Finite Group Databasebeta

← #74307

Subgroups — #74307

Order 384.

604 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 141 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×C4×D12C211yesmaximal, characteristic, complemented
2C2×#872C211yesmaximal, characteristic, complemented
3C3×#383C211yesmaximal, characteristic, complemented
4#4677C211yesmaximal, characteristic, complemented
5#4672C211yesmaximal, characteristic, complemented
6#4671C211yesmaximal, characteristic
7#4670C212yesmaximal, complemented
8#4668C212yesmaximal, complemented
9#4665C212yesmaximal, complemented
10#4665C212yesmaximal, complemented
11#4659C211yesmaximal, characteristic, complemented
12#2598—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×D12C2212yes—
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2212yes—
17C2×C3×C42C2211yescharacteristic
18C2×(C2×D3)⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊Dic3C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×C4×Dic3C2211yescharacteristic
30C4×D12—24no—
31#872—24no—
32C3×#95C2211yescharacteristic
33C3×#95C2212yes—
34C3×#95C2212yes—
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×D4—31no—
39C2×#117—31no—
40#400—31no—
41#399—31no—
42#394—31no—
43#392—31no—
44#388—32no—
45#386—32no—
46#386—32no—
47#385—32no—
48#383D311yescharacteristic, complemented
49C23×D3C2311yescharacteristic
50C23×D3C2312yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×D12—24no—
61C2×D12—22no—
62C2×D12—24no—
63C2×D12—24no—
64C2×D12—42no—
65C2×C4×D3—22no—
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3—24no—
69C2×C4×D3—42no—
70C2×C4×D3—42no—
71C3×C42D414yescomplemented
72(C2×D3)⋊C4—24no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—42no—
78(C2×D3)⋊C4—42no—
79C4⋊Dic3—22no—
80C4⋊Dic3—42no—
81Dic3⋊C4—22no—
82Dic3⋊C4—22no—
83C4×Dic3—22no—
84C22×D4—31no—
85C22×D4—32no—
86C23×C4—31no—
87C23×C4—32no—
88C4×D4—64no—
89#117—64no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—32no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C42—31no—
102C2×C42C2×D311yescharacteristic
103#95—32no—
104#95—32no—
105#95C2×D312yes—
106#95C2×D312yes—
107#95C2×D311yescharacteristic
108#95C2×D311yescharacteristic
109C23×C3C2411yescharacteristic
110C22×D3—21no—
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—21no—
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3#4114yes—
121C22×D3#4112yes—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D412yes—
134C2×C3×C4C2×D412yes—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3#4112yes—
145D12—28no—
146D12—48no—
147C4×D3—28no—
148C4×D3—48no—
149C24—32no—
150C24—31no—
151C2×D4—64no—
152C2×D4—64no—
153C2×D4—64no—
154C2×D4—62no—
155C2×D4—122no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—32no—
159C22×C4—31no—
160C22×C4—62no—
161C22×C4—62no—
162C22×C4—62no—
163C22×C4—64no—
164C22×C4—122no—
165C22×C4—122no—
166C22×C4C22×D312yes—
167C22×C4C22×D312yes—
168C22×C4C22×D311yescharacteristic
169C22×C4C22×D311yescharacteristic
170C22×C4C22×D311yescharacteristic
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—122no—
175C22⋊C4—64no—
176C22⋊C4—64no—
177C22⋊C4—62no—
178C22⋊C4—62no—
179C22⋊C4—64no—
180C22⋊C4—122no—
181C22⋊C4—122no—
182C42—62no—
183C42D1214yescomplemented
184C22×C3C22×D411yescharacteristic
185C22×C3C2×#4111yescharacteristic
186C22×C3C2×#4112yes—
187C22×C3C2×#4111yescharacteristic
188C22×C3H(5,𝔽2)12yes—
189C2×D3—22no—
190C2×D3—24no—
191C2×D3—24no—
192C2×D3—24no—
193C2×D3—24no—
194C2×D3—42no—
195C2×D3—41no—
196C2×D3—42no—
197C2×D3—41no—
198C2×D3—42no—
199C2×D3—42no—
200C2×D3—42no—
201C3×C4—22no—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—42no—
205C3×C4—42no—
206Dic3—22no—
207Dic3—41no—
208Dic3—42no—
209Dic3—41no—
210C23—34no—
211C23—32no—
212C23—61no—
213C23—62no—
214C23—62no—
215C23—61no—
216C23—62no—
217C23—62no—
218C23—62no—
219C23—62no—
220C23—62no—
221C23—62no—
222C23C23×D311yescharacteristic
223D4—68no—
224D4—128no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—32no—
236C2×C4—68no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—128no—
247C2×C4C2×D1212yes—
248C2×C4C2×D1212yes—
249C2×C4C2×D1212yes—
250C2×C3#52511yescharacteristic
251C2×C3#52511yescharacteristic
252C2×C3#53511yescharacteristic
253C2×C3#55112yes—
254C2×C3#55312yes—
255D3—42no—
256D3—44no—
257C22—64no—
258C22—64no—
259C22—64no—
260C22—62no—
261C22—64no—
262C22—122no—
263C22—122no—
264C22—122no—
265C22—122no—
266C22—121no—
267C22—122no—
268C22—121no—
269C22C22×D1211yescharacteristic
270C22D3×#4111yescharacteristic
271C22D3×#4112yes—
272C22D3×#4111yescharacteristic
273C22#100912yes—
274C4—22no—
275C4—22no—
276C4—22no—
277C4—42no—
278C4—42no—
279C4—62no—
280C4—121no—
281C4—122no—
282C4—121no—
283C3#259811yescharacteristic, complemented, Sylow
284C2—122no—
285C2—124no—
286C2#552411yescharacteristic
287C2#552411yescharacteristic
288C2#552511yescharacteristic
289C2#570512yes—
290C2#570912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#383C2
prime radical#4671C2
hypercenter#383D3
hypocenterC3#2598
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
1. C2×C4×D12C2semidirect
2. C2×#872C2semidirect
3. C3×#383C2semidirect
4. #4677C2semidirect
5. #4672C2semidirect
7. #4670C2semidirect
8. #4668C2semidirect
9. #4665C2semidirect
10. #4665C2semidirect
11. #4659C2semidirect
48. #383D3semidirect
71. C3×C42D4semidirect
183. C42D12semidirect
283. C3#2598semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D122
C2×#8722
C3×#3832
#46772
#46722
#46712
#46702
#46682
#46652
#46592
#25983