Finite Group Databasebeta

← #10823322

Subgroups — #10823322

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
2#82227C212yesmaximal
3#82226C211yesmaximal, characteristic
4C3×#1740C211yesmaximal, characteristic
5#71593C212yesmaximal
6#68958C218yesmaximal
7#23564—31nomaximal, Sylow
8C2×C3×#117C2212yes—
9C2×C3×#117C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C2×#926C2212yes—
12C22×C4×Dic3C2211yescharacteristic
13#5519—24no—
14#5517—24no—
15#5500—22no—
16#5483C2214yes—
17#5470—22no—
18#4984—24no—
19#4972—22no—
20#4950C2218yes—
21#4946C2214yes—
22C4×C4⋊Dic3—24no—
23C4×Dic3⋊C4—22no—
24C3×#342C2214yes—
25#4527C2218yes—
26#3458—31no—
27#2296—31no—
28#2296—32no—
29#2062—38no—
30#1740—32no—
31#1740D311yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C23×Dic3C2312yes—
34C3×#117—22no—
35C3×#117—22no—
36C3×#117—24no—
37C3×#117Q818yescomplemented
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4D412yes—
40C2×C3×C22⋊C4C2312yes—
41C2×C3×C22⋊C4C2314yes—
42C2×C3×C42—22no—
43C2×C3×C42D412yes—
44C2×#217C2314yes—
45C2×#217C2312yes—
46#926—22no—
47#926—22no—
48#926—22no—
49#926—22no—
50#926—22no—
51#926Q814yescomplemented
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—28no—
54C2×C4⋊Dic3D414yes—
55#924—28no—
56C2×Dic3⋊C4—24no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—22no—
60C2×C4×Dic3—21no—
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—31no—
78C2×#117C2×D312yes—
79C2×#117C2×D311yescharacteristic
80C22×C42—31no—
81#389—64no—
82#389—62no—
83#385—34no—
84#385—38no—
85C4×C4⋊C4—64no—
86C4×C4⋊C4—62no—
87#342—38no—
88#342C2×D314yes—
89C24×C3C2411yescharacteristic
90C22×C3×C4—24no—
91C22×C3×C4C2×D414yes—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D412yes—
95C22×C3×C4C2×D412yes—
96C22×C3×C4C2×Q814yes—
97C22×C3×C4C2×Q812yes—
98C22×C3×C4#4112yes—
99C22×Dic3—22no—
100C22×Dic3—24no—
101C22×Dic3—22no—
102C22×Dic3—44no—
103C22×Dic3C2×D418yes—
104C2×Dic6—28no—
105C3×C4⋊C4—24no—
106C3×C4⋊C4—28no—
107C3×C22⋊C4—24no—
108C3×C22⋊C4—24no—
109C3×C22⋊C4C2×Q818yes—
110C3×C42—22no—
111C3×C42—22no—
112C3×C42—28no—
113#217—24no—
114#217—24no—
115#217—22no—
116#217C2×Q814yes—
117C4⋊Dic3—28no—
118C4⋊Dic3—24no—
119C4⋊Dic3—24no—
120C4⋊Dic3—24no—
121C4⋊Dic3—416no—
122Dic3⋊C4—216no—
123Dic3⋊C4—48no—
124C4×Dic3—24no—
125C4×Dic3—24no—
126C4×Dic3—24no—
127C4×Dic3—42no—
128C4×Dic3—41no—
129C4×Dic3—41no—
130C23×C4—32no—
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—24no—
142#117—34no—
143#117—62no—
144#117—62no—
145#117—62no—
146#117—62no—
147#117—62no—
148#117Dic618yescomplemented
149C2×C4⋊C4—22no—
150C2×C4⋊C4—34no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—68no—
153C2×C4⋊C4—64no—
154C2×C4⋊C4D1212yes—
155C2×C22⋊C4—32no—
156C2×C22⋊C4—34no—
157C2×C22⋊C4C22×D312yes—
158C2×C22⋊C4C22×D314yes—
159C2×C42—22no—
160C2×C42—34no—
161C2×C42—61no—
162C2×C42—61no—
163C2×C42—61no—
164C2×C42—62no—
165C2×C42—62no—
166C2×C42D1212yes—
167#95—68no—
168#95—64no—
169C23×C3—42no—
170C23×C3C22×D411yescharacteristic
171C23×C3C22×D412yes—
172C23×C3C22×Q812yes—
173C23×C3C22×Q811yescharacteristic
174C23×C3C2×#4111yescharacteristic
175C2×C3×C4—24no—
176C2×C3×C4—28no—
177C2×C3×C4—48no—
178C2×C3×C4—44no—
179C2×C3×C4#12012yes—
180C2×C3×C4#12011yescharacteristic
181C2×C3×C4#12011yescharacteristic
182C2×C3×C4C22⋊Q814yes—
183C2×C3×C4C22⋊Q814yes—
184C2×C3×C4C22⋊Q812yes—
185C2×C3×C4C22⋊Q812yes—
186C2×C3×C4C4⋊Q812yes—
187C2×C3×C4C4⋊Q812yes—
188C2×C3×C4C4⋊Q814yes—
189C2×C3×C4C4⋊Q814yes—
190C2×Dic3—24no—
191C2×Dic3—28no—
192C2×Dic3—28no—
193C2×Dic3—24no—
194C2×Dic3—44no—
195C2×Dic3—42no—
196C2×Dic3—44no—
197C2×Dic3—42no—
198C2×Dic3—48no—
199Dic6—88no—
200C24C23×D311yescharacteristic
201C2×Q8—68no—
202C22×C4—24no—
203C22×C4—38no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4—64no—
207C22×C4—124no—
208C22×C4C2×Dic614yes—
209C22×C4C2×D1214yes—
210C22×C4C2×D1211yescharacteristic
211C22×C4C2×D1211yescharacteristic
212C22×C4D3×D412yes—
213C22×C4D3×D412yes—
214C22×C4#23712yes—
215C22×C4D3×Q812yes—
216C4⋊C4—28no—
217C4⋊C4—24no—
218C4⋊C4—616no—
219C4⋊C4—68no—
220C4⋊C4—64no—
221C4⋊C4—64no—
222C4⋊C4—64no—
223C4⋊C4—128no—
224C4⋊C4—1216no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—34no—
228C22⋊C4—64no—
229C22⋊C4—64no—
230C22⋊C4—62no—
231C22⋊C4C2×Dic618yes—
232C42—28no—
233C42—22no—
234C42—22no—
235C42—64no—
236C42—64no—
237C42—64no—
238C42—121no—
239C42—122no—
240C42—121no—
241C22×C3—22no—
242C22×C3—24no—
243C22×C3—42no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3C2×#12011yescharacteristic
247C22×C3C2×C22⋊Q812yes—
248C22×C3C2×C22⋊Q811yescharacteristic
249C22×C3C2×C4⋊Q811yescharacteristic
250C22×C3C2×C4⋊Q812yes—
251C3×C4—48no—
252C3×C4—44no—
253C3×C4#39514yes—
254C3×C4#39512yes—
255C3×C4#39512yes—
256Dic3—44no—
257Dic3—48no—
258Dic3—44no—
259C23—42no—
260C23C22×Dic612yes—
261C23C22×D1211yescharacteristic
262C23C2×D3×D412yes—
263C23C2×#23711yescharacteristic
264C23C2×D3×Q811yescharacteristic
265Q8—248no—
266C2×C4—28no—
267C2×C4—24no—
268C2×C4—48no—
269C2×C4—44no—
270C2×C4—64no—
271C2×C4—68no—
272C2×C4—68no—
273C2×C4—64no—
274C2×C4—124no—
275C2×C4—122no—
276C2×C4—124no—
277C2×C4—122no—
278C2×C4—128no—
279C2×C4#86912yes—
280C2×C4#86912yes—
281C2×C4#87814yes—
282C2×C4#87814yes—
283C2×C4#88211yescharacteristic
284C2×C4#88212yes—
285C2×C4#88211yescharacteristic
286C2×C4#88814yes—
287C2×C4#88814yes—
288C2×C4#89712yes—
289C2×C4#89712yes—
290C2×C3—21no—
291C2×C3—22no—
292C2×C3—21no—
293C2×C3—22no—
294C2×C3C2×#39511yescharacteristic
295C2×C3#302512yes—
296C22—22no—
297C22—24no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22C2×#86911yescharacteristic
302C22C2×#87812yes—
303C22C2×#88211yescharacteristic
304C22C2×#88812yes—
305C22C2×#89711yescharacteristic
306C4—48no—
307C4—44no—
308C4—124no—
309C4—124no—
310C4—128no—
311C4#465614yes—
312C4#465612yes—
313C4#465612yes—
314C3#2356411yescharacteristic, complemented, Sylow
315C2—22no—
316C2—21no—
317C2—21no—
318C2—22no—
319C2C2×#465611yescharacteristic
320C2#7869012yes—
subgroupG/H
centerC2×C4#882
commutator, prime radical, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#1740C2
socleC22×C3C2×#120
hypercenter#1740D3
hypocenterC3#23564
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
37. C3×#117Q8semidirect
51. #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
#822272
#822262
C3×#17402
#715932
#689582
#235643