Finite Group Databasebeta

← #76868

Subgroups — #76868

Order 384.

672 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 139 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×#946C211yesmaximal, characteristic, complemented
2C2×D3×C22⋊C4C211yesmaximal, characteristic
3C3×#393C211yesmaximal, characteristic, complemented
4#5231C211yesmaximal, characteristic, complemented
5#4955C212yesmaximal
6#4953C211yesmaximal, characteristic, complemented
7#4944C212yesmaximal, complemented
8#4674C211yesmaximal, characteristic, complemented
9#4669C212yesmaximal, complemented
10#4666C211yesmaximal, characteristic, complemented
11#4664C212yesmaximal, complemented
12#2561—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C22×C3×Q8C2211yescharacteristic, complemented
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2212yes—
19C2×#217C2211yescharacteristic
20C2×#217C2212yes—
21#946—24no—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2211yescharacteristic, complemented
29C2×Dic3⋊C4C2212yescomplemented
30C2×C4×Dic3C2212yescomplemented
31D3×C22⋊C4—24no—
32C3×#95C2212yescomplemented
33C3×#95C2211yescharacteristic, complemented
34C3⋊#95C2212yescomplemented
35C3⋊#95C2212yescomplemented
36C3⋊#95C2212yescomplemented
37C3⋊#95C2211yescharacteristic, complemented
38C2×C22⋊Q8—31no—
39C22×C22⋊C4—31no—
40#397—31no—
41#397—31no—
42#393—31no—
43#393D311yescharacteristic, complemented
44#388—32no—
45#388—32no—
46#381—32no—
47#379—32no—
48C4×C22⋊C4—31no—
49C24×C3C2311yescharacteristic
50C23×D3—22no—
51C23×D3—22no—
52C23×D3—22no—
53C23×D3—24no—
54C23×D3—42no—
55C23×D3C2311yescharacteristic
56C23×D3C2311yescharacteristic
57C2×C3×Q8—22no—
58C2×C3×Q8—42no—
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2312yes—
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C2×C4×D3—22no—
68C2×C4×D3—24no—
69C2×C4×D3—42no—
70C2×C4×D3D414yes—
71C3×C22⋊C4—22no—
72C3×C22⋊C4—42no—
73#217—24no—
74#217—22no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—42no—
81C4⋊Dic3—22no—
82Dic3⋊C4—24no—
83C4×Dic3—42no—
84C25—31no—
85C22×Q8C2×D311yescharacteristic, complemented
86C23×C4—31no—
87C23×C4—31no—
88C22⋊Q8—64no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—31no—
91C2×C22⋊C4—31no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—64no—
100C2×C22⋊C4C2×D311yescharacteristic
101C2×C22⋊C4C2×D312yes—
102C2×C42—32no—
103#95—31no—
104#95—32no—
105#95—32no—
106#95—32no—
107#95C2×D311yescharacteristic, complemented
108#95C2×D312yescomplemented
109C23×C3—21no—
110C23×C3—21no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3#4112yes—
114C23×C3C2411yescharacteristic
115C22×D3—21no—
116C22×D3—21no—
117C22×D3—22no—
118C22×D3—28no—
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—48no—
124C22×D3—48no—
125C22×D3—44no—
126C22×D3—44no—
127C22×D3C2×D412yes—
128C22×D3#4112yes—
129C22×D3#4114yes—
130C3×Q8—44no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4C2×D412yes—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3C2×D412yes—
151C4×D3—44no—
152C4×D3—44no—
153C24—31no—
154C24—31no—
155C24—62no—
156C24—62no—
157C24—64no—
158C24—62no—
159C24—122no—
160C24C22×D311yescharacteristic
161C2×Q8—22no—
162C2×Q8—42no—
163C22×C4—31no—
164C22×C4—34no—
165C22×C4—31no—
166C22×C4—32no—
167C22×C4—32no—
168C22×C4—62no—
169C22×C4—64no—
170C22×C4—122no—
171C22×C4C22×D311yescharacteristic
172C22×C4C22×D312yes—
173C22×C4C22×D312yes—
174C22×C4C22×D311yescharacteristic
175C4⋊C4—64no—
176C4⋊C4—62no—
177C22⋊C4—22no—
178C22⋊C4—42no—
179C22⋊C4—62no—
180C22⋊C4—64no—
181C22⋊C4—64no—
182C22⋊C4—64no—
183C22⋊C4—62no—
184C22⋊C4—64no—
185C22⋊C4—62no—
186C22⋊C4—122no—
187C42—122no—
188C22×C3—22no—
189C22×C3—24no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—42no—
193C22×C3C22×D411yescharacteristic
194C22×C3C2×#4111yescharacteristic
195C22×C3C2×#4111yescharacteristic
196C22×C3C2×#4112yes—
197C22×C3H(5,𝔽2)12yes—
198C2×D3—22no—
199C2×D3—22no—
200C2×D3—24no—
201C2×D3—24no—
202C2×D3—22no—
203C2×D3—24no—
204C2×D3—416no—
205C2×D3—41no—
206C2×D3—41no—
207C2×D3—42no—
208C2×D3#12414yes—
209C3×C4—22no—
210C3×C4—41no—
211C3×C4—42no—
212C3×C4—42no—
213Dic3—22no—
214Dic3—42no—
215Dic3—41no—
216Dic3—42no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—32no—
222C23—34no—
223C23—32no—
224C23—68no—
225C23—62no—
226C23—61no—
227C23—61no—
228C23—61no—
229C23—62no—
230C23—61no—
231C23—62no—
232C23—128no—
233C23—124no—
234C23—128no—
235C23—124no—
236C23#23912yes—
237C23C23×D311yescharacteristic
238Q8—44no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—32no—
249C2×C4—62no—
250C2×C4—61no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—124no—
259C2×C4—124no—
260C2×C4D3×D412yes—
261C2×C3—42no—
262C2×C3#52511yescharacteristic
263C2×C3C2×#12411yescharacteristic
264C2×C3#53411yescharacteristic
265C2×C3#55112yes—
266C2×C3#55112yes—
267D3—24no—
268D3—42no—
269C22—24no—
270C22—22no—
271C22—34no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—64no—
276C22—64no—
277C22—62no—
278C22—62no—
279C22—64no—
280C22—62no—
281C22—1216no—
282C22—122no—
283C22—121no—
284C22—121no—
285C22C2×D3×D411yescharacteristic
286C22#100412yes—
287C22C2×#23911yescharacteristic
288C22D3×#4112yes—
289C22D3×#4111yescharacteristic
290C4—22no—
291C4—41no—
292C4—42no—
293C4—42no—
294C4—62no—
295C4—122no—
296C4—121no—
297C4—122no—
298C3#256111yescharacteristic, complemented, Sylow
299C2—42no—
300C2—64no—
301C2—122no—
302C2#558911yescharacteristic
303C2#562711yescharacteristic
304C2#563712yes—
305C2D3×#12411yescharacteristic
306C2#567912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC24×D3C22
hypercenter#393D3
hypocenterC3#2561
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#946C2semidirect
3. C3×#393C2semidirect
4. #5231C2semidirect
6. #4953C2semidirect
7. #4944C2semidirect
8. #4674C2semidirect
9. #4669C2semidirect
10. #4666C2semidirect
11. #4664C2semidirect
14. C22×C3×Q8C22semidirect
28. C2×C4⋊Dic3C22semidirect
29. C2×Dic3⋊C4C22semidirect
30. C2×C4×Dic3C22semidirect
32. C3×#95C22semidirect
33. C3×#95C22semidirect
34. C3⋊#95C22semidirect
35. C3⋊#95C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
43. #393D3semidirect
85. C22×Q8C2×D3semidirect
107. #95C2×D3semidirect
108. #95C2×D3semidirect
298. C3#2561semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C2×D3×C22⋊C42
C3×#3932
#52312
#49552
#49532
#49442
#46742
#46692
#46662
#46642
#25613