Finite Group Databasebeta

← #67698

Subgroups — #67698

Order 384.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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×C3×#100C211yesmaximal, characteristic
2#5220C211yesmaximal, characteristic
3C2×#834C211yesmaximal, characteristic
4#1390—31nomaximal, Sylow
5C22×C3×D4C2211yescharacteristic
6C23×Dic3C411yescharacteristic, complemented
7C2×C3×M16—21no—
8C2×#217—21no—
9C2×#217—21no—
10C2×#217C411yescharacteristic, complemented
11C3×#100—21no—
12C3×#100C412yescomplemented
13#834—21no—
14#834C412yescomplemented
15C3⋊#95—21no—
16C2×#100D311yescharacteristic
17C2×#99—31no—
18#386—31no—
19C24×C3C2×C411yescharacteristic
20C24×C3D411yescharacteristic
21C2×C3×D4—41no—
22C2×C3×D4—41no—
23C2×C3×D4C2×C411yescharacteristic
24C2×C3×D4C2×C411yescharacteristic
25C2×C3×D4D411yescharacteristic
26C2×C3×D4Q811yescharacteristic
27C22×C3×C4D411yescharacteristic
28C22×Dic3—21no—
29C22×Dic3—21no—
30C22×Dic3—42no—
31C22×Dic3—41no—
32C22×Dic3—41no—
33C3×M16—22no—
34C3×M16—41no—
35C2×C3×C8—41no—
36#217—22no—
37#217—42no—
38#217—41no—
39#217—41no—
40C22×D4C2×D311yescharacteristic
41C23×C4—31no—
42C2×M16—21no—
43C2×C22⋊C4—31no—
44C2×C22⋊C4—61no—
45C2×C22⋊C4—61no—
46#100—21no—
47#100Dic312yescomplemented
48#99—32no—
49#99—61no—
50#95—61no—
51C23×C3—21no—
52C23×C3—21no—
53C23×C3—21no—
54C23×C3—21no—
55C23×C3—41no—
56C23×C3—41no—
57C23×C3—41no—
58C23×C3—41no—
59C23×C3C4211yescharacteristic
60C23×C3C22⋊C411yescharacteristic
61C23×C3C22⋊C411yescharacteristic
62C23×C3C4⋊C411yescharacteristic
63C23×C3C4⋊C411yescharacteristic
64C3×D4—41no—
65C3×D4—41no—
66C3×D4—41no—
67C3×D4—41no—
68C2×C3×C4—21no—
69C2×C3×C4—21no—
70C2×C3×C4C22⋊C411yescharacteristic
71C2×C3×C4C4⋊C411yescharacteristic
72C2×Dic3—22no—
73C2×Dic3—41no—
74C2×Dic3—42no—
75C2×Dic3—41no—
76C2×Dic3—42no—
77C2×Dic3—42no—
78C2×Dic3—41no—
79C2×Dic3—41no—
80C2×Dic3—41no—
81C2×Dic3—81no—
82C2×Dic3—81no—
83C3×C8—42no—
84C24C4×D311yescharacteristic
85C24D1211yescharacteristic
86C2×D4—41no—
87C2×D4—41no—
88C2×D4Dic611yescharacteristic
89C2×D4C4×D311yescharacteristic
90C2×D4C2×Dic311yescharacteristic
91C2×D4#6711yescharacteristic
92C22×C4—61no—
93C22×C4—61no—
94C22×C4—122no—
95C22×C4—121no—
96C22×C4—121no—
97C22×C4#6711yescharacteristic
98M16—22no—
99M16—41no—
100C2×C8—41no—
101C22⋊C4—62no—
102C22⋊C4—122no—
103C22⋊C4—121no—
104C22⋊C4—121no—
105C22×C3—21no—
106C22×C3—21no—
107C22×C3—21no—
108C22×C3—21no—
109C22×C3—21no—
110C22×C3—21no—
111C22×C3—41no—
112C22×C3—41no—
113C22×C3—41no—
114C22×C3—41no—
115C22×C3—42no—
116C22×C3—41no—
117C22×C3—41no—
118C22×C3—41no—
119C22×C3—41no—
120C22×C3—81no—
121C22×C3#9511yescharacteristic
122C22×C3#9911yescharacteristic
123C22×C3#9911yescharacteristic
124C3×C4—21no—
125C3×C4—21no—
126Dic3—42no—
127Dic3—81no—
128Dic3—82no—
129C23—21no—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23C4×Dic311yescharacteristic
138C23Dic3⋊C411yescharacteristic
139C23C4⋊Dic311yescharacteristic
140C23(C2×D3)⋊C411yescharacteristic
141C23(C2×D3)⋊C411yescharacteristic
142D4—41no—
143D4—41no—
144D4—41no—
145D4—41no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—62no—
149C2×C4—121no—
150C2×C4—122no—
151C2×C4—121no—
152C2×C4—122no—
153C2×C4—122no—
154C2×C4—121no—
155C2×C4—121no—
156C2×C4—121no—
157C2×C4—241no—
158C2×C4—241no—
159C2×C4Dic3⋊C411yescharacteristic
160C2×C4#21711yescharacteristic
161C8—42no—
162C2×C3—21no—
163C2×C3—21no—
164C2×C3—41no—
165C2×C3—41no—
166C2×C3—41no—
167C2×C3—41no—
168C2×C3#34211yescharacteristic
169C2×C3AΣL(1,𝔽16)11yescharacteristic
170C2×C3#35211yescharacteristic
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—81no—
187C22#80611yescharacteristic
188C22#80611yescharacteristic
189C22C3⋊#9511yescharacteristic
190C4—21no—
191C4—21no—
192C4—122no—
193C4—241no—
194C4—242no—
195C3#139011yescharacteristic, complemented, Sylow
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2—41no—
201C2—41no—
202C2#447511yescharacteristic
203C2#447611yescharacteristic
204C2#452711yescharacteristic
subgroupG/H
centerC22#806
commutatorC23×C3C42
FrattiniC22×D4C2×D3
FittingC2×C3×#100C2
socleC22×C3#99
prime radical, cosocleC22×C3×D4C22
hypercenterC2×#100D3
hypocenterC3#1390
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
6. C23×Dic3C4semidirect
10. C2×#217C4semidirect
12. C3×#100C4semidirect
14. #834C4semidirect
47. #100Dic3semidirect
195. C3#1390semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1002
#52202
C2×#8342
#13903