Finite Group Databasebeta

← #10848776

Subgroups — #10848776

Order 768.

1180 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic
2C2×#5076C211yesmaximal, characteristic, complemented
3C2×#5069C211yesmaximal, characteristic, complemented
4#82262C211yesmaximal, characteristic, complemented
5#71765C211yesmaximal, characteristic, complemented
6#71740C211yesmaximal, characteristic, complemented
7#71685C214yesmaximal, complemented
8#71612C214yesmaximal, complemented
9#71582C211yesmaximal, characteristic, complemented
10#25606—31nomaximal, Sylow
11C3×#540—24no—
12C3×#540—24no—
13C2×C3×#127C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C2×C3×#123—22no—
16C2×C3×#121—22no—
17C2×C3×#121C2212yes—
18C2×C3×C4×D4C2212yes—
19C2×C3×#117C2211yescharacteristic
20C22×#214C2214yes—
21C2×#924C2212yes—
22C2×#869C2211yescharacteristic
23C2×#832C2214yes—
24C2×#830C2212yes—
25#5076—24no—
26#5069—24no—
27#4981—22no—
28C2×#810C2214yes—
29#4953—22no—
30#4936C2211yescharacteristic
31#4929C2212yes—
32C2×#803C2211yescharacteristic
33C2×C4×C3⋊C8C2211yescharacteristic
34#4529C2214yes—
35#4524C2212yes—
36C2×#540D311yescharacteristic
37C2×#492—31no—
38C2×#491—31no—
39#2621—31no—
40#2000—34no—
41#1991—31no—
42#1961—31no—
43#1877—34no—
44#1750—31no—
45C22×C3×D4—22no—
46C22×C3×D4—22no—
47C22×C3×D4C2312yes—
48C23×C3×C4—22no—
49C23×C3×C4C2311yescharacteristic
50C22×Dic6C2312yes—
51C3×#127—22no—
52C3×#127D414yes—
53C3×#125—22no—
54C3×#125D414yes—
55C3×#123—216no—
56C3×#121—216no—
57C3×#121—24no—
58C3×#121—24no—
59C3×C4×D4—24no—
60C3×C4×D4—24no—
61C3×#117—22no—
62C3×#117—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4D412yes—
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72C2×#214—28no—
73C2×#214—28no—
74C2×#214—44no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3C2312yes—
77#924—44no—
78C2×Dic3⋊C4—22no—
79C22×C3⋊C8C2312yes—
80C22×C3⋊C8C2312yes—
81#869—22no—
82#869—22no—
83#832—44no—
84C3⋊#95—24no—
85C3⋊#95—22no—
86#830—24no—
87#810—28no—
88#803—22no—
89C4×C3⋊C8—22no—
90C22×SD16—34no—
91#540—24no—
92#540—24no—
93C2×C4⋊Q8—31no—
94C2×#127C2×D311yescharacteristic
95C2×#125C2×D311yescharacteristic
96C2×#123—22no—
97C2×C22⋊Q8—32no—
98C2×#121—22no—
99C2×#121C2×D312yes—
100C2×C4×D4C2×D312yes—
101C2×#117C2×D311yescharacteristic
102#492—64no—
103#491—64no—
104C2×#103—34no—
105C2×#102—34no—
106C2×C22⋊C8—32no—
107C2×#97—31no—
108C2×C4×C8—31no—
109#398—62no—
110#397—62no—
111#391—32no—
112#382—31no—
113#340—34no—
114#336—32no—
115C24×C3—22no—
116C24×C3C2×D411yescharacteristic
117C2×C3×D4—28no—
118C2×C3×D4—28no—
119C2×C3×D4—22no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—84no—
123C2×C3×D4—84no—
124C2×C3×D4C2×D414yes—
125C22×C3×C4—22no—
126C22×C3×C4—28no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—84no—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2411yescharacteristic
136C22×Dic3—22no—
137C22×Dic3—22no—
138C2×Dic6—22no—
139C2×Dic6—22no—
140C2×Dic6—24no—
141C2×Dic6—24no—
142C3×C4⋊C4—28no—
143C3×C4⋊C4—24no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4C2×D414yes—
146C3×C22⋊C4—28no—
147C3×C22⋊C4—28no—
148C3×C22⋊C4—24no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42C2×D412yes—
152C3×C42C2×D412yes—
153#217—44no—
154#214—416no—
155C4⋊Dic3—24no—
156C4⋊Dic3—44no—
157Dic3⋊C4—44no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8#4114yes—
164C22×Q8—32no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4C22×D312yes—
168C23×C4—22no—
169C23×C4C22×D311yescharacteristic
170C2×SD16—68no—
171C2×SD16—68no—
172C2×SD16—124no—
173C22×C8—32no—
174C22×C8—32no—
175C4⋊Q8—62no—
176C4⋊Q8—62no—
177#127—22no—
178#127#6714yes—
179#125—22no—
180#125#6714yes—
181#123—216no—
182C22⋊Q8—124no—
183#121—216no—
184#121—24no—
185#121—24no—
186C4×D4—24no—
187C4×D4—24no—
188#117—22no—
189#117—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4#6712yes—
195C2×C4⋊C4C22×D312yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4#6712yes—
200C2×C42C22×D311yescharacteristic
201C2×C42C22×D311yescharacteristic
202#103—68no—
203#102—124no—
204C22⋊C8—64no—
205#97—62no—
206C4×C8—62no—
207#95—62no—
208#95—64no—
209C23×C3—21no—
210C23×C3—24no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—24no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—44no—
217C23×C3—44no—
218C23×C3C22×D411yescharacteristic
219C3×D4—28no—
220C3×D4—44no—
221C3×D4—88no—
222C3×D4—88no—
223C2×C3×C4—24no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—24no—
230C2×C3×C4—44no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—88no—
234C2×C3×C4#12014yes—
235C2×C3×C4C22×D411yescharacteristic
236C2×C3×C4C22×D411yescharacteristic
237C2×C3×C4C2×#4112yes—
238C2×C3×C4H(5,𝔽2)12yes—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—44no—
244Dic6—44no—
245Dic6—44no—
246C3⋊C8—24no—
247C3⋊C8—42no—
248C24—22no—
249C24C2×#6711yescharacteristic
250C2×Q8—64no—
251C2×Q8—64no—
252C2×Q8—62no—
253C2×Q8—62no—
254C2×D4—28no—
255C2×D4—28no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4—84no—
260C2×D4—84no—
261C2×D4C2×#6714yes—
262C22×C4—22no—
263C22×C4—28no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—84no—
271C22×C4C2×#6712yes—
272C22×C4C2×#6711yescharacteristic
273C22×C4C2×#6712yes—
274C22×C4C23×D311yescharacteristic
275SD16—1216no—
276C2×C8—34no—
277C2×C8—62no—
278C2×C8—62no—
279C2×C8—62no—
280C2×C8—62no—
281C2×C8—62no—
282C4⋊C4—28no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—64no—
286C4⋊C4—124no—
287C4⋊C4—124no—
288C4⋊C4C2×#6714yes—
289C22⋊C4—28no—
290C22⋊C4—28no—
291C22⋊C4—24no—
292C22⋊C4—124no—
293C42—21no—
294C42—21no—
295C42C2×#6712yes—
296C42C2×#6712yes—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—44no—
300C22×C3—42no—
301C22×C3—44no—
302C22×C3—42no—
303C22×C3—44no—
304C22×C3—84no—
305C22×C3—84no—
306C22×C3C2×#12011yescharacteristic
307C22×C3#53211yescharacteristic
308C22×C3#53511yescharacteristic
309C22×C3#57714yes—
310C3×C4—24no—
311C3×C4—41no—
312C3×C4—42no—
313C3×C4—44no—
314C3×C4#54612yes—
315C3×C4#55012yes—
316Dic3—82no—
317Dic3—82no—
318C23—21no—
319C23—21no—
320C23—24no—
321C23—21no—
322C23—24no—
323C23—42no—
324C23—42no—
325C23—44no—
326C23—44no—
327C23C22×#6711yescharacteristic
328Q8—124no—
329Q8—124no—
330D4—28no—
331D4—44no—
332D4—88no—
333D4—88no—
334C2×C4—24no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—88no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—124no—
349C2×C4—122no—
350C2×C4#95314yes—
351C2×C4D3×#4112yes—
352C2×C4#100912yes—
353C2×C4C22×#6711yescharacteristic
354C2×C4C22×#6711yescharacteristic
355C8—64no—
356C8—122no—
357C2×C3—81no—
358C2×C3—84no—
359C2×C3#243111yescharacteristic
360C2×C3#300912yes—
361C2×C3#317412yes—
362C2×C3#322112yes—
363C22—41no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—44no—
368C22—42no—
369C22—42no—
370C22—84no—
371C22—84no—
372C22#583011yescharacteristic
373C22#583211yescharacteristic
374C22#583614yes—
375C22C2×#95311yescharacteristic
376C4—24no—
377C4—41no—
378C4—42no—
379C4—44no—
380C4—242no—
381C4—242no—
382C4#582912yes—
383C4#583412yes—
384C3#2560611yescharacteristic, complemented, Sylow
385C2—81no—
386C2—84no—
387C2#8343112yes—
388C2#8344012yes—
389C2#8409211yescharacteristic
390C2#8410512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#540C2
socleC23×C3C22×D4
hypercenterC2×#540D3
hypocenterC3#25606
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
2. C2×#5076C2semidirect
3. C2×#5069C2semidirect
4. #82262C2semidirect
5. #71765C2semidirect
6. #71740C2semidirect
7. #71685C2semidirect
8. #71612C2semidirect
9. #71582C2semidirect
384. C3#25606semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×#50762
C2×#50692
#822622
#717652
#717402
#716852
#716122
#715822
#256063