Finite Group Databasebeta

← #76572

Subgroups — #76572

Order 384.

492 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#947C211yesmaximal, characteristic
2C2×#944C211yesmaximal, characteristic, complemented
3C3×#391C211yesmaximal, characteristic, complemented
4#4994C212yesmaximal, complemented
5#4940C212yesmaximal, complemented
6#4673C212yesmaximal, complemented
7#4672C212yesmaximal
8#4670C212yesmaximal, complemented
9#4653C212yesmaximal, complemented
10#2731—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×D12C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C42C2211yescharacteristic
16#947—24no—
17#944—24no—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×C4⋊Dic3C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C3×#95C2212yes—
30C3×#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C2×C4⋊Q8—31no—
35C2×#124—31no—
36#396—32no—
37#394—32no—
38#393—32no—
39#391D311yescharacteristic, complemented
40#388—32no—
41#388—32no—
42#387—32no—
43C23×D3C2312yes—
44C2×C3×Q8—41no—
45C2×C3×Q8—41no—
46C2×C3×Q8D414yes—
47C22×C3×C4C2312yes—
48C22×C3×C4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C2×D12—22no—
56C2×D12—41no—
57C2×D12—41no—
58C2×Dic6—22no—
59C2×Dic6—41no—
60C2×Dic6—41no—
61C3×C4⋊C4—41no—
62C3×C42—41no—
63(C2×D3)⋊C4—24no—
64(C2×D3)⋊C4—24no—
65(C2×D3)⋊C4—24no—
66(C2×D3)⋊C4—24no—
67(C2×D3)⋊C4—42no—
68(C2×D3)⋊C4—42no—
69C4⋊Dic3—24no—
70Dic3⋊C4—24no—
71Dic3⋊C4—24no—
72C4×Dic3—22no—
73C4×Dic3—22no—
74C22×Q8—31no—
75C22×Q8C2×D311yescharacteristic
76C22×D4—31no—
77C4⋊Q8—64no—
78#124—64no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4C2×D311yescharacteristic
83C2×C22⋊C4—32no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C42—31no—
90C2×C42—31no—
91C2×C42C2×D311yescharacteristic
92#95—32no—
93#95—32no—
94#95—32no—
95#95C2×D312yes—
96#95C2×D312yes—
97C23×C3C2411yescharacteristic
98C22×D3—22no—
99C22×D3—22no—
100C22×D3—22no—
101C22×D3—22no—
102C22×D3—22no—
103C22×D3—22no—
104C22×D3—22no—
105C3×Q8—44no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4C2×D412yes—
119C2×C3×C4C2×D412yes—
120C2×C3×C4C2×D412yes—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3#4114yes—
130D12—44no—
131Dic6—44no—
132C24—32no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—62no—
136C2×Q8—121no—
137C2×Q8—121no—
138C2×Q8D1214yes—
139C2×D4—62no—
140C2×D4—121no—
141C2×D4—121no—
142C22×C4—32no—
143C22×C4—32no—
144C22×C4—32no—
145C22×C4C22×D312yes—
146C22×C4C22×D312yes—
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D311yescharacteristic
150C4⋊C4—41no—
151C4⋊C4—64no—
152C4⋊C4—64no—
153C4⋊C4—64no—
154C22⋊C4—64no—
155C22⋊C4—64no—
156C22⋊C4—64no—
157C22⋊C4—64no—
158C22⋊C4—122no—
159C22⋊C4—122no—
160C42—41no—
161C42—62no—
162C42—62no—
163C22×C3C22×D411yescharacteristic
164C22×C3C2×#4112yes—
165C22×C3H(5,𝔽2)11yescharacteristic
166C22×C3H(5,𝔽2)11yescharacteristic
167C22×C32−1+412yes—
168C2×D3—42no—
169C2×D3—42no—
170C2×D3—42no—
171C2×D3—42no—
172C2×D3—42no—
173C2×D3—42no—
174C2×D3—42no—
175C3×C4—22no—
176C3×C4—22no—
177C3×C4—22no—
178C3×C4—42no—
179C3×C4—42no—
180Dic3—24no—
181Dic3—42no—
182Dic3—42no—
183C23—62no—
184C23—62no—
185C23—62no—
186C23—62no—
187C23—62no—
188C23—62no—
189C23—62no—
190C23C23×D311yescharacteristic
191Q8—44no—
192Q8—124no—
193D4—124no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—34no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4C2×D1212yes—
216C2×C4C2×D1212yes—
217C2×C4C2×D1212yes—
218C2×C3#53411yescharacteristic
219C2×C3#54212yes—
220C2×C3#54812yes—
221C2×C3#55511yescharacteristic
222C2×C3#56311yescharacteristic
223D3—82no—
224C22—122no—
225C22—122no—
226C22—122no—
227C22—122no—
228C22—122no—
229C22—122no—
230C22—122no—
231C22C22×D1211yescharacteristic
232C22#100411yescharacteristic
233C22#100411yescharacteristic
234C22#100712yes—
235C22D3×#4112yes—
236C4—22no—
237C4—22no—
238C4—22no—
239C4—42no—
240C4—42no—
241C4—64no—
242C4—122no—
243C4—122no—
244C3#273111yescharacteristic, complemented, Sylow
245C2—242no—
246C2#549311yescharacteristic
247C2#557612yes—
248C2#564012yes—
249C2#567011yescharacteristic
250C2#573511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#391C2
prime radicalC22×D12C22
hypercenter#391D3
hypocenterC3#2731
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#944C2semidirect
3. C3×#391C2semidirect
4. #4994C2semidirect
5. #4940C2semidirect
6. #4673C2semidirect
8. #4670C2semidirect
9. #4653C2semidirect
39. #391D3semidirect
244. C3#2731semidirect

Maximal subgroups

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

subgroupindex
C2×#9472
C2×#9442
C3×#3912
#49942
#49402
#46732
#46722
#46702
#46532
#27313