Finite Group Databasebeta

← #76864

Subgroups — #76864

Order 384.

914 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 157 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×#239C211yesmaximal, characteristic, complemented
2C2×#884C212yesmaximal, complemented
3C2×#883C211yesmaximal, characteristic, complemented
4C3×#393C211yesmaximal, characteristic, complemented
5#5230C211yesmaximal, characteristic, complemented
6#4960C211yesmaximal, characteristic, complemented
7#4956C212yesmaximal, complemented
8#4672C211yesmaximal, characteristic, complemented
9#4668C212yesmaximal, complemented
10#4665C212yesmaximal, complemented
11D3×#95C211yesmaximal, characteristic, complemented
12#2430—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×#67C2212yes—
16C2×#239—28no—
17C2×#239—24no—
18C22×D12C2211yescharacteristic
19C22×D12C2212yes—
20C22×C4×D3C2212yes—
21C22×C4×D3C2212yes—
22C22×C4×D3C2211yescharacteristic
23C2×C3×C22⋊C4C2212yes—
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2211yescharacteristic
33#884—28no—
34#883—24no—
35C3×#95C2211yescharacteristic
36C3×#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C3⋊#95C2212yes—
40C22×#41—31no—
41C2×#123—31no—
42C2×#121—32no—
43#395—31no—
44#394—31no—
45#393D311yescharacteristic, complemented
46#392—31no—
47#386—32no—
48#386—32no—
49#385—32no—
50C2×#95—31no—
51C24×C3C2311yescharacteristic
52C23×D3C2312yes—
53C23×D3C2311yescharacteristic
54C2×C3×Q8—24no—
55C2×C3×Q8—22no—
56C22×C3×C4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2312yes—
59C22×C3×C4C2311yescharacteristic
60C2×#67—22no—
61C2×#67—24no—
62C2×#67—24no—
63C2×#67—24no—
64C2×#67—42no—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69#239—232no—
70C2×D12—24no—
71C2×D12—24no—
72C2×D12—24no—
73C2×D12—22no—
74C2×D12D418yescomplemented
75C2×C4×D3—24no—
76C2×C4×D3—24no—
77C2×C4×D3—24no—
78C2×C4×D3—24no—
79C2×C4×D3—24no—
80C2×C4×D3—24no—
81C2×C4×D3—42no—
82C2×C4×D3D414yescomplemented
83C3×C22⋊C4—22no—
84C3×C22⋊C4—24no—
85#217—22no—
86(C2×D3)⋊C4—22no—
87(C2×D3)⋊C4—24no—
88(C2×D3)⋊C4—24no—
89(C2×D3)⋊C4—42no—
90C4⋊Dic3—22no—
91Dic3⋊C4—22no—
92Dic3⋊C4—24no—
93C2×#41—64no—
94C2×#41—68no—
95C22×Q8C2×D311yescharacteristic
96C22×D4—31no—
97C22×D4—32no—
98C22×D4—32no—
99C22×D4—31no—
100C23×C4—31no—
101C23×C4—32no—
102C23×C4—32no—
103#123—64no—
104#121—68no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—31no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4C2×D312yes—
114C2×C22⋊C4C2×D311yescharacteristic
115#95—32no—
116#95—31no—
117#95—32no—
118#95C2×D311yescharacteristic
119#95C2×D312yes—
120C23×C3—21no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3C2411yescharacteristic
126C22×D3—22no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—22no—
133C22×D3—22no—
134C22×D3—22no—
135C22×D3C2×D414yes—
136C22×D3C2×D412yes—
137C22×D3C2×D414yes—
138C3×Q8—28no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4C2×D414yes—
148C2×C3×C4C2×D412yes—
149#67—48no—
150#67—44no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3C2×D412yes—
160C2×Dic3#4114yescomplemented
161D12—28no—
162D12—216no—
163C4×D3—216no—
164C4×D3—28no—
165C4×D3—48no—
166C24—31no—
167C24—32no—
168C24C22×D311yescharacteristic
169#41—632no—
170C2×Q8—22no—
171C2×Q8—24no—
172C2×D4—38no—
173C2×D4—64no—
174C2×D4—64no—
175C2×D4—64no—
176C2×D4—64no—
177C2×D4—64no—
178C2×D4—64no—
179C2×D4—62no—
180C2×D4—62no—
181C2×D4—122no—
182C22×C4—32no—
183C22×C4—31no—
184C22×C4—34no—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—64no—
188C22×C4—64no—
189C22×C4—64no—
190C22×C4—64no—
191C22×C4—64no—
192C22×C4—64no—
193C22×C4—122no—
194C22×C4C22×D312yes—
195C22×C4C22×D311yescharacteristic
196C22×C4C22×D311yescharacteristic
197C22×C4C22×D312yes—
198C4⋊C4—62no—
199C4⋊C4—64no—
200C4⋊C4—62no—
201C22⋊C4—24no—
202C22⋊C4—22no—
203C22⋊C4—64no—
204C22⋊C4—64no—
205C22⋊C4—62no—
206C22⋊C4—62no—
207C22⋊C4—122no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3C22×D411yescharacteristic
214C22×C3C22×D412yes—
215C22×C3C2×#4112yes—
216C22×C3H(5,𝔽2)11yescharacteristic
217C22×C32−1+411yescharacteristic
218C2×D3—22no—
219C2×D3—22no—
220C2×D3—24no—
221C2×D3—24no—
222C2×D3—24no—
223C2×D3—24no—
224C2×D3—24no—
225C2×D3—42no—
226C2×D3—42no—
227C2×D3—42no—
228C2×D3—42no—
229C2×D3#12014yes—
230C3×C4—24no—
231C3×C4—22no—
232C3×C4—41no—
233C3×C4—42no—
234Dic3—22no—
235Dic3—24no—
236Dic3—41no—
237Dic3—41no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—22no—
242C23—21no—
243C23—32no—
244C23—34no—
245C23—34no—
246C23—61no—
247C23—62no—
248C23—61no—
249C23—62no—
250C23—62no—
251C23—62no—
252C23—62no—
253C23—62no—
254C23—62no—
255C23C23×D311yescharacteristic
256Q8—28no—
257D4—616no—
258D4—68no—
259D4—128no—
260D4—124no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—32no—
270C2×C4—34no—
271C2×C4—68no—
272C2×C4—616no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—61no—
277C2×C4—62no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—128no—
282C2×C4D3×D412yes—
283C2×C4D3×D414yes—
284C2×C3—81no—
285C2×C3C2×#12011yescharacteristic
286C2×C3#53211yescharacteristic
287C2×C3#53711yescharacteristic
288C2×C3#54612yes—
289C2×C3#54712yes—
290D3—24no—
291D3—44no—
292C22—34no—
293C22—41no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—62no—
299C22—64no—
300C22—62no—
301C22—64no—
302C22—64no—
303C22—64no—
304C22—64no—
305C22—122no—
306C22—122no—
307C22—122no—
308C22—122no—
309C22C2×D3×D411yescharacteristic
310C22C2×D3×D412yes—
311C22#100711yescharacteristic
312C22D3×#4112yes—
313C22#100911yescharacteristic
314C4—22no—
315C4—24no—
316C4—41no—
317C4—42no—
318C4—64no—
319C4—62no—
320C4—121no—
321C4—121no—
322C3#243011yescharacteristic, complemented, Sylow
323C2—64no—
324C2—81no—
325C2—124no—
326C2D3×#12011yescharacteristic
327C2#562911yescharacteristic
328C2#563112yes—
329C2#566811yescharacteristic
330C2#567612yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
hypercenter#393D3
hypocenterC3#2430
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#239C2semidirect
2. C2×#884C2semidirect
3. C2×#883C2semidirect
4. C3×#393C2semidirect
5. #5230C2semidirect
6. #4960C2semidirect
7. #4956C2semidirect
8. #4672C2semidirect
9. #4668C2semidirect
10. #4665C2semidirect
11. D3×#95C2semidirect
45. #393D3semidirect
74. C2×D12D4semidirect
82. C2×C4×D3D4semidirect
160. C2×Dic3#41semidirect
322. C3#2430semidirect

Maximal subgroups

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

subgroupindex
C22×#2392
C2×#8842
C2×#8832
C3×#3932
#52302
#49602
#49562
#46722
#46682
#46652
D3×#952
#24303