Finite Group Databasebeta

← #68372

Subgroups — #68372

Order 384.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#5699C211yesmaximal, characteristic
2#5064C211yesmaximal, characteristic, complemented
3#4715C211yesmaximal, characteristic
4#4702C211yesmaximal, characteristic, complemented
5C3×#330C211yesmaximal, characteristic, complemented
6#4465C211yesmaximal, characteristic, complemented
7#4464C211yesmaximal, characteristic
8#1660—31nomaximal, Sylow
9C3×#125C2211yescharacteristic
10C2×Dic12—21no—
11C2×#204—21no—
12#898—21no—
13#898—21no—
14#897—21no—
15#896—21no—
16#894—21no—
17#888—21no—
18#875C2211yescharacteristic
19#870C2211yescharacteristic
20#869C2211yescharacteristic
21C3×#97C2211yescharacteristic
22C3×#97C2211yescharacteristic
23#818—21no—
24#817—21no—
25#808—21no—
26#807—21no—
27#803C2211yescharacteristic
28#563—31no—
29#501—31no—
30#501—31no—
31#497—31no—
32#330—31no—
33#330D311yescharacteristic, complemented
34#329—31no—
35C2×D12D411yescharacteristic
36C2×C4×D3—21no—
37C2×Dic6—21no—
38C2×Dic6—21no—
39C2×Dic6D411yescharacteristic
40C2×C3×C8—21no—
41C2×C3×C8—21no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4—21no—
44C3×C4⋊C4D411yescharacteristic
45C3×C4⋊C4D411yescharacteristic
46C3×C42C2311yescharacteristic
47(C2×D3)⋊C4—21no—
48(C2×D3)⋊C4—21no—
49(C2×D3)⋊C4—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52C4⋊Dic3—21no—
53C4⋊Dic3—21no—
54C4⋊Dic3—21no—
55C4⋊Dic3D411yescharacteristic
56C4⋊Dic3D411yescharacteristic
57Dic3⋊C4—21no—
58Dic3⋊C4—21no—
59Dic3⋊C4—21no—
60C4×Dic3—21no—
61C2×C3⋊C8—21no—
62Dic12—42no—
63#204—42no—
64C2×Q16—61no—
65C2×SD16—61no—
66C4⋊Q8—31no—
67C4⋊Q8—61no—
68#126—61no—
69#126—61no—
70#125—31no—
71#125C2×D311yescharacteristic
72#124—31no—
73#123—61no—
74C22⋊Q8—61no—
75C22⋊Q8—61no—
76#107—61no—
77#107—61no—
78#106—61no—
79#106—61no—
80#97—31no—
81#97C2×D311yescharacteristic
82#97C2×D311yescharacteristic
83C22×D3—21no—
84C2×C3×C4—21no—
85C2×C3×C4—21no—
86C2×C3×C4C2×D411yescharacteristic
87C2×C3×C4C2×D411yescharacteristic
88C2×C3×C4C2×D411yescharacteristic
89C2×Dic3—21no—
90C2×Dic3—21no—
91C2×Dic3—21no—
92C2×Dic3—41no—
93D12—41no—
94C4×D3—81no—
95Dic6—41no—
96Dic6—41no—
97Dic6—41no—
98Dic6—81no—
99C3×C8—22no—
100C3×C8—22no—
101C3⋊C8—22no—
102C2×Q8—31no—
103C2×Q8—61no—
104C2×Q8—61no—
105C2×D4—31no—
106C22×C4—61no—
107Q16—122no—
108SD16—122no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—61no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—31no—
115C4⋊C4—31no—
116C4⋊C4—61no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4D1211yescharacteristic
123C4⋊C4D1211yescharacteristic
124C22⋊C4—61no—
125C22⋊C4—61no—
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C42—61no—
130C42C22×D311yescharacteristic
131C22×C3#12011yescharacteristic
132C2×D3—41no—
133C2×D3—41no—
134C2×D3—41no—
135C3×C4—21no—
136C3×C4—21no—
137C3×C4—21no—
138C3×C4—41no—
139C3×C4—41no—
140Dic3—41no—
141Dic3—41no—
142Dic3—41no—
143Dic3—81no—
144C23—61no—
145Q8—121no—
146Q8—121no—
147Q8—121no—
148Q8—241no—
149D4—121no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—61no—
153C2×C4—61no—
154C2×C4—61no—
155C2×C4—121no—
156C2×C4—241no—
157C2×C4C2×D1211yescharacteristic
158C2×C4D3×D411yescharacteristic
159C2×C4D3×D411yescharacteristic
160C8—22no—
161C8—22no—
162C8—62no—
163C2×C3#45311yescharacteristic
164C2×C3#45611yescharacteristic
165C2×C3#45611yescharacteristic
166D3—81no—
167C22—121no—
168C22—121no—
169C22—121no—
170C22#88211yescharacteristic
171C4—21no—
172C4—21no—
173C4—21no—
174C4—41no—
175C4—41no—
176C4—121no—
177C4—121no—
178C4—121no—
179C4—241no—
180C3#166011yescharacteristic, complemented, Sylow
181C2—241no—
182C2#475211yescharacteristic
183C2#482211yescharacteristic
184C2#482611yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#330C2
socleC22×C3#120
prime radicalC2×D12D4
hypercenter#330D3
hypocenterC3#1660
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #5064C2semidirect
4. #4702C2semidirect
5. C3×#330C2semidirect
6. #4465C2semidirect
33. #330D3semidirect
180. C3#1660semidirect

Maximal subgroups

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

subgroupindex
#56992
#50642
#47152
#47022
C3×#3302
#44652
#44642
#16603