Finite Group Databasebeta

← #69150

Subgroups — #69150

Order 384.

286 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 55 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#5715C211yesmaximal, characteristic
2D3×#99C212yesmaximal, complemented
3C3×#353C211yesmaximal, characteristic, complemented
4#4545C211yesmaximal, characteristic, complemented
5#4473C212yesmaximal, complemented
6#2127—31nomaximal, Sylow
7C2×#237C411yescharacteristic, complemented
8C2×D3×D4C2211yescharacteristic
9C3×#127C2211yescharacteristic
10#940C411yescharacteristic, complemented
11#939—22no—
12#937—22no—
13#936C2211yescharacteristic
14D3×C22⋊C4—22no—
15#875C411yescharacteristic, complemented
16#872C411yescharacteristic, complemented
17C3×#99C2212yes—
18#834C2212yes—
19#806—22no—
20#535—31no—
21C2×#99—32no—
22#354—32no—
23#353—31no—
24#353D311yescharacteristic, complemented
25C23×D3D412yes—
26C2×C3×D4—22no—
27C2×C3×D4C2×C411yescharacteristic, complemented
28C2×C3×D4C2311yescharacteristic
29C2×#67—21no—
30C2×#67—22no—
31C2×#67D412yes—
32C22×Dic3—21no—
33#237—42no—
34D3×D4—42no—
35C2×D12C2×C411yescharacteristic
36C2×C4×D3—42no—
37C2×C4×D3C2×C411yescharacteristic
38C2×Dic6C2×C411yescharacteristic, complemented
39C3×C22⋊C4—22no—
40C3×C42C2×C411yescharacteristic, complemented
41#217—22no—
42#217—22no—
43(C2×D3)⋊C4—21no—
44(C2×D3)⋊C4—22no—
45(C2×D3)⋊C4—42no—
46Dic3⋊C4—21no—
47C4×Dic3C2×C411yescharacteristic, complemented
48C2×#41—31no—
49C22×D4—31no—
50#127—31no—
51#127C2×D311yescharacteristic
52#124—31no—
53#124—31no—
54#121—62no—
55#120—62no—
56#117—31no—
57C2×C22⋊C4—62no—
58#99—32no—
59#99—62no—
60#99C2×D312yes—
61C23×C3—21no—
62C23×C3C2×D412yes—
63C22×D3—22no—
64C22×D3—22no—
65C22×D3—82no—
66C22×D3C22⋊C412yes—
67C22×D3C22⋊C411yescharacteristic
68C3×D4—41no—
69C3×D4—41no—
70C3×D4—82no—
71C2×C3×C4—21no—
72C2×C3×C4—42no—
73C2×C3×C4C22×C411yescharacteristic
74#67—42no—
75#67—42no—
76#67—82no—
77C2×Dic3—21no—
78C2×Dic3—21no—
79C2×Dic3—42no—
80C2×Dic3—42no—
81C2×Dic3C22⋊C411yescharacteristic
82D12—41no—
83C4×D3—22no—
84C4×D3—84no—
85Dic6—41no—
86C24—32no—
87#41—122no—
88C2×Q8—31no—
89C2×D4—22no—
90C2×D4—32no—
91C2×D4—31no—
92C2×D4—61no—
93C2×D4—62no—
94C2×D4—122no—
95C2×D4C4×D311yescharacteristic, complemented
96C2×D4C22×D311yescharacteristic
97C22×C4—31no—
98C22×C4—61no—
99C22×C4—122no—
100C4⋊C4—61no—
101C22⋊C4—22no—
102C22⋊C4—61no—
103C22⋊C4—62no—
104C22⋊C4—62no—
105C22⋊C4—62no—
106C22⋊C4—122no—
107C42—31no—
108C42C4×D311yescharacteristic, complemented
109C22×C3—22no—
110C22×C3—42no—
111C22×C3—42no—
112C22×C3—41no—
113C22×C3C2×C22⋊C411yescharacteristic
114C2×D3—22no—
115C2×D3—41no—
116C2×D3—44no—
117C2×D3—42no—
118C2×D3—82no—
119C2×D3#9912yes—
120C3×C4—21no—
121C3×C4—41no—
122C3×C4—82no—
123Dic3—21no—
124Dic3—41no—
125Dic3—41no—
126Dic3—82no—
127C23—21no—
128C23—31no—
129C23—32no—
130C23—62no—
131C23—62no—
132C23—242no—
133C23D3×D412yes—
134Q8—121no—
135D4—41no—
136D4—41no—
137D4—82no—
138D4—121no—
139D4—122no—
140D4—122no—
141D4—242no—
142C2×C4—21no—
143C2×C4—31no—
144C2×C4—42no—
145C2×C4—62no—
146C2×C4—61no—
147C2×C4—61no—
148C2×C4—122no—
149C2×C4—122no—
150C2×C4—244no—
151C2×C4C2×C4×D311yescharacteristic
152C2×C3—21no—
153C2×C3—42no—
154C2×C3—81no—
155C2×C3C2×#9911yescharacteristic
156D3—22no—
157D3—42no—
158C22—22no—
159C22—32no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22—62no—
164C22—121no—
165C22—122no—
166C22—124no—
167C22—242no—
168C22D3×C22⋊C411yescharacteristic
169C4—21no—
170C4—41no—
171C4—61no—
172C4—82no—
173C4—121no—
174C4—121no—
175C4—242no—
176C3#212711yescharacteristic, complemented, Sylow
177C2—21no—
178C2—42no—
179C2—62no—
180C2—81no—
181C2—122no—
182C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC2×C3×C4C22×C4
FrattiniC2×D4C22×D3
FittingC3×#353C2
socleC2×C3C2×#99
prime radical#5715C2
cosocleC2×C3×D4C23
hypercenter#353D3
hypocenterC3#2127
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. D3×#99C2semidirect
3. C3×#353C2semidirect
4. #4545C2semidirect
5. #4473C2semidirect
7. C2×#237C4semidirect
10. #940C4semidirect
15. #875C4semidirect
16. #872C4semidirect
24. #353D3semidirect
27. C2×C3×D4C2×C4semidirect
38. C2×Dic6C2×C4semidirect
40. C3×C42C2×C4semidirect
47. C4×Dic3C2×C4semidirect
95. C2×D4C4×D3semidirect
108. C42C4×D3semidirect
176. C3#2127semidirect

Maximal subgroups

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

subgroupindex
#57152
D3×#992
C3×#3532
#45452
#44732
#21273