Finite Group Databasebeta

← #71196

Subgroups — #71196

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#5726C211yesmaximal, characteristic, complemented
2C3×#486C211yesmaximal, characteristic, complemented
3#5072C211yesmaximal, characteristic, complemented
4#5072C211yesmaximal, characteristic, complemented
5#4547C211yesmaximal, characteristic, complemented
6#4459C211yesmaximal, characteristic, complemented
7#4459C211yesmaximal, characteristic, complemented
8#1625—31nomaximal, Sylow
9C2×#239—21no—
10C3×C4⋊Q8C2211yescharacteristic
11C3×#127C2211yescharacteristic
12#946—21no—
13C2×#213—21no—
14C2×#213—21no—
15#895—21no—
16#894—22no—
17D3×C4⋊C4—21no—
18C4×D12C2211yescharacteristic
19C4×D12C2211yescharacteristic
20C3×#102—22no—
21C3×C4×C8C2211yescharacteristic
22#832—21no—
23#832—21no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4⋊(C3⋊C8)C2211yescharacteristic
26#547—31no—
27#486D311yescharacteristic, complemented
28#459—31no—
29#459—31no—
30#332—31no—
31D4⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34C2×C3×D4—21no—
35C2×C3×D4—21no—
36C2×C3×D4—21no—
37#239—42no—
38#239—42no—
39C2×D12—22no—
40C2×D12D411yescharacteristic
41C2×D12D411yescharacteristic
42C2×C4×D3—22no—
43C2×C4×D3—21no—
44C2×C4×D3—21no—
45C2×C3×C8—21no—
46C3×C4⋊C4—21no—
47C3×C4⋊C4D412yes—
48C3×C42C2311yescharacteristic
49#213—42no—
50#213—42no—
51(C2×D3)⋊C4—22no—
52(C2×D3)⋊C4—21no—
53(C2×D3)⋊C4—21no—
54C4⋊Dic3D411yescharacteristic
55C4⋊Dic3D411yescharacteristic
56Dic3⋊C4—22no—
57C2×C3⋊C8—21no—
58C2×C3⋊C8—21no—
59C2×#41—61no—
60C2×D8—61no—
61C2×D8—61no—
62C4⋊Q8C2×D311yescharacteristic
63#127C2×D311yescharacteristic
64#123—62no—
65C22⋊Q8—61no—
66#121—61no—
67C4×D4—31no—
68C4×D4—31no—
69C2×C4⋊C4—61no—
70C4⋊C8—31no—
71C4⋊C8—31no—
72#102—22no—
73#102—61no—
74#102—61no—
75C4×C8C2×D311yescharacteristic
76C23×C3—41no—
77C22×D3—21no—
78C22×D3—21no—
79C3×Q8—41no—
80C3×Q8—41no—
81C3×D4—42no—
82C3×D4—41no—
83C3×D4—41no—
84C3×D4—41no—
85C3×D4—41no—
86C2×C3×C4—22no—
87C2×C3×C4C2×D411yescharacteristic
88C2×C3×C4C2×D411yescharacteristic
89C2×C3×C4C2×D411yescharacteristic
90C2×Dic3—21no—
91C2×Dic3—21no—
92D12—21no—
93D12—21no—
94D12—21no—
95D12—44no—
96D12D812yescomplemented
97C4×D3—44no—
98C4×D3—41no—
99C4×D3—41no—
100C4×D3—41no—
101C4×D3—41no—
102C3×C8—41no—
103C3⋊C8—41no—
104C3⋊C8—41no—
105#41—122no—
106#41—122no—
107C2×Q8—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—31no—
112C2×D4—31no—
113C2×D4—62no—
114C22×C4—62no—
115C22×C4—61no—
116C22×C4—61no—
117D8—122no—
118D8—122no—
119C2×C8—21no—
120C2×C8—61no—
121C2×C8—61no—
122C4⋊C4—21no—
123C4⋊C4—31no—
124C4⋊C4—31no—
125C4⋊C4—62no—
126C4⋊C4#6712yes—
127C22⋊C4—62no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C42C22×D311yescharacteristic
131C22×C3—81no—
132C22×C3—81no—
133C22×C3—81no—
134C22×C3#12011yescharacteristic
135C2×D3—22no—
136C2×D3—41no—
137C2×D3—41no—
138C2×D3—41no—
139C2×D3—41no—
140C2×D3—41no—
141C3×C4—21no—
142C3×C4—42no—
143C3×C4C2×D811yescharacteristic
144C3×C4#13511yescharacteristic
145C3×C4AGL(1,ℤ/8)11yescharacteristic
146C3×C4AGL(1,ℤ/8)11yescharacteristic
147Dic3—41no—
148Dic3—41no—
149C23—41no—
150C23—61no—
151C23—61no—
152Q8—41no—
153Q8—41no—
154D4—32no—
155D4—42no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—61no—
161D4—61no—
162D4—61no—
163D4—124no—
164C2×C4—22no—
165C2×C4—61no—
166C2×C4—61no—
167C2×C4—124no—
168C2×C4—121no—
169C2×C4—121no—
170C2×C4—121no—
171C2×C4—121no—
172C2×C4D3×D411yescharacteristic
173C2×C4D3×D411yescharacteristic
174C2×C4C2×#6711yescharacteristic
175C8—41no—
176C8—121no—
177C8—121no—
178C2×C3—161no—
179C2×C3#44711yescharacteristic
180C2×C3#44911yescharacteristic
181C2×C3#45411yescharacteristic
182D3—42no—
183D3—81no—
184C22—62no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—121no—
189C22—121no—
190C22—121no—
191C22—121no—
192C22—121no—
193C22#93711yescharacteristic
194C4—21no—
195C4—42no—
196C4—121no—
197C4—121no—
198C4D3×D811yescharacteristic
199C4#91111yescharacteristic
200C4#91411yescharacteristic
201C4#91611yescharacteristic
202C3#162511yescharacteristic, complemented, Sylow
203C2—122no—
204C2—161no—
205C2—241no—
206C2#506211yescharacteristic
207C2#515611yescharacteristic
208C2#517211yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#486C2
socleC22×C3#120
hypercenter#486D3
hypocenterC3#1625
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #5726C2semidirect
2. C3×#486C2semidirect
3. #5072C2semidirect
4. #5072C2semidirect
5. #4547C2semidirect
6. #4459C2semidirect
7. #4459C2semidirect
27. #486D3semidirect
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
#57262
C3×#4862
#50722
#45472
#44592
#16253