Finite Group Databasebeta

← #71264

Subgroups — #71264

Order 384.

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

normal subgroups only

This group has 236 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#5730C211yesmaximal, characteristic, complemented
2C3×#492C211yesmaximal, characteristic, complemented
3#5075C211yesmaximal, characteristic, complemented
4#5072C211yesmaximal, characteristic
5#4546C211yesmaximal, characteristic, complemented
6#4459C211yesmaximal, characteristic, complemented
7#4456C211yesmaximal, characteristic, complemented
8#1622—31nomaximal, Sylow
9C2×C3×SD16—22no—
10C3×C4⋊Q8C2211yescharacteristic
11C3×#127C2211yescharacteristic
12#946—22no—
13C2×#214—21no—
14C2×#213—21no—
15#897—21no—
16#893—21no—
17D3×C4⋊C4—21no—
18#890—21no—
19C4×D12C2211yescharacteristic
20C4×Dic6C2211yescharacteristic
21C3×C4×C8C2211yescharacteristic
22#832—21no—
23#832—21no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4⋊(C3⋊C8)C2211yescharacteristic
26#554—31no—
27#492D311yescharacteristic, complemented
28#460—31no—
29#459—31no—
30#333—31no—
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8D412yes—
34C2×C3×D4—21no—
35C2×C3×D4—21no—
36C2×C3×D4—21no—
37C2×D12D411yescharacteristic
38C2×C4×D3—21no—
39C2×C4×D3—22no—
40C2×Dic6D411yescharacteristic
41C3×SD16—44no—
42C2×C3×C8—21no—
43C3×C4⋊C4—21no—
44C3×C4⋊C4—21no—
45C3×C42C2311yescharacteristic
46#214—42no—
47#213—42no—
48(C2×D3)⋊C4—22no—
49(C2×D3)⋊C4—21no—
50C4⋊Dic3—22no—
51C4⋊Dic3D411yescharacteristic
52C4⋊Dic3D411yescharacteristic
53Dic3⋊C4—22no—
54Dic3⋊C4—21no—
55C4×Dic3—21no—
56C2×C3⋊C8—21no—
57C2×C3⋊C8—21no—
58C2×SD16—22no—
59C2×SD16—61no—
60C2×D8—61no—
61C4⋊Q8C2×D311yescharacteristic
62#127C2×D311yescharacteristic
63#125—61no—
64C22⋊Q8—61no—
65C22⋊Q8—62no—
66C4×Q8—31no—
67C4×D4—31no—
68C4×D4—61no—
69C2×C4⋊C4—61no—
70C4⋊C8—31no—
71C4⋊C8—31no—
72#102—61no—
73#102—61no—
74C4×C8C2×D311yescharacteristic
75C23×C3—41no—
76C22×D3—21no—
77C3×Q8—42no—
78C3×D4—42no—
79C3×D4—41no—
80C3×D4—41no—
81C3×D4—41no—
82C3×D4—41no—
83C2×C3×C4—22no—
84C2×C3×C4C2×D411yescharacteristic
85C2×C3×C4C2×D411yescharacteristic
86C2×C3×C4C2×D411yescharacteristic
87C2×Dic3—21no—
88C2×Dic3—21no—
89C2×Dic3—21no—
90D12—21no—
91D12SD1612yescomplemented
92C4×D3—41no—
93C4×D3—41no—
94C4×D3—82no—
95Dic6—21no—
96Dic6—21no—
97C3×C8—22no—
98C3⋊C8—41no—
99C3⋊C8—41no—
100C2×Q8—31no—
101C2×Q8#6712yes—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—31no—
106C22×C4—62no—
107C22×C4—61no—
108SD16—44no—
109SD16—122no—
110D8—122no—
111C2×C8—21no—
112C2×C8—61no—
113C2×C8—61no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—31no—
117C4⋊C4—31no—
118C4⋊C4—62no—
119C4⋊C4—62no—
120C4⋊C4—61no—
121C22⋊C4—62no—
122C22⋊C4—61no—
123C42—61no—
124C42C22×D311yescharacteristic
125C22×C3—81no—
126C22×C3—81no—
127C22×C3—81no—
128C22×C3#12011yescharacteristic
129C2×D3—22no—
130C2×D3—41no—
131C2×D3—41no—
132C3×C4—21no—
133C3×C4—42no—
134C3×C4C2×SD1611yescharacteristic
135C3×C4#13511yescharacteristic
136C3×C4AGL(1,ℤ/8)11yescharacteristic
137C3×C4#13711yescharacteristic
138Dic3—41no—
139Dic3—41no—
140Dic3—41no—
141C23—41no—
142C23—61no—
143Q8—42no—
144Q8—61no—
145Q8—61no—
146D4—32no—
147D4—42no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152D4—61no—
153C2×C4—22no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—121no—
158C2×C4—121no—
159C2×C4—242no—
160C2×C4D3×D411yescharacteristic
161C2×C4D3×D411yescharacteristic
162C2×C4C2×#6711yescharacteristic
163C8—22no—
164C8—121no—
165C8—121no—
166C2×C3—161no—
167C2×C3#44811yescharacteristic
168C2×C3#44911yescharacteristic
169C2×C3#45311yescharacteristic
170D3—42no—
171C22—62no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—121no—
176C22—121no—
177C22#93711yescharacteristic
178C4—21no—
179C4—42no—
180C4—121no—
181C4—121no—
182C4—121no—
183C4D3×SD1611yescharacteristic
184C4#91411yescharacteristic
185C4#91511yescharacteristic
186C4#91611yescharacteristic
187C3#162211yescharacteristic, complemented, Sylow
188C2—122no—
189C2—161no—
190C2#507711yescharacteristic
191C2#517011yescharacteristic
192C2#517211yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#492C2
socleC22×C3#120
prime radical#5072C2
hypercenter#492D3
hypocenterC3#1622
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #5730C2semidirect
2. C3×#492C2semidirect
3. #5075C2semidirect
5. #4546C2semidirect
6. #4459C2semidirect
7. #4456C2semidirect
27. #492D3semidirect
91. D12SD16semidirect
187. C3#1622semidirect

Maximal subgroups

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

subgroupindex
#57302
C3×#4922
#50752
#50722
#45462
#44592
#44562
#16223