Finite Group Databasebeta

← #10900922

Subgroups — #10900922

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#869C211yesmaximal, characteristic, complemented
2C3×#2854C211yesmaximal, characteristic
3C2×#5033C212yesmaximal
4#71531C211yesmaximal, characteristic, complemented
5#69638C214yesmaximal, complemented
6#69604C214yesmaximal, complemented
7#69501C212yesmaximal, complemented
8#29638—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#5033—28no—
26C2×#807C2214yes—
27#4529C2218yes—
28#4524C2214yes—
29C22×C4⋊Q8—31no—
30C2×#480—32no—
31#2854D311yescharacteristic
32#2057—34no—
33#2023—34no—
34#1823—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—44no—
45C22×Dic6—44no—
46C3×#123—44no—
47C3×#121—24no—
48C2×C3×C4⋊C4—22no—
49C2×C3×C4⋊C4C2311yescharacteristic
50C2×C3×C4⋊C4C2312yes—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—22no—
55C2×C3×C42—24no—
56C2×C3×C42D412yes—
57C2×C3×C42C2312yes—
58C2×C4⋊Dic3—28no—
59C2×C4⋊Dic3—24no—
60C2×C4⋊Dic3—22no—
61C2×C4⋊Dic3—24no—
62C2×C4⋊Dic3D418yes—
63C2×C4⋊Dic3C2314yes—
64C22×C3⋊C8C2314yes—
65#869—216no—
66#869—216no—
67C3×#95—21no—
68C3×#95—22no—
69#832—28no—
70#830—24no—
71#807—28no—
72C23×Q8—61no—
73C2×C4⋊Q8—34no—
74C2×C4⋊Q8—68no—
75C2×C4⋊Q8—62no—
76C2×#123—21no—
77C2×#121C2×D311yescharacteristic
78C22×C4⋊C4—32no—
79C22×C4⋊C4—61no—
80C22×C42C2×D311yescharacteristic
81#480—68no—
82C2×#107—34no—
83C2×#102—34no—
84C2×C22⋊C8—32no—
85#390C2×D312yes—
86#389C2×D312yes—
87#388—22no—
88#386—21no—
89#380C2×D311yescharacteristic
90#340—38no—
91#336—34no—
92C24×C3—21no—
93C24×C3C2×D411yescharacteristic
94C2×C3×D4—22no—
95C2×C3×D4—24no—
96C2×C3×D4—42no—
97C2×C3×D4—42no—
98C2×C3×D4—42no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—22no—
102C22×C3×C4—24no—
103C22×C3×C4—24no—
104C22×C3×C4—22no—
105C22×C3×C4—21no—
106C22×C3×C4D814yes—
107C22×C3×C4C2×D414yes—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D412yes—
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2411yescharacteristic
112C22×Dic3—28no—
113C22×Dic3—24no—
114C22×Dic3—44no—
115C22×Dic3—44no—
116C2×Dic6—216no—
117C2×Dic6—48no—
118C2×Dic6—432no—
119C2×Dic6—48no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—24no—
122C3×C4⋊C4—44no—
123C3×C22⋊C4—42no—
124C3×C22⋊C4—42no—
125C3×C22⋊C4—44no—
126C3×C22⋊C4—44no—
127C3×C42—24no—
128C3×C42—24no—
129C4⋊Dic3—216no—
130C4⋊Dic3—28no—
131C4⋊Dic3—24no—
132C4⋊Dic3C2×D418yes—
133C2×C3⋊C8—24no—
134C2×C3⋊C8—28no—
135C22×Q8—68no—
136C22×Q8—64no—
137C22×Q8—124no—
138C22×Q8—124no—
139C22×D4—21no—
140C22×D4C22×D311yescharacteristic
141C23×C4—62no—
142C23×C4#6711yescharacteristic
143C23×C4#6711yescharacteristic
144C23×C4C22×D311yescharacteristic
145C22×C8—34no—
146C4⋊Q8—616no—
147C4⋊Q8—616no—
148#123—44no—
149#121—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—38no—
152C2×C4⋊C4—34no—
153C2×C4⋊C4—68no—
154C2×C4⋊C4—64no—
155C2×C4⋊C4—62no—
156C2×C4⋊C4—64no—
157C2×C4⋊C4C22×D311yescharacteristic
158C2×C4⋊C4C22×D312yes—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C42—24no—
164C2×C42#6712yes—
165C2×C42C22×D312yes—
166#107—68no—
167#102—68no—
168C22⋊C8—64no—
169#95—21no—
170#95—22no—
171C23×C3—22no—
172C23×C3—22no—
173C23×C3—41no—
174C23×C3—42no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—42no—
178C23×C3#12012yes—
179C23×C3C2×D812yes—
180C23×C3AGL(1,ℤ/8)12yes—
181C23×C3C22×D411yescharacteristic
182C3×D4—44no—
183C3×D4—84no—
184C2×C3×C4—28no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—28no—
191C2×C3×C4—28no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4#12012yes—
199C2×C3×C4C2×D814yes—
200C2×C3×C4#13714yes—
201C2×C3×C4C22×D412yes—
202C2×C3×C42−1+414yes—
203C2×Dic3—28no—
204C2×Dic3—44no—
205C2×Dic3—44no—
206C2×Dic3—416no—
207Dic6—432no—
208C3⋊C8—44no—
209C24—21no—
210C24C2×#6711yescharacteristic
211C2×Q8—616no—
212C2×Q8—1232no—
213C2×Q8—128no—
214C2×Q8—128no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—68no—
228C22×C4—64no—
229C22×C4—124no—
230C22×C4—124no—
231C22×C4#21314yes—
232C22×C4D3×D414yes—
233C22×C4C2×#6712yes—
234C22×C4C2×#6711yescharacteristic
235C22×C4C2×#6712yes—
236C22×C4C23×D311yescharacteristic
237C2×C8—64no—
238C2×C8—68no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—38no—
242C4⋊C4—44no—
243C4⋊C4—616no—
244C4⋊C4—68no—
245C4⋊C4—64no—
246C22⋊C4—42no—
247C22⋊C4—44no—
248C22⋊C4—42no—
249C22⋊C4—44no—
250C42—24no—
251C42—24no—
252C22×C3—22no—
253C22×C3—24no—
254C22×C3—24no—
255C22×C3—22no—
256C22×C3—81no—
257C22×C3—82no—
258C22×C3—82no—
259C22×C3—81no—
260C22×C3—81no—
261C22×C3#44714yes—
262C22×C3C2×#12011yescharacteristic
263C22×C3#53612yes—
264C22×C3C22×D811yescharacteristic
265C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C3C2×#13712yes—
267C3×C4—22no—
268C3×C4—24no—
269C3×C4—24no—
270C3×C4—82no—
271C3×C4—81no—
272C3×C4#53614yes—
273Dic3—48no—
274C23—22no—
275C23—22no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—42no—
280C23—42no—
281C23C2×#21312yes—
282C23#93712yes—
283C23#94912yes—
284C23C22×#6711yescharacteristic
285Q8—1232no—
286D4—44no—
287D4—84no—
288C2×C4—28no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—28no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—28no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—68no—
303C2×C4—124no—
304C2×C4—1216no—
305C2×C4—124no—
306C2×C4C2×#21314yes—
307C2×C4#93712yes—
308C2×C4#95114yes—
309C2×C4C2×D3×D412yes—
310C2×C4#101014yes—
311C8—124no—
312C2×C3—24no—
313C2×C3—161no—
314C2×C3#261411yescharacteristic
315C2×C3C2×#44711yescharacteristic
316C2×C3#300411yescharacteristic
317C2×C3#322612yes—
318C2×C3#323812yes—
319C22—24no—
320C22—22no—
321C22—24no—
322C22—22no—
323C22—82no—
324C22—81no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22#503614yes—
329C22C22×#21311yescharacteristic
330C22C2×#93711yescharacteristic
331C22C2×#94911yescharacteristic
332C22C2×#95112yes—
333C22#582512yes—
334C4—22no—
335C4—24no—
336C4—24no—
337C4—82no—
338C4—81no—
339C4—128no—
340C4#564814yes—
341C3#2963811yescharacteristic, complemented, Sylow
342C2—24no—
343C2—161no—
344C2C2×#503611yescharacteristic
345C2#8286411yescharacteristic
346C2#8293311yescharacteristic
347C2#8312312yes—
348C2#8327012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2854C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2854D3
hypocenterC3#29638
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. #69604C2semidirect
7. #69501C2semidirect
341. C3#29638semidirect

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×#50332
#715312
#696382
#696042
#695012
#296383