Finite Group Databasebeta

← #10796014

Subgroups — #10796014

Order 768.

1330 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5812C211yesmaximal, characteristic
2C3×#2310C211yesmaximal, characteristic
3C22×#836C211yesmaximal, characteristic
4#82207C212yesmaximal
5#71578C212yesmaximal
6#69672C218yesmaximal, complemented
7#20857—31nomaximal, Sylow
8C23×C3×Q8C2211yescharacteristic
9C2×C3×#117C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11#5812—22no—
12#5812—24no—
13#5812C2214yes—
14C2×Q8×Dic3—24no—
15C2×#926—22no—
16C2×#926C2212yes—
17C22×#208C2212yes—
18C3×#391—22no—
19C3×#391C2214yes—
20C3×#381—22no—
21C2×#836C418yescomplemented
22C2×#836C2214yes—
23#4998—24no—
24#4995—24no—
25#4964C2218yes—
26#4523C2218yes—
27C2×#519—31no—
28#2911—32no—
29C22×#101—31no—
30#2310D311yescharacteristic
31#1857—38no—
32#1733—32no—
33C22×C3×Q8—22no—
34C22×C3×Q8—22no—
35C22×C3×Q8—24no—
36C22×C3×Q8C2×C414yes—
37C22×C3×Q8D412yes—
38C22×C3×Q8Q814yes—
39C22×C3×Q8C2312yes—
40C23×C3×C4D412yes—
41C23×C3×C4C2311yescharacteristic
42C3×#117—24no—
43C3×#117—22no—
44C3×#117—22no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4C2312yes—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2312yes—
51C2×#217—22no—
52Q8×Dic3—216no—
53Q8×Dic3—216no—
54#926—28no—
55#926—28no—
56#926—22no—
57#926—22no—
58#926D414yes—
59#926C2314yes—
60C2×C4⋊Dic3—24no—
61C2×C4⋊Dic3D414yes—
62C2×C4×Dic3—24no—
63C2×C4×Dic3—22no—
64C2×#208—22no—
65C2×#208—24no—
66C2×#208C2×C418yes—
67C2×#208C2314yes—
68C22×C3⋊C8—22no—
69C3×#95—28no—
70#836C2×C4116yescomplemented
71#808—28no—
72#807—28no—
73C23×Q8C2×D311yescharacteristic
74C22×M16—32no—
75#519—34no—
76#519—64no—
77#519—62no—
78C2×C4×Q8—64no—
79C2×#117—32no—
80C2×#117—62no—
81C2×#117C2×D311yescharacteristic
82C2×#117C2×D311yescharacteristic
83#428—38no—
84#428—64no—
85#427—64no—
86C2×#101—38no—
87C2×#101—34no—
88#391—22no—
89#391C2×D314yes—
90#381—22no—
91#339—38no—
92C24×C3C2×D411yescharacteristic
93C2×C3×Q8—28no—
94C2×C3×Q8—28no—
95C2×C3×Q8—24no—
96C2×C3×Q8—28no—
97C2×C3×Q8—28no—
98C2×C3×Q8—24no—
99C2×C3×Q8—24no—
100C2×C3×Q8—24no—
101C2×C3×Q8C4⋊C418yes—
102C2×C3×Q8C22×C414yes—
103C2×C3×Q8C2×Q814yes—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—24no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4C4⋊C418yes—
110C22×C3×C4C22×C412yes—
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2×D414yes—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×Q812yes—
116C22×C3×C4#4114yes—
117C22×C3×C4C2411yescharacteristic
118C22×Dic3—24no—
119C3×C4⋊C4—24no—
120C3×C4⋊C4—42no—
121C3×C22⋊C4—41no—
122C3×C22⋊C4—41no—
123C3×C22⋊C4—42no—
124C3×C42—24no—
125C3×C42—42no—
126#217—24no—
127#217—24no—
128C4⋊Dic3—216no—
129C4⋊Dic3—24no—
130C4⋊Dic3C2×D418yes—
131C4×Dic3—216no—
132C4×Dic3—24no—
133C4×Dic3—24no—
134#208—28no—
135#208—24no—
136#208C22×C418yes—
137C2×C3⋊C8—28no—
138C2×C3⋊C8—24no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—24no—
142C22×Q8Dic614yes—
143C22×Q8C4×D314yes—
144C22×Q8#6712yes—
145C22×Q8C22×D312yes—
146C23×C4#6712yes—
147C23×C4C22×D311yescharacteristic
148C2×M16—38no—
149C2×M16—34no—
150C2×M16—62no—
151C2×M16—64no—
152C22×C8—62no—
153C4×Q8—616no—
154C4×Q8—616no—
155#117—24no—
156#117—22no—
157#117—22no—
158#117—34no—
159#117—34no—
160#117—68no—
161#117—62no—
162#117—62no—
163#117—68no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—34no—
166C2×C4⋊C4—64no—
167C2×C4⋊C4C22×D312yes—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—62no—
171C2×C42—21no—
172C2×C42—64no—
173C2×C42—62no—
174C2×C42C22×D312yes—
175#107—68no—
176#106—68no—
177#101—316no—
178#95—28no—
179C23×C3—42no—
180C23×C3C2×C4⋊C412yes—
181C23×C3#12011yescharacteristic
182C23×C3#12011yescharacteristic
183C23×C3#12312yes—
184C23×C3C22×D411yescharacteristic
185C3×Q8—28no—
186C3×Q8—28no—
187C3×Q8—216no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×C3×C4—28no—
191C2×C3×C4—28no—
192C2×C3×C4—44no—
193C2×C3×C4—44no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4C2×C4⋊C418yes—
197C2×C3×C4C2×C4⋊C412yes—
198C2×C3×C4C4×D414yes—
199C2×C3×C4C4×D414yes—
200C2×C3×C4#12012yes—
201C2×C3×C4C22⋊Q814yes—
202C2×C3×C4C22⋊Q814yes—
203C2×C3×C4#12312yes—
204C2×C3×C4C23×C411yescharacteristic
205C2×C3×C4C22×D412yes—
206C2×C3×C4C22×Q811yescharacteristic
207C2×C3×C4C2×#4112yes—
208C2×Dic3—28no—
209C2×Dic3—48no—
210C3⋊C8—28no—
211C24C2×#6711yescharacteristic
212C2×Q8—28no—
213C2×Q8—28no—
214C2×Q8—28no—
215C2×Q8—28no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8Dic3⋊C418yes—
221C2×Q8C2×Dic614yes—
222C2×Q8C2×C4×D314yes—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—64no—
229C22×C4Dic3⋊C418yes—
230C22×C4C2×Dic612yes—
231C22×C4C2×C4×D312yes—
232C22×C4D3×D412yes—
233C22×C4D3×D412yes—
234C22×C4#23714yes—
235C22×C4C2×#6714yes—
236C22×C4C2×#6711yescharacteristic
237C22×C4C23×D311yescharacteristic
238M16—38no—
239M16—68no—
240M16—64no—
241C2×C8—68no—
242C2×C8—64no—
243C4⋊C4—24no—
244C4⋊C4—38no—
245C4⋊C4—42no—
246C4⋊C4—616no—
247C4⋊C4—64no—
248C22⋊C4—42no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—64no—
252C22⋊C4—64no—
253C42—24no—
254C42—42no—
255C42—616no—
256C42—64no—
257C42—64no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—42no—
262C22×C3—42no—
263C22×C3#38512yes—
264C22×C3#38512yes—
265C22×C3C22×C4⋊C411yescharacteristic
266C22×C3C2×#12011yescharacteristic
267C22×C3C2×#12311yescharacteristic
268C3×C4—28no—
269C3×C4—82no—
270C3×C4—82no—
271C3×C4#38514yes—
272C3×C4C2×C4×D412yes—
273C3×C4C2×C22⋊Q812yes—
274Dic3—48no—
275C23—42no—
276C23C2×Dic3⋊C412yes—
277C23#93512yes—
278C23#93711yescharacteristic
279C23#93711yescharacteristic
280C23C22×#6711yescharacteristic
281Q8—216no—
282Q8—28no—
283Q8—28no—
284C2×C4—28no—
285C2×C4—28no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—68no—
293C2×C4—128no—
294C2×C4#87814yes—
295C2×C4#87814yes—
296C2×C4#67⋊C414yes—
297C2×C4#67⋊C414yes—
298C2×C4C2×Dic3⋊C418yes—
299C2×C4C2×Dic3⋊C412yes—
300C2×C4#93512yes—
301C2×C4#93712yes—
302C2×C4C22×Dic611yescharacteristic
303C2×C4C22×C4×D311yescharacteristic
304C2×C4C2×D3×D412yes—
305C2×C4C2×#23712yes—
306C8—68no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—22no—
310C2×C3—22no—
311C2×C3C2×#38511yescharacteristic
312C2×C3#302012yes—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—42no—
317C22—42no—
318C22#494612yes—
319C22#494612yes—
320C22C22×Dic3⋊C411yescharacteristic
321C22C2×#93511yescharacteristic
322C22C2×#93711yescharacteristic
323C4—28no—
324C4—82no—
325C4—82no—
326C4—128no—
327C4#494614yes—
328C4C2×#87812yes—
329C4C2×#67⋊C412yes—
330C3#2085711yescharacteristic, complemented, Sylow
331C2—21no—
332C2—22no—
333C2—22no—
334C2—21no—
335C2C2×#494611yescharacteristic
336C2#8341512yes—
subgroupG/H
centerC22C2×#937
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2310C2
socleC22×C3C2×#120
prime radicalC24×C3C2×D4
cosocleC22×C3×C4C24
hypercenter#2310D3
hypocenterC3#20857
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
6. #69672C2semidirect
21. C2×#836C4semidirect
70. #836C2×C4semidirect
330. C3#20857semidirect

Maximal subgroups

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

subgroupindex
C2×#58122
C3×#23102
C22×#8362
#822072
#715782
#696722
#208573