Finite Group Databasebeta

← #71012

Subgroups — #71012

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#5551C211yesmaximal, characteristic, complemented
2C3×#459C211yesmaximal, characteristic, complemented
3#5072C211yesmaximal, characteristic, complemented
4#5070C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#4486C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic, complemented
8#1625—31nomaximal, Sylow
9C2×#235—21no—
10C2×C3×D8—21no—
11C3×#127C2211yescharacteristic
12C3×C4×D4C2211yescharacteristic
13#938—21no—
14C2×#213—21no—
15#924—21no—
16D3×C4⋊C4—21no—
17#883—22no—
18C4×D12C2211yescharacteristic
19#869C2211yescharacteristic
20C3×C4⋊C8C2211yescharacteristic
21C3×#102—21no—
22#832—22no—
23#832—21no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C3⋊C8C2211yescharacteristic
26#547—31no—
27#486—31no—
28#459—31no—
29#459D311yescharacteristic, complemented
30#332—31no—
31D4⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×D4—21no—
34C2×C3×D4—21no—
35C2×C3×D4—21no—
36C2×C3×D4D411yescharacteristic
37C22×C3×C4—21no—
38C2×#67—22no—
39#235—42no—
40#235—42no—
41C2×D12D411yescharacteristic
42C2×C4×D3—22no—
43C2×C4×D3—21no—
44C2×Dic6—21no—
45C3×D8—42no—
46C2×C3×C8—21no—
47C3×C4⋊C4D411yescharacteristic
48C3×C22⋊C4—21no—
49C3×C42C2311yescharacteristic
50#217—22no—
51#213—42no—
52(C2×D3)⋊C4—21no—
53C4⋊Dic3—21no—
54C4⋊Dic3D412yes—
55C4⋊Dic3D411yescharacteristic
56Dic3⋊C4—22no—
57C2×C3⋊C8—21no—
58C2×C3⋊C8—21no—
59C2×#41—61no—
60C2×D8—21no—
61C2×D8—61no—
62C4⋊Q8—31no—
63#127C2×D311yescharacteristic
64#123—62no—
65C22⋊Q8—61no—
66#121—61no—
67C4×D4—31no—
68C4×D4C2×D311yescharacteristic
69C2×C4⋊C4—61no—
70C4⋊C8—31no—
71C4⋊C8C2×D311yescharacteristic
72#102—21no—
73#102—61no—
74#102—62no—
75C4×C8—31no—
76C23×C3—21no—
77C23×C3—41no—
78C22×D3—21no—
79C3×D4—21no—
80C3×D4—21no—
81C3×D4—42no—
82C3×D4—41no—
83C3×D4—41no—
84C3×D4—41no—
85C3×D4—41no—
86C2×C3×C4—21no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4C2×D411yescharacteristic
90C2×C3×C4C2×D411yescharacteristic
91C2×C3×C4C2×D411yescharacteristic
92#67—44no—
93C2×Dic3—21no—
94C2×Dic3—22no—
95D12—21no—
96D12D812yescomplemented
97C4×D3—44no—
98C4×D3—41no—
99C4×D3—41no—
100Dic6—41no—
101Dic6—41no—
102C3×C8—41no—
103C3⋊C8—41no—
104C3⋊C8—41no—
105#41—122no—
106#41—122no—
107C2×Q8—61no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—31no—
112C2×D4—62no—
113C2×D4#6711yescharacteristic
114C22×C4—21no—
115C22×C4—62no—
116C22×C4—61no—
117D8—42no—
118D8—122no—
119C2×C8—21no—
120C2×C8—61no—
121C2×C8—61no—
122C4⋊C4—31no—
123C4⋊C4—32no—
124C4⋊C4—62no—
125C4⋊C4—61no—
126C4⋊C4#6711yescharacteristic
127C22⋊C4—21no—
128C22⋊C4—62no—
129C22⋊C4—61no—
130C42C22×D311yescharacteristic
131C22×C3—41no—
132C22×C3—41no—
133C22×C3—41no—
134C22×C3—81no—
135C22×C3—81no—
136C22×C3—81no—
137C22×C3#12011yescharacteristic
138C2×D3—22no—
139C2×D3—41no—
140C2×D3—41no—
141C3×C4—21no—
142C3×C4—41no—
143C3×C4C2×D811yescharacteristic
144C3×C4#13511yescharacteristic
145C3×C4AGL(1,ℤ/8)11yescharacteristic
146C3×C4AGL(1,ℤ/8)11yescharacteristic
147Dic3—42no—
148Dic3—41no—
149C23—21no—
150C23—41no—
151C23—61no—
152Q8—121no—
153Q8—121no—
154D4—21no—
155D4—21no—
156D4—32no—
157D4—42no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—61no—
163D4—124no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—61no—
168C2×C4—62no—
169C2×C4—124no—
170C2×C4—121no—
171C2×C4—121no—
172C2×C4D3×D411yescharacteristic
173C2×C4D3×D411yescharacteristic
174C2×C4C2×#6711yescharacteristic
175C8—41no—
176C8—121no—
177C8—121no—
178C2×C3—81no—
179C2×C3—161no—
180C2×C3#44711yescharacteristic
181C2×C3#44911yescharacteristic
182C2×C3#45411yescharacteristic
183D3—42no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—62no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—121no—
192C22—121no—
193C22#93711yescharacteristic
194C4—21no—
195C4—41no—
196C4—122no—
197C4—121no—
198C4D3×D811yescharacteristic
199C4#91111yescharacteristic
200C4#93211yescharacteristic
201C4#95011yescharacteristic
202C3#162511yescharacteristic, complemented, Sylow
203C2—81no—
204C2—122no—
205C2—161no—
206C2#503711yescharacteristic
207C2#515611yescharacteristic
208C2#520111yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#459C2
socleC22×C3#120
hypercenter#459D3
hypocenterC3#1625
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5551C2semidirect
2. C3×#459C2semidirect
3. #5072C2semidirect
4. #5070C2semidirect
5. D4⋊(C3⋊C8)C2semidirect
6. #4486C2semidirect
7. #4483C2semidirect
29. #459D3semidirect
96. D12D8semidirect
202. C3#1625semidirect

Maximal subgroups

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

subgroupindex
#55512
C3×#4592
#50722
#50702
D4⋊(C3⋊C8)2
#44862
#44832
#16253