Finite Group Databasebeta

← #10776455

Subgroups — #10776455

Order 768.

850 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C2×#5672C211yesmaximal, characteristic
2C3×#3140C211yesmaximal, characteristic
3#83326C212yesmaximal, complemented
4#71719C211yesmaximal, characteristic
5#71247C218yesmaximal, complemented
6#69863C212yesmaximal, complemented
7#18999—31nomaximal, Sylow
8C3×#533C2211yescharacteristic
9C2×C3×#124C2211yescharacteristic
10C2×#944—21no—
11C2×#935—21no—
12#5672—22no—
13#5672—24no—
14#5672C2214yes—
15C2×#879—22no—
16C2×#878—22no—
17C2×#870C2212yes—
18C3×#489C2214yes—
19C3×#488—22no—
20C3×#419—21no—
21C3×#418—21no—
22C3×C4×M16C2211yescharacteristic
23#5082C2214yescomplemented
24#5081C2214yescomplemented
25#5044—22no—
26#5043—22no—
27#5022—24no—
28#4926C2212yes—
29#4540C2214yes—
30#4465C2218yescomplemented
31C2×#563—31no—
32#3236—32no—
33#3140D311yescharacteristic
34#1664—38no—
35#1533—31no—
36#1528—32no—
37C22×C3×Q8D411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C23×Dic3—21no—
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C3×C4⋊Q8C2312yes—
43C3×#125—21no—
44C3×#124—22no—
45C3×#124D412yes—
46C3×#124C2312yes—
47C3×C22⋊Q8—22no—
48C3×C4×Q8—22no—
49C3×#117—21no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C42C2311yescharacteristic
52C2×#217—22no—
53C2×#217—21no—
54#944—24no—
55#944—24no—
56#935—24no—
57#935—24no—
58C2×C4⋊Dic3D412yes—
59C2×C4⋊Dic3D412yes—
60C2×Dic3⋊C4—22no—
61C2×Dic3⋊C4—22no—
62C2×Dic3⋊C4—22no—
63C2×C4×Dic3—21no—
64#879—28no—
65#879—28no—
66#878—28no—
67#878—28no—
68#870—22no—
69#870—22no—
70#870D414yes—
71#870C2314yes—
72C3×#103—24no—
73C3×#102—24no—
74C3×#97C2312yes—
75C3×C4×C8—21no—
76#835—24no—
77#835—24no—
78#830—22no—
79#808—24no—
80#807—24no—
81C4⋊(C3⋊C8)—22no—
82#803C2314yes—
83#563—34no—
84#563—64no—
85#563—62no—
86#533C2×D311yescharacteristic
87C2×C4⋊Q8—61no—
88C2×#126—62no—
89C2×#125—32no—
90C2×#124C2×D311yescharacteristic
91C2×#123—61no—
92C2×C22⋊Q8—62no—
93#501—34no—
94#491—34no—
95#489C2×D314yes—
96#488—22no—
97#484—62no—
98#483—62no—
99#479—64no—
100#432—32no—
101#419—21no—
102#418—21no—
103C4×M16C2×D311yescharacteristic
104#330—38no—
105#329—34no—
106C24×C3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—42no—
110C2×C3×Q8C2×D412yes—
111C2×C3×D4—21no—
112C2×C3×D4—42no—
113C2×C3×D4C2×D412yes—
114C22×C3×C4—21no—
115C22×C3×C4C2×D412yes—
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—22no—
118C22×Dic3—22no—
119C22×Dic3—24no—
120C22×Dic3—82no—
121C2×Dic6—24no—
122C2×Dic6—82no—
123C3×M16—42no—
124C2×C3×C8—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—42no—
128C3×C22⋊C4—24no—
129C3×C22⋊C4—24no—
130C3×C22⋊C4—41no—
131C3×C42—41no—
132C3×C42C2×D412yes—
133C3×C42C2×D411yescharacteristic
134C3×C42C2411yescharacteristic
135#217—28no—
136#217—24no—
137C4⋊Dic3—22no—
138C4⋊Dic3—22no—
139C4⋊Dic3C2×D414yes—
140C4⋊Dic3C2×D414yes—
141Dic3⋊C4—28no—
142Dic3⋊C4—24no—
143Dic3⋊C4—24no—
144Dic3⋊C4—24no—
145Dic3⋊C4—24no—
146C4×Dic3—24no—
147C2×C3⋊C8—24no—
148C22×Q8—61no—
149C22×Q8#6711yescharacteristic
150C22×D4#6711yescharacteristic
151C23×C4—61no—
152C2×M16—21no—
153C4⋊Q8—64no—
154C4⋊Q8—64no—
155C4⋊Q8C22×D312yes—
156#126—68no—
157#126—68no—
158#125—21no—
159#125—34no—
160#125—34no—
161#125—62no—
162#125—62no—
163#124—22no—
164#124#6712yes—
165#124C22×D312yes—
166#123—64no—
167#123—64no—
168C22⋊Q8—22no—
169C22⋊Q8—68no—
170C22⋊Q8—68no—
171C4×Q8—22no—
172#117—21no—
173C2×C4⋊C4—32no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—62no—
176C2×C4⋊C4—62no—
177C2×C4⋊C4—62no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—61no—
181C2×C42—61no—
182C2×C42C22×D311yescharacteristic
183#107—64no—
184#106—64no—
185C4⋊C8—62no—
186#103—24no—
187#103—64no—
188#103—64no—
189#102—24no—
190C22⋊C8—62no—
191#97—34no—
192#97C22×D312yes—
193C4×C8—21no—
194C23×C3—22no—
195C23×C3—44no—
196C23×C3—42no—
197C23×C3#12011yescharacteristic
198C3×Q8—42no—
199C3×Q8—44no—
200C3×Q8—81no—
201C3×D4—42no—
202C3×D4—81no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12012yes—
210C2×C3×C4C22×D412yes—
211C2×C3×C4C22×D411yescharacteristic
212C2×Dic3—24no—
213C2×Dic3—24no—
214C2×Dic3—44no—
215C2×Dic3—44no—
216C2×Dic3—84no—
217Dic6—84no—
218C3×C8—42no—
219C3⋊C8—44no—
220C24—21no—
221C2×Q8—21no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×Q8—64no—
225C2×Q8—242no—
226C2×Q8C2×#6712yes—
227C2×D4—21no—
228C2×D4—42no—
229C2×D4C2×#6712yes—
230C22×C4—21no—
231C22×C4—62no—
232C22×C4—62no—
233C22×C4—64no—
234C22×C4—242no—
235C22×C4D3×D412yes—
236C22×C4C2×#6711yescharacteristic
237M16—42no—
238C2×C8—22no—
239C2×C8—64no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—34no—
243C4⋊C4—34no—
244C4⋊C4—42no—
245C4⋊C4—68no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—64no—
249C4⋊C4—64no—
250C4⋊C4—64no—
251C4⋊C4—64no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—41no—
255C22⋊C4—68no—
256C22⋊C4—64no—
257C42—41no—
258C42—64no—
259C42D3×D412yes—
260C42C2×#6711yescharacteristic
261C42C23×D311yescharacteristic
262C22×C3—22no—
263C22×C3—42no—
264C22×C3—44no—
265C22×C3—84no—
266C22×C3#45512yes—
267C22×C3C2×#12011yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—42no—
273C3×C4—82no—
274Dic3—44no—
275Dic3—44no—
276C23—22no—
277C23—44no—
278C23—42no—
279C23#93711yescharacteristic
280Q8—42no—
281Q8—44no—
282Q8—81no—
283Q8—244no—
284D4—42no—
285D4—81no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—64no—
292C2×C4—64no—
293C2×C4—124no—
294C2×C4—124no—
295C2×C4—244no—
296C2×C4#93711yescharacteristic
297C2×C4#93712yes—
298C2×C4C2×D3×D412yes—
299C2×C4C22×#6711yescharacteristic
300C8—42no—
301C8—124no—
302C2×C3—22no—
303C2×C3—82no—
304C2×C3C2×#45511yescharacteristic
305C2×C3#302012yes—
306C22—22no—
307C22—42no—
308C22—44no—
309C22—84no—
310C22#506112yes—
311C22C2×#93711yescharacteristic
312C4—21no—
313C4—21no—
314C4—22no—
315C4—22no—
316C4—42no—
317C4—82no—
318C4—124no—
319C4—124no—
320C3#1899911yescharacteristic, complemented, Sylow
321C2—22no—
322C2—82no—
323C2C2×#506111yescharacteristic
324C2#8400612yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3140C2
socleC22×C3C2×#120
prime radicalC22×C3×D4D4
hypercenter#3140D3
hypocenterC3#18999
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3140

Complements

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

NH ≅ G/Ntype
3. #83326C2semidirect
5. #71247C2semidirect
6. #69863C2semidirect
23. #5082C22semidirect
24. #5081C22semidirect
30. #4465C22semidirect
320. C3#18999semidirect

Maximal subgroups

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

subgroupindex
C2×#56722
C3×#31402
#833262
#717192
#712472
#698632
#189993