Finite Group Databasebeta

← #68312

Subgroups — #68312

Order 384.

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

normal subgroups only

This group has 222 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#4719C211yesmaximal, characteristic, complemented
3#4650C211yesmaximal, characteristic, complemented
4C3×#328C211yesmaximal, characteristic, complemented
5#4538C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic
7#4451C211yesmaximal, characteristic
8#1512—31nomaximal, Sylow
9C3×C22⋊Q8C2211yescharacteristic
10#947C411yescharacteristic, complemented
11Q8×Dic3C411yescharacteristic, complemented
12#930C2211yescharacteristic
13C2×Dic3⋊C4—21no—
14C2×C4×Dic3C2211yescharacteristic
15#898—21no—
16#893C411yescharacteristic, complemented
17#892—21no—
18#890C411yescharacteristic, complemented
19#886—21no—
20#67⋊C4—21no—
21C3×C22⋊C8C2211yescharacteristic
22C3×#95C2211yescharacteristic
23C3⋊#95—21no—
24C3⋊#95C2211yescharacteristic
25#830C2211yescharacteristic
26#815—21no—
27Dic3⋊C8—21no—
28#529—31no—
29#432—31no—
30#382—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—
56Dic3⋊C4—42no—
57C4×Dic3—21no—
58C4×Dic3C2×C411yescharacteristic
59C4×Dic3C2×C411yescharacteristic
60C4×Dic3D411yescharacteristic
61C4×Dic3D411yescharacteristic
62C2×C3⋊C8—21no—
63#126—61no—
64#125—31no—
65#124—31no—
66#123—61no—
67C22⋊Q8C2×D311yescharacteristic
68#121—31no—
69C4×Q8—31no—
70C4×D4—31no—
71C4×D4—61no—
72#117—61no—
73C2×C4⋊C4—61no—
74C2×C42—31no—
75C4⋊C8—61no—
76C22⋊C8—31no—
77C22⋊C8C2×D311yescharacteristic
78#97—61no—
79#95—31no—
80#95—61no—
81#95C2×D311yescharacteristic
82C23×C3C2×D411yescharacteristic
83C22×D3—21no—
84C3×Q8—41no—
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×C4C22×C411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93#67—41no—
94#67—41no—
95C2×Dic3—21no—
96C2×Dic3—21no—
97C2×Dic3—21no—
98C2×Dic3—41no—
99C2×Dic3—41no—
100C2×Dic3—41no—
101C2×Dic3—41no—
102C2×Dic3C22⋊C411yescharacteristic
103C2×Dic3C22⋊C411yescharacteristic
104C2×Dic3C22⋊C411yescharacteristic
105C2×Dic3C22⋊C411yescharacteristic
106D12—41no—
107C4×D3—42no—
108C3×C8—41no—
109C3⋊C8—41no—
110C2×Q8C4×D311yescharacteristic
111C2×D4—31no—
112C2×D4—61no—
113C22×C4—21no—
114C22×C4—31no—
115C22×C4—31no—
116C22×C4—61no—
117C22×C4—61no—
118C22×C4C22×D311yescharacteristic
119C2×C8—21no—
120C2×C8—61no—
121C4⋊C4—21no—
122C4⋊C4—31no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—122no—
127C4⋊C4C4×D311yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—61no—
130C22⋊C4—61no—
131C22⋊C4—61no—
132C22⋊C4—61no—
133C42—31no—
134C42—31no—
135C42—31no—
136C42—31no—
137C42—61no—
138C22×C3—21no—
139C22×C3—21no—
140C22×C3C2×C22⋊C411yescharacteristic
141C22×C3AGL(1,ℤ/8)11yescharacteristic
142C22×C3#13711yescharacteristic
143C2×D3—41no—
144C2×D3—41no—
145C2×D3—41no—
146C3×C4—21no—
147C3×C4—21no—
148C3×C4—41no—
149C3×C4—41no—
150C3×C4—81no—
151Dic3—21no—
152Dic3—21no—
153Dic3—21no—
154Dic3—21no—
155Dic3—41no—
156Dic3—81no—
157C23—61no—
158C23D3×D411yescharacteristic
159Q8—41no—
160D4—121no—
161D4—121no—
162D4—121no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—61no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—122no—
177C2×C4—121no—
178C2×C4—121no—
179C2×C4—121no—
180C2×C4—121no—
181C2×C4C2×C4×D311yescharacteristic
182C2×C4D3×D411yescharacteristic
183C8—41no—
184C8—121no—
185C2×C3—21no—
186C2×C3—21no—
187C2×C3#41011yescharacteristic
188C2×C3#41711yescharacteristic
189C2×C3#42111yescharacteristic
190D3—81no—
191C22—21no—
192C22—21no—
193C22—121no—
194C22—121no—
195C22—121no—
196C22D3×C22⋊C411yescharacteristic
197C22#91411yescharacteristic
198C22#91811yescharacteristic
199C4—21no—
200C4—21no—
201C4—41no—
202C4—41no—
203C4—61no—
204C4—61no—
205C4—61no—
206C4—61no—
207C4—81no—
208C4—121no—
209C4—241no—
210C3#151211yescharacteristic, complemented, Sylow
211C2—21no—
212C2—21no—
213C2—241no—
214C2#474011yescharacteristic
215C2#480311yescharacteristic
216C2#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#1512
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. #4719C2semidirect
3. #4650C2semidirect
4. C3×#328C2semidirect
5. #4538C2semidirect
10. #947C4semidirect
11. Q8×Dic3C4semidirect
16. #893C4semidirect
18. #890C4semidirect
32. #328D3semidirect
210. C3#1512semidirect

Maximal subgroups

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

subgroupindex
#56242
#47192
#46502
C3×#3282
#45382
#44702
#44512
#15123