Finite Group Databasebeta

← #10692749

Subgroups — #10692749

Order 768.

444 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#83231C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic
3C3×#1556C211yesmaximal, characteristic, complemented
4#70835C211yesmaximal, characteristic
5#70835C211yesmaximal, characteristic
6#67752C211yesmaximal, characteristic, complemented
7#67752C211yesmaximal, characteristic, complemented
8#9430—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13#5229—22no—
14#5093C412yescomplemented
15#5089—22no—
16#5088C412yescomplemented
17#4980—22no—
18#4980—22no—
19#4966—22no—
20#4937C2211yescharacteristic
21#4937C2211yescharacteristic
22C4×#208C2211yescharacteristic
23C3×#332—22no—
24#4470—22no—
25#4470—22no—
26#3141—31no—
27#2381—31no—
28#1556D311yescharacteristic, complemented
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⋊C4D412yes—
41C2×C3×C42C2311yescharacteristic
42#930—21no—
43#930—21no—
44C2×C4⋊Dic3—22no—
45C2×C4×Dic3—22no—
46C2×#208—21no—
47#875—21no—
48#874C2×C411yescharacteristic
49C4×D12—21no—
50C4×D12—21no—
51#869C2×C411yescharacteristic
52C3×C4⋊C8—21no—
53C3×C4⋊C8—21no—
54C3×C22⋊C8—21no—
55C3×C22⋊C8—21no—
56C3×#97—21no—
57C3×#97—21no—
58C3⋊#95—24no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95D412yes—
64C3⋊#95D412yes—
65#810—24no—
66#809—24no—
67#803—21no—
68C4×C3⋊C8C2×C412yes—
69#532—31no—
70C2×C4⋊Q8C2×D311yescharacteristic
71#489—62no—
72#487—32no—
73#486—32no—
74#432C2×D311yescharacteristic
75#432C2×D311yescharacteristic
76#417—62no—
77C4×M16—31no—
78#391—62no—
79#387—62no—
80#387—62no—
81#383—31no—
82#383—31no—
83#332—22no—
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×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C2×#67—41no—
95C22×Dic3—22no—
96C22×Dic3—22no—
97#235—44no—
98C2×D12—21no—
99C2×D12—21no—
100C2×D12—21no—
101C2×C4×D3—41no—
102C2×Dic6—21no—
103C2×C3×C8—41no—
104C2×C3×C8—41no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4C22⋊C414yes—
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—
117C4⋊Dic3—82no—
118C4×Dic3—82no—
119#208—42no—
120C2×C3⋊C8—22no—
121C2×#41—61no—
122C22×Q8—21no—
123C2×M16—61no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—31no—
127C4⋊Q8C4×D312yes—
128C4⋊Q8#6712yes—
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—62no—
137C2×C4⋊C4D1212yes—
138C2×C42—62no—
139C2×C42C22×D311yescharacteristic
140C4⋊C8—21no—
141C4⋊C8—21no—
142#103—64no—
143#102—64no—
144C22⋊C8—21no—
145C22⋊C8—21no—
146#97—21no—
147#97—21no—
148#97—61no—
149C4×C8—32no—
150#95—32no—
151#95—32no—
152#95—64no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
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×C4C2×C22⋊C411yescharacteristic
168C2×C3×C4C4×D411yescharacteristic
169C2×C3×C4C4×D411yescharacteristic
170C2×C3×C4#12111yescharacteristic
171C2×C3×C4#12111yescharacteristic
172C2×C3×C4#12311yescharacteristic
173#67—82no—
174C2×Dic3—42no—
175C2×Dic3—42no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—41no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181D12—42no—
182D12—81no—
183D12—81no—
184C4×D3—82no—
185Dic6—42no—
186C3×C8—81no—
187C3×C8—81no—
188C3⋊C8—42no—
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—62no—
201C22×C4—62no—
202C22×C4—121no—
203C22×C4C2×D1211yescharacteristic
204C22×C4D3×D411yescharacteristic
205C22×C4D3×D411yescharacteristic
206M16—122no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—62no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—61no—
214C4⋊C4—61no—
215C4⋊C4—242no—
216C4⋊C4(C2×D3)⋊C414yes—
217C22⋊C4—121no—
218C22⋊C4—121no—
219C42—242no—
220C42C2×C4×D311yescharacteristic
221C42#23711yescharacteristic
222C42#23711yescharacteristic
223C42C2×#6711yescharacteristic
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—
238Dic3—82no—
239Dic3—82no—
240Dic3—81no—
241C23—121no—
242C23#88211yescharacteristic
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—122no—
257C2×C4—122no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4—121no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—242no—
264C2×C4#67⋊C411yescharacteristic
265C2×C4#67⋊C411yescharacteristic
266C2×C4#88611yescharacteristic
267C2×C4C2×(C2×D3)⋊C411yescharacteristic
268C2×C4#93911yescharacteristic
269C2×C4#93911yescharacteristic
270C8—81no—
271C8—81no—
272C8—122no—
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#475312yes—
284C22#495811yescharacteristic
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—44no—
292C4—242no—
293C4—242no—
294C4—241no—
295C3#943011yescharacteristic, complemented, Sylow
296C2—22no—
297C2—481no—
298C2#7010411yescharacteristic
299C2#7031411yescharacteristic
300C2#7033411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1556D3
hypocenterC3#9430
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
1. #83231C2semidirect
3. C3×#1556C2semidirect
6. #67752C2semidirect
7. #67752C2semidirect
14. #5093C4semidirect
16. #5088C4semidirect
28. #1556D3semidirect
295. C3#9430semidirect

Maximal subgroups

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

subgroupindex
#832312
#831992
C3×#15562
#708352
#677522
#94303