Finite Group Databasebeta

← #77087

Subgroups — #77087

Order 384.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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×#885C212yesmaximal, complemented
2C3×#394C211yesmaximal, characteristic, complemented
3#5223C212yesmaximal, complemented
4#4991C211yesmaximal, characteristic, complemented
5#4977C211yesmaximal, characteristic, complemented
6#4961C212yesmaximal, complemented
7#4960C212yesmaximal, complemented
8#4947C212yesmaximal, complemented
9#4673C212yesmaximal, complemented
10#2819—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×#67C2212yes—
13C22×#67C2212yes—
14C22×D12C2211yescharacteristic
15C22×Dic6C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×C4⋊Dic3C2211yescharacteristic
26C2×C4×Dic3C2211yescharacteristic
27#885—28no—
28C3×#95C2211yescharacteristic
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32C3⋊#95C2212yes—
33C2×#124—32no—
34#394—31no—
35#394—32no—
36#394D311yescharacteristic, complemented
37#393—32no—
38#393—32no—
39#392—32no—
40#392—32no—
41#382—31no—
42C24×C3C2312yes—
43C23×D3C2312yes—
44C2×C3×D4—42no—
45C2×C3×D4—41no—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C2×#67—24no—
51C2×#67—42no—
52C2×#67—42no—
53C2×#67—42no—
54C2×#67—42no—
55C2×#67—42no—
56C22×Dic3C2312yes—
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C2×D12—42no—
60C2×D12—41no—
61C2×Dic6—24no—
62C2×Dic6—42no—
63C2×Dic6—42no—
64C3×C4⋊C4—41no—
65C3×C22⋊C4—24no—
66C3×C22⋊C4—42no—
67#217—24no—
68#217—42no—
69(C2×D3)⋊C4—24no—
70(C2×D3)⋊C4—24no—
71(C2×D3)⋊C4—42no—
72(C2×D3)⋊C4—42no—
73C4⋊Dic3—41no—
74C4×Dic3D414yescomplemented
75C22×Q8—32no—
76C22×D4—32no—
77C22×D4—32no—
78C22×D4—31no—
79C22×D4C2×D311yescharacteristic
80#124—68no—
81C2×C4⋊C4—31no—
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×C22⋊C4C2×D312yes—
90C2×C22⋊C4C2×D312yes—
91C2×C42—31no—
92#95—32no—
93#95—32no—
94#95—32no—
95#95—31no—
96#95C2×D311yescharacteristic
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—22no—
100C23×C3—22no—
101C23×C3—22no—
102C23×C3—22no—
103C23×C3—22no—
104C23×C3C2411yescharacteristic
105C22×D3—22no—
106C22×D3—22no—
107C22×D3—22no—
108C22×D3—22no—
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—22no—
112C3×D4—44no—
113C2×C3×C4—21no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4C2×D412yes—
123#67—48no—
124#67—48no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3C2×D414yes—
134D12—44no—
135Dic6—48no—
136C24—32no—
137C24C22×D312yes—
138C2×Q8—64no—
139C2×Q8—122no—
140C2×Q8—122no—
141C2×D4—42no—
142C2×D4—41no—
143C2×D4—64no—
144C2×D4—122no—
145C2×D4—121no—
146C2×D4—122no—
147C2×D4—122no—
148C2×D4—122no—
149C2×D4—122no—
150C2×D4—122no—
151C22×C4—32no—
152C22×C4—32no—
153C22×C4—32no—
154C22×C4C22×D311yescharacteristic
155C22×C4C22×D311yescharacteristic
156C22×C4C22×D312yes—
157C22×C4C22×D311yescharacteristic
158C4⋊C4—41no—
159C4⋊C4—121no—
160C22⋊C4—24no—
161C22⋊C4—42no—
162C22⋊C4—64no—
163C22⋊C4—64no—
164C22⋊C4—64no—
165C22⋊C4—122no—
166C22⋊C4—122no—
167C22⋊C4—122no—
168C42—34no—
169C22×C3—42no—
170C22×C3—42no—
171C22×C3—42no—
172C22×C3—42no—
173C22×C3—42no—
174C22×C3—42no—
175C22×C3—42no—
176C22×C3C22×D411yescharacteristic
177C22×C3H(5,𝔽2)12yes—
178C22×C3H(5,𝔽2)11yescharacteristic
179C22×C3H(5,𝔽2)12yes—
180C22×C32−1+411yescharacteristic
181C2×D3—42no—
182C2×D3—42no—
183C2×D3—42no—
184C2×D3—42no—
185C2×D3—42no—
186C2×D3—42no—
187C2×D3—42no—
188C3×C4—22no—
189C3×C4—42no—
190C3×C4—41no—
191C3×C4—41no—
192Dic3—24no—
193Dic3—42no—
194Dic3—42no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—62no—
203C23—62no—
204C23—62no—
205C23—62no—
206C23—62no—
207C23—62no—
208C23—62no—
209C23C23×D311yescharacteristic
210Q8—128no—
211D4—44no—
212D4—124no—
213D4—128no—
214D4—128no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—34no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4D3×D412yes—
234C2×C3—82no—
235C2×C3#53512yes—
236C2×C3#53711yescharacteristic
237C2×C3H(3,𝔽4)11yescharacteristic
238C2×C3H(3,𝔽4)11yescharacteristic
239C2×C3#56212yes—
240D3—82no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—122no—
249C22—122no—
250C22—122no—
251C22—122no—
252C22—122no—
253C22—122no—
254C22—122no—
255C22C2×D3×D411yescharacteristic
256C22#100412yes—
257C22#100411yescharacteristic
258C22#100912yes—
259C22#101011yescharacteristic
260C4—22no—
261C4—41no—
262C4—41no—
263C4—42no—
264C4—64no—
265C4—122no—
266C4—122no—
267C3#281911yescharacteristic, complemented, Sylow
268C2—82no—
269C2—242no—
270C2#559411yescharacteristic
271C2#559711yescharacteristic
272C2#560712yes—
273C2#566712yes—
274C2#568311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2819
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#885C2semidirect
2. C3×#394C2semidirect
3. #5223C2semidirect
4. #4991C2semidirect
5. #4977C2semidirect
6. #4961C2semidirect
7. #4960C2semidirect
8. #4947C2semidirect
9. #4673C2semidirect
36. #394D3semidirect
74. C4×Dic3D4semidirect
267. C3#2819semidirect

Maximal subgroups

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

subgroupindex
C2×#8852
C3×#3942
#52232
#49912
#49772
#49612
#49602
#49472
#46732
#28193