Finite Group Databasebeta

← #68321

Subgroups — #68321

Order 384.

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

normal subgroups only

This group has 232 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#5644C211yesmaximal, characteristic
2#5043C211yesmaximal, characteristic, complemented
3#4736C211yesmaximal, characteristic, complemented
4#4674C211yesmaximal, characteristic, complemented
5C3×#328C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1612—31nomaximal, Sylow
9C3×C22⋊Q8C2211yescharacteristic
10#947—21no—
11#930C2211yescharacteristic
12C2×(C2×D3)⋊C4—21no—
13C2×(C2×D3)⋊C4C2211yescharacteristic
14C2×C4⋊Dic3C2211yescharacteristic
15#898—21no—
16#895—21no—
17#894—21no—
18#882—21no—
19#67⋊C4—21no—
20C3×C22⋊C8C2211yescharacteristic
21C3×#95C2211yescharacteristic
22#835—21no—
23C3⋊#95—21no—
24#830C2211yescharacteristic
25#821—21no—
26#818—21no—
27#808—21no—
28#538—31no—
29#483—31no—
30#480—31no—
31#397—31no—
32#328D311yescharacteristic, complemented
33#327—31no—
34#323—31no—
35C23×D3D411yescharacteristic
36C2×C3×Q8D411yescharacteristic
37C22×C3×C4—21no—
38C22×C3×C4C2311yescharacteristic
39C2×#67—21no—
40C22×Dic3—21no—
41C22×Dic3D411yescharacteristic
42C2×D12—21no—
43C2×D12—21no—
44C2×D12D411yescharacteristic
45C2×C4×D3—21no—
46C2×C3×C8—21no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4D411yescharacteristic
49C3×C22⋊C4—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52(C2×D3)⋊C4—21no—
53(C2×D3)⋊C4—21no—
54(C2×D3)⋊C4—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—42no—
57C4⋊Dic3—21no—
58C4⋊Dic3—21no—
59C4⋊Dic3D411yescharacteristic
60Dic3⋊C4—21no—
61C4×Dic3—21no—
62C2×C3⋊C8—21no—
63#126—61no—
64#124—61no—
65#123—61no—
66C22⋊Q8C2×D311yescharacteristic
67#121—31no—
68#121—61no—
69#120—61no—
70C4×D4—61no—
71C2×C4⋊C4—31no—
72C2×C22⋊C4—31no—
73C2×C22⋊C4—61no—
74#107—61no—
75#106—61no—
76#103—61no—
77#102—61no—
78C22⋊C8—31no—
79C22⋊C8C2×D311yescharacteristic
80#95—61no—
81#95C2×D311yescharacteristic
82C23×C3C2×D411yescharacteristic
83C22×D3—21no—
84C22×D3—21no—
85C22×D3—21no—
86C22×D3—21no—
87C22×D3—41no—
88C22×D3—41no—
89C3×Q8—41no—
90C2×C3×C4—21no—
91C2×C3×C4—21no—
92C2×C3×C4—41no—
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×C3×C4C2×D411yescharacteristic
97C2×C3×C4C2×D411yescharacteristic
98#67—41no—
99#67—41no—
100C2×Dic3—21no—
101C2×Dic3—21no—
102C2×Dic3—41no—
103C2×Dic3—41no—
104C2×Dic3—41no—
105C2×Dic3—41no—
106D12—41no—
107D12—81no—
108D12—81no—
109C4×D3—81no—
110C3×C8—41no—
111C3⋊C8—41no—
112C24—31no—
113C2×Q8D1211yescharacteristic
114C2×D4—31no—
115C2×D4—61no—
116C2×D4—61no—
117C2×D4—61no—
118C22×C4—21no—
119C22×C4—31no—
120C22×C4—61no—
121C22×C4—61no—
122C22×C4C22×D311yescharacteristic
123C2×C8—21no—
124C2×C8—61no—
125C4⋊C4—21no—
126C4⋊C4—31no—
127C4⋊C4—61no—
128C4⋊C4—61no—
129C4⋊C4—61no—
130C4⋊C4D1211yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—61no—
133C22⋊C4—61no—
134C22⋊C4—61no—
135C22⋊C4—61no—
136C22⋊C4—61no—
137C22⋊C4—61no—
138C22⋊C4—122no—
139C42—61no—
140C22×C3—21no—
141C22×C3—21no—
142C22×C3#12011yescharacteristic
143C22×C3#13511yescharacteristic
144C22×C3AGL(1,ℤ/8)11yescharacteristic
145C2×D3—42no—
146C2×D3—41no—
147C2×D3—41no—
148C2×D3—41no—
149C2×D3—41no—
150C2×D3—41no—
151C2×D3—41no—
152C2×D3—81no—
153C3×C4—21no—
154C3×C4—21no—
155C3×C4—41no—
156C3×C4—41no—
157C3×C4—81no—
158Dic3—41no—
159Dic3—41no—
160Dic3—81no—
161C23—61no—
162C23—61no—
163C23—61no—
164C23—61no—
165C23—121no—
166C23—121no—
167C23D3×D411yescharacteristic
168Q8—41no—
169D4—121no—
170D4—121no—
171D4—121no—
172D4—241no—
173D4—241no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—61no—
181C2×C4—61no—
182C2×C4—121no—
183C2×C4—121no—
184C2×C4—121no—
185C2×C4—121no—
186C2×C4—241no—
187C2×C4C2×D1211yescharacteristic
188C2×C4D3×D411yescharacteristic
189C8—41no—
190C8—121no—
191C2×C3—21no—
192C2×C3—21no—
193C2×C3#44911yescharacteristic
194C2×C3#45511yescharacteristic
195C2×C3#45811yescharacteristic
196D3—81no—
197D3—81no—
198C22—21no—
199C22—21no—
200C22—122no—
201C22—121no—
202C22—121no—
203C22—121no—
204C22—121no—
205C22—121no—
206C22—121no—
207C22—241no—
208C22#88211yescharacteristic
209C22#91411yescharacteristic
210C22#91911yescharacteristic
211C4—21no—
212C4—21no—
213C4—41no—
214C4—41no—
215C4—81no—
216C4—121no—
217C4—121no—
218C4—241no—
219C3#161211yescharacteristic, complemented, Sylow
220C2—21no—
221C2—21no—
222C2—241no—
223C2—241no—
224C2#474211yescharacteristic
225C2#481011yescharacteristic
226C2#482511yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3#120
prime radical#5644C2
hypercenter#328D3
hypocenterC3#1612
LCSG ▷ C22×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. #5043C2semidirect
3. #4736C2semidirect
4. #4674C2semidirect
5. C3×#328C2semidirect
6. #4468C2semidirect
7. #4451C2semidirect
32. #328D3semidirect
219. C3#1612semidirect

Maximal subgroups

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

subgroupindex
#56442
#50432
#47362
#46742
C3×#3282
#44682
#44512
#16123