Finite Group Databasebeta

← #68328

Subgroups — #68328

Order 384.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#5645C211yesmaximal, characteristic
2#5043C211yesmaximal, characteristic, complemented
3#4736C211yesmaximal, characteristic
4#4661C211yesmaximal, characteristic, complemented
5C3×#328C211yesmaximal, characteristic, complemented
6#4469C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic
8#1609—31nomaximal, Sylow
9C3×C22⋊Q8C2211yescharacteristic
10#946—21no—
11#930C2211yescharacteristic
12C2×C4⋊Dic3—21no—
13C2×C4⋊Dic3C2211yescharacteristic
14C2×Dic3⋊C4C2211yescharacteristic
15#898—21no—
16#894—21no—
17#890—21no—
18#886—21no—
19#67⋊C4—21no—
20C3×C22⋊C8C2211yescharacteristic
21C3×#95C2211yescharacteristic
22#835—21no—
23C3⋊#95—21no—
24#830C2211yescharacteristic
25#821—21no—
26#818—21no—
27#808—21no—
28#539—31no—
29#483—31no—
30#480—31no—
31#398—31no—
32#328D311yescharacteristic, complemented
33#327—31no—
34#324—31no—
35C2×C3×Q8D411yescharacteristic
36C22×C3×C4—21no—
37C22×C3×C4C2311yescharacteristic
38C2×#67—21no—
39C22×Dic3—21no—
40C22×Dic3D411yescharacteristic
41C22×Dic3D411yescharacteristic
42C2×D12D411yescharacteristic
43C2×C4×D3—21no—
44C2×C3×C8—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4D411yescharacteristic
47C3×C22⋊C4—21no—
48(C2×D3)⋊C4—21no—
49(C2×D3)⋊C4—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52C4⋊Dic3—21no—
53C4⋊Dic3—21no—
54C4⋊Dic3—21no—
55C4⋊Dic3—21no—
56C4⋊Dic3—42no—
57C4⋊Dic3D411yescharacteristic
58Dic3⋊C4—21no—
59Dic3⋊C4—21no—
60Dic3⋊C4—21no—
61C4×Dic3—21no—
62C2×C3⋊C8—21no—
63#126—61no—
64#125—61no—
65#123—61no—
66#123—61no—
67C22⋊Q8—61no—
68C22⋊Q8C2×D311yescharacteristic
69#121—31no—
70C4×D4—61no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4—31no—
73C2×C4⋊C4—61no—
74#107—61no—
75#106—61no—
76#103—61no—
77#102—61no—
78C22⋊C8—31no—
79C22⋊C8C2×D311yescharacteristic
80#95—61no—
81#95C2×D311yescharacteristic
82C23×C3C2×D411yescharacteristic
83C22×D3—21no—
84C3×Q8—41no—
85C2×C3×C4—22no—
86C2×C3×C4—21no—
87C2×C3×C4—21no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4C2×D411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93#67—41no—
94#67—41no—
95C2×Dic3—21no—
96C2×Dic3—21no—
97C2×Dic3—21no—
98C2×Dic3—21no—
99C2×Dic3—21no—
100C2×Dic3—41no—
101C2×Dic3—41no—
102C2×Dic3—41no—
103C2×Dic3—41no—
104D12—41no—
105C4×D3—81no—
106C3×C8—41no—
107C3⋊C8—41no—
108C2×Q8D1211yescharacteristic
109C2×D4—31no—
110C2×D4—61no—
111C22×C4—21no—
112C22×C4—31no—
113C22×C4—31no—
114C22×C4—61no—
115C22×C4—61no—
116C22×C4C22×D311yescharacteristic
117C2×C8—21no—
118C2×C8—61no—
119C4⋊C4—21no—
120C4⋊C4—31no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C4⋊C4—61no—
128C4⋊C4—122no—
129C4⋊C4D1211yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—61no—
132C22⋊C4—61no—
133C22⋊C4—61no—
134C22⋊C4—61no—
135C42—61no—
136C22×C3—21no—
137C22×C3—21no—
138C22×C3#12011yescharacteristic
139C22×C3#13511yescharacteristic
140C22×C3AGL(1,ℤ/8)11yescharacteristic
141C2×D3—41no—
142C2×D3—41no—
143C2×D3—41no—
144C3×C4—21no—
145C3×C4—21no—
146C3×C4—42no—
147C3×C4—41no—
148C3×C4—41no—
149Dic3—41no—
150Dic3—41no—
151Dic3—41no—
152Dic3—81no—
153C23—61no—
154C23D3×D411yescharacteristic
155Q8—41no—
156D4—121no—
157D4—121no—
158D4—121no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—61no—
166C2×C4—61no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—61no—
170C2×C4—121no—
171C2×C4—121no—
172C2×C4—121no—
173C2×C4—121no—
174C2×C4—241no—
175C2×C4C2×D1211yescharacteristic
176C2×C4D3×D411yescharacteristic
177C8—41no—
178C8—121no—
179C2×C3—21no—
180C2×C3—21no—
181C2×C3#44911yescharacteristic
182C2×C3#45611yescharacteristic
183C2×C3#45811yescharacteristic
184D3—81no—
185C22—21no—
186C22—21no—
187C22—121no—
188C22—121no—
189C22—121no—
190C22#88211yescharacteristic
191C22#91411yescharacteristic
192C22#91911yescharacteristic
193C4—21no—
194C4—21no—
195C4—42no—
196C4—41no—
197C4—41no—
198C4—121no—
199C4—121no—
200C4—121no—
201C4—241no—
202C3#160911yescharacteristic, complemented, Sylow
203C2—21no—
204C2—21no—
205C2—241no—
206C2#474211yescharacteristic
207C2#481011yescharacteristic
208C2#482611yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3#120
prime radical#930C22
hypercenter#328D3
hypocenterC3#1609
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #5043C2semidirect
4. #4661C2semidirect
5. C3×#328C2semidirect
6. #4469C2semidirect
32. #328D3semidirect
202. C3#1609semidirect

Maximal subgroups

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

subgroupindex
#56452
#50432
#47362
#46612
C3×#3282
#44692
#44512
#16093