Finite Group Databasebeta

← #10822605

Subgroups — #10822605

Order 768.

1178 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85725C211yesmaximal, characteristic
2C3×#2310C211yesmaximal, characteristic
3#84062C212yesmaximal
4#71593C212yesmaximal
5#71592C211yesmaximal, characteristic
6#69536C218yesmaximal
7#23470—31nomaximal, Sylow
8C23×C3×Q8C2211yescharacteristic
9C2×C3×#117C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×#926C2212yes—
12C2×#926C2212yes—
13#5519—24no—
14#5517—24no—
15#5500—22no—
16#5483C2214yes—
17#5470—22no—
18C3×#391—22no—
19C3×#391—22no—
20C3×#381C2214yes—
21#5229—24no—
22#5229—24no—
23#5211C2218yes—
24#4527C2218yes—
25#4526C2214yes—
26#3458—31no—
27#2310—32no—
28#2310D311yescharacteristic
29#2030—38no—
30#1740—31no—
31#1740—32no—
32C22×C3×Q8—22no—
33C22×C3×Q8—22no—
34C22×C3×Q8—21no—
35C22×C3×Q8—21no—
36C22×C3×Q8—22no—
37C22×C3×Q8—24no—
38C22×C3×Q8Q814yescomplemented
39C23×C3×C4C2312yes—
40C23×C3×C4C2311yescharacteristic
41C3×#117—22no—
42C3×#117—22no—
43C3×#117—24no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D412yes—
46C2×C3×C22⋊C4C2312yes—
47C2×C3×C22⋊C4C2312yes—
48C2×C3×C42—21no—
49C2×C3×C42D412yes—
50C2×#217C2314yes—
51C2×#217C2314yes—
52#926—22no—
53#926—22no—
54#926—22no—
55#926—22no—
56#926—24no—
57#926—24no—
58C2×C4⋊Dic3—22no—
59C2×C4⋊Dic3D414yes—
60#924—28no—
61C2×C4×Dic3—22no—
62C2×C4×Dic3D414yes—
63#890—28no—
64#888—28no—
65#878—216no—
66#870—24no—
67#869—24no—
68C4×Dic6—28no—
69C3×#95—28no—
70C3⋊#95—216no—
71C23×Q8C2×D311yescharacteristic
72#544—64no—
73#544—62no—
74#543—34no—
75#533—64no—
76#533—62no—
77C2×#117—32no—
78C2×#117—32no—
79C2×#117C2×D311yescharacteristic
80C2×#117C2×D311yescharacteristic
81#391—22no—
82#391—22no—
83#391—64no—
84#391—64no—
85#381—38no—
86#381C2×D314yes—
87#342—34no—
88#342—38no—
89C24×C3C2411yescharacteristic
90C2×C3×Q8—28no—
91C2×C3×Q8—28no—
92C2×C3×Q8—24no—
93C2×C3×Q8—24no—
94C2×C3×Q8—41no—
95C2×C3×Q8—42no—
96C2×C3×Q8—41no—
97C2×C3×Q8—44no—
98C2×C3×Q8—44no—
99C2×C3×Q8—42no—
100C2×C3×Q8—42no—
101C22×C3×C4—24no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—24no—
105C22×C3×C4—22no—
106C22×C3×C4C2×D412yes—
107C22×C3×C4C2×D412yes—
108C22×C3×C4C2×D414yes—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×Q814yes—
112C22×C3×C4C2×Q812yes—
113C22×C3×C4#4112yes—
114C22×C3×C4#4114yes—
115C22×Dic3—24no—
116C22×Dic3C2×D418yes—
117C2×Dic6—28no—
118C3×C4⋊C4—24no—
119C3×C4⋊C4—42no—
120C3×C22⋊C4—24no—
121C3×C22⋊C4—24no—
122C3×C42—22no—
123C3×C42—22no—
124C3×C42—42no—
125#217—24no—
126#217—24no—
127#217—24no—
128#217—24no—
129C4⋊Dic3—28no—
130C4⋊Dic3—24no—
131C4⋊Dic3—24no—
132C4⋊Dic3—44no—
133Dic3⋊C4—216no—
134C4×Dic3—24no—
135C4×Dic3—24no—
136C4×Dic3—44no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—24no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8Dic614yescomplemented
144C23×C4C22×D312yes—
145C23×C4C22×D311yescharacteristic
146C4⋊Q8—64no—
147C4⋊Q8—68no—
148#125—64no—
149#125—68no—
150C22⋊Q8—616no—
151C22⋊Q8—68no—
152C4×Q8—68no—
153#117—22no—
154#117—22no—
155#117—24no—
156#117—62no—
157#117—62no—
158#117—62no—
159#117—62no—
160#117—64no—
161#117—64no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—34no—
164C2×C4⋊C4—62no—
165C2×C4⋊C4#6712yes—
166C2×C22⋊C4—34no—
167C2×C22⋊C4—34no—
168C2×C22⋊C4C22×D312yes—
169C2×C22⋊C4C22×D312yes—
170C2×C42—21no—
171C2×C42—34no—
172C2×C42—62no—
173C2×C42#6712yes—
174#95—28no—
175#95—616no—
176C23×C3—42no—
177C23×C3C22×D412yes—
178C23×C3C22×D411yescharacteristic
179C23×C3C22×Q811yescharacteristic
180C23×C3C2×#4111yescharacteristic
181C23×C3C2×#4112yes—
182C3×Q8—44no—
183C3×Q8—48no—
184C3×Q8—44no—
185C2×C3×C4—24no—
186C2×C3×C4—28no—
187C2×C3×C4—24no—
188C2×C3×C4—42no—
189C2×C3×C4—44no—
190C2×C3×C4—42no—
191C2×C3×C4—44no—
192C2×C3×C4—44no—
193C2×C3×C4—42no—
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12012yes—
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4C22⋊Q814yes—
198C2×C3×C4C22⋊Q814yes—
199C2×C3×C4C22⋊Q812yes—
200C2×C3×C4C22⋊Q812yes—
201C2×C3×C4#12414yes—
202C2×C3×C4#12414yes—
203C2×C3×C4#12412yes—
204C2×C3×C4#12412yes—
205C2×Dic3—28no—
206C2×Dic3—48no—
207C2×Dic3—48no—
208C2×Dic3—44no—
209Dic6—88no—
210C24C23×D311yescharacteristic
211C2×Q8—28no—
212C2×Q8—28no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—44no—
216C2×Q8—44no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—42no—
222C2×Q8—68no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—38no—
229C22×C4—64no—
230C22×C4C2×Dic614yes—
231C22×C4C2×Dic612yes—
232C22×C4#23512yes—
233C22×C4D3×D412yes—
234C22×C4D3×D412yes—
235C22×C4#23714yes—
236C22×C4C2×#6714yes—
237C22×C4C2×#6711yescharacteristic
238C22×C4C2×#6711yescharacteristic
239C4⋊C4—24no—
240C4⋊C4—42no—
241C4⋊C4—616no—
242C4⋊C4—68no—
243C4⋊C4—64no—
244C4⋊C4—64no—
245C4⋊C4—124no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—64no—
249C22⋊C4—64no—
250C22⋊C4—64no—
251C22⋊C4—64no—
252C42—22no—
253C42—22no—
254C42—42no—
255C42—64no—
256C42—64no—
257C42—124no—
258C22×C3—24no—
259C22×C3—22no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—42no—
263C22×C3C2×#12011yescharacteristic
264C22×C3C2×C22⋊Q812yes—
265C22×C3C2×C22⋊Q811yescharacteristic
266C22×C3C2×#12412yes—
267C22×C3C2×#12411yescharacteristic
268C3×C4—44no—
269C3×C4—44no—
270C3×C4—82no—
271C3×C4#39312yes—
272C3×C4#39314yes—
273C3×C4#39312yes—
274Dic3—48no—
275Dic3—84no—
276C23—42no—
277C23C22×Dic611yescharacteristic
278C23C2×#23511yescharacteristic
279C23C2×D3×D412yes—
280C23C2×#23712yes—
281C23C22×#6711yescharacteristic
282Q8—44no—
283Q8—48no—
284Q8—44no—
285Q8—248no—
286C2×C4—28no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—68no—
296C2×C4—128no—
297C2×C4—128no—
298C2×C4—124no—
299C2×C4#87814yes—
300C2×C4#87814yes—
301C2×C4#88514yes—
302C2×C4#88514yes—
303C2×C4#92412yes—
304C2×C4#92412yes—
305C2×C4#93612yes—
306C2×C4#93612yes—
307C2×C4#93711yescharacteristic
308C2×C4#93711yescharacteristic
309C2×C4#93712yes—
310C2×C3—22no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—21no—
314C2×C3C2×#39311yescharacteristic
315C2×C3#302812yes—
316C22—22no—
317C22—24no—
318C22—41no—
319C22—41no—
320C22—42no—
321C22C2×#87812yes—
322C22C2×#88512yes—
323C22C2×#92411yescharacteristic
324C22C2×#93611yescharacteristic
325C22C2×#93711yescharacteristic
326C4—44no—
327C4—44no—
328C4—82no—
329C4—128no—
330C4—244no—
331C4#494712yes—
332C4#494714yes—
333C4#494712yes—
334C3#2347011yescharacteristic, complemented, Sylow
335C2—21no—
336C2—22no—
337C2—22no—
338C2—21no—
339C2C2×#494711yescharacteristic
340C2#8341712yes—
subgroupG/H
centerC22C2×#937
commutator, prime radical, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2310C2
socleC22×C3C2×#120
hypercenter#2310D3
hypocenterC3#23470
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
38. C22×C3×Q8Q8semidirect
143. C22×Q8Dic6semidirect
334. C3#23470semidirect

Maximal subgroups

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

subgroupindex
#857252
C3×#23102
#840622
#715932
#715922
#695362
#234703