Finite Group Databasebeta

← #69717

Subgroups — #69717

Order 384.

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

normal subgroups only

This group has 208 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#5515C211yesmaximal, characteristic
2C3×#400C211yesmaximal, characteristic, complemented
3#4970C211yesmaximal, characteristic, complemented
4#4968C211yesmaximal, characteristic
5#4528C211yesmaximal, characteristic, complemented
6#4452C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic
8#1615—31nomaximal, Sylow
9C2×C3×C4⋊C4C2211yescharacteristic
10C2×C3×C4⋊C4C2211yescharacteristic
11#930C2211yescharacteristic
12#929—21no—
13C4×#67—21no—
14#924C2211yescharacteristic
15#898—21no—
16#896—21no—
17#894—21no—
18#889—21no—
19C3×#95—21no—
20C3×#95—21no—
21C3×#95C2211yescharacteristic
22#830C2211yescharacteristic
23#830C2211yescharacteristic
24#810—21no—
25#809—21no—
26#808—21no—
27#807—21no—
28#539—31no—
29#484—31no—
30#481—31no—
31#400D311yescharacteristic, complemented
32#328—31no—
33#327—31no—
34#324—31no—
35C22×C3×C4—21no—
36C22×C3×C4—21no—
37C22×C3×C4D411yescharacteristic
38C22×C3×C4D411yescharacteristic
39C22×C3×C4C2311yescharacteristic
40C2×#67—21no—
41C2×D12D411yescharacteristic
42C2×C4×D3—21no—
43C2×Dic6D411yescharacteristic
44C3×C4⋊C4—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4—21no—
47C3×C4⋊C4—21no—
48#217—21no—
49(C2×D3)⋊C4—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52(C2×D3)⋊C4—21no—
53C4⋊Dic3D411yescharacteristic
54C4⋊Dic3D411yescharacteristic
55Dic3⋊C4—21no—
56Dic3⋊C4—21no—
57Dic3⋊C4—21no—
58Dic3⋊C4—21no—
59C4×Dic3—21no—
60C2×C3⋊C8—21no—
61C2×C3⋊C8—21no—
62#126—61no—
63#125—61no—
64#123—61no—
65#123—61no—
66C22⋊Q8—31no—
67C22⋊Q8—61no—
68#121—31no—
69C4×D4—61no—
70C2×C4⋊C4C2×D311yescharacteristic
71C2×C4⋊C4C2×D311yescharacteristic
72#107—61no—
73#106—61no—
74#103—61no—
75#102—61no—
76C22⋊C8—31no—
77C22⋊C8—31no—
78#95—21no—
79#95—21no—
80#95C2×D311yescharacteristic
81C23×C3C2×D411yescharacteristic
82C22×D3—21no—
83C2×C3×C4—21no—
84C2×C3×C4—21no—
85C2×C3×C4—21no—
86C2×C3×C4—21no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4—42no—
92C2×C3×C4—41no—
93C2×C3×C4—41no—
94C2×C3×C4C2×D411yescharacteristic
95C2×C3×C4C2×D411yescharacteristic
96#67—41no—
97#67—41no—
98C2×Dic3—21no—
99C2×Dic3—21no—
100C2×Dic3—21no—
101D12—41no—
102C4×D3—81no—
103Dic6—41no—
104C3⋊C8—41no—
105C3⋊C8—41no—
106C2×Q8—31no—
107C2×D4—31no—
108C2×D4—61no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—61no—
112C22×C4#6711yescharacteristic
113C22×C4#6711yescharacteristic
114C22×C4C22×D311yescharacteristic
115C2×C8—61no—
116C2×C8—61no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—31no—
122C4⋊C4—31no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4—61no—
131C22⋊C4—61no—
132C42—61no—
133C22×C3—21no—
134C22×C3—21no—
135C22×C3#12011yescharacteristic
136C22×C3#13511yescharacteristic
137C22×C3#13711yescharacteristic
138C2×D3—41no—
139C2×D3—41no—
140C2×D3—41no—
141C3×C4—21no—
142C3×C4—21no—
143C3×C4—41no—
144C3×C4—41no—
145C3×C4—81no—
146C3×C4—81no—
147Dic3—41no—
148Dic3—41no—
149Dic3—41no—
150C23—61no—
151C23C2×#6711yescharacteristic
152Q8—121no—
153D4—121no—
154D4—121no—
155D4—121no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—61no—
170C2×C4—241no—
171C2×C4D3×D411yescharacteristic
172C2×C4D3×D411yescharacteristic
173C8—121no—
174C8—121no—
175C2×C3—21no—
176C2×C3—21no—
177C2×C3#45211yescharacteristic
178C2×C3#45411yescharacteristic
179C2×C3#45811yescharacteristic
180D3—81no—
181C22—21no—
182C22—21no—
183C22—121no—
184C22—121no—
185C22—121no—
186C22#93711yescharacteristic
187C22#94211yescharacteristic
188C22#95011yescharacteristic
189C4—21no—
190C4—21no—
191C4—41no—
192C4—41no—
193C4—81no—
194C4—81no—
195C4—121no—
196C4—121no—
197C4—121no—
198C3#161511yescharacteristic, complemented, Sylow
199C2—21no—
200C2—21no—
201C2—241no—
202C2#504711yescharacteristic
203C2#505311yescharacteristic
204C2#506211yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#400C2
socleC22×C3#120
prime radical#930C22
hypercenter#400D3
hypocenterC3#1615
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C3×#400C2semidirect
3. #4970C2semidirect
5. #4528C2semidirect
6. #4452C2semidirect
31. #400D3semidirect
198. C3#1615semidirect

Maximal subgroups

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

subgroupindex
#55152
C3×#4002
#49702
#49682
#45282
#44522
#44512
#16153