Finite Group Databasebeta

← #78129

Subgroups — #78129

Order 384.

560 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 143 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×#893C211yesmaximal, characteristic
2C2×#892C211yesmaximal, characteristic, complemented
3C3×#400C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic, complemented
5#4981C211yesmaximal, characteristic, complemented
6#4675C212yesmaximal, complemented
7#4674C211yesmaximal, characteristic, complemented
8#4672C212yesmaximal
9#4666C212yesmaximal, complemented
10#4666C211yesmaximal, characteristic, complemented
11#4649C212yesmaximal, complemented
12#2395—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×C4⋊Dic3C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2212yes—
29C2×C4×Dic3C2212yes—
30#893—24no—
31#892—24no—
32C3×#95C2212yes—
33C3×#95C2211yescharacteristic
34C3×#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×C4×D4—31no—
39C2×#117—31no—
40#400D311yescharacteristic, complemented
41#399—31no—
42#398—31no—
43#397—31no—
44#397—32no—
45#394—32no—
46C4×C4⋊C4—32no—
47C4×C22⋊C4—32no—
48C4×C22⋊C4—31no—
49C23×D3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C2×D12—22no—
61C2×D12—22no—
62C2×D12—22no—
63C2×C4×D3—24no—
64C2×C4×D3—22no—
65C2×C4×D3—24no—
66C2×C4×D3D414yes—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—22no—
69(C2×D3)⋊C4—24no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—24no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—24no—
75C4⋊Dic3—22no—
76Dic3⋊C4—24no—
77Dic3⋊C4—24no—
78C4×Dic3—24no—
79C4×Dic3—22no—
80C4×Dic3—24no—
81C22×D4—31no—
82C23×C4—31no—
83C23×C4—31no—
84C4×D4—64no—
85#117—64no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4C2×D311yescharacteristic
90C2×C4⋊C4C2×D311yescharacteristic
91C2×C22⋊C4—32no—
92C2×C22⋊C4—31no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C42—32no—
98C2×C42—31no—
99C2×C42—32no—
100#95—32no—
101#95—31no—
102#95—32no—
103#95C2×D312yes—
104#95C2×D311yescharacteristic
105#95C2×D312yes—
106C23×C3C2411yescharacteristic
107C22×D3—21no—
108C22×D3—22no—
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—22no—
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—21no—
115C22×D3C2×D412yes—
116C22×D3#4112yes—
117C2×C3×C4—21no—
118C2×C3×C4—22no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4C2×D412yes—
129C2×C3×C4#4112yes—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3C2×D412yes—
138C2×Dic3#4114yes—
139C2×Dic3#4114yes—
140D12—28no—
141C4×D3—28no—
142C4×D3—44no—
143C24—31no—
144C24—31no—
145C2×D4—62no—
146C2×D4—62no—
147C2×D4—62no—
148C22×C4—31no—
149C22×C4—32no—
150C22×C4—34no—
151C22×C4—31no—
152C22×C4—32no—
153C22×C4—64no—
154C22×C4—62no—
155C22×C4—64no—
156C22×C4C22×D311yescharacteristic
157C22×C4C22×D312yes—
158C22×C4C22×D312yes—
159C22×C4C22×D311yescharacteristic
160C22×C4C22×D311yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—64no—
164C4⋊C4—62no—
165C4⋊C4—64no—
166C22⋊C4—64no—
167C22⋊C4—64no—
168C22⋊C4—64no—
169C22⋊C4—62no—
170C22⋊C4—64no—
171C22⋊C4—62no—
172C42—64no—
173C42—62no—
174C42—64no—
175C22×C3C22×D411yescharacteristic
176C22×C3C2×#4111yescharacteristic
177C22×C3C2×#4112yes—
178C22×C3C2×#4111yescharacteristic
179C22×C3C2×#4112yes—
180C2×D3—22no—
181C2×D3—22no—
182C2×D3—22no—
183C2×D3—24no—
184C2×D3—22no—
185C2×D3—41no—
186C2×D3—42no—
187C2×D3—42no—
188C2×D3—42no—
189C2×D3—41no—
190C3×C4—22no—
191C3×C4—22no—
192C3×C4—41no—
193C3×C4—42no—
194C3×C4—42no—
195Dic3—24no—
196Dic3—42no—
197Dic3—41no—
198Dic3#12314yes—
199C23—32no—
200C23—32no—
201C23—61no—
202C23—61no—
203C23—61no—
204C23—62no—
205C23—62no—
206C23—62no—
207C23—62no—
208C23—61no—
209C23C23×D311yescharacteristic
210D4—68no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—32no—
223C2×C4—34no—
224C2×C4—34no—
225C2×C4—68no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—61no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—124no—
234C2×C4D3×D412yes—
235C2×C4#23912yes—
236C2×C3C2×#12311yescharacteristic
237C2×C3#52511yescharacteristic
238C2×C3#52912yes—
239C2×C3#52912yes—
240C2×C3#53211yescharacteristic
241D3—42no—
242D3—42no—
243C22—62no—
244C22—62no—
245C22—64no—
246C22—62no—
247C22—62no—
248C22—121no—
249C22—121no—
250C22—122no—
251C22—122no—
252C22—122no—
253C22C2×D3×D411yescharacteristic
254C22C2×#23911yescharacteristic
255C22D3×#4112yes—
256C22D3×#4112yes—
257C22D3×#4111yescharacteristic
258C4—22no—
259C4—22no—
260C4—34no—
261C4—41no—
262C4—42no—
263C4—42no—
264C4—64no—
265C4—121no—
266C4—122no—
267C3#239511yescharacteristic, complemented, Sylow
268C2—122no—
269C2—122no—
270C2#562711yescharacteristic
271C2D3×#12311yescharacteristic
272C2#566811yescharacteristic
273C2#569112yes—
274C2#570612yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC22×D12C22
hypercenter#400D3
hypocenterC3#2395
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#892C2semidirect
3. C3×#400C2semidirect
4. #4993C2semidirect
5. #4981C2semidirect
6. #4675C2semidirect
7. #4674C2semidirect
9. #4666C2semidirect
10. #4666C2semidirect
11. #4649C2semidirect
40. #400D3semidirect
267. C3#2395semidirect

Maximal subgroups

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

subgroupindex
C2×#8932
C2×#8922
C3×#4002
#49932
#49812
#46752
#46742
#46722
#46662
#46492
#23953