Finite Group Databasebeta

← #76781

Subgroups — #76781

Order 384.

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

normal subgroups only

This group has 852 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×D3×Q8C211yesmaximal, characteristic, complemented
2C2×#885C212yesmaximal, complemented
3C2×#884C211yesmaximal, characteristic
4C3×#393C211yesmaximal, characteristic, complemented
5#5231C211yesmaximal, characteristic, complemented
6#4956C212yesmaximal, complemented
7#4947C211yesmaximal, characteristic, complemented
8#4673C211yesmaximal, characteristic, complemented
9#4666C211yesmaximal, characteristic, complemented
10#4665C212yesmaximal, complemented
11#4646C212yesmaximal, complemented
12#2436—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C2×D3×Q8—24no—
17C2×D3×Q8—28no—
18C22×D12C2211yescharacteristic
19C22×C4×D3C2212yes—
20C22×C4×D3C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22C22×Dic6C2212yes—
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2212yes—
25C2×#217C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×C4×Dic3C2212yes—
32#885—28no—
33#884—24no—
34C3×#95C2212yes—
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2211yescharacteristic
40C23×Q8—31no—
41C2×#124—32no—
42C2×#121—31no—
43#393—31no—
44#393—31no—
45#393—31no—
46#393D311yescharacteristic, complemented
47#391—32no—
48#386—32no—
49#386—32no—
50C4×C22⋊C4—31no—
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C23×D3C2311yescharacteristic
54C2×C3×Q8—22no—
55C2×C3×Q8—24no—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2312yes—
60C2×#67—22no—
61C2×#67—24no—
62C2×#67—22no—
63C2×#67—24no—
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68D3×Q8—232no—
69C2×D12—22no—
70C2×D12—24no—
71C2×C4×D3—24no—
72C2×C4×D3—24no—
73C2×C4×D3—24no—
74C2×C4×D3—24no—
75C2×C4×D3D414yes—
76C2×Dic6—22no—
77C2×Dic6—24no—
78C2×Dic6—24no—
79C2×Dic6—24no—
80C2×Dic6D418yescomplemented
81C3×C22⋊C4—22no—
82C3×C22⋊C4—24no—
83#217—24no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—24no—
87(C2×D3)⋊C4—24no—
88Dic3⋊C4—22no—
89C4×Dic3—24no—
90C22×Q8—32no—
91C22×Q8—31no—
92C22×Q8—64no—
93C22×Q8—68no—
94C22×Q8C2×D311yescharacteristic
95C22×D4—31no—
96C22×D4—31no—
97C22×D4—31no—
98C23×C4—31no—
99C23×C4—32no—
100#124—68no—
101#121—64no—
102C2×C4⋊C4—31no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—31no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4C2×D311yescharacteristic
109C2×C22⋊C4C2×D312yes—
110C2×C42—32no—
111#95—31no—
112#95—32no—
113#95—32no—
114#95—32no—
115#95C2×D311yescharacteristic
116#95C2×D312yes—
117C23×C3—22no—
118C23×C3—21no—
119C23×C3—22no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3C2411yescharacteristic
123C22×D3—22no—
124C22×D3—21no—
125C22×D3—21no—
126C22×D3—22no—
127C22×D3—21no—
128C22×D3—22no—
129C22×D3—21no—
130C22×D3—21no—
131C22×D3C2×D412yes—
132C22×D3#4114yes—
133C3×Q8—28no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4C2×D412yes—
143C2×C3×C4C2×D414yes—
144#67—44no—
145#67—44no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3C2×D414yes—
154C2×Dic3C2×D414yes—
155C2×Dic3C2×D412yes—
156D12—44no—
157C4×D3—216no—
158C4×D3—28no—
159Dic6—28no—
160Dic6—216no—
161C24—31no—
162C24—31no—
163C24C22×D311yescharacteristic
164C2×Q8—22no—
165C2×Q8—24no—
166C2×Q8—38no—
167C2×Q8—64no—
168C2×Q8—64no—
169C2×Q8—64no—
170C2×Q8—632no—
171C2×Q8—62no—
172C2×D4—62no—
173C2×D4—64no—
174C2×D4—62no—
175C2×D4—64no—
176C2×D4—62no—
177C2×D4—64no—
178C22×C4—31no—
179C22×C4—32no—
180C22×C4—34no—
181C22×C4—31no—
182C22×C4—32no—
183C22×C4—64no—
184C22×C4—64no—
185C22×C4—64no—
186C22×C4—64no—
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D312yes—
189C22×C4C22×D311yescharacteristic
190C22×C4C22×D312yes—
191C4⋊C4—62no—
192C22⋊C4—22no—
193C22⋊C4—24no—
194C22⋊C4—64no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—62no—
198C22⋊C4—64no—
199C42—64no—
200C22×C3—41no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3C22×D411yescharacteristic
206C22×C3C22×D412yes—
207C22×C3C2×#4112yes—
208C22×C32−1+411yescharacteristic
209C22×C32−1+411yescharacteristic
210C2×D3—22no—
211C2×D3—22no—
212C2×D3—24no—
213C2×D3—24no—
214C2×D3—41no—
215C2×D3—41no—
216C2×D3—41no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3#12414yescomplemented
220C3×C4—22no—
221C3×C4—24no—
222C3×C4—41no—
223C3×C4—42no—
224Dic3—24no—
225Dic3—41no—
226Dic3—42no—
227Dic3#12014yes—
228C23—22no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—21no—
233C23—32no—
234C23—34no—
235C23—62no—
236C23—61no—
237C23—61no—
238C23—62no—
239C23—61no—
240C23—61no—
241C23—62no—
242C23—61no—
243C23C23×D311yescharacteristic
244Q8—28no—
245Q8—616no—
246Q8—68no—
247D4—124no—
248D4—124no—
249D4—124no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—34no—
259C2×C4—32no—
260C2×C4—34no—
261C2×C4—68no—
262C2×C4—616no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4D3×D412yes—
271C2×C4D3×D414yes—
272C2×C3—81no—
273C2×C3C2×#12011yescharacteristic
274C2×C3C2×#12411yescharacteristic
275C2×C3#53611yescharacteristic
276C2×C3#54812yes—
277C2×C3#54812yes—
278D3—24no—
279D3—81no—
280C22—34no—
281C22—42no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22—64no—
287C22—64no—
288C22—62no—
289C22—62no—
290C22—122no—
291C22—121no—
292C22—122no—
293C22—121no—
294C22—121no—
295C22C2×D3×D412yes—
296C22C2×D3×D411yescharacteristic
297C22#100711yescharacteristic
298C22D3×#4112yes—
299C22#101011yescharacteristic
300C4—22no—
301C4—24no—
302C4—34no—
303C4—41no—
304C4—42no—
305C4—64no—
306C4—122no—
307C4—121no—
308C3#243611yescharacteristic, complemented, Sylow
309C2—64no—
310C2—81no—
311C2—241no—
312C2D3×#12011yescharacteristic
313C2#562811yescharacteristic
314C2#563312yes—
315C2D3×#12411yescharacteristic
316C2#567712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC2×#884C2
hypercenter#393D3
hypocenterC3#2436
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×D3×Q8C2semidirect
2. C2×#885C2semidirect
4. C3×#393C2semidirect
5. #5231C2semidirect
6. #4956C2semidirect
7. #4947C2semidirect
8. #4673C2semidirect
9. #4666C2semidirect
10. #4665C2semidirect
11. #4646C2semidirect
46. #393D3semidirect
80. C2×Dic6D4semidirect
219. C2×D3#124semidirect
308. C3#2436semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Q82
C2×#8852
C2×#8842
C3×#3932
#52312
#49562
#49472
#46732
#46662
#46652
#46462
#24363