Finite Group Databasebeta

← #10900929

Subgroups — #10900929

Order 768.

1292 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
1C22×#869C211yesmaximal, characteristic, complemented
2C3×#2854C211yesmaximal, characteristic
3C2×#5034C212yesmaximal
4#71531C211yesmaximal, characteristic, complemented
5#69638C214yesmaximal, complemented
6#69605C214yesmaximal, complemented
7#69502C212yesmaximal, complemented
8#29639—31nomaximal, Sylow
9C23×Dic6—21no—
10C2×C3×#123—21no—
11C2×C3×#121C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C22×C4⋊Dic3—21no—
14C22×C4⋊Dic3C2212yes—
15C2×#869—22no—
16C2×#869—28no—
17C2×#869C2214yes—
18C3×#390C2212yes—
19C3×#389C2212yes—
20C3×#388—22no—
21C3×#386—21no—
22C3×#380C2211yescharacteristic
23C2×#832C2214yes—
24C2×#830C2212yes—
25#5034—28no—
26C2×#808C2214yes—
27#4529C2218yes—
28#4524C2214yes—
29C22×C4⋊Q8—31no—
30C2×#481—32no—
31#2854D311yescharacteristic
32#2058—34no—
33#2023—34no—
34#1824—32no—
35#1817—31no—
36C22×C3×D4—21no—
37C22×C3×D4C2311yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C23×Dic3—22no—
42C22×Dic6—28no—
43C22×Dic6—24no—
44C22×Dic6—48no—
45C3×#123—44no—
46C3×#121—24no—
47C2×C3×C4⋊C4—22no—
48C2×C3×C4⋊C4C2311yescharacteristic
49C2×C3×C4⋊C4C2312yes—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C42—24no—
55C2×C3×C42D412yes—
56C2×C3×C42C2312yes—
57C2×C4⋊Dic3—28no—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3—22no—
60C2×C4⋊Dic3—24no—
61C2×C4⋊Dic3D418yes—
62C2×C4⋊Dic3C2314yes—
63C22×C3⋊C8C2314yes—
64#869—216no—
65#869—216no—
66C3×#95—21no—
67C3×#95—22no—
68#832—28no—
69#830—24no—
70#808—28no—
71C23×Q8—61no—
72C2×C4⋊Q8—34no—
73C2×C4⋊Q8—68no—
74C2×C4⋊Q8—62no—
75C2×#123—21no—
76C2×#121C2×D311yescharacteristic
77C22×C4⋊C4—32no—
78C22×C4⋊C4—61no—
79C22×C42C2×D311yescharacteristic
80#481—68no—
81C2×#106—34no—
82C2×#102—34no—
83C2×C22⋊C8—32no—
84#390C2×D312yes—
85#389C2×D312yes—
86#388—22no—
87#386—21no—
88#380C2×D311yescharacteristic
89#340—38no—
90#336—34no—
91C24×C3—21no—
92C24×C3C2×D411yescharacteristic
93C2×C3×D4—22no—
94C2×C3×D4—24no—
95C2×C3×D4—42no—
96C2×C3×D4—42no—
97C2×C3×D4—42no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—24no—
102C22×C3×C4—24no—
103C22×C3×C4—22no—
104C22×C3×C4—21no—
105C22×C3×C4SD1614yes—
106C22×C3×C4C2×D414yes—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D412yes—
109C22×C3×C4C2×D412yes—
110C22×C3×C4C2411yescharacteristic
111C22×Dic3—28no—
112C22×Dic3—24no—
113C22×Dic3—48no—
114C2×Dic6—216no—
115C2×Dic6—432no—
116C2×Dic6—48no—
117C2×Dic6—48no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—24no—
120C3×C4⋊C4—44no—
121C3×C22⋊C4—42no—
122C3×C22⋊C4—42no—
123C3×C22⋊C4—44no—
124C3×C22⋊C4—44no—
125C3×C42—24no—
126C3×C42—24no—
127C4⋊Dic3—216no—
128C4⋊Dic3—28no—
129C4⋊Dic3—24no—
130C4⋊Dic3C2×D418yes—
131C2×C3⋊C8—24no—
132C2×C3⋊C8—28no—
133C22×Q8—68no—
134C22×Q8—64no—
135C22×Q8—128no—
136C22×D4—21no—
137C22×D4C22×D311yescharacteristic
138C23×C4—62no—
139C23×C4#6711yescharacteristic
140C23×C4#6711yescharacteristic
141C23×C4C22×D311yescharacteristic
142C22×C8—34no—
143C4⋊Q8—616no—
144C4⋊Q8—616no—
145#123—44no—
146#121—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—38no—
149C2×C4⋊C4—34no—
150C2×C4⋊C4—68no—
151C2×C4⋊C4—64no—
152C2×C4⋊C4—62no—
153C2×C4⋊C4—64no—
154C2×C4⋊C4C22×D311yescharacteristic
155C2×C4⋊C4C22×D312yes—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C42—24no—
161C2×C42#6712yes—
162C2×C42C22×D312yes—
163#106—68no—
164#102—68no—
165C22⋊C8—64no—
166#95—21no—
167#95—22no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—41no—
171C23×C3—42no—
172C23×C3—41no—
173C23×C3—42no—
174C23×C3—41no—
175C23×C3#12012yes—
176C23×C3C2×SD1612yes—
177C23×C3#13712yes—
178C23×C3C22×D411yescharacteristic
179C3×D4—44no—
180C3×D4—84no—
181C2×C3×C4—28no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—28no—
188C2×C3×C4—28no—
189C2×C3×C4—41no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—41no—
195C2×C3×C4#12012yes—
196C2×C3×C4C2×SD1614yes—
197C2×C3×C4AGL(1,ℤ/8)14yes—
198C2×C3×C4C22×D412yes—
199C2×C3×C42−1+414yes—
200C2×Dic3—28no—
201C2×Dic3—416no—
202C2×Dic3—48no—
203Dic6—432no—
204C3⋊C8—44no—
205C24—21no—
206C24C2×#6711yescharacteristic
207C2×Q8—616no—
208C2×Q8—1232no—
209C2×Q8—128no—
210C2×Q8—128no—
211C2×D4—22no—
212C2×D4—24no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—68no—
224C22×C4—64no—
225C22×C4—128no—
226C22×C4#21414yes—
227C22×C4D3×D414yes—
228C22×C4C2×#6712yes—
229C22×C4C2×#6711yescharacteristic
230C22×C4C2×#6712yes—
231C22×C4C23×D311yescharacteristic
232C2×C8—64no—
233C2×C8—68no—
234C4⋊C4—22no—
235C4⋊C4—24no—
236C4⋊C4—38no—
237C4⋊C4—44no—
238C4⋊C4—616no—
239C4⋊C4—68no—
240C4⋊C4—64no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—42no—
244C22⋊C4—44no—
245C42—24no—
246C42—24no—
247C22×C3—22no—
248C22×C3—24no—
249C22×C3—24no—
250C22×C3—22no—
251C22×C3—81no—
252C22×C3—82no—
253C22×C3—82no—
254C22×C3—81no—
255C22×C3—81no—
256C22×C3#44814yes—
257C22×C3C2×#12011yescharacteristic
258C22×C3#53612yes—
259C22×C3C22×SD1611yescharacteristic
260C22×C3C2×AGL(1,ℤ/8)12yes—
261C22×C3C2×#13711yescharacteristic
262C3×C4—22no—
263C3×C4—24no—
264C3×C4—24no—
265C3×C4—82no—
266C3×C4—81no—
267C3×C4#53614yes—
268Dic3—48no—
269C23—22no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—42no—
274C23—41no—
275C23—42no—
276C23C2×#21412yes—
277C23#93712yes—
278C23#95112yes—
279C23C22×#6711yescharacteristic
280Q8—1232no—
281D4—44no—
282D4—84no—
283C2×C4—28no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—28no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—28no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—68no—
298C2×C4—1216no—
299C2×C4—128no—
300C2×C4C2×#21414yes—
301C2×C4#93712yes—
302C2×C4#94914yes—
303C2×C4C2×D3×D412yes—
304C2×C4#101014yes—
305C8—124no—
306C2×C3—24no—
307C2×C3—161no—
308C2×C3#261411yescharacteristic
309C2×C3C2×#44811yescharacteristic
310C2×C3#300311yescharacteristic
311C2×C3#322712yes—
312C2×C3#323912yes—
313C22—24no—
314C22—22no—
315C22—24no—
316C22—22no—
317C22—82no—
318C22—81no—
319C22—82no—
320C22—81no—
321C22—81no—
322C22#504014yes—
323C22C22×#21411yescharacteristic
324C22C2×#93711yescharacteristic
325C22C2×#94912yes—
326C22C2×#95111yescharacteristic
327C22#582512yes—
328C4—22no—
329C4—24no—
330C4—24no—
331C4—82no—
332C4—81no—
333C4—128no—
334C4#564814yes—
335C3#2963911yescharacteristic, complemented, Sylow
336C2—24no—
337C2—161no—
338C2C2×#504011yescharacteristic
339C2#8286411yescharacteristic
340C2#8292711yescharacteristic
341C2#8312412yes—
342C2#8327112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2854C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2854D3
hypocenterC3#29639
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C22×#869C2semidirect
4. #71531C2semidirect
5. #69638C2semidirect
6. #69605C2semidirect
7. #69502C2semidirect
335. C3#29639semidirect

Maximal subgroups

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

subgroupindex
C22×#8692
C3×#28542
C2×#50342
#715312
#696382
#696052
#695022
#296393