Finite Group Databasebeta

← #71115

Subgroups — #71115

Order 384.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#5542C211yesmaximal, characteristic
2C3×#474C211yesmaximal, characteristic
3#5090C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊C8)C211yesmaximal, characteristic
6#4484C211yesmaximal, characteristic, complemented
7#4482C211yesmaximal, characteristic, complemented
8#1652—31nomaximal, Sylow
9C3×C4⋊Q8C2211yescharacteristic
10C3×C4×D4C2211yescharacteristic
11C2×#216—21no—
12#936—21no—
13#926—21no—
14#924—21no—
15#887—21no—
16#879—22no—
17#869C2211yescharacteristic
18C4×Dic6C2211yescharacteristic
19C3×#107—21no—
20C3×C4⋊C8C2211yescharacteristic
21C3×#102—21no—
22#835—21no—
23#808—22no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C3⋊C8C2211yescharacteristic
26#556—31no—
27#498—31no—
28#474D311yescharacteristic
29#462—31no—
30#332—31no—
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34C2×C3×D4D411yescharacteristic
35C22×C3×C4—21no—
36C2×Dic6—21no—
37C2×Dic6D411yescharacteristic
38C2×C3×C8—21no—
39C3×C4⋊C4—21no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4D411yescharacteristic
42C3×C22⋊C4—21no—
43C3×C42C2311yescharacteristic
44#217—22no—
45#217—22no—
46#216—42no—
47C4⋊Dic3—21no—
48C4⋊Dic3D412yes—
49C4⋊Dic3D411yescharacteristic
50Dic3⋊C4—22no—
51Dic3⋊C4—21no—
52C4×Dic3—22no—
53C4×Dic3—21no—
54C2×C3⋊C8—21no—
55C2×C3⋊C8—21no—
56C2×Q16—61no—
57C4⋊Q8—31no—
58C4⋊Q8C2×D311yescharacteristic
59#126—62no—
60#124—61no—
61C22⋊Q8—61no—
62C4×Q8—31no—
63C4×Q8—61no—
64C4×D4C2×D311yescharacteristic
65#117—61no—
66#107—21no—
67#106—62no—
68C4⋊C8—31no—
69C4⋊C8C2×D311yescharacteristic
70#103—61no—
71#102—21no—
72C4×C8—31no—
73C23×C3—21no—
74C3×Q8—41no—
75C3×Q8—41no—
76C3×D4—21no—
77C3×D4—21no—
78C2×C3×C4—21no—
79C2×C3×C4—41no—
80C2×C3×C4—41no—
81C2×C3×C4—41no—
82C2×C3×C4C2×D411yescharacteristic
83C2×C3×C4C2×D411yescharacteristic
84C2×C3×C4C2×D411yescharacteristic
85C2×Dic3—21no—
86C2×Dic3—21no—
87C2×Dic3—22no—
88Dic6—21no—
89Dic6—21no—
90Dic6—41no—
91Dic6—41no—
92C3×C8—41no—
93C3⋊C8—22no—
94C3⋊C8—41no—
95C2×Q8—21no—
96C2×Q8—31no—
97C2×Q8—61no—
98C2×D4#6711yescharacteristic
99C22×C4—21no—
100Q16—122no—
101C2×C8—21no—
102C2×C8—61no—
103C2×C8—61no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—31no—
107C4⋊C4—32no—
108C4⋊C4—62no—
109C4⋊C4—61no—
110C4⋊C4—61no—
111C4⋊C4#6711yescharacteristic
112C22⋊C4—21no—
113C22⋊C4—62no—
114C22⋊C4—62no—
115C42—62no—
116C42—61no—
117C42C22×D311yescharacteristic
118C22×C3—41no—
119C22×C3—41no—
120C22×C3—41no—
121C22×C3#12011yescharacteristic
122C3×C4—21no—
123C3×C4—41no—
124C3×C4—81no—
125C3×C4#13511yescharacteristic
126C3×C4#13511yescharacteristic
127C3×C4AGL(1,ℤ/8)11yescharacteristic
128C3×C4#13711yescharacteristic
129Dic3—41no—
130Dic3—42no—
131Dic3—41no—
132C23—21no—
133Q8—41no—
134Q8—41no—
135Q8—61no—
136Q8—61no—
137Q8—121no—
138Q8—121no—
139D4—21no—
140D4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—61no—
146C2×C4—62no—
147C2×C4—61no—
148C2×C4D3×D411yescharacteristic
149C2×C4D3×D411yescharacteristic
150C2×C4C2×#6711yescharacteristic
151C8—41no—
152C8—62no—
153C8—121no—
154C2×C3—81no—
155C2×C3#44911yescharacteristic
156C2×C3#45211yescharacteristic
157C2×C3#45611yescharacteristic
158C22—41no—
159C22—41no—
160C22—41no—
161C22#93711yescharacteristic
162C4—21no—
163C4—41no—
164C4—81no—
165C4—121no—
166C4—121no—
167C4—122no—
168C4#91111yescharacteristic
169C4#91211yescharacteristic
170C4#94211yescharacteristic
171C4#95011yescharacteristic
172C3#165211yescharacteristic, complemented, Sylow
173C2—81no—
174C2#504711yescharacteristic
175C2#515811yescharacteristic
176C2#520511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#474C2
socleC22×C3#120
prime radicalC2×C3×D4D4
hypercenter#474D3
hypocenterC3#1652
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
3. #5090C2semidirect
4. #5080C2semidirect
6. #4484C2semidirect
7. #4482C2semidirect
172. C3#1652semidirect

Maximal subgroups

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

subgroupindex
#55422
C3×#4742
#50902
#50802
D4⋊(C3⋊C8)2
#44842
#44822
#16523