C2211yescharacteristic
43C4×Dic211C2211yescharacteristic
44C2×Q8×C211C2211yescharacteristic
45C22×C4×C211C2211yescharacteristic
46C4⋊C4×C211C2211yescharacteristic
47C4⋊C4×C211C2211yescharacteristic
48C4⋊C4×C211C2211yescharacteristic
49C22⋊C4×C211C2211yescharacteristic
50C22⋊C4×C211C2211yescharacteristic
51C22×D211C2311yescharacteristic
52C22×D211C2311yescharacteristic
53Dic422—22no—
54Dic422—41no—
55D844—41no—
56C22⋊D211—22no—
57C22⋊D211—41no—
58C2×Dic211C2311yescharacteristic
59C2×Dic211C2311yescharacteristic
60C2×Dic211C2311yescharacteristic
61C2×Dic211C2311yescharacteristic
62C2×Dic211C2311yescharacteristic
63C2×Dic211C2311yescharacteristic
64C4×D211—22no—
65C4×D211—41no—
66C23×C211C2311yescharacteristic
67Q8×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211C2311yescharacteristic
70C2×C4×C211C2311yescharacteristic
71C2×C4×C211C2311yescharacteristic
72C2×C4×C211C2311yescharacteristic
73C2×C4×C211C2311yescharacteristic
74C2×C4×C211C2311yescharacteristic
75C22×C211—21no—
76C22×C211—21no—
77C22×C211—21no—
78C22×C211C2411yescharacteristic
79C2×D211—21no—
80C2×D211—21no—
81C2×D211—21no—
82C2×D211—21no—
83C2×D211—21no—
84C2×D211—21no—
85C4×C211—21no—
86C4×C211—21no—
87C4×C211—21no—
88C4×C211—21no—
89C4×C211—21no—
90C4×C211—21no—
91Dic211—21no—
92Dic211—21no—
93Dic211—21no—
94Dic211—21no—
95Dic211—21no—
96Dic211#4112yes—
97C2×C211—41no—
98C2×C211C2×#4111yescharacteristic
99C2×C211H(5,𝔽2)11yescharacteristic
100C2×C2112−1+411yescharacteristic
101D211—41no—
102D211—41no—
103C211#54211yescharacteristic, complemented, Sylow
104#542—2111nomaximal, Sylow
105C4⋊Q8—2111no—
106#126—2111no—
107#126—2111no—
108#124—2111no—
109#124—2111no—
110#124—2111no—
111#123—2111no—
112#123—2111no—
113C22⋊Q8—2111no—
114C22⋊Q8—2111no—
115C22⋊Q8D21111yescharacteristic, complemented
116#121—2111no—
117C4×Q8—2111no—
118C4×D4—2111no—
119#117—2111no—
120C2×Q8—2111no—
121C2×Q8—2111no—
122C2×Q8C2×D21111yescharacteristic
123C2×D4—2111no—
124C2×D4—2111no—
125C2×D4—2111no—
126C22×C4—2111no—
127C22×C4—2111no—
128C22×C4C2×D21111yescharacteristic
129C4⋊C4—2111no—
130C4⋊C4—2111no—
131C4⋊C4—2111no—
132C4⋊C4—2111no—
133C4⋊C4—2111no—
134C4⋊C4—2111no—
135C4⋊C4—2111no—
136C4⋊C4C2×D21111yescharacteristic
137C4⋊C4C2×D21111yescharacteristic
138C4⋊C4C2×D21111yescharacteristic
139C22⋊C4—2111no—
140C22⋊C4—2111no—
141C22⋊C4—2111no—
142C22⋊C4—2111no—
143C22⋊C4—2111no—
144C22⋊C4—2111no—
145C22⋊C4—2111no—
146C22⋊C4—2111no—
147C22⋊C4—2111no—
148C22⋊C4—2111no—
149C22⋊C4C2×D21111yescharacteristic
150C22⋊C4C2×D21111yescharacteristic
151C42—2111no—
152C42—2111no—
153C42—2111no—
154C42—2111no—
155C23—2111no—
156C23—2111no—
157C23C22×D21111yescharacteristic
158Q8—41no—
159Q8—8441no—
160Q8—4222no—
161D4—8441no—
162D4—8441no—
163D4—4222no—
164C2×C4—41no—
165C2×C4—8441no—
166C2×C4—4222no—
167C2×C4—2111no—
168C2×C4—2111no—
169C2×C4—2111no—
170C2×C4—2111no—
171C2×C4—2111no—
172C2×C4—2111no—
173C2×C4C22×D21111yescharacteristic
174C2×C4C22×D21111yescharacteristic
175C2×C4C22×D21111yescharacteristic
176C2×C4C22×D21111yescharacteristic
177C2×C4C22×D21111yescharacteristic
178C2×C4C22×D21111yescharacteristic
179C22—21no—
180C22—21no—
181C22—21no—
182C22—4221no—
183C22—4221no—
184C22—4221no—
185C22—4221no—
186C22—4221no—
187C22—4221no—
188C22C23×D21111yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—4221no—
196C4—4221no—
197C4—4221no—
198C4—4221no—
199C4—4221no—
200C4—2112no—
201C2—41no—
202C2—8441no—
203C2—8441no—
204C2#41×D21111yescharacteristic
205C2#2027002211yescharacteristic
206C2#2027002511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D211
commutator, socle, cosocleC22×C211C24
FittingC22⋊Q8×C211C2
prime radical#20269950C2
hypercenterC22⋊Q8D211
hypocenterC211#542
LCSG ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20269975C2semidirect
2. #20269972C2semidirect
3. #20269970C2semidirect
4. #20269966C2semidirect
5. #20269963C2semidirect
6. #20269963C2semidirect
7. #20269962C2semidirect
8. #20269960C2semidirect
9. #20269955C2semidirect
10. #20269955C2semidirect
12. #20269942C2semidirect
13. C4×C22⋊D211C2semidirect
14. #20269935C2semidirect
15. C22⋊Q8×C211C2semidirect
103. C211#542semidirect
115. C22⋊Q8D211semidirect

Maximal subgroups

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

subgroupindex
#202699752
#202699722
#202699702
#202699662
#202699632
#202699622
#202699602
#202699552
#202699502
#202699422
C4×C22⋊D2112
#202699352
C22⋊Q8×C2112
#542211