Finite Group Databasebeta

← #68121

Subgroups — #68121

Order 384.

316 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 69 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×#67⋊C4C211yesmaximal, characteristic
2#4726C212yesmaximal, complemented
3C3×#323C211yesmaximal, characteristic, complemented
4#4525C211yesmaximal, characteristic, complemented
5#4469C212yesmaximal, complemented
6#1465—31nomaximal, Sylow
7C22×#67C411yescharacteristic
8C23×Dic3C411yescharacteristic
9C22×C4×D3C2211yescharacteristic
10C2×C3×C22⋊C4C2211yescharacteristic
11C2×(C2×D3)⋊C4C411yescharacteristic
12C2×Dic3⋊C4C2211yescharacteristic
13C2×C4×Dic3C411yescharacteristic
14C2×ΓL(1,𝔽25)—22no—
15#67⋊C4—22no—
16#67⋊C4—22no—
17C3×C22⋊C8C2212yes—
18#830C2212yes—
19(C2×D3)⋊C8—22no—
20C2×C4×D4—31no—
21#407—32no—
22#324—32no—
23#323—31no—
24#323D311yescharacteristic, complemented
25C24×C3C2×C411yescharacteristic
26C23×D3C2×C411yescharacteristic
27C22×C3×C4C2×C411yescharacteristic
28C22×C3×C4C2311yescharacteristic
29C2×#67—22no—
30C2×#67—22no—
31C2×#67—22no—
32C22×Dic3—21no—
33C22×Dic3—21no—
34C22×Dic3—42no—
35C22×Dic3C2×C411yescharacteristic
36C22×Dic3C2×C411yescharacteristic
37C22×Dic3C2×C411yescharacteristic
38C2×C4×D3—42no—
39C2×C4×D3D412yes—
40C2×C4×D3D412yes—
41C2×C3×C8—22no—
42C3×C22⋊C4—22no—
43(C2×D3)⋊C4—22no—
44Dic3⋊C4—22no—
45C4×Dic3—21no—
46C4×Dic3—21no—
47C2×C3⋊C8—22no—
48ΓL(1,𝔽25)—44no—
49C22×D4—31no—
50C23×C4—31no—
51C23×C4—31no—
52C2×M16—62no—
53C4×D4—62no—
54C4×D4—62no—
55C2×C4⋊C4—31no—
56C2×C22⋊C4—31no—
57C2×C22⋊C4C2×D311yescharacteristic
58C2×C42—31no—
59C22⋊C8—32no—
60C22⋊C8—62no—
61C22⋊C8C2×D312yes—
62C23×C3—21no—
63C23×C3—21no—
64C23×C3—41no—
65C23×C3—41no—
66C23×C3M1612yescomplemented
67C23×C3C22×C411yescharacteristic
68C22×D3—21no—
69C22×D3—21no—
70C22×D3—41no—
71C22×D3—41no—
72C22×D3C22⋊C412yes—
73C2×C3×C4—21no—
74C2×C3×C4—41no—
75C2×C3×C4—41no—
76C2×C3×C4C2×D412yes—
77#67—28no—
78C2×Dic3—21no—
79C2×Dic3—21no—
80C2×Dic3—22no—
81C2×Dic3—21no—
82C2×Dic3—42no—
83C2×Dic3—42no—
84C2×Dic3—41no—
85C2×Dic3C22⋊C411yescharacteristic
86C2×Dic3C22⋊C411yescharacteristic
87C2×Dic3M1612yescomplemented
88C4×D3—24no—
89C4×D3—42no—
90C3×C8—42no—
91C3⋊C8—42no—
92C24—31no—
93C24C4×D311yescharacteristic
94C2×D4—62no—
95C2×D4—62no—
96C2×D4—62no—
97C22×C4—31no—
98C22×C4—31no—
99C22×C4—32no—
100C22×C4—32no—
101C22×C4—31no—
102C22×C4—61no—
103C22×C4—61no—
104C22×C4—122no—
105C22×C4—122no—
106C22×C4C4×D311yescharacteristic
107C22×C4C22×D311yescharacteristic
108M16—124no—
109C2×C8—22no—
110C2×C8—62no—
111C4⋊C4—62no—
112C22⋊C4—22no—
113C22⋊C4—62no—
114C42—61no—
115C42—61no—
116C22×C3—21no—
117C22×C3—21no—
118C22×C3—22no—
119C22×C3—42no—
120C22×C3—42no—
121C22×C3—41no—
122C22×C3—41no—
123C22×C3C2×C22⋊C411yescharacteristic
124C22×C3C2×M1611yescharacteristic
125C22×C3C2×M1611yescharacteristic
126C2×D3—22no—
127C2×D3—22no—
128C2×D3—22no—
129C2×D3—42no—
130C2×D3—42no—
131C3×C4—22no—
132C3×C4—41no—
133Dic3—22no—
134Dic3—21no—
135Dic3—41no—
136Dic3#9912yes—
137C23—21no—
138C23—21no—
139C23—32no—
140C23—41no—
141C23—41no—
142C23—61no—
143C23—61no—
144C23—121no—
145C23—121no—
146C23ΓL(1,𝔽25)12yescomplemented
147C23C2×C4×D311yescharacteristic
148D4—68no—
149C2×C4—21no—
150C2×C4—31no—
151C2×C4—31no—
152C2×C4—32no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—64no—
156C2×C4—62no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C4—122no—
161C2×C4—122no—
162C2×C4—122no—
163C2×C4—121no—
164C2×C4D3×D412yes—
165C8—42no—
166C8—122no—
167C2×C3—21no—
168C2×C3—21no—
169C2×C3—42no—
170C2×C3#40711yescharacteristic
171C2×C3C2×#9911yescharacteristic
172C2×C3#41311yescharacteristic
173D3—42no—
174C22—21no—
175C22—21no—
176C22—22no—
177C22—42no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—62no—
182C22—62no—
183C22—62no—
184C22—122no—
185C22—122no—
186C22D3×C22⋊C411yescharacteristic
187C22C2×ΓL(1,𝔽25)11yescharacteristic
188C22D3×M1611yescharacteristic
189C4—22no—
190C4—32no—
191C4—41no—
192C4—62no—
193C4—61no—
194C4—121no—
195C3#146511yescharacteristic, complemented, Sylow
196C2—21no—
197C2—21no—
198C2—42no—
199C2—122no—
200C2#472611yescharacteristic
201C2D3×#9911yescharacteristic
202C2#474511yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#323C2
socleC22×C3C2×C22⋊C4
prime radicalC22×#67C4
cosocleC22×C3×C4C23
hypercenter#323D3
hypocenterC3#1465
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #4726C2semidirect
3. C3×#323C2semidirect
4. #4525C2semidirect
5. #4469C2semidirect
24. #323D3semidirect
66. C23×C3M16semidirect
87. C2×Dic3M16semidirect
146. C23ΓL(1,𝔽25)semidirect
195. C3#1465semidirect

Maximal subgroups

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

subgroupindex
C2×#67⋊C42
#47262
C3×#3232
#45252
#44692
#14653