Finite Group Databasebeta

← #68358

Subgroups — #68358

Order 384.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#5684C211yesmaximal, characteristic
2#5055C211yesmaximal, characteristic, complemented
3#4714C211yesmaximal, characteristic, complemented
4#4703C211yesmaximal, characteristic, complemented
5C3×#329C211yesmaximal, characteristic, complemented
6#4465C211yesmaximal, characteristic, complemented
7#4464C211yesmaximal, characteristic, complemented
8#1662—31nomaximal, Sylow
9C3×#124C2211yescharacteristic
10#946—21no—
11#939—21no—
12#886—21no—
13#884—21no—
14#883—21no—
15#878—21no—
16#875C2211yescharacteristic
17#870C2211yescharacteristic
18#869C2211yescharacteristic
19C3×#97C2211yescharacteristic
20C3×#97C2211yescharacteristic
21#835—21no—
22#832—21no—
23#821—21no—
24#816—21no—
25#816—21no—
26#816—21no—
27#803C2211yescharacteristic
28#562—31no—
29#491—31no—
30#489—31no—
31#489—31no—
32#330—31no—
33#329—31no—
34#329D311yescharacteristic, complemented
35C2×C3×Q8D411yescharacteristic
36C2×C3×D4D411yescharacteristic
37C2×#67—21no—
38C2×#67—21no—
39C22×Dic3—21no—
40C2×D12D411yescharacteristic
41C2×C4×D3—21no—
42C2×Dic6—21no—
43C2×Dic6D411yescharacteristic
44C2×C3×C8—21no—
45C2×C3×C8—21no—
46C3×C22⋊C4—21no—
47C3×C22⋊C4—21no—
48C3×C42C2311yescharacteristic
49#217—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52(C2×D3)⋊C4—21no—
53C4⋊Dic3—21no—
54C4⋊Dic3—21no—
55C4⋊Dic3—21no—
56C4⋊Dic3D411yescharacteristic
57C4⋊Dic3D411yescharacteristic
58Dic3⋊C4—21no—
59Dic3⋊C4—21no—
60Dic3⋊C4—21no—
61C2×C3⋊C8—21no—
62C4⋊Q8—31no—
63#125—31no—
64#124—31no—
65#124C2×D311yescharacteristic
66#123—61no—
67#123—61no—
68C22⋊Q8—61no—
69C22⋊Q8—61no—
70#121—61no—
71#121—61no—
72#103—61no—
73#103—61no—
74#103—61no—
75#103—61no—
76#102—61no—
77#102—61no—
78#97—31no—
79#97C2×D311yescharacteristic
80#97C2×D311yescharacteristic
81C23×C3—21no—
82C22×D3—21no—
83C3×Q8—41no—
84C3×D4—41no—
85C2×C3×C4—21no—
86C2×C3×C4C2×D411yescharacteristic
87C2×C3×C4C2×D411yescharacteristic
88C2×C3×C4C2×D411yescharacteristic
89#67—81no—
90#67—81no—
91C2×Dic3—21no—
92C2×Dic3—21no—
93C2×Dic3—21no—
94C2×Dic3—41no—
95C2×Dic3—81no—
96D12—41no—
97C4×D3—81no—
98Dic6—41no—
99Dic6—42no—
100C3×C8—41no—
101C3×C8—41no—
102C3⋊C8—41no—
103C2×Q8—31no—
104C2×Q8—61no—
105C2×Q8D1211yescharacteristic
106C2×D4—31no—
107C2×D4—61no—
108C2×D4—61no—
109C2×D4D1211yescharacteristic
110C22×C4—61no—
111C22×C4—61no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—61no—
115C4⋊C4—31no—
116C4⋊C4—31no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C22⋊C4—21no—
124C22⋊C4—21no—
125C22⋊C4—61no—
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C42C22×D311yescharacteristic
130C22×C3—41no—
131C22×C3—41no—
132C22×C3—41no—
133C22×C3#12011yescharacteristic
134C2×D3—41no—
135C2×D3—41no—
136C2×D3—41no—
137C3×C4—21no—
138C3×C4—21no—
139C3×C4—21no—
140C3×C4—41no—
141Dic3—41no—
142Dic3—41no—
143Dic3—41no—
144Dic3—81no—
145C23—21no—
146C23—61no—
147Q8—41no—
148Q8—121no—
149Q8—122no—
150D4—41no—
151D4—121no—
152D4—241no—
153D4—241no—
154C2×C4—21no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—61no—
158C2×C4—121no—
159C2×C4—241no—
160C2×C4—241no—
161C2×C4C2×D1211yescharacteristic
162C2×C4D3×D411yescharacteristic
163C2×C4D3×D411yescharacteristic
164C8—41no—
165C8—41no—
166C8—121no—
167C2×C3—81no—
168C2×C3#45411yescharacteristic
169C2×C3#45511yescharacteristic
170C2×C3#45511yescharacteristic
171D3—81no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—121no—
176C22—121no—
177C22—121no—
178C22#88211yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—41no—
183C4—121no—
184C4—121no—
185C4—121no—
186C4—241no—
187C3#166211yescharacteristic, complemented, Sylow
188C2—81no—
189C2—241no—
190C2#474611yescharacteristic
191C2#482511yescharacteristic
192C2#482711yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#329C2
socleC22×C3#120
prime radical#5684C2
hypercenter#329D3
hypocenterC3#1662
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #5055C2semidirect
3. #4714C2semidirect
4. #4703C2semidirect
5. C3×#329C2semidirect
6. #4465C2semidirect
7. #4464C2semidirect
34. #329D3semidirect
187. C3#1662semidirect

Maximal subgroups

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

subgroupindex
#56842
#50552
#47142
#47032
C3×#3292
#44652
#44642
#16623