Finite Group Databasebeta

← #5712

Subgroups — #5712

Order 192.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C3×#126C211yesmaximal, characteristic, complemented
2#898C211yesmaximal, characteristic, complemented
3#897C211yesmaximal, characteristic, complemented
4#896C211yesmaximal, characteristic, complemented
5#890C211yesmaximal, characteristic, complemented
6#889C211yesmaximal, characteristic, complemented
7#888C211yesmaximal, characteristic, complemented
8#886C211yesmaximal, characteristic
9#885C211yesmaximal, characteristic
10#883C211yesmaximal, characteristic
11#879C211yesmaximal, characteristic, complemented
12#878C211yesmaximal, characteristic, complemented
13#878C211yesmaximal, characteristic, complemented
14#876C211yesmaximal, characteristic, complemented
15#869C211yesmaximal, characteristic, complemented
16#563—31nomaximal, Sylow
17C2×#67C2211yescharacteristic
18C22×Dic3C2211yescharacteristic
19C2×C4×D3C2211yescharacteristic
20C2×Dic6C2211yescharacteristic
21C2×Dic6C2211yescharacteristic
22C2×Dic6C2211yescharacteristic
23C3×C4⋊C4C2211yescharacteristic
24C3×C4⋊C4C2211yescharacteristic
25C3×C4⋊C4C2211yescharacteristic
26C3×C22⋊C4C2211yescharacteristic
27C3×C22⋊C4C2211yescharacteristic
28C3×C22⋊C4C2211yescharacteristic
29C3×C42C2211yescharacteristic
30#217C2211yescharacteristic
31#217C2211yescharacteristic
32(C2×D3)⋊C4C2211yescharacteristic
33(C2×D3)⋊C4C2211yescharacteristic
34(C2×D3)⋊C4C2211yescharacteristic
35(C2×D3)⋊C4C2211yescharacteristic
36(C2×D3)⋊C4C2211yescharacteristic
37C4⋊Dic3C2211yescharacteristic
38C4⋊Dic3C2211yescharacteristic
39C4⋊Dic3C2211yescharacteristic
40C4⋊Dic3C2211yescharacteristic
41C4⋊Dic3C2211yescharacteristic
42C4⋊Dic3C2211yescharacteristic
43Dic3⋊C4C2211yescharacteristic
44Dic3⋊C4C2211yescharacteristic
45Dic3⋊C4C2211yescharacteristic
46Dic3⋊C4C2211yescharacteristic
47Dic3⋊C4C2211yescharacteristic
48Dic3⋊C4C2211yescharacteristic
49Dic3⋊C4C2211yescharacteristic
50C4×Dic3C2211yescharacteristic
51C4×Dic3C2211yescharacteristic
52C4⋊Q8—31no—
53C4⋊Q8—31no—
54#126—31no—
55#126—31no—
56#126—31no—
57#126D311yescharacteristic, complemented
58#125—31no—
59#125—31no—
60#124—31no—
61#123—31no—
62#123—31no—
63C22⋊Q8—31no—
64C22⋊Q8—31no—
65C22⋊Q8—31no—
66C22⋊Q8—31no—
67C23×C3C2311yescharacteristic
68C22×D3C2311yescharacteristic
69C2×C3×C4C2311yescharacteristic
70C2×C3×C4C2311yescharacteristic
71C2×C3×C4C2311yescharacteristic
72C2×C3×C4C2311yescharacteristic
73C2×C3×C4C2311yescharacteristic
74C2×C3×C4C2311yescharacteristic
75#67—41no—
76C2×Dic3—41no—
77C2×Dic3C2311yescharacteristic
78C2×Dic3C2311yescharacteristic
79C2×Dic3C2311yescharacteristic
80C2×Dic3C2311yescharacteristic
81C2×Dic3C2311yescharacteristic
82C2×Dic3C2311yescharacteristic
83C2×Dic3C2311yescharacteristic
84C4×D3—41no—
85Dic6—41no—
86Dic6—41no—
87Dic6—41no—
88C2×Q8—31no—
89C2×Q8—31no—
90C2×Q8—31no—
91C2×D4—31no—
92C22×C4—31no—
93C22×C4—31no—
94C4⋊C4—31no—
95C4⋊C4—31no—
96C4⋊C4—31no—
97C4⋊C4—31no—
98C4⋊C4—31no—
99C4⋊C4—31no—
100C4⋊C4—31no—
101C4⋊C4—31no—
102C4⋊C4—31no—
103C4⋊C4—31no—
104C4⋊C4—31no—
105C4⋊C4—31no—
106C4⋊C4—31no—
107C4⋊C4C2×D311yescharacteristic
108C4⋊C4C2×D311yescharacteristic
109C4⋊C4C2×D311yescharacteristic
110C22⋊C4—31no—
111C22⋊C4—31no—
112C22⋊C4—31no—
113C22⋊C4—31no—
114C22⋊C4—31no—
115C22⋊C4—31no—
116C22⋊C4—31no—
117C22⋊C4C2×D311yescharacteristic
118C22⋊C4C2×D311yescharacteristic
119C22⋊C4C2×D311yescharacteristic
120C42—31no—
121C42—31no—
122C42C2×D311yescharacteristic
123C22×C3—21no—
124C22×C3—21no—
125C22×C3—21no—
126C22×C3C2411yescharacteristic
127C2×D3—21no—
128C2×D3—21no—
129C2×D3—21no—
130C3×C4—21no—
131C3×C4—21no—
132C3×C4—21no—
133C3×C4—21no—
134C3×C4—21no—
135C3×C4—21no—
136Dic3—21no—
137Dic3—21no—
138Dic3—21no—
139Dic3—21no—
140Dic3—21no—
141Dic3—21no—
142Dic3—21no—
143C23—31no—
144C23C22×D311yescharacteristic
145Q8—121no—
146Q8—121no—
147Q8—121no—
148D4—121no—
149C2×C4—31no—
150C2×C4—31no—
151C2×C4—31no—
152C2×C4—31no—
153C2×C4—31no—
154C2×C4—31no—
155C2×C4—31no—
156C2×C4—121no—
157C2×C4—121no—
158C2×C4C22×D311yescharacteristic
159C2×C4C22×D311yescharacteristic
160C2×C4C22×D311yescharacteristic
161C2×C4C22×D311yescharacteristic
162C2×C4C22×D311yescharacteristic
163C2×C4C22×D311yescharacteristic
164C2×C3—41no—
165C2×C3H(5,𝔽2)11yescharacteristic
166C2×C32−1+411yescharacteristic
167C2×C32−1+411yescharacteristic
168D3—41no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—61no—
173C22—61no—
174C22—61no—
175C22C23×D311yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—61no—
183C4—61no—
184C4—61no—
185C4—61no—
186C4—61no—
187C4—61no—
188C4—61no—
189C3#56311yescharacteristic, complemented, Sylow
190C2—41no—
191C2—121no—
192C2#100911yescharacteristic
193C2#101011yescharacteristic
194C2#101011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3
commutator, socle, cosocleC22×C3C24
FittingC3×#126C2
prime radicalC2×#67C22
hypercenter#126D3
hypocenterC3#563
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C3×#126C2semidirect
2. #898C2semidirect
3. #897C2semidirect
4. #896C2semidirect
5. #890C2semidirect
6. #889C2semidirect
7. #888C2semidirect
11. #879C2semidirect
12. #878C2semidirect
13. #878C2semidirect
14. #876C2semidirect
15. #869C2semidirect
57. #126D3semidirect
189. C3#563semidirect

Maximal subgroups

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

subgroupindex
C3×#1262
#8982
#8972
#8962
#8902
#8892
#8882
#8862
#8852
#8832
#8792
#8782
#8762
#8692
#5633