Finite Group Databasebeta

← #71048

Subgroups — #71048

Order 384.

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

normal subgroups only

This group has 240 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#5555C211yesmaximal, characteristic
2C3×#461C211yesmaximal, characteristic, complemented
3#5087C211yesmaximal, characteristic, complemented
4#5078C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#4483C211yesmaximal, characteristic, complemented
7#4482C211yesmaximal, characteristic, complemented
8#1647—31nomaximal, Sylow
9C2×#237—21no—
10C2×C3×SD16—21no—
11C3×C4⋊Q8C2211yescharacteristic
12C3×C4×D4C2211yescharacteristic
13#930—21no—
14C2×C4⋊Dic3—21no—
15#897—21no—
16#886—22no—
17C4×D12C2211yescharacteristic
18#869C2211yescharacteristic
19C3×C4⋊C8C2211yescharacteristic
20C3×#103—21no—
21#809—21no—
22#807—22no—
23#807—21no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C3⋊C8C2211yescharacteristic
26#547—31no—
27#500—31no—
28#474—31no—
29#461D311yescharacteristic, complemented
30#332—31no—
31D4⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34C2×C3×D4D411yescharacteristic
35C22×C3×C4—21no—
36C2×#67—22no—
37C22×Dic3—22no—
38#237—42no—
39#237—42no—
40C2×D12D411yescharacteristic
41C2×C4×D3—21no—
42C2×Dic6—21no—
43C3×SD16—42no—
44C2×C3×C8—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4—21no—
47C3×C4⋊C4D411yescharacteristic
48C3×C22⋊C4—21no—
49C3×C42C2311yescharacteristic
50(C2×D3)⋊C4—22no—
51(C2×D3)⋊C4—21no—
52C4⋊Dic3—22no—
53C4⋊Dic3—21no—
54C4⋊Dic3D412yes—
55C4⋊Dic3D411yescharacteristic
56C2×C3⋊C8—21no—
57C2×C3⋊C8—21no—
58C2×#41—61no—
59C2×SD16—21no—
60C4⋊Q8—31no—
61C4⋊Q8C2×D311yescharacteristic
62#123—62no—
63C22⋊Q8—61no—
64#121—61no—
65C4×D4—31no—
66C4×D4C2×D311yescharacteristic
67C2×C4⋊C4—61no—
68#107—61no—
69#107—62no—
70C4⋊C8—31no—
71C4⋊C8C2×D311yescharacteristic
72#103—21no—
73#102—61no—
74C4×C8—31no—
75C23×C3—21no—
76C22×D3—21no—
77C3×Q8—41no—
78C3×Q8—41no—
79C3×D4—21no—
80C3×D4D812yes—
81C2×C3×C4—21no—
82C2×C3×C4—41no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×C3×C4C2×D411yescharacteristic
86C2×C3×C4C2×D411yescharacteristic
87C2×C3×C4C2×D411yescharacteristic
88#67—44no—
89C2×Dic3—22no—
90C2×Dic3—21no—
91C2×Dic3—44no—
92D12—21no—
93D12—21no—
94C4×D3—41no—
95C4×D3—41no—
96Dic6—41no—
97Dic6—41no—
98C3×C8—41no—
99C3⋊C8—22no—
100C3⋊C8—41no—
101#41—122no—
102#41—122no—
103C2×Q8—21no—
104C2×Q8—61no—
105C2×D4—31no—
106C2×D4—62no—
107C2×D4#6711yescharacteristic
108C22×C4—21no—
109C22×C4—62no—
110C22×C4—61no—
111SD16—42no—
112C2×C8—21no—
113C2×C8—61no—
114C2×C8—61no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—31no—
118C4⋊C4—32no—
119C4⋊C4—62no—
120C4⋊C4—61no—
121C4⋊C4#6711yescharacteristic
122C22⋊C4—21no—
123C22⋊C4—62no—
124C22⋊C4—61no—
125C42C22×D311yescharacteristic
126C22×C3—22no—
127C22×C3—41no—
128C22×C3—41no—
129C22×C3#12011yescharacteristic
130C2×D3—41no—
131C2×D3—41no—
132C2×D3—41no—
133C3×C4—21no—
134C3×C4—41no—
135C3×C4—81no—
136C3×C4C2×D811yescharacteristic
137C3×C4#13511yescharacteristic
138C3×C4AGL(1,ℤ/8)11yescharacteristic
139C3×C4#13711yescharacteristic
140Dic3—42no—
141Dic3—41no—
142C23—21no—
143C23—61no—
144Q8—41no—
145Q8—41no—
146Q8—121no—
147Q8—121no—
148D4—21no—
149D4—61no—
150D4—61no—
151D4—124no—
152D4#21312yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—61no—
158C2×C4—62no—
159C2×C4—124no—
160C2×C4—121no—
161C2×C4—121no—
162C2×C4D3×D411yescharacteristic
163C2×C4D3×D411yescharacteristic
164C2×C4C2×#6711yescharacteristic
165C8—41no—
166C8—62no—
167C8—121no—
168C2×C3—42no—
169C2×C3#44711yescharacteristic
170C2×C3#45211yescharacteristic
171C2×C3#45611yescharacteristic
172D3—81no—
173C22—22no—
174C22—41no—
175C22—41no—
176C22—121no—
177C22—121no—
178C22—121no—
179C22#93711yescharacteristic
180C4—21no—
181C4—41no—
182C4—81no—
183C4—121no—
184C4—122no—
185C4#91511yescharacteristic
186C4#91611yescharacteristic
187C4C2×#21311yescharacteristic
188C4#94911yescharacteristic
189C3#164711yescharacteristic, complemented, Sylow
190C2—42no—
191C2—241no—
192C2#503611yescharacteristic
193C2#517311yescharacteristic
194C2#520511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#461C2
socleC22×C3#120
prime radical#5555C2
hypercenter#461D3
hypocenterC3#1647
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. C3×#461C2semidirect
3. #5087C2semidirect
4. #5078C2semidirect
5. D4⋊(C3⋊C8)C2semidirect
6. #4483C2semidirect
7. #4482C2semidirect
29. #461D3semidirect
189. C3#1647semidirect

Maximal subgroups

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

subgroupindex
#55552
C3×#4612
#50872
#50782
D4⋊(C3⋊C8)2
#44832
#44822
#16473