Finite Group Databasebeta

← #69431

Subgroups — #69431

Order 384.

238 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 73 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#5479C211yesmaximal, characteristic
2C3×#376C211yesmaximal, characteristic, complemented
3#4925C211yesmaximal, characteristic, complemented
4#4452C212yesmaximal, complemented
5#4451C212yesmaximal
6#1500—31nomaximal, Sylow
7C2×C3×C42—21no—
8C2×C3×C42C2211yescharacteristic
9#930C2211yescharacteristic
10#929—21no—
11C4×#67—21no—
12#924C2211yescharacteristic
13#876—21no—
14#875C411yescharacteristic, complemented
15C4×D12C411yescharacteristic, complemented
16#872—21no—
17#870C411yescharacteristic, complemented
18C4×Dic6C411yescharacteristic, complemented
19C3×#95—21no—
20C3×#95C2212yes—
21#830C2212yes—
22C4⋊(C3⋊C8)—21no—
23C4×C3⋊C8—21no—
24#529—31no—
25#431—31no—
26#376D311yescharacteristic, complemented
27#328—32no—
28#327—32no—
29C22×C3×C4—22no—
30C22×C3×C4D411yescharacteristic
31C22×C3×C4D411yescharacteristic
32C22×C3×C4C2311yescharacteristic
33C2×#67—21no—
34C2×D12C2×C411yescharacteristic
35C2×C4×D3—21no—
36C2×Dic6C2×C411yescharacteristic
37C3×C42—24no—
38C3×C42C2×C411yescharacteristic
39C3×C42C2×C411yescharacteristic
40C3×C42D411yescharacteristic
41C3×C42D411yescharacteristic
42#217—21no—
43(C2×D3)⋊C4—21no—
44(C2×D3)⋊C4—21no—
45(C2×D3)⋊C4—21no—
46(C2×D3)⋊C4—21no—
47C4⋊Dic3C2×C411yescharacteristic
48C4⋊Dic3C2×C411yescharacteristic
49Dic3⋊C4—21no—
50Dic3⋊C4—21no—
51Dic3⋊C4—21no—
52Dic3⋊C4—21no—
53C4×Dic3—21no—
54C2×C3⋊C8—22no—
55#126—61no—
56#125—31no—
57#124—31no—
58#123—61no—
59C22⋊Q8—31no—
60#121—31no—
61C4×Q8—31no—
62C4×D4—31no—
63C4×D4—61no—
64#117—61no—
65C2×C42—21no—
66C2×C42C2×D311yescharacteristic
67C4⋊C8—61no—
68C22⋊C8—32no—
69C4×C8—61no—
70#95—21no—
71#95C2×D312yes—
72C23×C3C2×D411yescharacteristic
73C22×D3—21no—
74C2×C3×C4—22no—
75C2×C3×C4—22no—
76C2×C3×C4—24no—
77C2×C3×C4—41no—
78C2×C3×C4—42no—
79C2×C3×C4—42no—
80C2×C3×C4C22⋊C411yescharacteristic
81C2×C3×C4C22⋊C411yescharacteristic
82C2×C3×C4C22⋊C411yescharacteristic
83C2×C3×C4C22⋊C411yescharacteristic
84C2×C3×C4C22×C411yescharacteristic
85C2×C3×C4C2×D411yescharacteristic
86#67—42no—
87C2×Dic3—21no—
88C2×Dic3—21no—
89C2×Dic3—21no—
90D12—41no—
91C4×D3—42no—
92Dic6—41no—
93C3⋊C8—42no—
94C2×Q8—31no—
95C2×D4—31no—
96C2×D4—61no—
97C22×C4—22no—
98C22×C4—61no—
99C22×C4#6711yescharacteristic
100C22×C4#6711yescharacteristic
101C22×C4C22×D311yescharacteristic
102C2×C8—62no—
103C4⋊C4—31no—
104C4⋊C4—31no—
105C4⋊C4—61no—
106C4⋊C4—61no—
107C4⋊C4—61no—
108C4⋊C4—61no—
109C22⋊C4—61no—
110C22⋊C4—61no—
111C22⋊C4—61no—
112C22⋊C4—61no—
113C22⋊C4—61no—
114C42—24no—
115C42—61no—
116C42C4×D311yescharacteristic
117C42C4×D311yescharacteristic
118C42D1211yescharacteristic
119C42D1211yescharacteristic
120C22×C3—21no—
121C22×C3—21no—
122C22×C3C2×C22⋊C411yescharacteristic
123C22×C3#13512yes—
124C2×D3—41no—
125C2×D3—41no—
126C2×D3—41no—
127C3×C4—21no—
128C3×C4—21no—
129C3×C4—21no—
130C3×C4—21no—
131C3×C4—21no—
132C3×C4—44no—
133C3×C4#10412yes—
134Dic3—41no—
135Dic3—41no—
136Dic3—41no—
137C23—61no—
138C23C2×#6711yescharacteristic
139Q8—121no—
140D4—121no—
141D4—122no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—61no—
149C2×C4—61no—
150C2×C4—61no—
151C2×C4—122no—
152C2×C4(C2×D3)⋊C411yescharacteristic
153C2×C4(C2×D3)⋊C411yescharacteristic
154C2×C4(C2×D3)⋊C411yescharacteristic
155C2×C4(C2×D3)⋊C411yescharacteristic
156C2×C4C2×C4×D311yescharacteristic
157C2×C4C2×D1211yescharacteristic
158C8—122no—
159C2×C3—22no—
160C2×C3#41011yescharacteristic
161C2×C3#41611yescharacteristic
162C2×C3C2×#10411yescharacteristic
163D3—81no—
164C22—21no—
165C22—21no—
166C22—121no—
167C22—121no—
168C22—121no—
169C22C2×(C2×D3)⋊C411yescharacteristic
170C22#95012yes—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—44no—
177C4—121no—
178C4—121no—
179C4—121no—
180C4#80512yes—
181C3#150011yescharacteristic, complemented, Sylow
182C2—22no—
183C2—241no—
184C2C2×#80511yescharacteristic
185C2#500211yescharacteristic
186C2#500611yescharacteristic
subgroupG/H
centerC2×C4(C2×D3)⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#376C2
socleC22×C3C2×C22⋊C4
prime radical#930C22
cosocleC22×C3×C4C23
hypercenter#376D3
hypocenterC3#1500
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
2. C3×#376C2semidirect
3. #4925C2semidirect
4. #4452C2semidirect
14. #875C4semidirect
15. C4×D12C4semidirect
17. #870C4semidirect
18. C4×Dic6C4semidirect
26. #376D3semidirect
181. C3#1500semidirect

Maximal subgroups

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

subgroupindex
#54792
C3×#3762
#49252
#44522
#44512
#15003