Finite Group Databasebeta

← #10692751

Subgroups — #10692751

Order 768.

444 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#2381C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic
3#71776C211yesmaximal, characteristic, complemented
4#69564C211yesmaximal, characteristic
5#69564C211yesmaximal, characteristic
6#67633C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9430—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#391—22no—
12C3×#387—22no—
13C3×#387—22no—
14C3×#383C2211yescharacteristic
15C3×#383C2211yescharacteristic
16#5093C412yescomplemented
17#5089—22no—
18#5088C412yescomplemented
19#4966—22no—
20#4926C2211yescharacteristic
21#4926C2211yescharacteristic
22C4×#208C2211yescharacteristic
23#4547—22no—
24#4451—22no—
25#4451—22no—
26#3141—31no—
27#2381D311yescharacteristic, complemented
28#1556—31no—
29#1514—31no—
30#1514—31no—
31#1288—31no—
32#1288—31no—
33C22×C3×Q8—21no—
34C2×#235—21no—
35C3×C4⋊Q8—21no—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8C2×C412yes—
38C3×C4⋊Q8D412yes—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4D412yes—
42C2×C3×C42—22no—
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930—21no—
46C2×#208—21no—
47#875—21no—
48#874C2×C411yescharacteristic
49C4×D12—21no—
50C4×D12—21no—
51#869C2×C411yescharacteristic
52C3×#95—24no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95D412yes—
58C3×#95D412yes—
59#830—21no—
60#830—21no—
61#810—24no—
62#809—24no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)—21no—
65#803—21no—
66#803—21no—
67#803—21no—
68C4×C3⋊C8C2×C412yes—
69#532—31no—
70C2×C4⋊Q8C2×D311yescharacteristic
71#489—62no—
72#487—32no—
73#486—32no—
74#432—31no—
75#432—31no—
76#417—62no—
77C4×M16—31no—
78#391—22no—
79#387—22no—
80#387—22no—
81#383C2×D311yescharacteristic
82#383C2×D311yescharacteristic
83#332—62no—
84#327—62no—
85#327—62no—
86C2×C3×Q8—22no—
87C2×C3×Q8—22no—
88C2×C3×Q8—81no—
89C2×C3×Q8—81no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C2×#67—41no—
97#235—44no—
98C2×D12—21no—
99C2×D12—21no—
100C2×D12—21no—
101C2×C4×D3—41no—
102C2×Dic6—21no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—82no—
107C3×C4⋊C4C22⋊C414yes—
108C3×C42—82no—
109C3×C42C22×C411yescharacteristic
110C3×C42C2×D411yescharacteristic
111C3×C42#4111yescharacteristic
112C3×C42#4111yescharacteristic
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117#208—42no—
118C2×C3⋊C8—22no—
119C2×C3⋊C8—41no—
120C2×C3⋊C8—41no—
121C2×#41—61no—
122C22×Q8—21no—
123C2×M16—61no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—31no—
127C4⋊Q8C4×D312yes—
128C4⋊Q8D1212yes—
129#127—31no—
130#124—61no—
131#121—61no—
132#121—61no—
133C4×D4—61no—
134C4×D4—61no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4#6712yes—
138C2×C42—22no—
139C2×C42C22×D311yescharacteristic
140C4⋊C8—61no—
141C4⋊C8—61no—
142#103—64no—
143#102—64no—
144C22⋊C8—61no—
145C22⋊C8—61no—
146#97—61no—
147#97—61no—
148#97—61no—
149C4×C8—32no—
150#95—24no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95#6712yes—
156#95#6712yes—
157C23×C3#12011yescharacteristic
158C22×D3—41no—
159C3×Q8—82no—
160C3×Q8—82no—
161C2×C3×C4—22no—
162C2×C3×C4—24no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4C2×C22⋊C411yescharacteristic
174C2×C3×C4C4×D411yescharacteristic
175C2×C3×C4C4×D411yescharacteristic
176C2×C3×C4#12111yescharacteristic
177C2×C3×C4#12111yescharacteristic
178C2×C3×C4#12311yescharacteristic
179#67—82no—
180C2×Dic3—41no—
181D12—42no—
182D12—81no—
183D12—81no—
184C4×D3—82no—
185Dic6—42no—
186C3⋊C8—42no—
187C3⋊C8—81no—
188C3⋊C8—81no—
189#41—124no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—61no—
193C2×Q8—81no—
194C2×Q8—81no—
195C2×D4—61no—
196C2×D4—61no—
197C2×D4—61no—
198C2×D4—121no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—121no—
203C22×C4C2×#6711yescharacteristic
204C22×C4C2×#6711yescharacteristic
205C22×C4C2×#6711yescharacteristic
206M16—122no—
207C2×C8—62no—
208C2×C8—121no—
209C2×C8—121no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—61no—
214C4⋊C4—61no—
215C4⋊C4—82no—
216C4⋊C4(C2×D3)⋊C414yes—
217C22⋊C4—121no—
218C22⋊C4—121no—
219C42—82no—
220C42C2×C4×D311yescharacteristic
221C42C2×D1211yescharacteristic
222C42#23511yescharacteristic
223C42#23511yescharacteristic
224C22×C3—21no—
225C22×C3—21no—
226C22×C3#38611yescharacteristic
227C22×C3#45412yes—
228C2×D3—81no—
229C2×D3—81no—
230C2×D3—81no—
231C3×C4—21no—
232C3×C4—21no—
233C3×C4—21no—
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—44no—
238C3×C4—82no—
239C3×C4—82no—
240Dic3—81no—
241C23—121no—
242C23#95311yescharacteristic
243Q8—82no—
244Q8—82no—
245Q8—122no—
246D4—122no—
247D4—241no—
248D4—241no—
249D4—242no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—121no—
263C2×C4—242no—
264C2×C4C2×(C2×D3)⋊C411yescharacteristic
265C2×C4C4×#6711yescharacteristic
266C2×C4C4×#6711yescharacteristic
267C2×C4#92911yescharacteristic
268C2×C4#93011yescharacteristic
269C2×C4#93011yescharacteristic
270C8—122no—
271C8—241no—
272C8—241no—
273C2×C3—22no—
274C2×C3#188811yescharacteristic
275C2×C3#190411yescharacteristic
276C2×C3#190511yescharacteristic
277D3—161no—
278C22—21no—
279C22—21no—
280C22—241no—
281C22—241no—
282C22—241no—
283C22#521311yescharacteristic
284C22#524512yes—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—44no—
292C4—82no—
293C4—82no—
294C4—241no—
295C3#943011yescharacteristic, complemented, Sylow
296C2—22no—
297C2—481no—
298C2#7165111yescharacteristic
299C2#7166111yescharacteristic
300C2#7166211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2381C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2381D3
hypocenterC3#9430
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2381

Complements

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

NH ≅ G/Ntype
1. C3×#2381C2semidirect
3. #71776C2semidirect
6. #67633C2semidirect
7. #67633C2semidirect
16. #5093C4semidirect
18. #5088C4semidirect
27. #2381D3semidirect
295. C3#9430semidirect

Maximal subgroups

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

subgroupindex
C3×#23812
#831992
#717762
#695642
#676332
#94303