Finite Group Databasebeta

← #68315

Subgroups — #68315

Order 384.

218 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 61 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#5624C211yesmaximal, characteristic
2#4718C211yesmaximal, characteristic, complemented
3#4651C211yesmaximal, characteristic, complemented
4C3×#328C211yesmaximal, characteristic, complemented
5#4538C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic
7#4451C211yesmaximal, characteristic
8#1521—31nomaximal, Sylow
9C3×C22⋊Q8C2211yescharacteristic
10#947C411yescharacteristic, complemented
11Q8×Dic3C411yescharacteristic, complemented
12#930C2211yescharacteristic
13C2×C4×Dic3C2211yescharacteristic
14#898—21no—
15#893C411yescharacteristic, complemented
16#892—21no—
17#890C411yescharacteristic, complemented
18#886—21no—
19#67⋊C4—21no—
20C3×C22⋊C8C2211yescharacteristic
21C3×#95C2211yescharacteristic
22C3⋊#95—21no—
23C3⋊#95—21no—
24C3⋊#95C2211yescharacteristic
25#830C2211yescharacteristic
26Dic3⋊C8—21no—
27C8×Dic3—21no—
28#529—31no—
29#431—31no—
30#383—31no—
31#328—31no—
32#328D311yescharacteristic, complemented
33#327—31no—
34#327—31no—
35C2×C3×Q8C2×C411yescharacteristic
36C22×C3×C4—21no—
37C22×C3×C4C2311yescharacteristic
38C2×#67—21no—
39C22×Dic3—21no—
40C22×Dic3D411yescharacteristic
41C22×Dic3D411yescharacteristic
42C2×D12C2×C411yescharacteristic
43C2×C4×D3—21no—
44C2×C3×C8—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4C2×C411yescharacteristic
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⋊Dic3C2×C411yescharacteristic
55Dic3⋊C4—21no—
56C4×Dic3—21no—
57C4×Dic3C2×C411yescharacteristic
58C4×Dic3C2×C411yescharacteristic
59C4×Dic3D411yescharacteristic
60C4×Dic3D411yescharacteristic
61C2×C3⋊C8—21no—
62#126—61no—
63#125—31no—
64#124—31no—
65#123—61no—
66C22⋊Q8C2×D311yescharacteristic
67#121—31no—
68C4×Q8—31no—
69C4×D4—31no—
70C4×D4—61no—
71#117—61no—
72C2×C42—31no—
73C4⋊C8—61no—
74C22⋊C8—31no—
75C22⋊C8C2×D311yescharacteristic
76C4×C8—61no—
77#95—31no—
78#95—61no—
79#95—61no—
80#95C2×D311yescharacteristic
81C23×C3C2×D411yescharacteristic
82C22×D3—21no—
83C3×Q8—41no—
84C2×C3×C4—21no—
85C2×C3×C4—21no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4C22×C411yescharacteristic
91C2×C3×C4C2×D411yescharacteristic
92#67—41no—
93#67—41no—
94C2×Dic3—21no—
95C2×Dic3—21no—
96C2×Dic3—21no—
97C2×Dic3—41no—
98C2×Dic3—41no—
99C2×Dic3—41no—
100C2×Dic3—41no—
101C2×Dic3C22⋊C411yescharacteristic
102C2×Dic3C22⋊C411yescharacteristic
103C2×Dic3C22⋊C411yescharacteristic
104C2×Dic3C22⋊C411yescharacteristic
105D12—41no—
106C4×D3—42no—
107C3×C8—41no—
108C3⋊C8—41no—
109C2×Q8C4×D311yescharacteristic
110C2×D4—31no—
111C2×D4—61no—
112C22×C4—21no—
113C22×C4—31no—
114C22×C4—31no—
115C22×C4—61no—
116C22×C4—61no—
117C22×C4C22×D311yescharacteristic
118C2×C8—21no—
119C2×C8—61no—
120C4⋊C4—21no—
121C4⋊C4—31no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4C4×D311yescharacteristic
126C22⋊C4—21no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4—61no—
131C42—31no—
132C42—31no—
133C42—31no—
134C42—31no—
135C42—61no—
136C22×C3—21no—
137C22×C3—21no—
138C22×C3C2×C22⋊C411yescharacteristic
139C22×C3#13511yescharacteristic
140C22×C3#13511yescharacteristic
141C2×D3—41no—
142C2×D3—41no—
143C2×D3—41no—
144C3×C4—21no—
145C3×C4—21no—
146C3×C4—41no—
147C3×C4—41no—
148C3×C4—81no—
149Dic3—21no—
150Dic3—21no—
151Dic3—21no—
152Dic3—21no—
153Dic3—41no—
154Dic3—81no—
155C23—61no—
156C23D3×D411yescharacteristic
157Q8—41no—
158D4—121no—
159D4—121no—
160D4—121no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—31no—
164C2×C4—31no—
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—122no—
175C2×C4—121no—
176C2×C4—121no—
177C2×C4—121no—
178C2×C4—121no—
179C2×C4C2×C4×D311yescharacteristic
180C2×C4D3×D411yescharacteristic
181C8—41no—
182C8—121no—
183C2×C3—21no—
184C2×C3—21no—
185C2×C3#41011yescharacteristic
186C2×C3#41611yescharacteristic
187C2×C3#42111yescharacteristic
188D3—81no—
189C22—21no—
190C22—21no—
191C22—121no—
192C22—121no—
193C22—121no—
194C22D3×C22⋊C411yescharacteristic
195C22#91611yescharacteristic
196C22#91911yescharacteristic
197C4—21no—
198C4—21no—
199C4—41no—
200C4—41no—
201C4—61no—
202C4—61no—
203C4—61no—
204C4—61no—
205C4—81no—
206C4—121no—
207C4—241no—
208C3#152111yescharacteristic, complemented, Sylow
209C2—21no—
210C2—21no—
211C2—241no—
212C2#474011yescharacteristic
213C2#480411yescharacteristic
214C2#482111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3C2×C22⋊C4
prime radical#930C22
cosocleC22×C3×C4C23
hypercenter#328D3
hypocenterC3#1521
LCSG ▷ C2×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. #4718C2semidirect
3. #4651C2semidirect
4. C3×#328C2semidirect
5. #4538C2semidirect
10. #947C4semidirect
11. Q8×Dic3C4semidirect
15. #893C4semidirect
17. #890C4semidirect
32. #328D3semidirect
208. C3#1521semidirect

Maximal subgroups

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

subgroupindex
#56242
#47182
#46512
C3×#3282
#45382
#44702
#44512
#15213