Finite Group Databasebeta

← #73611

Subgroups — #73611

Order 384.

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

normal subgroups only

This group has 598 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×#930C211yesmaximal, characteristic, complemented
2C2×#929C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
4#4961C212yesmaximal, complemented
5#4960C212yesmaximal, complemented
6#4940C212yesmaximal, complemented
7#4939C212yesmaximal, complemented
8#4674C211yesmaximal, characteristic, complemented
9#4672C211yesmaximal, characteristic, complemented
10#4671C211yesmaximal, characteristic, complemented
11#4661C211yesmaximal, characteristic, complemented
12#2661—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×#67C2212yes—
16C22×D12C2211yescharacteristic
17C22×D12C2212yes—
18C2×C3×C22⋊C4C2212yes—
19C2×C3×C42C2212yes—
20C2×#217C2211yescharacteristic
21#930—24no—
22#929—24no—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2212yes—
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C3×#95C2211yescharacteristic
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#123—31no—
39C2×#121—31no—
40#398—31no—
41#397—31no—
42#394—32no—
43#394—31no—
44#392—32no—
45#392—31no—
46#390—32no—
47#388—32no—
48C4×C22⋊C4D311yescharacteristic, complemented
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C23×D3C2312yes—
52C22×C3×C4—42no—
53C22×C3×C4D414yescomplemented
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2312yes—
58C2×#67—22no—
59C2×#67—24no—
60C2×#67—42no—
61C2×#67—42no—
62C2×#67—42no—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C2×D12—22no—
68C2×D12—24no—
69C2×D12—42no—
70C2×D12—42no—
71C2×D12—42no—
72C3×C22⋊C4—24no—
73C3×C42—42no—
74#217—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—42no—
80(C2×D3)⋊C4—42no—
81(C2×D3)⋊C4—42no—
82C4⋊Dic3—22no—
83C4⋊Dic3—42no—
84Dic3⋊C4—22no—
85Dic3⋊C4—22no—
86C22×D4—32no—
87C22×D4—31no—
88C22×D4—31no—
89C22×D4—32no—
90C23×C4C2×D311yescharacteristic
91#123—64no—
92#121—64no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—32no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—32no—
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⋊C4C2×D312yes—
106C2×C42C2×D312yes—
107#95—32no—
108#95—32no—
109#95C2×D311yescharacteristic
110#95C2×D311yescharacteristic
111C23×C3—22no—
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—22no—
115C23×C3C2×D412yes—
116C23×C3C2411yescharacteristic
117C22×D3—21no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—21no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—21no—
127C22×D3—22no—
128C22×D3—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—44no—
137C2×C3×C4C2×D412yes—
138C2×C3×C4C2×D412yes—
139C2×C3×C4#4114yes—
140#67—44no—
141#67—48no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152D12—44no—
153D12—48no—
154C24—31no—
155C24—32no—
156C24C22×D311yescharacteristic
157C2×D4—64no—
158C2×D4—62no—
159C2×D4—62no—
160C2×D4—64no—
161C2×D4—122no—
162C2×D4—122no—
163C2×D4—122no—
164C2×D4—122no—
165C2×D4—122no—
166C2×D4—122no—
167C22×C4—31no—
168C22×C4—32no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—42no—
172C22×C4D1214yescomplemented
173C22×C4C22×D312yes—
174C22×C4C22×D311yescharacteristic
175C22×C4C22×D311yescharacteristic
176C22×C4C22×D312yes—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—122no—
181C22⋊C4—24no—
182C22⋊C4—62no—
183C22⋊C4—64no—
184C22⋊C4—62no—
185C22⋊C4—62no—
186C22⋊C4—64no—
187C22⋊C4—122no—
188C22⋊C4—122no—
189C22⋊C4—122no—
190C42—42no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3C22×D411yescharacteristic
197C22×C3C2×#4112yes—
198C22×C3H(5,𝔽2)12yes—
199C22×C3H(5,𝔽2)11yescharacteristic
200C22×C3H(5,𝔽2)11yescharacteristic
201C2×D3—41no—
202C2×D3—42no—
203C2×D3—42no—
204C2×D3—41no—
205C2×D3—42no—
206C2×D3—41no—
207C2×D3—42no—
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—42no—
211C2×D3—42no—
212C2×D3—42no—
213C3×C4—22no—
214C3×C4—22no—
215C3×C4—24no—
216C3×C4—42no—
217Dic3—41no—
218Dic3—41no—
219Dic3—41no—
220Dic3—42no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—61no—
226C23—62no—
227C23—62no—
228C23—62no—
229C23—62no—
230C23—62no—
231C23—62no—
232C23—62no—
233C23—61no—
234C23—62no—
235C23—62no—
236C23—61no—
237C23C2×D1212yes—
238C23C23×D311yescharacteristic
239D4—124no—
240D4—128no—
241D4—124no—
242D4—128no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—44no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—62no—
254C2×C4—61no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—62no—
261C2×C4C2×D1212yes—
262C2×C4C2×D1212yes—
263C2×C4#23514yes—
264C2×C3—42no—
265C2×C3#53211yescharacteristic
266C2×C3#53411yescharacteristic
267C2×C3#53511yescharacteristic
268C2×C3#53812yes—
269C2×C3#54012yes—
270D3—82no—
271D3—81no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—121no—
278C22—122no—
279C22—122no—
280C22—121no—
281C22—122no—
282C22—121no—
283C22—122no—
284C22—122no—
285C22—122no—
286C22—122no—
287C22—122no—
288C22—122no—
289C22C22×D1211yescharacteristic
290C22C2×#23512yes—
291C22#100412yes—
292C22#100911yescharacteristic
293C22#100911yescharacteristic
294C4—22no—
295C4—22no—
296C4—24no—
297C4—42no—
298C4—121no—
299C4—121no—
300C4—122no—
301C4—121no—
302C3#266111yescharacteristic, complemented, Sylow
303C2—42no—
304C2—242no—
305C2—241no—
306C2#547711yescharacteristic
307C2#549311yescharacteristic
308C2#552511yescharacteristic
309C2#556012yes—
310C2#556212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×C4×C22⋊C4C2
hypercenterC4×C22⋊C4D3
hypocenterC3#2661
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#930C2semidirect
2. C2×#929C2semidirect
3. C3×C4×C22⋊C4C2semidirect
4. #4961C2semidirect
5. #4960C2semidirect
6. #4940C2semidirect
7. #4939C2semidirect
8. #4674C2semidirect
9. #4672C2semidirect
10. #4671C2semidirect
11. #4661C2semidirect
48. C4×C22⋊C4D3semidirect
53. C22×C3×C4D4semidirect
172. C22×C4D12semidirect
302. C3#2661semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C2×#9292
C3×C4×C22⋊C42
#49612
#49602
#49402
#49392
#46742
#46722
#46712
#46612
#26613