Finite Group Databasebeta

← #77574

Subgroups — #77574

Order 384.

568 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 129 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×#883C212yesmaximal, complemented
2C3×#397C211yesmaximal, characteristic, complemented
3#4994C212yesmaximal, complemented
4#4960C211yesmaximal, characteristic
5#4958C212yesmaximal, complemented
6#4946C211yesmaximal, characteristic, complemented
7#4667C212yesmaximal, complemented
8#4665C212yesmaximal, complemented
9#4660C212yesmaximal, complemented
10#2693—31nomaximal, Sylow
11C22×#67C2212yes—
12C23×Dic3C2211yescharacteristic
13C22×D12C2211yescharacteristic
14C22×C4×D3C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×#217C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×C4⋊Dic3C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26#883—28no—
27C3×#95C2212yes—
28C3×#95C2211yescharacteristic
29C3⋊#95C2211yescharacteristic
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×#123—32no—
34#400—32no—
35#397D311yescharacteristic, complemented
36#396—32no—
37#392—31no—
38#386—32no—
39#386—32no—
40#385—31no—
41#380—32no—
42C24×C3C2311yescharacteristic
43C23×D3C2312yes—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C2×#67—24no—
49C2×#67—24no—
50C2×#67—42no—
51C22×Dic3—42no—
52C22×Dic3—41no—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C2×D12—24no—
57C2×D12—41no—
58C2×C4×D3—24no—
59C2×C4×D3—24no—
60C2×C4×D3—42no—
61C3×C4⋊C4—41no—
62C3×C22⋊C4—24no—
63C3×C22⋊C4—41no—
64#217D414yescomplemented
65(C2×D3)⋊C4—24no—
66(C2×D3)⋊C4—42no—
67(C2×D3)⋊C4—42no—
68(C2×D3)⋊C4—42no—
69C4⋊Dic3—24no—
70Dic3⋊C4—24no—
71Dic3⋊C4—42no—
72C22×D4—32no—
73C22×D4—31no—
74C23×C4—31no—
75C23×C4—32no—
76#123—68no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4C2×D311yescharacteristic
81C2×C22⋊C4—31no—
82C2×C22⋊C4—32no—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4C2×D312yes—
87C2×C22⋊C4C2×D311yescharacteristic
88#95—32no—
89#95—32no—
90#95—32no—
91#95—31no—
92#95C2×D311yescharacteristic
93#95C2×D312yes—
94C23×C3—22no—
95C23×C3—21no—
96C23×C3—22no—
97C23×C3—21no—
98C23×C3C2×D412yes—
99C23×C3C2411yescharacteristic
100C22×D3—22no—
101C22×D3—22no—
102C22×D3—22no—
103C22×D3—22no—
104C22×D3—22no—
105C22×D3—22no—
106C22×D3#4114yes—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—22no—
117#67—48no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—44no—
127C2×Dic3C2×D414yes—
128D12—44no—
129C4×D3—48no—
130C24—32no—
131C24C22×D311yescharacteristic
132C2×D4—64no—
133C2×D4—64no—
134C2×D4—64no—
135C2×D4—121no—
136C2×D4—122no—
137C22×C4—32no—
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—64no—
141C22×C4—64no—
142C22×C4—122no—
143C22×C4—121no—
144C22×C4—122no—
145C22×C4C22×D311yescharacteristic
146C22×C4C22×D312yes—
147C22×C4C22×D312yes—
148C22×C4C22×D311yescharacteristic
149C4⋊C4—41no—
150C4⋊C4—64no—
151C4⋊C4—64no—
152C4⋊C4—122no—
153C22⋊C4—24no—
154C22⋊C4—34no—
155C22⋊C4—41no—
156C22⋊C4—64no—
157C22⋊C4—122no—
158C22⋊C4—122no—
159C22⋊C4—122no—
160C22×C3—22no—
161C22×C3—24no—
162C22×C3—42no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3C22×D411yescharacteristic
166C22×C3C2×#4112yes—
167C22×C3H(5,𝔽2)11yescharacteristic
168C22×C3H(5,𝔽2)12yes—
169C22×C32−1+411yescharacteristic
170C2×D3—24no—
171C2×D3—24no—
172C2×D3—24no—
173C2×D3—42no—
174C2×D3—42no—
175C2×D3—42no—
176C2×D3—42no—
177C3×C4—42no—
178C3×C4—42no—
179C3×C4—41no—
180C3×C4—41no—
181Dic3—24no—
182Dic3—42no—
183Dic3—42no—
184C23—21no—
185C23—22no—
186C23—22no—
187C23—21no—
188C23—34no—
189C23—62no—
190C23—62no—
191C23—62no—
192C23—62no—
193C23—62no—
194C23—62no—
195C23D3×D412yes—
196C23C23×D311yescharacteristic
197D4—124no—
198D4—128no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—34no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—128no—
219C2×C4—124no—
220C2×C3—42no—
221C2×C3#53711yescharacteristic
222C2×C3#53912yes—
223C2×C3#54612yes—
224C2×C3#55111yescharacteristic
225C2×C3#56011yescharacteristic
226D3—44no—
227C22—24no—
228C22—22no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—64no—
233C22—64no—
234C22—64no—
235C22—122no—
236C22—122no—
237C22—122no—
238C22—122no—
239C22C2×D3×D411yescharacteristic
240C22#100411yescharacteristic
241C22D3×#4112yes—
242C22#100912yes—
243C22#101011yescharacteristic
244C4—42no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—64no—
249C4—122no—
250C4—122no—
251C3#269311yescharacteristic, complemented, Sylow
252C2—42no—
253C2—124no—
254C2#559711yescharacteristic
255C2#565412yes—
256C2#566611yescharacteristic
257C2#567911yescharacteristic
258C2#570712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radical#4960C2
hypercenter#397D3
hypocenterC3#2693
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#883C2semidirect
2. C3×#397C2semidirect
3. #4994C2semidirect
5. #4958C2semidirect
6. #4946C2semidirect
7. #4667C2semidirect
8. #4665C2semidirect
9. #4660C2semidirect
35. #397D3semidirect
64. #217D4semidirect
251. C3#2693semidirect

Maximal subgroups

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

subgroupindex
C2×#8832
C3×#3972
#49942
#49602
#49582
#49462
#46672
#46652
#46602
#26933