Finite Group Databasebeta

← #74137

Subgroups — #74137

Order 384.

620 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 137 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×C4×#67C211yesmaximal, characteristic, complemented
3C3×#381C211yesmaximal, characteristic, complemented
4#4961C212yesmaximal, complemented
5#4960C212yesmaximal, complemented
6#4677C211yesmaximal, characteristic, complemented
7#4672C211yesmaximal, characteristic, complemented
8#4671C211yesmaximal, characteristic, complemented
9#4670C212yesmaximal, complemented
10#4665C212yesmaximal, complemented
11#4659C211yesmaximal, characteristic, complemented
12#2635—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×#67C2212yes—
16C22×D12C2212yes—
17C22×D12C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2211yescharacteristic
21#930—24no—
22C4×#67—24no—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2212yes—
31C2×Dic3⋊C4C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33C3×#95C2211yescharacteristic
34C3×#95C2212yes—
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#121—31no—
39C2×C4×D4—31no—
40#400—31no—
41#399—31no—
42#394—32no—
43#394—31no—
44#392—32no—
45#392—31no—
46#388—32no—
47#386—32no—
48#381D311yescharacteristic, complemented
49C24×C3C2311yescharacteristic
50C23×D3C2312yes—
51C23×D3C2311yescharacteristic
52C22×C3×C4—42no—
53C22×C3×C4D414yescomplemented
54C22×C3×C4C2312yes—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C2×#67—22no—
59C2×#67—22no—
60C2×#67—24no—
61C2×#67—22no—
62C2×#67—42no—
63C2×#67—42no—
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C2×D12—24no—
69C2×D12—22no—
70C2×D12—42no—
71C2×D12—42no—
72C2×D12—42no—
73C2×C4×D3—22no—
74C2×C4×D3—42no—
75C3×C22⋊C4—42no—
76#217—22no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—42no—
81(C2×D3)⋊C4—42no—
82(C2×D3)⋊C4—42no—
83C4⋊Dic3—22no—
84C4⋊Dic3—42no—
85Dic3⋊C4—22no—
86C4×Dic3—22no—
87C22×D4—31no—
88C22×D4—32no—
89C22×D4—31no—
90C22×D4—32no—
91C23×C4—31no—
92C23×C4C2×D311yescharacteristic
93#121—64no—
94C4×D4—64no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4—32no—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4C2×D312yes—
106C2×C42—31no—
107#95—32no—
108#95—32no—
109#95C2×D312yes—
110#95C2×D311yescharacteristic
111#95C2×D311yescharacteristic
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3C2×D412yes—
117C23×C3C2411yescharacteristic
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3#4112yes—
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—22no—
137C2×C3×C4—22no—
138C2×C3×C4—44no—
139C2×C3×C4C2×D412yes—
140C2×C3×C4C2×D412yes—
141#67—28no—
142#67—48no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3#4112yes—
153D12—44no—
154D12—48no—
155C4×D3—44no—
156C24—31no—
157C24—32no—
158C24C22×D311yescharacteristic
159C2×D4—64no—
160C2×D4—62no—
161C2×D4—62no—
162C2×D4—62no—
163C2×D4—64no—
164C2×D4—62no—
165C2×D4—122no—
166C2×D4—122no—
167C2×D4—122no—
168C2×D4—122no—
169C2×D4—122no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—32no—
173C22×C4—31no—
174C22×C4—42no—
175C22×C4—62no—
176C22×C4—122no—
177C22×C4D1214yescomplemented
178C22×C4C22×D312yes—
179C22×C4C22×D312yes—
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182C4⋊C4—62no—
183C4⋊C4—62no—
184C4⋊C4—122no—
185C22⋊C4—42no—
186C22⋊C4—64no—
187C22⋊C4—62no—
188C22⋊C4—62no—
189C22⋊C4—64no—
190C22⋊C4—122no—
191C22⋊C4—122no—
192C22⋊C4—122no—
193C42—62no—
194C22×C3—22no—
195C22×C3—22no—
196C22×C3—22no—
197C22×C3—42no—
198C22×C3—42no—
199C22×C3C22×D411yescharacteristic
200C22×C3C2×#4111yescharacteristic
201C22×C3C2×#4111yescharacteristic
202C22×C3H(5,𝔽2)12yes—
203C22×C3H(5,𝔽2)12yes—
204C2×D3—22no—
205C2×D3—24no—
206C2×D3—41no—
207C2×D3—41no—
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—42no—
211C2×D3—42no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—42no—
215C2×D3—42no—
216C3×C4—22no—
217C3×C4—22no—
218C3×C4—42no—
219C3×C4—42no—
220Dic3—22no—
221Dic3—41no—
222Dic3—41no—
223Dic3—42no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—32no—
229C23—61no—
230C23—62no—
231C23—61no—
232C23—62no—
233C23—62no—
234C23—62no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—62no—
239C23—62no—
240C23C2×D1212yes—
241C23C23×D311yescharacteristic
242D4—68no—
243D4—124no—
244D4—128no—
245D4—128no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—32no—
255C2×C4—44no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—124no—
266C2×C4C2×D1212yes—
267C2×C4C2×D1212yes—
268C2×C3—42no—
269C2×C3#52511yescharacteristic
270C2×C3#53411yescharacteristic
271C2×C3#53511yescharacteristic
272C2×C3#53812yes—
273C2×C3#54012yes—
274D3—42no—
275D3—82no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22—64no—
282C22—62no—
283C22—122no—
284C22—121no—
285C22—121no—
286C22—122no—
287C22—122no—
288C22—122no—
289C22—122no—
290C22—122no—
291C22—122no—
292C22—122no—
293C22C22×D1211yescharacteristic
294C22#100412yes—
295C22D3×#4111yescharacteristic
296C22D3×#4111yescharacteristic
297C22#100912yes—
298C4—22no—
299C4—22no—
300C4—42no—
301C4—42no—
302C4—62no—
303C4—121no—
304C4—122no—
305C4—121no—
306C3#263511yescharacteristic, complemented, Sylow
307C2—42no—
308C2—122no—
309C2—242no—
310C2#549311yescharacteristic
311C2#552411yescharacteristic
312C2#552511yescharacteristic
313C2#565712yes—
314C2#566312yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#381C2
hypercenter#381D3
hypocenterC3#2635
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#930C2semidirect
2. C2×C4×#67C2semidirect
3. C3×#381C2semidirect
4. #4961C2semidirect
5. #4960C2semidirect
6. #4677C2semidirect
7. #4672C2semidirect
8. #4671C2semidirect
9. #4670C2semidirect
10. #4665C2semidirect
11. #4659C2semidirect
48. #381D3semidirect
53. C22×C3×C4D4semidirect
177. C22×C4D12semidirect
306. C3#2635semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C2×C4×#672
C3×#3812
#49612
#49602
#46772
#46722
#46712
#46702
#46652
#46592
#26353