Finite Group Databasebeta

← #68459

Subgroups — #68459

Order 384.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#5727C211yesmaximal, characteristic
2#5093C211yesmaximal, characteristic, complemented
3#4850C211yesmaximal, characteristic, complemented
4#4842C211yesmaximal, characteristic
5C3×#332C211yesmaximal, characteristic, complemented
6#4484C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic
8#1637—31nomaximal, Sylow
9C2×D3×Q8—21no—
10C3×C4⋊Q8C2211yescharacteristic
11#946—21no—
12#896—22no—
13#893—21no—
14#888—21no—
15C4×D12C2211yescharacteristic
16#869C2211yescharacteristic
17C4×Dic6C2211yescharacteristic
18C3×C4⋊C8C2211yescharacteristic
19C3×C4⋊C8C2211yescharacteristic
20#821—21no—
21#818—21no—
22#817—21no—
23#816—21no—
24#810—22no—
25C4×C3⋊C8C2211yescharacteristic
26D4×Q8—31no—
27#487—31no—
28#477—31no—
29#476—31no—
30#332D311yescharacteristic, complemented
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34D3×Q8—42no—
35D3×Q8—42no—
36C2×D12D411yescharacteristic
37C2×C4×D3—22no—
38C2×C4×D3—21no—
39C2×Dic6—22no—
40C2×Dic6—21no—
41C2×Dic6D411yescharacteristic
42C2×C3×C8—21no—
43C2×C3×C8—21no—
44C3×C4⋊C4—21no—
45C3×C4⋊C4D412yes—
46C3×C42C2311yescharacteristic
47(C2×D3)⋊C4—22no—
48(C2×D3)⋊C4—21no—
49C4⋊Dic3—21no—
50C4⋊Dic3—21no—
51C4⋊Dic3D411yescharacteristic
52C4⋊Dic3D411yescharacteristic
53Dic3⋊C4—22no—
54Dic3⋊C4—21no—
55C4×Dic3—21no—
56C2×C3⋊C8—21no—
57C22×Q8—61no—
58C4⋊Q8—31no—
59C4⋊Q8—61no—
60C4⋊Q8C2×D311yescharacteristic
61C22⋊Q8—61no—
62C22⋊Q8—62no—
63C4×Q8—31no—
64C4×D4—31no—
65C4×D4—61no—
66#107—61no—
67#106—61no—
68C4⋊C8C2×D311yescharacteristic
69C4⋊C8C2×D311yescharacteristic
70#103—61no—
71#103—62no—
72#102—61no—
73C4×C8—31no—
74C22×D3—21no—
75C3×Q8—41no—
76C3×Q8—41no—
77C2×C3×C4—22no—
78C2×C3×C4C2×D411yescharacteristic
79C2×C3×C4C2×D411yescharacteristic
80C2×C3×C4C2×D411yescharacteristic
81C2×Dic3—21no—
82C2×Dic3—21no—
83C2×Dic3—21no—
84C2×Dic3—41no—
85D12—21no—
86D12Q1612yes—
87C4×D3—44no—
88C4×D3—41no—
89C4×D3—41no—
90Dic6—21no—
91Dic6—44no—
92Dic6—42no—
93Dic6SD1612yes—
94C3×C8—41no—
95C3×C8—41no—
96C3⋊C8—41no—
97C2×Q8—21no—
98C2×Q8—31no—
99C2×Q8—62no—
100C2×Q8—61no—
101C2×Q8—122no—
102C2×Q8—122no—
103C2×D4—31no—
104C22×C4—62no—
105C22×C4—61no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—61no—
109C4⋊C4—21no—
110C4⋊C4—31no—
111C4⋊C4—31no—
112C4⋊C4—62no—
113C4⋊C4—61no—
114C4⋊C4—61no—
115C4⋊C4—61no—
116C4⋊C4D1212yes—
117C22⋊C4—62no—
118C22⋊C4—61no—
119C42—61no—
120C42C22×D311yescharacteristic
121C22×C3#12011yescharacteristic
122C2×D3—22no—
123C2×D3—41no—
124C2×D3—41no—
125C3×C4—21no—
126C3×C4—42no—
127C3×C4C2×SD1611yescharacteristic
128C3×C4C2×Q1611yescharacteristic
129C3×C4AGL(1,ℤ/8)11yescharacteristic
130C3×C4#13711yescharacteristic
131Dic3—22no—
132Dic3—41no—
133Dic3—41no—
134Dic3—81no—
135C23—61no—
136Q8—32no—
137Q8—41no—
138Q8—41no—
139Q8—61no—
140Q8—122no—
141Q8—124no—
142D4—32no—
143D4—61no—
144C2×C4—22no—
145C2×C4—61no—
146C2×C4—61no—
147C2×C4—61no—
148C2×C4—124no—
149C2×C4—121no—
150C2×C4—121no—
151C2×C4—121no—
152C2×C4C2×D1211yescharacteristic
153C2×C4D3×D411yescharacteristic
154C2×C4D3×D411yescharacteristic
155C8—41no—
156C8—41no—
157C8—121no—
158C2×C3#45011yescharacteristic
159C2×C3#45111yescharacteristic
160C2×C3#45411yescharacteristic
161D3—42no—
162C22—62no—
163C22—121no—
164C22—121no—
165C22#88211yescharacteristic
166C4—21no—
167C4—42no—
168C4—62no—
169C4—121no—
170C4—121no—
171C4—241no—
172C4#91111yescharacteristic
173C4D3×SD1611yescharacteristic
174C4#91511yescharacteristic
175C4D3×Q1611yescharacteristic
176C3#163711yescharacteristic, complemented, Sylow
177C2—122no—
178C2#475311yescharacteristic
179C2#477611yescharacteristic
180C2#480911yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#332C2
socleC22×C3#120
prime radicalC2×D12D4
hypercenter#332D3
hypocenterC3#1637
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #5093C2semidirect
3. #4850C2semidirect
5. C3×#332C2semidirect
6. #4484C2semidirect
30. #332D3semidirect
176. C3#1637semidirect

Maximal subgroups

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

subgroupindex
#57272
#50932
#48502
#48422
C3×#3322
#44842
#44832
#16373