Finite Group Databasebeta

← #77737

Subgroups — #77737

Order 384.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#893C212yesmaximal, complemented
2C2×#893C211yesmaximal, characteristic, complemented
3C2×#892C211yesmaximal, characteristic, complemented
4C3×#398C211yesmaximal, characteristic, complemented
5#4994C212yesmaximal, complemented
6#4992C211yesmaximal, characteristic, complemented
7#4991C211yesmaximal, characteristic, complemented
8#4672C212yesmaximal, complemented
9#4671C211yesmaximal, characteristic
10#4666C212yesmaximal, complemented
11#4649C211yesmaximal, characteristic, complemented
12#2392—31nomaximal, Sylow
13C22×D12C2212yes—
14C22×D12C2211yescharacteristic
15C22×C4×D3C2212yes—
16C22×C4×D3C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31C2×C4×Dic3C2211yescharacteristic
32#893—28no—
33#893—24no—
34#892—24no—
35C3×#95C2211yescharacteristic
36C3×#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C2×C4×D4—31no—
40C2×C4×D4—32no—
41C2×#117—31no—
42#398D311yescharacteristic, complemented
43#397—31no—
44#396—32no—
45#394—31no—
46#394—32no—
47#392—31no—
48C4×C4⋊C4—31no—
49C4×C22⋊C4—32no—
50C23×D3C2311yescharacteristic
51C23×D3C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C2×D12—24no—
62C2×D12—24no—
63C2×D12—24no—
64C2×D12—22no—
65C2×D12—24no—
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3—24no—
69C2×C4×D3—24no—
70C2×C4×D3—24no—
71C2×C4×D3D414yescomplemented
72C3×C4⋊C4—22no—
73C3×C4⋊C4—24no—
74C3×C4⋊C4—22no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—24no—
82C4⋊Dic3—22no—
83Dic3⋊C4—24no—
84C4×Dic3—22no—
85C4×Dic3—24no—
86C4×Dic3D414yescomplemented
87C22×D4—31no—
88C22×D4—32no—
89C23×C4—31no—
90C23×C4—32no—
91C4×D4—64no—
92C4×D4—68no—
93#117—64no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4C2×D311yescharacteristic
97C2×C4⋊C4C2×D312yes—
98C2×C4⋊C4C2×D311yescharacteristic
99C2×C22⋊C4—31no—
100C2×C22⋊C4—31no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C42—31no—
107C2×C42—31no—
108C2×C42—32no—
109#95—32no—
110#95—31no—
111#95C2×D312yes—
112#95C2×D311yescharacteristic
113C23×C3C2411yescharacteristic
114C22×D3—21no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—21no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3C2×D412yes—
125C22×D3#4114yes—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4C2×D412yes—
137C2×C3×C4C2×D412yes—
138C2×C3×C4#4114yes—
139C2×Dic3—22no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3C2×D412yes—
146C2×Dic3C2×D412yes—
147C2×Dic3C2×D412yes—
148C2×Dic3#4112yes—
149D12—28no—
150D12—216no—
151C4×D3—28no—
152C4×D3—48no—
153C24—31no—
154C24—32no—
155C2×D4—64no—
156C2×D4—64no—
157C2×D4—64no—
158C2×D4—62no—
159C2×D4—64no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—34no—
163C22×C4—32no—
164C22×C4—31no—
165C22×C4—62no—
166C22×C4—62no—
167C22×C4—64no—
168C22×C4—64no—
169C22×C4—64no—
170C22×C4C22×D312yes—
171C22×C4C22×D311yescharacteristic
172C22×C4C22×D311yescharacteristic
173C22×C4C22×D311yescharacteristic
174C22×C4C22×D312yes—
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—22no—
178C4⋊C4—62no—
179C4⋊C4—64no—
180C22⋊C4—62no—
181C22⋊C4—64no—
182C22⋊C4—64no—
183C22⋊C4—64no—
184C22⋊C4—62no—
185C22⋊C4—64no—
186C22⋊C4—64no—
187C42—34no—
188C42—62no—
189C42—64no—
190C22×C3C22×D411yescharacteristic
191C22×C3C22×D411yescharacteristic
192C22×C3C2×#4112yes—
193C22×C3C2×#4111yescharacteristic
194C22×C3C2×#4112yes—
195C2×D3—22no—
196C2×D3—24no—
197C2×D3—24no—
198C2×D3—24no—
199C2×D3—24no—
200C2×D3—42no—
201C2×D3—41no—
202C2×D3—42no—
203C2×D3—42no—
204C2×D3—42no—
205C2×D3—41no—
206C2×D3—42no—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—24no—
210C3×C4—42no—
211C3×C4—41no—
212Dic3—22no—
213Dic3—42no—
214Dic3—41no—
215Dic3#12114yes—
216C23—32no—
217C23—34no—
218C23—61no—
219C23—62no—
220C23—62no—
221C23—62no—
222C23—62no—
223C23—62no—
224C23—62no—
225C23—61no—
226C23—62no—
227C23—62no—
228C23C23×D311yescharacteristic
229D4—68no—
230D4—616no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—32no—
242C2×C4—32no—
243C2×C4—32no—
244C2×C4—32no—
245C2×C4—68no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—61no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—128no—
253C2×C4D3×D412yes—
254C2×C4D3×D412yes—
255C2×C4#23914yes—
256C2×C3C2×#12111yescharacteristic
257C2×C3#52511yescharacteristic
258C2×C3#52512yes—
259C2×C3#52912yes—
260C2×C3#53211yescharacteristic
261D3—42no—
262D3—44no—
263C22—64no—
264C22—64no—
265C22—62no—
266C22—64no—
267C22—64no—
268C22—121no—
269C22—122no—
270C22—122no—
271C22—121no—
272C22—122no—
273C22—122no—
274C22—122no—
275C22C2×D3×D411yescharacteristic
276C22C2×D3×D411yescharacteristic
277C22C2×#23912yes—
278C22D3×#4112yes—
279C22D3×#4111yescharacteristic
280C4—22no—
281C4—22no—
282C4—24no—
283C4—34no—
284C4—41no—
285C4—42no—
286C4—62no—
287C4—121no—
288C4—122no—
289C3#239211yescharacteristic, complemented, Sylow
290C2—124no—
291C2—122no—
292C2D3×#12111yescharacteristic
293C2#562712yes—
294C2#565611yescharacteristic
295C2#569112yes—
296C2#572511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radical#4671C2
hypercenter#398D3
hypocenterC3#2392
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#893C2semidirect
2. C2×#893C2semidirect
3. C2×#892C2semidirect
4. C3×#398C2semidirect
5. #4994C2semidirect
6. #4992C2semidirect
7. #4991C2semidirect
8. #4672C2semidirect
10. #4666C2semidirect
11. #4649C2semidirect
42. #398D3semidirect
71. C2×C4×D3D4semidirect
86. C4×Dic3D4semidirect
289. C3#2392semidirect

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×#3982
#49942
#49922
#49912
#46722
#46712
#46662
#46492
#23923