Finite Group Databasebeta

← #68101

Subgroups — #68101

Order 384.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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#4718C212yesmaximal, complemented
3C3×#323C211yesmaximal, characteristic, complemented
4#4525C211yesmaximal, characteristic, complemented
5#4468C212yesmaximal, complemented
6#1460—31nomaximal, Sylow
7C22×#67C411yescharacteristic
8C23×Dic3C411yescharacteristic
9C22×C4×D3C411yescharacteristic
10C2×C3×C22⋊C4C2211yescharacteristic
11C2×(C2×D3)⋊C4C2211yescharacteristic
12C2×Dic3⋊C4C411yescharacteristic
13C2×C4×Dic3C2211yescharacteristic
14#67⋊C4—22no—
15#67⋊C4—22no—
16C3×C22⋊C8C2212yes—
17#830C2212yes—
18Dic3⋊C8—22no—
19C8×Dic3—22no—
20C2×C4×D4—31no—
21#431—32no—
22#323—31no—
23#323—32no—
24#323D311yescharacteristic, complemented
25C24×C3C2×C411yescharacteristic
26C23×D3C2×C411yescharacteristic
27C22×C3×C4C2×C411yescharacteristic
28C22×C3×C4C2311yescharacteristic
29C2×#67—22no—
30C2×#67—22no—
31C2×#67C814yescomplemented
32C22×Dic3—21no—
33C22×Dic3—21no—
34C22×Dic3—42no—
35C22×Dic3C2×C411yescharacteristic
36C22×Dic3C2×C411yescharacteristic
37C22×Dic3C2×C411yescharacteristic
38C2×C4×D3—21no—
39C2×C4×D3—21no—
40C2×C4×D3—42no—
41C2×C3×C8—22no—
42C3×C22⋊C4—22no—
43(C2×D3)⋊C4—22no—
44Dic3⋊C4—22no—
45C4×Dic3D412yes—
46C4×Dic3D412yes—
47C2×C3⋊C8—22no—
48C22×D4—31no—
49C23×C4—31no—
50C23×C4—31no—
51C4×D4—62no—
52C4×D4—62no—
53C2×C4⋊C4—31no—
54C2×C22⋊C4—31no—
55C2×C22⋊C4C2×D311yescharacteristic
56C2×C42—31no—
57C4⋊C8—62no—
58C22⋊C8—32no—
59C22⋊C8C2×D312yes—
60C4×C8—62no—
61C23×C3—21no—
62C23×C3—21no—
63C23×C3—41no—
64C23×C3—41no—
65C23×C3C2×C812yes—
66C23×C3C22×C411yescharacteristic
67C22×D3—21no—
68C22×D3—21no—
69C22×D3—41no—
70C22×D3—41no—
71C22×D3C2×C812yes—
72C2×C3×C4—21no—
73C2×C3×C4—41no—
74C2×C3×C4—41no—
75C2×C3×C4C2×D412yes—
76#67—28no—
77C2×Dic3—22no—
78C2×Dic3—21no—
79C2×Dic3—42no—
80C2×Dic3—42no—
81C2×Dic3—41no—
82C2×Dic3C22⋊C411yescharacteristic
83C2×Dic3C22⋊C412yes—
84C2×Dic3C22⋊C411yescharacteristic
85C2×Dic3C2×C812yes—
86C2×Dic3M1612yes—
87C4×D3—42no—
88C4×D3—42no—
89C3×C8—42no—
90C3⋊C8—42no—
91C24—31no—
92C24C4×D311yescharacteristic
93C2×D4—34no—
94C2×D4—62no—
95C2×D4—62no—
96C22×C4—31no—
97C22×C4—31no—
98C22×C4—31no—
99C22×C4—61no—
100C22×C4—61no—
101C22×C4—61no—
102C22×C4—61no—
103C22×C4—122no—
104C22×C4—122no—
105C22×C4C4×D311yescharacteristic
106C22×C4C22×D311yescharacteristic
107C2×C8—22no—
108C2×C8—62no—
109C4⋊C4—62no—
110C22⋊C4—22no—
111C22⋊C4—62no—
112C42—32no—
113C42—32no—
114C22×C3—21no—
115C22×C3—21no—
116C22×C3—22no—
117C22×C3—42no—
118C22×C3—42no—
119C22×C3—41no—
120C22×C3—41no—
121C22×C3C2×C22⋊C411yescharacteristic
122C22×C3C22×C811yescharacteristic
123C22×C3C2×M1611yescharacteristic
124C2×D3—22no—
125C2×D3—42no—
126C2×D3—41no—
127C2×D3—41no—
128C2×D3—42no—
129C3×C4—22no—
130C3×C4—41no—
131Dic3—21no—
132Dic3—41no—
133Dic3C22⋊C814yes—
134Dic3#10012yes—
135C23—21no—
136C23—21no—
137C23—32no—
138C23—41no—
139C23—41no—
140C23—61no—
141C23—61no—
142C23—121no—
143C23—121no—
144C23D3×C812yes—
145C23C2×C4×D311yescharacteristic
146D4—68no—
147C2×C4—21no—
148C2×C4—32no—
149C2×C4—31no—
150C2×C4—32no—
151C2×C4—32no—
152C2×C4—31no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—62no—
156C2×C4—61no—
157C2×C4—122no—
158C2×C4—122no—
159C2×C4—122no—
160C2×C4—122no—
161C2×C4—121no—
162C2×C4D3×D412yes—
163C8—42no—
164C8—122no—
165C2×C3—21no—
166C2×C3—21no—
167C2×C3—42no—
168C2×C3C2×C22⋊C811yescharacteristic
169C2×C3#41011yescharacteristic
170C2×C3C2×#10011yescharacteristic
171D3—42no—
172C22—21no—
173C22—21no—
174C22—22no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22—62no—
180C22—122no—
181C22—122no—
182C22—121no—
183C22—121no—
184C22D3×C22⋊C411yescharacteristic
185C22C2×D3×C811yescharacteristic
186C22D3×M1611yescharacteristic
187C4—22no—
188C4—34no—
189C4—32no—
190C4—41no—
191C4—61no—
192C4—121no—
193C3#146011yescharacteristic, complemented, Sylow
194C2—21no—
195C2—21no—
196C2—42no—
197C2—122no—
198C2D3×C22⋊C811yescharacteristic
199C2#474011yescharacteristic
200C2D3×#10011yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC22×C3C22×C8
FrattiniC22×C4C22×D3
FittingC3×#323C2
socleC22×C3C2×C22⋊C4
prime radicalC22×#67C4
cosocleC22×C3×C4C23
hypercenter#323D3
hypocenterC3#1460
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #4718C2semidirect
3. C3×#323C2semidirect
4. #4525C2semidirect
5. #4468C2semidirect
24. #323D3semidirect
31. C2×#67C8semidirect
193. C3#1460semidirect

Maximal subgroups

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

subgroupindex
C2×#67⋊C42
#47182
C3×#3232
#45252
#44682
#14603