Finite Group Databasebeta

← #10692731

Subgroups — #10692731

Order 768.

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

normal subgroups only

This group has 422 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#83341C211yesmaximal, characteristic
2#74265C211yesmaximal, characteristic
3C3×#1515C211yesmaximal, characteristic
4#71837C211yesmaximal, characteristic
5#70822C211yesmaximal, characteristic
6#67752C211yesmaximal, characteristic
7#67633C211yesmaximal, characteristic
8#9427—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#383C2211yescharacteristic
13#5080C412yescomplemented
14#5078C412yescomplemented
15#5064—22no—
16#4995—22no—
17#4937C2211yescharacteristic
18#4926C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4654—22no—
21#4653—22no—
22#4646—22no—
23C3×#328—22no—
24#4538—22no—
25#4486—22no—
26#3165—31no—
27#2381—31no—
28#1556—31no—
29#1515—31no—
30#1515D311yescharacteristic
31#1288—31no—
32#1288—31no—
33C22×Dic6—21no—
34C3×C4⋊Q8C2×C411yescharacteristic
35C3×C4⋊Q8C2×C411yescharacteristic
36C3×#125—21no—
37C3×C22⋊Q8—21no—
38C3×C22⋊Q8—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×#117—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—
58C3⋊#95—24no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95D412yes—
62#830—21no—
63#808—24no—
64#807—24no—
65C4⋊(C3⋊C8)—21no—
66#803—21no—
67#803—21no—
68C4×C3⋊C8C2×C412yes—
69#533C2×D311yescharacteristic
70C2×C4⋊Q8—31no—
71#501—62no—
72#500—32no—
73#498—32no—
74#432—31no—
75#432C2×D311yescharacteristic
76#428—62no—
77C4×M16—31no—
78#391—62no—
79#387—62no—
80#387—62no—
81#383—31no—
82#383C2×D311yescharacteristic
83#332—62no—
84#328—22no—
85#328—62no—
86C2×C3×Q8—21no—
87C2×C3×Q8—21no—
88C22×C3×C4—22no—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×Dic3—22no—
93C22×Dic3—22no—
94C2×Dic6—22no—
95C2×Dic6—22no—
96C2×Dic6—81no—
97C2×Dic6—81no—
98C2×C3×C8—41no—
99C3×C4⋊C4—21no—
100C3×C4⋊C4—21no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C42—41no—
107C3×C42C22×C411yescharacteristic
108C3×C42C2×D411yescharacteristic
109C3×C42#4111yescharacteristic
110C3×C42#4111yescharacteristic
111C4⋊Dic3—22no—
112C4⋊Dic3—22no—
113C4⋊Dic3—22no—
114C4⋊Dic3C22⋊C414yes—
115Dic3⋊C4—82no—
116C4×Dic3—82no—
117#208—24no—
118C2×C3⋊C8—22no—
119C2×C3⋊C8—41no—
120C22×Q8—61no—
121C2×M16—61no—
122C4⋊Q8—32no—
123C4⋊Q8—32no—
124C4⋊Q8—61no—
125C4⋊Q8—61no—
126C4⋊Q8C4×D311yescharacteristic
127C4⋊Q8C4×D311yescharacteristic
128#125—21no—
129C22⋊Q8—21no—
130C22⋊Q8—21no—
131C4×Q8—21no—
132C4×Q8—21no—
133#117—21no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—62no—
137C2×C42—62no—
138C2×C42C22×D311yescharacteristic
139#107—64no—
140#106—64no—
141C4⋊C8—21no—
142C4⋊C8—61no—
143C22⋊C8—21no—
144C22⋊C8—61no—
145#97—21no—
146#97—61no—
147#97—61no—
148C4×C8—32no—
149#95—21no—
150#95—21no—
151#95—32no—
152#95—64no—
153#95—61no—
154#95—61no—
155#95#6712yes—
156C23×C3#12011yescharacteristic
157C3×Q8—81no—
158C3×Q8—81no—
159C2×C3×C4—22no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
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
173C2×Dic3—24no—
174C2×Dic3—42no—
175C2×Dic3—42no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179Dic6—82no—
180Dic6—82no—
181C3×C8—81no—
182C3⋊C8—24no—
183C3⋊C8—81no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—62no—
187C2×Q8—62no—
188C2×Q8—241no—
189C2×Q8—241no—
190C22×C4—22no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4D3×D411yescharacteristic
194C22×C4D3×D411yescharacteristic
195C22×C4C2×#6711yescharacteristic
196M16—64no—
197C2×C8—41no—
198C2×C8—62no—
199C2×C8—121no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—34no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—62no—
208C4⋊C4—62no—
209C4⋊C4—62no—
210C4⋊C4—242no—
211C22⋊C4—41no—
212C42—41no—
213C42—242no—
214C42C2×C4×D311yescharacteristic
215C42#23511yescharacteristic
216C42D3×D411yescharacteristic
217C42#23711yescharacteristic
218C22×C3—21no—
219C22×C3—21no—
220C22×C3#38611yescharacteristic
221C22×C3#45612yes—
222C3×C4—21no—
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—82no—
229C3×C4—81no—
230C3×C4—81no—
231Dic3—44no—
232Dic3—82no—
233C23#93711yescharacteristic
234Q8—81no—
235Q8—81no—
236Q8—242no—
237Q8—242no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—64no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—122no—
252C2×C4D3×C22⋊C411yescharacteristic
253C2×C4#67⋊C411yescharacteristic
254C2×C4#88311yescharacteristic
255C2×C4#88411yescharacteristic
256C2×C4C4×#6711yescharacteristic
257C2×C4#93811yescharacteristic
258C8—64no—
259C8—81no—
260C8—241no—
261C2×C3—22no—
262C2×C3#188611yescharacteristic
263C2×C3#190411yescharacteristic
264C2×C3#190511yescharacteristic
265C22—21no—
266C22—21no—
267C22#495611yescharacteristic
268C22#520512yes—
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—82no—
276C4—81no—
277C4—81no—
278C4—124no—
279C4—242no—
280C3#942711yescharacteristic, complemented, Sylow
281C2—22no—
282C2#7005211yescharacteristic
283C2#7034511yescharacteristic
284C2#7040711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1515C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1515D3
hypocenterC3#9427
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
13. #5080C4semidirect
14. #5078C4semidirect
280. C3#9427semidirect

Maximal subgroups

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

subgroupindex
#833412
#742652
C3×#15152
#718372
#708222
#677522
#676332
#94273