Finite Group Databasebeta

← #10823321

Subgroups — #10823321

Order 768.

1162 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85725C211yesmaximal, characteristic
2C3×#2296C211yesmaximal, characteristic
3#82069C212yesmaximal
4#71593C212yesmaximal
5#71592C211yesmaximal, characteristic
6#69594C218yesmaximal
7#23564—31nomaximal, Sylow
8C2×C3×#117C2211yescharacteristic
9C2×C3×#117C2211yescharacteristic
10C22×C3×C42C2211yescharacteristic
11C2×#926C2212yes—
12C2×#926C2212yes—
13#5519—24no—
14#5517—24no—
15#5500—22no—
16#5483C2214yes—
17#5470—22no—
18C3×#389—22no—
19C3×#385C2214yes—
20C3×C4×C4⋊C4—22no—
21#5210C2218yes—
22C4×Dic3⋊C4—24no—
23#4928—24no—
24#4527C2218yes—
25#4526C2214yes—
26#3458—31no—
27#2296—32no—
28#2296D311yescharacteristic
29#2062—38no—
30#1740—31no—
31#1740—32no—
32C23×C3×C4C2312yes—
33C23×C3×C4C2311yescharacteristic
34C3×#117—22no—
35C3×#117—22no—
36C3×#117—22no—
37C3×#117Q814yescomplemented
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—24no—
40C2×C3×C4⋊C4D412yes—
41C2×C3×C22⋊C4C2312yes—
42C2×C3×C22⋊C4C2312yes—
43C2×C3×C42—21no—
44C2×C3×C42—22no—
45C2×C3×C42—22no—
46C2×C3×C42—21no—
47C2×C3×C42—21no—
48C2×C3×C42D412yes—
49C2×#217C2314yes—
50C2×#217C2314yes—
51#926—22no—
52#926—22no—
53#926—22no—
54#926—22no—
55#926—24no—
56#926Q818yescomplemented
57C2×C4⋊Dic3—22no—
58C2×C4⋊Dic3D414yes—
59#924—28no—
60C2×Dic3⋊C4—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—24no—
70C3⋊#95—28no—
71#544—64no—
72#544—62no—
73#543—34no—
74#533—64no—
75#533—62no—
76C2×#117—32no—
77C2×#117—32no—
78C2×#117C2×D311yescharacteristic
79C2×#117C2×D311yescharacteristic
80C22×C42C2×D311yescharacteristic
81#389—22no—
82#389—64no—
83#385—38no—
84#385C2×D314yes—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—64no—
87#342—34no—
88#342—38no—
89C24×C3C2411yescharacteristic
90C22×C3×C4—24no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—44no—
94C22×C3×C4C2×D412yes—
95C22×C3×C4C2×D412yes—
96C22×C3×C4C2×D414yes—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×Q812yes—
100C22×C3×C4C2×Q814yes—
101C22×C3×C4#4112yes—
102C22×Dic3—24no—
103C22×Dic3C2×D418yes—
104C2×Dic6—28no—
105C3×C4⋊C4—24no—
106C3×C4⋊C4—24no—
107C3×C4⋊C4—48no—
108C3×C22⋊C4—24no—
109C3×C22⋊C4—22no—
110C3×C22⋊C4C2×Q814yes—
111C3×C42—22no—
112C3×C42—22no—
113C3×C42—24no—
114C3×C42—42no—
115C3×C42—41no—
116C3×C42—41no—
117#217—24no—
118#217—24no—
119#217—24no—
120#217C2×Q818yes—
121C4⋊Dic3—28no—
122C4⋊Dic3—24no—
123C4⋊Dic3—24no—
124C4⋊Dic3—28no—
125Dic3⋊C4—216no—
126Dic3⋊C4—416no—
127C4×Dic3—24no—
128C4×Dic3—24no—
129C4×Dic3—28no—
130C23×C4C22×D312yes—
131C23×C4C22×D311yescharacteristic
132C4⋊Q8—64no—
133C4⋊Q8—68no—
134#125—64no—
135#125—68no—
136C22⋊Q8—616no—
137C22⋊Q8—68no—
138C4×Q8—68no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—38no—
143#117—64no—
144#117—62no—
145#117—62no—
146#117—62no—
147#117—62no—
148#117Dic614yescomplemented
149C2×C4⋊C4—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—34no—
152C2×C4⋊C4—62no—
153C2×C4⋊C4—68no—
154C2×C4⋊C4#6712yes—
155C2×C22⋊C4—34no—
156C2×C22⋊C4—34no—
157C2×C22⋊C4C22×D312yes—
158C2×C22⋊C4C22×D312yes—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—34no—
165C2×C42—62no—
166C2×C42#6712yes—
167#95—24no—
168#95—68no—
169C23×C3—42no—
170C23×C3C22×D412yes—
171C23×C3C22×D411yescharacteristic
172C23×C3C22×Q811yescharacteristic
173C23×C3C22×Q812yes—
174C23×C3C2×#4111yescharacteristic
175C2×C3×C4—28no—
176C2×C3×C4—24no—
177C2×C3×C4—24no—
178C2×C3×C4—24no—
179C2×C3×C4—44no—
180C2×C3×C4—44no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—44no—
184C2×C3×C4#12012yes—
185C2×C3×C4#12011yescharacteristic
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4C22⋊Q814yes—
188C2×C3×C4C22⋊Q814yes—
189C2×C3×C4C22⋊Q812yes—
190C2×C3×C4C22⋊Q812yes—
191C2×C3×C4C4⋊Q814yes—
192C2×C3×C4C4⋊Q814yes—
193C2×C3×C4C4⋊Q812yes—
194C2×C3×C4C4⋊Q812yes—
195C2×Dic3—28no—
196C2×Dic3—28no—
197C2×Dic3—48no—
198C2×Dic3—48no—
199Dic6—88no—
200C24C23×D311yescharacteristic
201C2×Q8—68no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—38no—
206C22×C4—44no—
207C22×C4—64no—
208C22×C4C2×Dic612yes—
209C22×C4#23512yes—
210C22×C4D3×D412yes—
211C22×C4D3×D412yes—
212C22×C4D3×Q814yes—
213C22×C4C2×#6714yes—
214C22×C4C2×#6711yescharacteristic
215C22×C4C2×#6711yescharacteristic
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—48no—
219C4⋊C4—616no—
220C4⋊C4—68no—
221C4⋊C4—68no—
222C4⋊C4—64no—
223C4⋊C4—64no—
224C4⋊C4—1216no—
225C22⋊C4—22no—
226C22⋊C4—24no—
227C22⋊C4—38no—
228C22⋊C4—64no—
229C22⋊C4—64no—
230C22⋊C4—64no—
231C22⋊C4C2×Dic614yes—
232C42—24no—
233C42—22no—
234C42—22no—
235C42—41no—
236C42—42no—
237C42—41no—
238C42—68no—
239C42—64no—
240C42—64no—
241C22×C3—24no—
242C22×C3—22no—
243C22×C3—41no—
244C22×C3—42no—
245C22×C3—41no—
246C22×C3C2×#12011yescharacteristic
247C22×C3C2×C22⋊Q812yes—
248C22×C3C2×C22⋊Q811yescharacteristic
249C22×C3C2×C4⋊Q812yes—
250C22×C3C2×C4⋊Q811yescharacteristic
251C3×C4—44no—
252C3×C4—44no—
253C3×C4—44no—
254C3×C4#39514yes—
255C3×C4#39512yes—
256C3×C4#39512yes—
257Dic3—48no—
258Dic3—48no—
259C23—42no—
260C23C22×Dic611yescharacteristic
261C23C2×#23511yescharacteristic
262C23C2×D3×D412yes—
263C23C2×D3×Q812yes—
264C23C22×#6711yescharacteristic
265Q8—248no—
266C2×C4—24no—
267C2×C4—28no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—68no—
276C2×C4—68no—
277C2×C4—128no—
278C2×C4—128no—
279C2×C4#88814yes—
280C2×C4#88814yes—
281C2×C4#89614yes—
282C2×C4#89614yes—
283C2×C4#92412yes—
284C2×C4#92412yes—
285C2×C4#93711yescharacteristic
286C2×C4#93712yes—
287C2×C4#93711yescharacteristic
288C2×C4#94412yes—
289C2×C4#94412yes—
290C2×C3—22no—
291C2×C3—21no—
292C2×C3—22no—
293C2×C3—21no—
294C2×C3C2×#39511yescharacteristic
295C2×C3#302512yes—
296C22—22no—
297C22—24no—
298C22—41no—
299C22—41no—
300C22—42no—
301C22C2×#88812yes—
302C22C2×#89612yes—
303C22C2×#92411yescharacteristic
304C22C2×#93711yescharacteristic
305C22C2×#94411yescharacteristic
306C4—44no—
307C4—44no—
308C4—44no—
309C4—128no—
310C4—128no—
311C4#497914yes—
312C4#497912yes—
313C4#497912yes—
314C3#2356411yescharacteristic, complemented, Sylow
315C2—21no—
316C2—22no—
317C2—22no—
318C2—21no—
319C2C2×#497911yescharacteristic
320C2#8293612yes—
subgroupG/H
centerC2×C4#937
commutator, prime radical, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2296C2
socleC22×C3C2×#120
hypercenter#2296D3
hypocenterC3#23564
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2296

Complements

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

NH ≅ G/Ntype
37. C3×#117Q8semidirect
56. #926Q8semidirect
148. #117Dic6semidirect
314. C3#23564semidirect

Maximal subgroups

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

subgroupindex
#857252
C3×#22962
#820692
#715932
#715922
#695942
#235643