Finite Group Databasebeta

← #10795139

Subgroups — #10795139

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic
2#82155C212yesmaximal, complemented
3#72139C212yesmaximal
4#71619C214yesmaximal, complemented
5C4×#834C214yesmaximal, complemented
6#71594C212yesmaximal, complemented
7#20725—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C3×#520—28no—
10C2×C3×C4×Q8C412yescomplemented
11C2×C3×C4×D4C412yescomplemented
12C2×C3×C4×D4C2212yes—
13C2×C3×C4×D4C2212yes—
14C2×C3×#117C412yescomplemented
15C2×C3×#117C2211yescharacteristic
16C22×C3×C4⋊C4C412yescomplemented
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#834C2218yes—
19#5212C2214yes—
20#5211C2214yes—
21C4×#217C2214yes—
22#4980—28no—
23#4526C2218yes—
24C2×#520D311yescharacteristic
25#2313—32no—
26#1820—32no—
27#1799—34no—
28#1782—32no—
29C4×#99—34no—
30C2×C3×#41—28no—
31C2×C3×#41—24no—
32C22×C3×Q8C2×C411yescharacteristic
33C22×C3×D4C2×C411yescharacteristic
34C22×C3×D4C2312yes—
35C23×C3×C4C2×C412yes—
36C23×C3×C4C2×C412yes—
37C23×C3×C4C2×C412yes—
38C23×C3×C4C2311yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C3×C4×Q8—28no—
42C3×C4×D4—28no—
43C3×C4×D4—28no—
44C3×C4×D4—28no—
45C3×#117—24no—
46C3×#117—28no—
47C2×C3×C4⋊C4—28no—
48C2×C3×C4⋊C4C2×C411yescharacteristic
49C2×C3×C4⋊C4C2×C412yes—
50C2×C3×C4⋊C4C2×C412yes—
51C2×C3×C4⋊C4C2×C411yescharacteristic
52C2×C3×C4⋊C4C2×C412yes—
53C2×C3×C4⋊C4C2×C412yes—
54C2×C3×C4⋊C4D418yes—
55C2×C3×C22⋊C4C2×C412yes—
56C2×C3×C22⋊C4C2×C412yes—
57C2×C3×C22⋊C4C2312yes—
58C2×C3×C42C2×C412yes—
59C2×C3×C42C2×C412yes—
60C2×C3×C42C2×C412yes—
61C2×#217C2318yes—
62C2×C4⋊Dic3—24no—
63C2×C4×Dic3—24no—
64#834—216no—
65C3⋊#95—24no—
66C3⋊#95—28no—
67C3⋊#95—24no—
68C22×#41C2×D311yescharacteristic
69#520—28no—
70C2×C4×Q8Dic312yescomplemented
71C2×C4×D4Dic312yescomplemented
72C2×C4×D4C2×D312yes—
73C2×C4×D4C2×D312yes—
74C2×#117Dic312yescomplemented
75C2×#117C2×D311yescharacteristic
76C22×C4⋊C4Dic312yescomplemented
77C22×C4⋊C4C2×D311yescharacteristic
78C2×#99—38no—
79#387—68no—
80#381—34no—
81#380—34no—
82C4×C22⋊C4—34no—
83#342—38no—
84C24×C3C22×C412yes—
85C24×C3C2411yescharacteristic
86C3×#41—232no—
87C2×C3×Q8—22no—
88C2×C3×Q8—24no—
89C2×C3×D4—24no—
90C2×C3×D4—24no—
91C2×C3×D4—24no—
92C2×C3×D4—22no—
93C2×C3×D4#4118yes—
94C22×C3×C4—24no—
95C22×C3×C4—24no—
96C22×C3×C4—24no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—24no—
100C22×C3×C4—81no—
101C22×C3×C4—81no—
102C22×C3×C4—82no—
103C22×C3×C4—82no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C22×C412yes—
107C22×C3×C4C22×C412yes—
108C22×C3×C4C22×C412yes—
109C22×C3×C4C22×C412yes—
110C22×C3×C4C22×C412yes—
111C22×C3×C4C2×D414yes—
112C22×C3×C4C2×D414yes—
113C22×C3×C4C2×D414yes—
114C22×Dic3—28no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—24no—
117C3×C4⋊C4—24no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4C22⋊C418yes—
120C3×C4⋊C4C22⋊C418yes—
121C3×C22⋊C4—24no—
122C3×C22⋊C4—24no—
123C3×C22⋊C4—24no—
124C3×C42—24no—
125C3×C42—24no—
126C3×C42—24no—
127#217—216no—
128C4⋊Dic3—84no—
129C4×Dic3—84no—
130C2×#41—28no—
131C2×#41—24no—
132C22×Q8C2×Dic311yescharacteristic
133C22×D4C2×Dic311yescharacteristic
134C22×D4C22×D312yes—
135C23×C4C2×Dic312yes—
136C23×C4C2×Dic312yes—
137C23×C4C2×Dic312yes—
138C23×C4C22×D311yescharacteristic
139C23×C4C22×D311yescharacteristic
140C23×C4C22×D311yescharacteristic
141C4×Q8—28no—
142C4×D4—28no—
143C4×D4—28no—
144C4×D4—28no—
145#117—24no—
146#117—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—64no—
149C2×C4⋊C4C2×Dic311yescharacteristic
150C2×C4⋊C4C2×Dic312yes—
151C2×C4⋊C4C2×Dic312yes—
152C2×C4⋊C4C2×Dic311yescharacteristic
153C2×C4⋊C4C2×Dic312yes—
154C2×C4⋊C4C2×Dic312yes—
155C2×C4⋊C4#6718yes—
156C2×C22⋊C4—38no—
157C2×C22⋊C4C2×Dic312yes—
158C2×C22⋊C4C2×Dic312yes—
159C2×C22⋊C4C22×D312yes—
160C2×C42—64no—
161C2×C42C2×Dic312yes—
162C2×C42C2×Dic312yes—
163C2×C42C2×Dic312yes—
164#99—616no—
165#95—64no—
166#95—64no—
167#95—68no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3#11714yes—
175C23×C3C23×C411yescharacteristic
176C23×C3C22×D412yes—
177C23×C3C2×#4114yes—
178C3×Q8—28no—
179C3×D4—28no—
180C3×D4—216no—
181C2×C3×C4—216no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—28no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—44no—
190C2×C3×C4—44no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—84no—
196C2×C3×C4—84no—
197C2×C3×C4C2×C22⋊C412yes—
198C2×C3×C4C2×C22⋊C414yes—
199C2×C3×C4C2×C22⋊C414yes—
200C2×C3×C4C2×C22⋊C412yes—
201C2×C3×C4#11714yes—
202C2×Dic3—48no—
203C2×Dic3—48no—
204C2×Dic3—48no—
205C24C22×Dic312yes—
206C24C23×D311yescharacteristic
207#41—232no—
208C2×Q8—24no—
209C2×Q8—22no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4#23518yes—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—68no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4C22×Dic312yes—
227C22×C4C22×Dic311yescharacteristic
228C22×C4C22×Dic311yescharacteristic
229C22×C4C22×Dic312yes—
230C22×C4C22×Dic312yes—
231C22×C4C22×Dic312yes—
232C22×C4C22×Dic312yes—
233C22×C4C2×#6714yes—
234C22×C4C2×#6714yes—
235C22×C4C2×#6714yes—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—244no—
241C4⋊C4#21718yes—
242C4⋊C4#21718yes—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—616no—
247C42—24no—
248C42—24no—
249C42—24no—
250C42—244no—
251C22×C3—22no—
252C22×C3—21no—
253C22×C3—22no—
254C22×C3—21no—
255C22×C3—24no—
256C22×C3—44no—
257C22×C3—44no—
258C22×C3—44no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3C22×C22⋊C411yescharacteristic
262C22×C3C2×#11712yes—
263C22×C3#52514yes—
264C3×C4—24no—
265C3×C4—24no—
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—24no—
269C3×C4—42no—
270C3×C4—42no—
271Dic3—88no—
272C23—22no—
273C23—22no—
274C23—41no—
275C23—41no—
276C23—42no—
277C23—42no—
278C23#92614yes—
279C23C2×#23514yes—
280C23C23×Dic311yescharacteristic
281C23C22×#6712yes—
282Q8—28no—
283D4—28no—
284D4—216no—
285C2×C4—216no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—28no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—128no—
302C2×C4—128no—
303C2×C4—128no—
304C2×C4#92614yes—
305C2×C4C2×#21712yes—
306C2×C4C2×#21714yes—
307C2×C4C2×#21714yes—
308C2×C4C2×#21712yes—
309C2×C3—24no—
310C2×C3—22no—
311C2×C3—44no—
312C2×C3#228011yescharacteristic
313C2×C3#288311yescharacteristic
314C2×C3#288411yescharacteristic
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—24no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—42no—
324C22—42no—
325C22C2×#92612yes—
326C22#579114yes—
327C22C22×#21711yescharacteristic
328C4—22no—
329C4—24no—
330C4—22no—
331C4—24no—
332C4—24no—
333C4—42no—
334C4—42no—
335C4—248no—
336C3#2072511yescharacteristic, complemented, Sylow
337C2—24no—
338C2—22no—
339C2—44no—
340C2#8402711yescharacteristic
341C2#8410711yescharacteristic
342C2#8411911yescharacteristic
subgroupG/H
centerC22C22×#217
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC2×C3×#520C2
socleC22×C3C22×C22⋊C4
prime radicalC22×C3×#41C22
cosocleC24×C3C24
hypercenterC2×#520D3
hypocenterC3#20725
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
4. #71619C2semidirect
5. C4×#834C2semidirect
6. #71594C2semidirect
10. C2×C3×C4×Q8C4semidirect
11. C2×C3×C4×D4C4semidirect
14. C2×C3×#117C4semidirect
16. C22×C3×C4⋊C4C4semidirect
70. C2×C4×Q8Dic3semidirect
71. C2×C4×D4Dic3semidirect
74. C2×#117Dic3semidirect
76. C22×C4⋊C4Dic3semidirect
336. C3#20725semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5202
#821552
#721392
#716192
C4×#8342
#715942
#207253