Finite Group Databasebeta

← #10900945

Subgroups — #10900945

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#869C211yesmaximal, characteristic
2C3×#2855C211yesmaximal, characteristic
3C2×#5043C212yesmaximal
4#71531C211yesmaximal, characteristic
5#69668C214yesmaximal, complemented
6#69605C214yesmaximal, complemented
7#69501C212yesmaximal
8#29641—31nomaximal, Sylow
9C23×Dic6—21no—
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C22×C4⋊Dic3—21no—
13C22×C4⋊Dic3C2212yes—
14C2×#869—22no—
15C2×#869—28no—
16C2×#869C2214yes—
17C3×#391C2212yes—
18C3×#389C2212yes—
19C3×#387—22no—
20C3×#385—21no—
21C3×#381—21no—
22C3×#380C2211yescharacteristic
23C2×#835C2214yes—
24C2×#830C2212yes—
25#5043—28no—
26C2×#808C2214yes—
27#4529C2218yes—
28#4524C2214yes—
29C22×C4⋊Q8—31no—
30C2×#483—32no—
31#2855D311yescharacteristic
32#2058—34no—
33#2024—34no—
34#1823—32no—
35#1817—31no—
36C22×C3×Q8C2311yescharacteristic
37C23×C3×C4D411yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—22no—
41C22×Dic6—28no—
42C22×Dic6—24no—
43C22×Dic6—44no—
44C22×Dic6—44no—
45C3×C22⋊Q8—24no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4C2311yescharacteristic
49C2×C3×C4⋊C4C2312yes—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—24no—
53C2×C3×C42D412yes—
54C2×C3×C42C2312yes—
55C2×C4⋊Dic3—28no—
56C2×C4⋊Dic3—24no—
57C2×C4⋊Dic3—22no—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3D418yes—
60C2×C4⋊Dic3C2314yes—
61C22×C3⋊C8C2314yes—
62#869—216no—
63#869—216no—
64C3×#95—21no—
65C3×#95—22no—
66C3×#95—22no—
67C3×#95—22no—
68#835—28no—
69#830—24no—
70#808—28no—
71C23×Q8—61no—
72C2×C4⋊Q8—34no—
73C2×C4⋊Q8—68no—
74C2×C4⋊Q8—62no—
75C2×C22⋊Q8C2×D311yescharacteristic
76C22×C4⋊C4—32no—
77C22×C4⋊C4—61no—
78C22×C42C2×D311yescharacteristic
79#483—68no—
80C2×#106—34no—
81C2×#103—34no—
82C2×C22⋊C8—32no—
83#391C2×D312yes—
84#389C2×D312yes—
85#387—22no—
86#385—21no—
87#381—21no—
88#380C2×D311yescharacteristic
89#340—38no—
90#336—34no—
91C24×C3C2×D411yescharacteristic
92C2×C3×Q8—22no—
93C2×C3×Q8—24no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4SD1614yes—
103C22×C3×C4C2×D414yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4C2×D412yes—
107C22×C3×C4C2411yescharacteristic
108C22×Dic3—28no—
109C22×Dic3—24no—
110C22×Dic3—44no—
111C22×Dic3—44no—
112C2×Dic6—216no—
113C2×Dic6—48no—
114C2×Dic6—432no—
115C2×Dic6—48no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—24no—
118C3×C4⋊C4—42no—
119C3×C4⋊C4—44no—
120C3×C22⋊C4—42no—
121C3×C22⋊C4—42no—
122C3×C42—24no—
123C3×C42—24no—
124C4⋊Dic3—216no—
125C4⋊Dic3—28no—
126C4⋊Dic3—24no—
127C4⋊Dic3C2×D418yes—
128C2×C3⋊C8—24no—
129C2×C3⋊C8—28no—
130C22×Q8—68no—
131C22×Q8—64no—
132C22×Q8—124no—
133C22×Q8—124no—
134C22×Q8C22×D311yescharacteristic
135C23×C4—62no—
136C23×C4#6711yescharacteristic
137C23×C4#6711yescharacteristic
138C23×C4C22×D311yescharacteristic
139C22×C8—34no—
140C4⋊Q8—616no—
141C4⋊Q8—616no—
142C22⋊Q8—24no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—38no—
146C2×C4⋊C4—34no—
147C2×C4⋊C4—68no—
148C2×C4⋊C4—64no—
149C2×C4⋊C4—62no—
150C2×C4⋊C4—64no—
151C2×C4⋊C4C22×D311yescharacteristic
152C2×C4⋊C4C22×D312yes—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C42—24no—
156C2×C42#6712yes—
157C2×C42C22×D312yes—
158#106—68no—
159#103—68no—
160C22⋊C8—64no—
161#95—21no—
162#95—22no—
163#95—22no—
164#95—22no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3#12012yes—
168C23×C3C2×SD1612yes—
169C23×C3AGL(1,ℤ/8)12yes—
170C23×C3C22×D411yescharacteristic
171C3×Q8—44no—
172C2×C3×C4—28no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—28no—
179C2×C3×C4—28no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4#12012yes—
188C2×C3×C4C2×SD1614yes—
189C2×C3×C4#13714yes—
190C2×C3×C4C22×D412yes—
191C2×C3×C42−1+414yes—
192C2×Dic3—28no—
193C2×Dic3—44no—
194C2×Dic3—44no—
195C2×Dic3—416no—
196Dic6—432no—
197C3⋊C8—44no—
198C24C2×#6711yescharacteristic
199C2×Q8—22no—
200C2×Q8—24no—
201C2×Q8—616no—
202C2×Q8—1232no—
203C2×Q8—128no—
204C2×Q8—128no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—68no—
214C22×C4—64no—
215C22×C4—124no—
216C22×C4—124no—
217C22×C4#21514yes—
218C22×C4D3×D414yes—
219C22×C4C2×#6711yescharacteristic
220C22×C4C2×#6712yes—
221C22×C4C2×#6712yes—
222C22×C4C23×D311yescharacteristic
223C2×C8—64no—
224C2×C8—68no—
225C4⋊C4—22no—
226C4⋊C4—24no—
227C4⋊C4—38no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—616no—
231C4⋊C4—68no—
232C4⋊C4—64no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C42—24no—
236C42—24no—
237C22×C3—22no—
238C22×C3—24no—
239C22×C3—24no—
240C22×C3—22no—
241C22×C3#45014yes—
242C22×C3C2×#12011yescharacteristic
243C22×C3#53612yes—
244C22×C3C22×SD1611yescharacteristic
245C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C3C2×#13712yes—
247C3×C4—22no—
248C3×C4—24no—
249C3×C4—24no—
250C3×C4—81no—
251C3×C4—82no—
252C3×C4—81no—
253C3×C4#53614yes—
254Dic3—48no—
255C23—22no—
256C23—22no—
257C23#93712yes—
258C23C2×#21512yes—
259C23#94912yes—
260C23C22×#6711yescharacteristic
261Q8—44no—
262Q8—1232no—
263C2×C4—28no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—28no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—68no—
279C2×C4—124no—
280C2×C4—1216no—
281C2×C4—124no—
282C2×C4#93712yes—
283C2×C4C2×#21514yes—
284C2×C4#95114yes—
285C2×C4C2×D3×D412yes—
286C2×C4#101014yes—
287C8—124no—
288C2×C3—24no—
289C2×C3#261411yescharacteristic
290C2×C3C2×#45011yescharacteristic
291C2×C3#300411yescharacteristic
292C2×C3#322912yes—
293C2×C3#323812yes—
294C22—24no—
295C22—22no—
296C22—24no—
297C22—22no—
298C22#504614yes—
299C22C2×#93711yescharacteristic
300C22C22×#21511yescharacteristic
301C22C2×#94911yescharacteristic
302C22C2×#95112yes—
303C22#582512yes—
304C4—24no—
305C4—22no—
306C4—24no—
307C4—81no—
308C4—82no—
309C4—81no—
310C4—128no—
311C4#564814yes—
312C3#2964111yescharacteristic, complemented, Sylow
313C2—24no—
314C2C2×#504611yescharacteristic
315C2#8286411yescharacteristic
316C2#8293311yescharacteristic
317C2#8312312yes—
318C2#8332312yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2855C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2855D3
hypocenterC3#29641
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
5. #69668C2semidirect
6. #69605C2semidirect
312. C3#29641semidirect

Maximal subgroups

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

subgroupindex
C22×#8692
C3×#28552
C2×#50432
#715312
#696682
#696052
#695012
#296413