Finite Group Databasebeta

← #74602

Subgroups — #74602

Order 384.

786 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C22×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
2C2×#886C211yesmaximal, characteristic, complemented
3C2×#882C211yesmaximal, characteristic
4C2×#877C211yesmaximal, characteristic, complemented
5C3×#385C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#4994C212yesmaximal, complemented
8#4988C212yesmaximal, complemented
9#4955C211yesmaximal, characteristic, complemented
10#4672C212yesmaximal, complemented
11#4670C212yesmaximal, complemented
12#2411—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C23×Dic3C2211yescharacteristic
17C22×D12C2212yes—
18C2×C3×C4⋊C4C2212yescomplemented
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×#217C2211yescharacteristic
22C2×(C2×D3)⋊C4—24no—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
30C2×(C2×D3)⋊C4C2212yescomplemented
31C2×C4⋊Dic3C2211yescharacteristic, complemented
32C2×C4⋊Dic3C2211yescharacteristic, complemented
33C2×Dic3⋊C4C2212yescomplemented
34C2×C4×Dic3C2212yescomplemented
35#886—24no—
36#882—24no—
37#877—24no—
38C3×#95C2212yescomplemented
39C3⋊#95C2212yescomplemented
40C2×#123—31no—
41C2×#120—31no—
42C2×#117—31no—
43C22×C22⋊C4—31no—
44#396—32no—
45#394—32no—
46#390—32no—
47#388—32no—
48#386—31no—
49#385D311yescharacteristic, complemented
50#379—31no—
51C24×C3C2311yescharacteristic
52C23×D3—22no—
53C23×D3—24no—
54C23×D3—22no—
55C23×D3—22no—
56C23×D3—42no—
57C23×D3C2311yescharacteristic
58C23×D3C2312yes—
59C22×C3×C4—22no—
60C22×C3×C4—42no—
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2312yes—
65C2×#67—22no—
66C2×#67—22no—
67C2×#67—22no—
68C22×Dic3—22no—
69C22×Dic3—22no—
70C22×Dic3D414yescomplemented
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2312yes—
73C22×Dic3C2311yescharacteristic
74C22×Dic3C2311yescharacteristic
75C2×D12—24no—
76C2×D12—24no—
77C2×D12—42no—
78C3×C4⋊C4—42no—
79C3×C22⋊C4—22no—
80C3×C22⋊C4D414yes—
81#217—22no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—24no—
87(C2×D3)⋊C4—24no—
88(C2×D3)⋊C4—42no—
89(C2×D3)⋊C4—42no—
90C4⋊Dic3—22no—
91C4⋊Dic3—22no—
92Dic3⋊C4—24no—
93C4×Dic3—24no—
94C25—31no—
95C22×D4—31no—
96C22×D4—32no—
97C23×C4—31no—
98C23×C4C2×D311yescharacteristic
99#123—64no—
100#120—64no—
101#117—64no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—31no—
105C2×C4⋊C4C2×D312yescomplemented
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—32no—
115C2×C22⋊C4—64no—
116C2×C22⋊C4C2×D311yescharacteristic
117C2×C22⋊C4C2×D311yescharacteristic
118C2×C42—32no—
119#95—32no—
120#95C2×D312yescomplemented
121C23×C3—22no—
122C23×C3—22no—
123C23×C3C2×D412yes—
124C23×C3C2×D412yes—
125C23×C3#4112yes—
126C23×C3C2411yescharacteristic
127C22×D3—28no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—22no—
133C22×D3—22no—
134C22×D3—21no—
135C22×D3—21no—
136C22×D3—22no—
137C22×D3—21no—
138C22×D3—22no—
139C22×D3—48no—
140C22×D3—48no—
141C22×D3—44no—
142C22×D3—44no—
143C22×D3#4114yes—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—44no—
154C2×C3×C4C2×D414yes—
155#67—44no—
156C2×Dic3—28no—
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3—21no—
160C2×Dic3—21no—
161C2×Dic3—22no—
162C2×Dic3—21no—
163C2×Dic3—21no—
164C2×Dic3—22no—
165C2×Dic3—21no—
166C2×Dic3—22no—
167C2×Dic3C2×D412yes—
168C2×Dic3C2×D412yes—
169D12—48no—
170C24—32no—
171C24—31no—
172C24—64no—
173C24—62no—
174C24—62no—
175C24—62no—
176C24—122no—
177C24C22×D311yescharacteristic
178C2×D4—64no—
179C2×D4—62no—
180C2×D4—62no—
181C2×D4—62no—
182C2×D4—64no—
183C2×D4—122no—
184C22×C4—22no—
185C22×C4—34no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—32no—
190C22×C4—42no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4C22×D312yes—
194C22×C4C22×D311yescharacteristic
195C22×C4C22×D311yescharacteristic
196C22×C4C22×D312yes—
197C4⋊C4—42no—
198C4⋊C4—62no—
199C4⋊C4—62no—
200C4⋊C4—64no—
201C22⋊C4—22no—
202C22⋊C4—64no—
203C22⋊C4—64no—
204C22⋊C4—62no—
205C22⋊C4—62no—
206C22⋊C4—64no—
207C22⋊C4—62no—
208C22⋊C4—64no—
209C22⋊C4—122no—
210C22⋊C4—122no—
211C22⋊C4D1214yes—
212C42—64no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—24no—
216C22×C3—24no—
217C22×C3#12114yes—
218C22×C3C22×D411yescharacteristic
219C22×C3C22×D411yescharacteristic
220C22×C3C2×#4111yescharacteristic
221C22×C3C2×#4112yes—
222C22×C3H(5,𝔽2)12yes—
223C2×D3—24no—
224C2×D3—24no—
225C2×D3—24no—
226C2×D3—416no—
227C2×D3—42no—
228C2×D3—42no—
229C2×D3—42no—
230C2×D3—42no—
231C2×D3—41no—
232C2×D3—41no—
233C2×D3—41no—
234C2×D3—42no—
235C2×D3—42no—
236C3×C4—24no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4—42no—
240Dic3—22no—
241Dic3—22no—
242Dic3—41no—
243Dic3—42no—
244C23—22no—
245C23—22no—
246C23—34no—
247C23—68no—
248C23—62no—
249C23—62no—
250C23—62no—
251C23—62no—
252C23—62no—
253C23—62no—
254C23—61no—
255C23—61no—
256C23—62no—
257C23—61no—
258C23—62no—
259C23—128no—
260C23—128no—
261C23—124no—
262C23—124no—
263C23C2×D1212yes—
264C23D3×D412yes—
265C23#23912yes—
266C23C23×D311yescharacteristic
267D4—128no—
268D4—124no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—32no—
279C2×C4—32no—
280C2×C4—44no—
281C2×C4—68no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—62no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—62no—
290C2×C4—61no—
291C2×C4—62no—
292C2×C4C2×D1214yes—
293C2×C3—24no—
294C2×C3C2×#12111yescharacteristic
295C2×C3#52511yescharacteristic
296C2×C3#53511yescharacteristic
297C2×C3#54612yes—
298C2×C3#55112yes—
299D3—44no—
300D3—81no—
301C22—24no—
302C22—22no—
303C22—24no—
304C22—22no—
305C22—64no—
306C22—64no—
307C22—64no—
308C22—1216no—
309C22—122no—
310C22—122no—
311C22—122no—
312C22—122no—
313C22—122no—
314C22—121no—
315C22—121no—
316C22—122no—
317C22—121no—
318C22#89514yes—
319C22C22×D1211yescharacteristic
320C22C2×D3×D411yescharacteristic
321C22#100412yes—
322C22C2×#23911yescharacteristic
323C22D3×#4112yes—
324C4—24no—
325C4—41no—
326C4—42no—
327C4—41no—
328C4—62no—
329C4—62no—
330C4—121no—
331C4—122no—
332C3#241111yescharacteristic, complemented, Sylow
333C2—24no—
334C2—124no—
335C2—241no—
336C2C2×#89511yescharacteristic
337C2#555412yes—
338C2#559211yescharacteristic
339C2#563712yes—
340C2#565611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC2×#882C2
hypercenter#385D3
hypocenterC3#2411
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×(C2×D3)⋊C4C2semidirect
2. C2×#886C2semidirect
4. C2×#877C2semidirect
5. C3×#385C2semidirect
6. #5213C2semidirect
7. #4994C2semidirect
8. #4988C2semidirect
9. #4955C2semidirect
10. #4672C2semidirect
11. #4670C2semidirect
18. C2×C3×C4⋊C4C22semidirect
28. C2×(C2×D3)⋊C4C22semidirect
29. C2×(C2×D3)⋊C4C22semidirect
30. C2×(C2×D3)⋊C4C22semidirect
31. C2×C4⋊Dic3C22semidirect
32. C2×C4⋊Dic3C22semidirect
33. C2×Dic3⋊C4C22semidirect
34. C2×C4×Dic3C22semidirect
38. C3×#95C22semidirect
39. C3⋊#95C22semidirect
49. #385D3semidirect
70. C22×Dic3D4semidirect
105. C2×C4⋊C4C2×D3semidirect
120. #95C2×D3semidirect
332. C3#2411semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D3)⋊C42
C2×#8862
C2×#8822
C2×#8772
C3×#3852
#52132
#49942
#49882
#49552
#46722
#46702
#24113