Finite Group Databasebeta

← #71144

Subgroups — #71144

Order 384.

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

normal subgroups only

This group has 214 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#5543C211yesmaximal, characteristic
2C3×#476C211yesmaximal, characteristic
3#5092C211yesmaximal, characteristic, complemented
4#5091C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊C8)C211yesmaximal, characteristic
6#4484C211yesmaximal, characteristic, complemented
7#4481C211yesmaximal, characteristic, complemented
8#1632—31nomaximal, Sylow
9C3×C4⋊Q8C2211yescharacteristic
10C3×C4×D4C2211yescharacteristic
11C2×#216—22no—
12D4×Dic3—21no—
13C2×C4⋊Dic3—21no—
14#924—21no—
15#890—21no—
16#878—22no—
17#869C2211yescharacteristic
18C4×Dic6C2211yescharacteristic
19C3×#106—21no—
20C3×C4⋊C8C2211yescharacteristic
21C3×#102—21no—
22#810—21no—
23#807—21no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C3⋊C8C2211yescharacteristic
26#554—31no—
27#494—31no—
28#477—31no—
29#476D311yescharacteristic
30#333—31no—
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34C2×C3×D4D411yescharacteristic
35C22×C3×C4—21no—
36C22×Dic3—22no—
37C2×Dic6D411yescharacteristic
38C2×Dic6D412yes—
39C2×C3×C8—21no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4—21no—
42C3×C4⋊C4D411yescharacteristic
43C3×C22⋊C4—21no—
44C3×C42C2311yescharacteristic
45#217—22no—
46#216—44no—
47C4⋊Dic3—22no—
48C4⋊Dic3—21no—
49C4⋊Dic3—21no—
50C4⋊Dic3D411yescharacteristic
51Dic3⋊C4—22no—
52Dic3⋊C4—21no—
53C4×Dic3—21no—
54C2×C3⋊C8—21no—
55C2×C3⋊C8—21no—
56C2×Q16—62no—
57C4⋊Q8—31no—
58C4⋊Q8C2×D311yescharacteristic
59#125—61no—
60C22⋊Q8—62no—
61C22⋊Q8—61no—
62C4×Q8—31no—
63C4×D4—61no—
64C4×D4C2×D311yescharacteristic
65C2×C4⋊C4—61no—
66#107—61no—
67#106—21no—
68C4⋊C8—31no—
69C4⋊C8C2×D311yescharacteristic
70#103—61no—
71#102—21no—
72C4×C8—31no—
73C23×C3—21no—
74C3×Q8—42no—
75C3×D4—21no—
76C3×D4Q1612yescomplemented
77C2×C3×C4—21no—
78C2×C3×C4—41no—
79C2×C3×C4—41no—
80C2×C3×C4—41no—
81C2×C3×C4C2×D411yescharacteristic
82C2×C3×C4C2×D411yescharacteristic
83C2×C3×C4C2×D411yescharacteristic
84C2×Dic3—21no—
85C2×Dic3—21no—
86C2×Dic3—22no—
87C2×Dic3—82no—
88Dic6—21no—
89Dic6—21no—
90Dic6—42no—
91C3×C8—41no—
92C3⋊C8—22no—
93C3⋊C8—41no—
94C2×Q8—21no—
95C2×Q8—31no—
96C2×Q8—32no—
97C2×D4#6711yescharacteristic
98C22×C4—21no—
99C22×C4—62no—
100Q16—124no—
101C2×C8—21no—
102C2×C8—61no—
103C2×C8—61no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—31no—
107C4⋊C4—62no—
108C4⋊C4—62no—
109C4⋊C4—61no—
110C4⋊C4—61no—
111C4⋊C4—61no—
112C4⋊C4#6711yescharacteristic
113C22⋊C4—21no—
114C22⋊C4—62no—
115C42—61no—
116C42C22×D311yescharacteristic
117C22×C3—22no—
118C22×C3—41no—
119C22×C3—41no—
120C22×C3#12011yescharacteristic
121C3×C4—21no—
122C3×C4—41no—
123C3×C4—81no—
124C3×C4C2×Q1611yescharacteristic
125C3×C4#13511yescharacteristic
126C3×C4#13711yescharacteristic
127C3×C4#13711yescharacteristic
128Dic3—41no—
129Dic3—42no—
130Dic3—41no—
131C23—21no—
132Q8—42no—
133Q8—61no—
134Q8—61no—
135Q8—122no—
136D4—21no—
137D4#21612yescomplemented
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—61no—
143C2×C4—61no—
144C2×C4—62no—
145C2×C4—242no—
146C2×C4D3×D411yescharacteristic
147C2×C4D3×D411yescharacteristic
148C2×C4C2×#6711yescharacteristic
149C8—41no—
150C8—62no—
151C8—121no—
152C2×C3—42no—
153C2×C3#45111yescharacteristic
154C2×C3#45211yescharacteristic
155C2×C3#45311yescharacteristic
156C22—22no—
157C22—41no—
158C22—41no—
159C22#93711yescharacteristic
160C4—21no—
161C4—41no—
162C4—81no—
163C4—121no—
164C4—121no—
165C4—122no—
166C4#91511yescharacteristic
167C4#91611yescharacteristic
168C4C2×#21611yescharacteristic
169C4#95111yescharacteristic
170C3#163211yescharacteristic, complemented, Sylow
171C2—42no—
172C2#505011yescharacteristic
173C2#517311yescharacteristic
174C2#519811yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#476C2
socleC22×C3#120
prime radicalC2×C3×D4D4
hypercenter#476D3
hypocenterC3#1632
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
3. #5092C2semidirect
4. #5091C2semidirect
6. #4484C2semidirect
7. #4481C2semidirect
76. C3×D4Q16semidirect
137. D4#216semidirect
170. C3#1632semidirect

Maximal subgroups

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

subgroupindex
#55432
C3×#4762
#50922
#50912
D4⋊(C3⋊C8)2
#44842
#44812
#16323