Finite Group Databasebeta

← #77684

Subgroups — #77684

Order 384.

824 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 143 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×#895C211yesmaximal, characteristic, complemented
2C2×#882C212yesmaximal, complemented
3C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
4C3×#397C211yesmaximal, characteristic, complemented
5#4992C211yesmaximal, characteristic, complemented
6#4960C211yesmaximal, characteristic, complemented
7#4959C212yesmaximal, complemented
8#4672C212yesmaximal, complemented
9#4671C211yesmaximal, characteristic, complemented
10#4667C211yesmaximal, characteristic, complemented
11#4665C212yesmaximal, complemented
12#2617—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C22×#67C2212yes—
15C22×D12C2212yes—
16C22×D12C2212yes—
17C22×D12C2211yescharacteristic, complemented
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic, complemented
21C2×C3×C22⋊C4C2212yes—
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yescomplemented
29C2×(C2×D3)⋊C4C2212yescomplemented
30C2×(C2×D3)⋊C4C2212yescomplemented
31C2×Dic3⋊C4C2212yescomplemented
32#895—24no—
33#882—28no—
34D3×C22⋊C4—24no—
35C3×#95C2212yescomplemented
36C3×#95C2211yescharacteristic, complemented
37C3⋊#95C2212yescomplemented
38C3⋊#95C2211yescharacteristic, complemented
39C2×#121—31no—
40C2×#120—32no—
41C22×C22⋊C4—31no—
42#397—31no—
43#397D311yescharacteristic, complemented
44#394—32no—
45#394—32no—
46#392—31no—
47#392—31no—
48#386—32no—
49#380—31no—
50C24×C3C2311yescharacteristic
51C23×D3—22no—
52C23×D3—22no—
53C23×D3—42no—
54C23×D3—42no—
55C23×D3D414yescomplemented
56C23×D3C2312yes—
57C23×D3C2311yescharacteristic
58C23×D3C2311yescharacteristic
59C22×C3×C4C2312yes—
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C2×#67—24no—
64C2×#67—42no—
65C2×#67—42no—
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69C2×D12—24no—
70C2×D12—24no—
71C2×D12—24no—
72C2×D12—24no—
73C2×D12—22no—
74C2×D12—42no—
75C2×D12—42no—
76C2×D12—42no—
77C2×C4×D3—22no—
78C2×C4×D3—42no—
79C2×C4×D3—42no—
80C2×C4×D3D414yescomplemented
81C3×C4⋊C4—22no—
82C3×C22⋊C4—24no—
83C3×C22⋊C4—22no—
84#217—22no—
85(C2×D3)⋊C4—24no—
86(C2×D3)⋊C4—24no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—24no—
90(C2×D3)⋊C4—42no—
91(C2×D3)⋊C4—42no—
92Dic3⋊C4—42no—
93C25—31no—
94C22×D4—32no—
95C22×D4—32no—
96C22×D4—32no—
97C22×D4—31no—
98C23×C4—31no—
99C23×C4—31no—
100#121—64no—
101#120—68no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4C2×D311yescharacteristic, complemented
104C2×C22⋊C4—31no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—64no—
113C2×C22⋊C4C2×D311yescharacteristic
114C2×C22⋊C4C2×D312yes—
115#95—31no—
116#95—32no—
117#95C2×D311yescharacteristic, complemented
118#95C2×D312yescomplemented
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3C2×D412yes—
124C23×C3C2411yescharacteristic
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—28no—
132C22×D3—22no—
133C22×D3—22no—
134C22×D3—22no—
135C22×D3—22no—
136C22×D3—22no—
137C22×D3—22no—
138C22×D3—22no—
139C22×D3—48no—
140C22×D3—44no—
141C22×D3—44no—
142C22×D3—48no—
143C22×D3C2×D412yes—
144C22×D3C2×D412yes—
145C22×D3C2×D412yes—
146C22×D3#4112yes—
147C2×C3×C4—22no—
148C2×C3×C4—21no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4C2×D412yes—
157#67—48no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—21no—
161C2×Dic3—22no—
162C2×Dic3—22no—
163C2×Dic3—22no—
164C2×Dic3C2×D412yes—
165D12—44no—
166D12—48no—
167D12—48no—
168C4×D3—44no—
169C4×D3—44no—
170C24—31no—
171C24—31no—
172C24—34no—
173C24—32no—
174C24—62no—
175C24—62no—
176C24—122no—
177C24—122no—
178C24C22×D311yescharacteristic
179C2×D4—64no—
180C2×D4—64no—
181C2×D4—64no—
182C2×D4—64no—
183C2×D4—64no—
184C2×D4—62no—
185C2×D4—122no—
186C2×D4—122no—
187C2×D4—122no—
188C2×D4—122no—
189C2×D4—122no—
190C22×C4—31no—
191C22×C4—34no—
192C22×C4—31no—
193C22×C4—32no—
194C22×C4—62no—
195C22×C4—122no—
196C22×C4—122no—
197C22×C4C22×D312yes—
198C22×C4C22×D311yescharacteristic
199C22×C4C22×D312yes—
200C22×C4C22×D311yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—122no—
203C22⋊C4—24no—
204C22⋊C4—22no—
205C22⋊C4—62no—
206C22⋊C4—64no—
207C22⋊C4—64no—
208C22⋊C4—64no—
209C22⋊C4—62no—
210C22⋊C4—62no—
211C22⋊C4—122no—
212C22⋊C4—122no—
213C22×C3—22no—
214C22×C3—24no—
215C22×C3—41no—
216C22×C3—41no—
217C22×C3—42no—
218C22×C3C22×D411yescharacteristic
219C22×C3C22×D411yescharacteristic
220C22×C3C2×#4111yescharacteristic
221C22×C3H(5,𝔽2)12yes—
222C22×C3H(5,𝔽2)12yes—
223C2×D3—22no—
224C2×D3—24no—
225C2×D3—22no—
226C2×D3—24no—
227C2×D3—22no—
228C2×D3—24no—
229C2×D3—416no—
230C2×D3—41no—
231C2×D3—41no—
232C2×D3—42no—
233C2×D3—42no—
234C2×D3—42no—
235C2×D3—42no—
236C2×D3—42no—
237C2×D3—42no—
238C2×D3—42no—
239C2×D3—42no—
240C2×D3#12114yes—
241C3×C4—22no—
242C3×C4—42no—
243C3×C4—42no—
244C3×C4—41no—
245Dic3—22no—
246Dic3—42no—
247Dic3—41no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—32no—
253C23—32no—
254C23—32no—
255C23—32no—
256C23—68no—
257C23—62no—
258C23—62no—
259C23—61no—
260C23—61no—
261C23—62no—
262C23—62no—
263C23—62no—
264C23—62no—
265C23—62no—
266C23—62no—
267C23—62no—
268C23—62no—
269C23—62no—
270C23—128no—
271C23—124no—
272C23—124no—
273C23—128no—
274C23D3×D412yes—
275C23C23×D311yescharacteristic
276D4—124no—
277D4—128no—
278D4—128no—
279D4—128no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—32no—
290C2×C4—62no—
291C2×C4—61no—
292C2×C4—62no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—124no—
297C2×C4—124no—
298C2×C4D3×D412yes—
299C2×C3—42no—
300C2×C3C2×#12111yescharacteristic
301C2×C3#53411yescharacteristic
302C2×C3#53412yes—
303C2×C3#53511yescharacteristic
304C2×C3#53812yes—
305D3—24no—
306D3—42no—
307D3—82no—
308C22—24no—
309C22—22no—
310C22—34no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—62no—
315C22—64no—
316C22—62no—
317C22—64no—
318C22—64no—
319C22—62no—
320C22—1216no—
321C22—121no—
322C22—121no—
323C22—122no—
324C22—122no—
325C22—122no—
326C22—122no—
327C22—122no—
328C22—122no—
329C22—122no—
330C22—122no—
331C22C2×D3×D411yescharacteristic
332C22C2×D3×D411yescharacteristic
333C22D3×#4111yescharacteristic
334C22#100912yes—
335C22#100912yes—
336C4—22no—
337C4—42no—
338C4—41no—
339C4—42no—
340C4—62no—
341C4—122no—
342C4—121no—
343C3#261711yescharacteristic, complemented, Sylow
344C2—42no—
345C2—64no—
346C2—122no—
347C2—242no—
348C2D3×#12111yescharacteristic
349C2#565311yescharacteristic
350C2#565312yes—
351C2#567411yescharacteristic
352C2#570412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
hypercenter#397D3
hypocenterC3#2617
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#895C2semidirect
2. C2×#882C2semidirect
3. C2×D3×C22⋊C4C2semidirect
4. C3×#397C2semidirect
5. #4992C2semidirect
6. #4960C2semidirect
7. #4959C2semidirect
8. #4672C2semidirect
9. #4671C2semidirect
10. #4667C2semidirect
11. #4665C2semidirect
17. C22×D12C22semidirect
20. C2×C3×C4⋊C4C22semidirect
28. C2×(C2×D3)⋊C4C22semidirect
29. C2×(C2×D3)⋊C4C22semidirect
30. C2×(C2×D3)⋊C4C22semidirect
31. C2×Dic3⋊C4C22semidirect
35. C3×#95C22semidirect
36. C3×#95C22semidirect
37. C3⋊#95C22semidirect
38. C3⋊#95C22semidirect
43. #397D3semidirect
55. C23×D3D4semidirect
80. C2×C4×D3D4semidirect
103. C2×C4⋊C4C2×D3semidirect
117. #95C2×D3semidirect
118. #95C2×D3semidirect
343. C3#2617semidirect

Maximal subgroups

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

subgroupindex
C2×#8952
C2×#8822
C2×D3×C22⋊C42
C3×#3972
#49922
#49602
#49592
#46722
#46712
#46672
#46652
#26173