Finite Group Databasebeta

← #68357

Subgroups — #68357

Order 384.

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

normal subgroups only

This group has 208 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#4715C211yesmaximal, characteristic, complemented
4#4702C211yesmaximal, characteristic, complemented
5C3×#329C211yesmaximal, characteristic, complemented
6#4465C211yesmaximal, characteristic, complemented
7#4464C211yesmaximal, characteristic, complemented
8#1661—31nomaximal, Sylow
9C3×#124C2211yescharacteristic
10#946—21no—
11#939—21no—
12C2×Dic12—21no—
13C2×#204—21no—
14#886—21no—
15#884—21no—
16#883—21no—
17#878—21no—
18#875C2211yescharacteristic
19#870C2211yescharacteristic
20#869C2211yescharacteristic
21C3×#97C2211yescharacteristic
22C3×#97C2211yescharacteristic
23#835—21no—
24#832—21no—
25#818—21no—
26#817—21no—
27#803C2211yescharacteristic
28#562—31no—
29#501—31no—
30#497—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—
62Dic12—42no—
63#204—42no—
64C2×Q16—61no—
65C2×SD16—61no—
66C4⋊Q8—31no—
67#125—31no—
68#124—31no—
69#124C2×D311yescharacteristic
70#123—61no—
71#123—61no—
72C22⋊Q8—61no—
73C22⋊Q8—61no—
74#121—61no—
75#121—61no—
76#107—61no—
77#106—61no—
78#103—61no—
79#102—61no—
80#97—31no—
81#97C2×D311yescharacteristic
82#97C2×D311yescharacteristic
83C23×C3—21no—
84C22×D3—21no—
85C3×Q8—41no—
86C3×D4—41no—
87C2×C3×C4—21no—
88C2×C3×C4C2×D411yescharacteristic
89C2×C3×C4C2×D411yescharacteristic
90C2×C3×C4C2×D411yescharacteristic
91#67—81no—
92#67—81no—
93C2×Dic3—21no—
94C2×Dic3—21no—
95C2×Dic3—21no—
96C2×Dic3—41no—
97C2×Dic3—81no—
98D12—41no—
99C4×D3—81no—
100Dic6—41no—
101Dic6—41no—
102Dic6—41no—
103C3×C8—22no—
104C3×C8—22no—
105C3⋊C8—41no—
106C2×Q8—31no—
107C2×Q8—61no—
108C2×Q8D1211yescharacteristic
109C2×D4—31no—
110C2×D4—61no—
111C2×D4—61no—
112C2×D4D1211yescharacteristic
113C22×C4—61no—
114C22×C4—61no—
115Q16—122no—
116SD16—122no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—61no—
120C4⋊C4—31no—
121C4⋊C4—31no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C4⋊C4—61no—
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—61no—
131C22⋊C4—61no—
132C22⋊C4—61no—
133C22⋊C4—61no—
134C42C22×D311yescharacteristic
135C22×C3—41no—
136C22×C3—41no—
137C22×C3—41no—
138C22×C3#12011yescharacteristic
139C2×D3—41no—
140C2×D3—41no—
141C2×D3—41no—
142C3×C4—21no—
143C3×C4—21no—
144C3×C4—21no—
145C3×C4—41no—
146Dic3—41no—
147Dic3—41no—
148Dic3—41no—
149Dic3—81no—
150C23—21no—
151C23—61no—
152Q8—41no—
153Q8—121no—
154Q8—121no—
155Q8—121no—
156D4—41no—
157D4—121no—
158D4—241no—
159D4—241no—
160C2×C4—21no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—121no—
165C2×C4—241no—
166C2×C4—241no—
167C2×C4C2×D1211yescharacteristic
168C2×C4D3×D411yescharacteristic
169C2×C4D3×D411yescharacteristic
170C8—22no—
171C8—22no—
172C8—121no—
173C2×C3—81no—
174C2×C3#45311yescharacteristic
175C2×C3#45511yescharacteristic
176C2×C3#45611yescharacteristic
177D3—81no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—121no—
182C22—121no—
183C22—121no—
184C22#88211yescharacteristic
185C4—21no—
186C4—21no—
187C4—21no—
188C4—41no—
189C4—121no—
190C4—121no—
191C4—121no—
192C4—241no—
193C3#166111yescharacteristic, complemented, Sylow
194C2—81no—
195C2—241no—
196C2#474611yescharacteristic
197C2#482211yescharacteristic
198C2#482611yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#329C2
socleC22×C3#120
prime radical#5684C2
hypercenter#329D3
hypocenterC3#1661
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. #4715C2semidirect
4. #4702C2semidirect
5. C3×#329C2semidirect
6. #4465C2semidirect
7. #4464C2semidirect
34. #329D3semidirect
193. C3#1661semidirect

Maximal subgroups

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

subgroupindex
#56842
#50552
#47152
#47022
C3×#3292
#44652
#44642
#16613