Finite Group Databasebeta

← #10692748

Subgroups — #10692748

Order 768.

444 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 89 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#83283C211yesmaximal, characteristic
2#74265C211yesmaximal, characteristic, complemented
3C3×#1514C211yesmaximal, characteristic
4#71813C211yesmaximal, characteristic
5#70822C211yesmaximal, characteristic, complemented
6#67752C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9430—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#383C2211yescharacteristic
13#5232—22no—
14#5079C412yescomplemented
15#5070C412yescomplemented
16#5055—22no—
17#4937C2211yescharacteristic
18#4926C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4654—22no—
21#4653—22no—
22#4646—22no—
23C3×#327—22no—
24#4537—22no—
25#4486—22no—
26#3141—31no—
27#2381—31no—
28#1556—31no—
29#1514—31no—
30#1514D311yescharacteristic
31#1288—31no—
32#1288—31no—
33C2×C3×#41—21no—
34C22×Dic6—21no—
35C3×C4⋊Q8C2×C411yescharacteristic
36C3×#127C2×C411yescharacteristic
37C3×#124—21no—
38C3×#121—21no—
39C3×#121—21no—
40C3×C4×D4—21no—
41C3×C4×D4—21no—
42C2×C3×C42C2311yescharacteristic
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3D412yes—
45C2×Dic3⋊C4—22no—
46C2×C4×Dic3—22no—
47C2×#208—21no—
48#869—21no—
49#869—21no—
50#869C2×C412yes—
51#869D412yes—
52C3×C4⋊C8—21no—
53C3×C22⋊C8—21no—
54C3×#97—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95D412yes—
58#835—24no—
59#832—24no—
60C3⋊#95—24no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95D412yes—
64#830—21no—
65C4⋊(C3⋊C8)—21no—
66#803—21no—
67#803—21no—
68C4×C3⋊C8C2×C412yes—
69#532C2×D311yescharacteristic
70C2×C4⋊Q8—31no—
71#489—62no—
72#487—32no—
73#486—32no—
74#432—31no—
75#432C2×D311yescharacteristic
76#417—62no—
77C4×M16—31no—
78#391—62no—
79#387—62no—
80#387—62no—
81#383—31no—
82#383C2×D311yescharacteristic
83#332—62no—
84#327—22no—
85#327—62no—
86C3×#41—44no—
87C2×C3×Q8—21no—
88C2×C3×D4—21no—
89C2×C3×D4—21no—
90C2×C3×D4—21no—
91C2×C3×D4—41no—
92C22×C3×C4—22no—
93C22×C3×C4—41no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—22no—
98C22×Dic3—22no—
99C2×Dic6—22no—
100C2×Dic6—22no—
101C2×Dic6—81no—
102C2×Dic6—81no—
103C2×C3×C8—41no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C22⋊C4—41no—
107C3×C22⋊C4—41no—
108C3×C42C22×C411yescharacteristic
109C3×C42C2×D411yescharacteristic
110C3×C42#4111yescharacteristic
111C3×C42#4111yescharacteristic
112C4⋊Dic3—22no—
113C4⋊Dic3—22no—
114C4⋊Dic3—22no—
115C4⋊Dic3C22⋊C414yes—
116Dic3⋊C4—82no—
117C4×Dic3—82no—
118#208—42no—
119C2×C3⋊C8—22no—
120C2×C3⋊C8—41no—
121C2×#41—21no—
122C22×Q8—61no—
123C2×M16—61no—
124C4⋊Q8—32no—
125C4⋊Q8—32no—
126C4⋊Q8—61no—
127C4⋊Q8—61no—
128C4⋊Q8C4×D311yescharacteristic
129#127C4×D311yescharacteristic
130#124—21no—
131#121—21no—
132#121—21no—
133C4×D4—21no—
134C4×D4—21no—
135C2×C4⋊C4—32no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—62no—
138C2×C42—62no—
139C2×C42C22×D311yescharacteristic
140C4⋊C8—21no—
141C4⋊C8—61no—
142#103—64no—
143#102—64no—
144C22⋊C8—21no—
145C22⋊C8—61no—
146#97—21no—
147#97—61no—
148#97—61no—
149C4×C8—32no—
150#95—21no—
151#95—21no—
152#95—32no—
153#95—64no—
154#95—61no—
155#95—61no—
156#95#6712yes—
157C23×C3—41no—
158C23×C3#12011yescharacteristic
159C3×Q8—42no—
160C3×D4—42no—
161C3×D4—81no—
162C3×D4—81no—
163C3×D4—82no—
164C2×C3×C4—22no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—82no—
172C2×C3×C4C2×C22⋊C411yescharacteristic
173C2×C3×C4C4×D411yescharacteristic
174C2×C3×C4C4×D411yescharacteristic
175C2×C3×C4#12111yescharacteristic
176C2×C3×C4#12111yescharacteristic
177C2×C3×C4#12311yescharacteristic
178C2×Dic3—24no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181C2×Dic3—42no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184Dic6—82no—
185Dic6—82no—
186C3×C8—81no—
187C3⋊C8—42no—
188C3⋊C8—81no—
189#41—44no—
190C2×Q8—21no—
191C2×Q8—62no—
192C2×Q8—62no—
193C2×Q8—241no—
194C2×Q8—241no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—41no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—62no—
202C22×C4—62no—
203C22×C4D3×D411yescharacteristic
204C22×C4D3×D411yescharacteristic
205C22×C4C2×#6711yescharacteristic
206M16—122no—
207C2×C8—41no—
208C2×C8—62no—
209C2×C8—121no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—34no—
213C4⋊C4—62no—
214C4⋊C4—62no—
215C4⋊C4—62no—
216C4⋊C4—242no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C42—242no—
220C42C2×C4×D311yescharacteristic
221C42#23511yescharacteristic
222C42D3×D411yescharacteristic
223C42#23711yescharacteristic
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—81no—
227C22×C3—81no—
228C22×C3—81no—
229C22×C3#38611yescharacteristic
230C22×C3#45412yes—
231C3×C4—21no—
232C3×C4—21no—
233C3×C4—21no—
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—82no—
238C3×C4—81no—
239Dic3—44no—
240Dic3—82no—
241C23—41no—
242C23#93711yescharacteristic
243Q8—42no—
244Q8—242no—
245Q8—242no—
246D4—42no—
247D4—81no—
248D4—81no—
249D4—82no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—64no—
258C2×C4—82no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4D3×C22⋊C411yescharacteristic
265C2×C4#67⋊C411yescharacteristic
266C2×C4#88311yescharacteristic
267C2×C4#88411yescharacteristic
268C2×C4C4×#6711yescharacteristic
269C2×C4#93811yescharacteristic
270C8—81no—
271C8—122no—
272C8—241no—
273C2×C3—22no—
274C2×C3—161no—
275C2×C3#188811yescharacteristic
276C2×C3#190411yescharacteristic
277C2×C3#190511yescharacteristic
278C22—21no—
279C22—21no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22#495611yescharacteristic
284C22#520112yes—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—82no—
292C4—81no—
293C4—124no—
294C4—242no—
295C3#943011yescharacteristic, complemented, Sylow
296C2—22no—
297C2—161no—
298C2#7005211yescharacteristic
299C2#7032511yescharacteristic
300C2#7040711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1514C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1514D3
hypocenterC3#9430
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1514

Complements

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

NH ≅ G/Ntype
2. #74265C2semidirect
5. #70822C2semidirect
6. #67752C2semidirect
7. #67633C2semidirect
14. #5079C4semidirect
15. #5070C4semidirect
295. C3#9430semidirect

Maximal subgroups

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

subgroupindex
#832832
#742652
C3×#15142
#718132
#708222
#677522
#676332
#94303