Finite Group Databasebeta

← #68405

Subgroups — #68405

Order 384.

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

normal subgroups only

This group has 256 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#5718C211yesmaximal, characteristic
2#5076C211yesmaximal, characteristic, complemented
3#4850C211yesmaximal, characteristic, complemented
4#4842C211yesmaximal, characteristic, complemented
5C3×#331C211yesmaximal, characteristic, complemented
6#4484C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic, complemented
8#1631—31nomaximal, Sylow
9C2×#237—21no—
10C3×#127C2211yescharacteristic
11#940—21no—
12#939—22no—
13#938—21no—
14D4×Dic3—21no—
15C2×#214—22no—
16C4×D12C2211yescharacteristic
17#869C2211yescharacteristic
18C4×Dic6C2211yescharacteristic
19C3×C4⋊C8C2211yescharacteristic
20C3×C4⋊C8C2211yescharacteristic
21#821—21no—
22#818—21no—
23#817—21no—
24#816—21no—
25C4×C3⋊C8C2211yescharacteristic
26#550—31no—
27#492—31no—
28#477—31no—
29#476—31no—
30#331D311yescharacteristic, complemented
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×D4—21no—
34C2×C3×D4—21no—
35C2×C3×D4D412yes—
36C2×#67—22no—
37C2×#67—22no—
38C22×Dic3—22no—
39#237—42no—
40#237—42no—
41C2×D12D411yescharacteristic
42C2×C4×D3—21no—
43C2×Dic6—21no—
44C2×Dic6D411yescharacteristic
45C2×C3×C8—21no—
46C2×C3×C8—21no—
47C3×C42C2311yescharacteristic
48#217—22no—
49#214—44no—
50(C2×D3)⋊C4—21no—
51C4⋊Dic3—21no—
52C4⋊Dic3—21no—
53C4⋊Dic3D411yescharacteristic
54C4⋊Dic3D411yescharacteristic
55Dic3⋊C4—21no—
56C4×Dic3—21no—
57C2×C3⋊C8—21no—
58C2×#41—61no—
59C2×SD16—62no—
60C4⋊Q8—31no—
61#127—61no—
62#127C2×D311yescharacteristic
63#121—61no—
64#121—62no—
65C4×Q8—31no—
66C4×D4—31no—
67C4×D4—61no—
68#107—61no—
69#106—61no—
70C4⋊C8C2×D311yescharacteristic
71C4⋊C8C2×D311yescharacteristic
72#103—61no—
73#102—61no—
74C4×C8—31no—
75C23×C3—22no—
76C22×D3—21no—
77C3×D4—42no—
78C3×D4—41no—
79C3×D4—41no—
80C3×D4—41no—
81C3×D4—41no—
82C2×C3×C4C2×D411yescharacteristic
83C2×C3×C4C2×D411yescharacteristic
84C2×C3×C4C2×D411yescharacteristic
85#67—44no—
86#67—82no—
87C2×Dic3—21no—
88C2×Dic3—21no—
89C2×Dic3—21no—
90C2×Dic3—44no—
91C2×Dic3—41no—
92D12—21no—
93D12—21no—
94C4×D3—41no—
95C4×D3—41no—
96Dic6—21no—
97Dic6—42no—
98Dic6SD1612yescomplemented
99C3×C8—41no—
100C3×C8—41no—
101C3⋊C8—22no—
102#41—122no—
103#41—122no—
104C2×Q8—31no—
105C2×Q8—61no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—31no—
109C2×D4—62no—
110C2×D4—62no—
111C2×D4D1212yes—
112C22×C4—62no—
113C22×C4—61no—
114SD16—124no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—61no—
118C4⋊C4—31no—
119C4⋊C4—31no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C22⋊C4—62no—
124C22⋊C4—61no—
125C42—61no—
126C42C22×D311yescharacteristic
127C22×C3—42no—
128C22×C3—42no—
129C22×C3—42no—
130C22×C3#12011yescharacteristic
131C2×D3—41no—
132C2×D3—41no—
133C2×D3—41no—
134C3×C4—21no—
135C3×C4C2×SD1611yescharacteristic
136C3×C4#13511yescharacteristic
137C3×C4AGL(1,ℤ/8)11yescharacteristic
138C3×C4#13711yescharacteristic
139Dic3—22no—
140Dic3—41no—
141Dic3—41no—
142Dic3—81no—
143C23—22no—
144C23—61no—
145Q8—32no—
146Q8—61no—
147Q8—122no—
148D4—42no—
149D4—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—61no—
154D4—61no—
155D4—124no—
156D4—242no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C4—124no—
161C2×C4—121no—
162C2×C4—121no—
163C2×C4—121no—
164C2×C4C2×D1211yescharacteristic
165C2×C4D3×D411yescharacteristic
166C2×C4D3×D411yescharacteristic
167C8—41no—
168C8—41no—
169C8—62no—
170C2×C3—82no—
171C2×C3#45011yescharacteristic
172C2×C3#45211yescharacteristic
173C2×C3#45311yescharacteristic
174D3—81no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22—121no—
179C22—121no—
180C22—121no—
181C22#88211yescharacteristic
182C4—21no—
183C4—62no—
184C4—121no—
185C4—121no—
186C4—241no—
187C4#91111yescharacteristic
188C4#91211yescharacteristic
189C4D3×SD1611yescharacteristic
190C4#91511yescharacteristic
191C3#163111yescharacteristic, complemented, Sylow
192C2—82no—
193C2—241no—
194C2#474711yescharacteristic
195C2#477611yescharacteristic
196C2#478311yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#331C2
socleC22×C3#120
prime radical#5718C2
hypercenter#331D3
hypocenterC3#1631
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #5076C2semidirect
3. #4850C2semidirect
4. #4842C2semidirect
5. C3×#331C2semidirect
6. #4484C2semidirect
7. #4483C2semidirect
30. #331D3semidirect
98. Dic6SD16semidirect
191. C3#1631semidirect

Maximal subgroups

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

subgroupindex
#57182
#50762
#48502
#48422
C3×#3312
#44842
#44832
#16313