Finite Group Databasebeta

← #68403

Subgroups — #68403

Order 384.

278 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 278 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#5718C211yesmaximal, characteristic, complemented
2#5073C211yesmaximal, characteristic, complemented
3#4848C211yesmaximal, characteristic, complemented
4#4843C211yesmaximal, characteristic, complemented
5C3×#331C211yesmaximal, characteristic, complemented
6#4484C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic, complemented
8#1621—31nomaximal, Sylow
9C2×#237—21no—
10C3×#127C2211yescharacteristic
11#940—21no—
12#939—22no—
13#938—21no—
14D4×Dic3—21no—
15C2×#213—22no—
16C2×D24—21no—
17C2×#204—21no—
18#874C2211yescharacteristic
19C4×D12C2211yescharacteristic
20C4×Dic6C2211yescharacteristic
21C3×C4⋊C8C2211yescharacteristic
22C3×C4⋊C8C2211yescharacteristic
23#821—21no—
24#821—21no—
25C4×C3⋊C8C2211yescharacteristic
26#550—31no—
27#493—31no—
28#460—31no—
29#459—31no—
30#331D311yescharacteristic, complemented
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×D4—21no—
34C2×C3×D4—21no—
35C2×C3×D4D412yes—
36C2×#67—22no—
37C2×#67—22no—
38C22×Dic3—22no—
39#237—42no—
40#237—42no—
41C2×D12—21no—
42C2×D12—21no—
43C2×D12—21no—
44C2×D12D411yescharacteristic
45C2×C4×D3—21no—
46C2×Dic6D411yescharacteristic
47C2×C3×C8—21no—
48C2×C3×C8—21no—
49C3×C42C2311yescharacteristic
50#217—22no—
51#213—44no—
52(C2×D3)⋊C4—21no—
53C4⋊Dic3D411yescharacteristic
54C4⋊Dic3D411yescharacteristic
55Dic3⋊C4—21no—
56C4×Dic3—21no—
57C2×C3⋊C8—21no—
58D24—42no—
59#204—42no—
60C2×#41—61no—
61C2×SD16—61no—
62C2×D8—61no—
63C2×D8—62no—
64#127—31no—
65#127—61no—
66#127C2×D311yescharacteristic
67#121—61no—
68#121—62no—
69C4×Q8—31no—
70C4×D4—31no—
71C4×D4—61no—
72C4⋊C8C2×D311yescharacteristic
73C4⋊C8C2×D311yescharacteristic
74#102—61no—
75#102—61no—
76C4×C8—31no—
77C23×C3—22no—
78C22×D3—21no—
79C22×D3—41no—
80C3×D4—42no—
81C3×D4—41no—
82C3×D4—41no—
83C3×D4—41no—
84C3×D4—41no—
85C2×C3×C4C2×D411yescharacteristic
86C2×C3×C4C2×D411yescharacteristic
87C2×C3×C4C2×D411yescharacteristic
88#67—44no—
89#67—82no—
90C2×Dic3—21no—
91C2×Dic3—21no—
92C2×Dic3—21no—
93C2×Dic3—44no—
94D12—21no—
95D12—21no—
96D12—42no—
97D12—41no—
98D12—41no—
99D12—41no—
100D12—41no—
101C4×D3—41no—
102C4×D3—41no—
103Dic6—21no—
104Dic6D812yescomplemented
105C3×C8—41no—
106C3×C8—41no—
107C3⋊C8—22no—
108#41—122no—
109#41—122no—
110C2×Q8—31no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—31no—
114C2×D4—62no—
115C2×D4—62no—
116C2×D4—61no—
117C2×D4—61no—
118C2×D4—61no—
119C2×D4D1212yes—
120C22×C4—62no—
121C22×C4—61no—
122SD16—122no—
123D8—122no—
124D8—124no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—61no—
128C4⋊C4—31no—
129C4⋊C4—31no—
130C4⋊C4—61no—
131C22⋊C4—62no—
132C22⋊C4—61no—
133C42—61no—
134C42C22×D311yescharacteristic
135C22×C3—42no—
136C22×C3—42no—
137C22×C3—42no—
138C22×C3#12011yescharacteristic
139C2×D3—41no—
140C2×D3—41no—
141C2×D3—41no—
142C2×D3—81no—
143C2×D3—81no—
144C2×D3—81no—
145C3×C4—21no—
146C3×C4C2×D811yescharacteristic
147C3×C4#13511yescharacteristic
148C3×C4AGL(1,ℤ/8)11yescharacteristic
149C3×C4AGL(1,ℤ/8)11yescharacteristic
150Dic3—22no—
151Dic3—41no—
152Dic3—41no—
153C23—22no—
154C23—61no—
155C23—121no—
156Q8—32no—
157Q8—61no—
158D4—42no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—61no—
164D4—61no—
165D4—122no—
166D4—124no—
167D4—121no—
168D4—121no—
169D4—121no—
170D4—121no—
171D4—242no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—61no—
175C2×C4—124no—
176C2×C4—121no—
177C2×C4—121no—
178C2×C4C2×D1211yescharacteristic
179C2×C4D3×D411yescharacteristic
180C2×C4D3×D411yescharacteristic
181C8—41no—
182C8—41no—
183C8—62no—
184C2×C3—82no—
185C2×C3#44711yescharacteristic
186C2×C3#44911yescharacteristic
187C2×C3#45311yescharacteristic
188D3—81no—
189D3—161no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—121no—
194C22—121no—
195C22—121no—
196C22—241no—
197C22—241no—
198C22—241no—
199C22#88211yescharacteristic
200C4—21no—
201C4—62no—
202C4—121no—
203C4—121no—
204C4D3×D811yescharacteristic
205C4#91111yescharacteristic
206C4#91411yescharacteristic
207C4#91611yescharacteristic
208C3#162111yescharacteristic, complemented, Sylow
209C2—82no—
210C2—241no—
211C2—481no—
212C2#474711yescharacteristic
213C2#477311yescharacteristic
214C2#478111yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#331C2
socleC22×C3#120
hypercenter#331D3
hypocenterC3#1621
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #5718C2semidirect
2. #5073C2semidirect
3. #4848C2semidirect
4. #4843C2semidirect
5. C3×#331C2semidirect
6. #4484C2semidirect
7. #4483C2semidirect
30. #331D3semidirect
104. Dic6D8semidirect
208. C3#1621semidirect

Maximal subgroups

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

subgroupindex
#57182
#50732
#48482
#48432
C3×#3312
#44842
#44832
#16213