Finite Group Databasebeta

← #10828477

Subgroups — #10828477

Order 768.

1144 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×#5043C212yesmaximal
3C2×#4928C211yesmaximal, characteristic
4#71783C212yesmaximal, complemented
5#71781C212yesmaximal, complemented
6#71692C212yesmaximal
7#71629C214yesmaximal, complemented
8#71531C211yesmaximal, characteristic
9#24063—31nomaximal, Sylow
10C3×#533—28no—
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#125—21no—
14C2×C3×C22⋊Q8—21no—
15C2×C3×C22⋊Q8C2212yes—
16C2×C3×C4×Q8—21no—
17C2×C3×C4×Q8C2212yes—
18C2×C3×#117—21no—
19C22×C3×C42C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4—21no—
23C2×#835C2214yes—
24C2×#835C2214yes—
25C2×C3⋊#95—21no—
26C2×#830C2212yes—
27#5043—28no—
28C2×#808C2214yes—
29#4928—21no—
30#4928—21no—
31#4928C2212yes—
32#4928C2212yes—
33#4529C2214yes—
34#4524C2212yes—
35#4524C2212yes—
36C2×#533D311yescharacteristic
37C2×#483—32no—
38C2×#389—31no—
39#2058—32no—
40#2043—32no—
41#1872—32no—
42#1865—34no—
43#1817—31no—
44C22×C3×Q8—21no—
45C22×C3×Q8C2312yes—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×Dic3—21no—
50C23×Dic3—21no—
51C3×C4⋊Q8—24no—
52C3×C4⋊Q8—24no—
53C3×#125—28no—
54C3×C22⋊Q8—28no—
55C3×C22⋊Q8—24no—
56C3×C22⋊Q8D418yes—
57C3×C4×Q8—28no—
58C3×C4×Q8—24no—
59C3×C4×Q8—24no—
60C3×#117—28no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C22⋊C4—22no—
68C2×C3×C42—24no—
69C2×C3×C42—22no—
70C2×C3×C42D411yescharacteristic
71C2×C3×C42D411yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2312yes—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—44no—
83C22×C3⋊C8C2314yes—
84#835—28no—
85#835—28no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88#830—24no—
89#808—28no—
90#533—28no—
91C2×C4⋊Q8C2×D311yescharacteristic
92C2×C4⋊Q8C2×D311yescharacteristic
93C2×#125—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8C2×D312yes—
96C2×C4×Q8—21no—
97C2×C4×Q8C2×D312yes—
98C2×#117—21no—
99C22×C4⋊C4—31no—
100C22×C4⋊C4—31no—
101C22×C4⋊C4—61no—
102C22×C42C2×D311yescharacteristic
103#483—68no—
104C2×#106—34no—
105C2×#103—34no—
106C2×#103—34no—
107C2×C22⋊C8—32no—
108#389—32no—
109#389—32no—
110#389—61no—
111#389—61no—
112C2×#95—61no—
113#340—34no—
114#336—32no—
115#336—32no—
116C24×C3C2×D411yescharacteristic
117C2×C3×Q8—24no—
118C2×C3×Q8—22no—
119C2×C3×Q8—24no—
120C2×C3×Q8—24no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×Q8C2×D414yes—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—41no—
133C22×C3×C4SD1614yes—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D414yes—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2411yescharacteristic
141C22×Dic3—24no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—42no—
145C22×Dic3—42no—
146C22×Dic3—42no—
147C22×Dic3—42no—
148C22×Dic3—42no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—28no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—28no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4C2×D414yes—
156C3×C22⋊C4—28no—
157C3×C42—28no—
158C3×C42—22no—
159C3×C42—22no—
160C3×C42—22no—
161C3×C42—22no—
162C4⋊Dic3—24no—
163C4⋊Dic3—22no—
164C4⋊Dic3—24no—
165C4⋊Dic3—42no—
166C4⋊Dic3#4114yes—
167Dic3⋊C4—44no—
168Dic3⋊C4—44no—
169C2×C3⋊C8—24no—
170C2×C3⋊C8—24no—
171C2×C3⋊C8—24no—
172C22×Q8—21no—
173C22×Q8C22×D312yes—
174C23×C4—61no—
175C23×C4—61no—
176C23×C4#6711yescharacteristic
177C23×C4#6711yescharacteristic
178C23×C4C22×D311yescharacteristic
179C22×C8—34no—
180C4⋊Q8—24no—
181C4⋊Q8—24no—
182#125—28no—
183C22⋊Q8—28no—
184C22⋊Q8—24no—
185C22⋊Q8#6718yes—
186C4×Q8—28no—
187C4×Q8—24no—
188C4×Q8—24no—
189#117—28no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—64no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—64no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4—124no—
204C2×C4⋊C4C22×D312yes—
205C2×C22⋊C4—22no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42#6711yescharacteristic
209C2×C42#6711yescharacteristic
210C2×C42C22×D311yescharacteristic
211C2×C42C22×D311yescharacteristic
212#106—68no—
213#103—68no—
214#103—68no—
215C22⋊C8—64no—
216#95—62no—
217#95—62no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3#12012yes—
223C23×C3C2×SD1612yes—
224C23×C3#13712yes—
225C23×C3C22×D411yescharacteristic
226C3×Q8—28no—
227C3×Q8—48no—
228C3×Q8—44no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—28no—
238C2×C3×C4—28no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—44no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4#12012yes—
250C2×C3×C4C2×SD1612yes—
251C2×C3×C4C2×SD1612yes—
252C2×C3×C4#13712yes—
253C2×C3×C4C22×D412yes—
254C2×C3×C4C2×#4112yes—
255C2×C3×C42−1+412yes—
256C2×Dic3—24no—
257C2×Dic3—48no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—84no—
264C3⋊C8—44no—
265C24C2×#6711yescharacteristic
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8C2×#6714yes—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—41no—
282C22×C4—64no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4—122no—
286C22×C4—122no—
287C22×C4—122no—
288C22×C4—122no—
289C22×C4—122no—
290C22×C4#21514yes—
291C22×C4C2×#6711yescharacteristic
292C22×C4C2×#6711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294C22×C4C2×#6711yescharacteristic
295C22×C4C2×#6711yescharacteristic
296C22×C4C2×#6714yes—
297C22×C4C23×D311yescharacteristic
298C2×C8—64no—
299C2×C8—64no—
300C2×C8—64no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—28no—
304C4⋊C4—24no—
305C4⋊C4—28no—
306C4⋊C4—22no—
307C4⋊C4—34no—
308C4⋊C4—64no—
309C4⋊C4—62no—
310C4⋊C4—64no—
311C4⋊C4—124no—
312C4⋊C4—124no—
313C4⋊C4—122no—
314C4⋊C4C2×#6714yes—
315C22⋊C4—28no—
316C42—28no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3#44814yes—
328C22×C3C2×#12011yescharacteristic
329C22×C3#52511yescharacteristic
330C22×C3#53611yescharacteristic
331C22×C3C22×SD1611yescharacteristic
332C22×C3C2×#13711yescharacteristic
333C22×C3C2×#13711yescharacteristic
334C22×C3#57511yescharacteristic
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—22no—
338C3×C4—22no—
339C3×C4—44no—
340C3×C4—41no—
341C3×C4—44no—
342C3×C4#54714yes—
343Dic3—44no—
344Dic3—82no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23C2×#21512yes—
350C23#94212yes—
351C23#95312yes—
352C23C22×#6711yescharacteristic
353Q8—28no—
354Q8—48no—
355Q8—44no—
356C2×C4—28no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—28no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—64no—
377C2×C4—128no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—244no—
384C2×C4C2×#21512yes—
385C2×C4C2×#21512yes—
386C2×C4#95112yes—
387C2×C4#95312yes—
388C2×C4D3×#4112yes—
389C2×C4#101012yes—
390C2×C4C22×#6712yes—
391C8—124no—
392C2×C3—24no—
393C2×C3#240611yescharacteristic
394C2×C3C2×#44811yescharacteristic
395C2×C3#300711yescharacteristic
396C2×C3#310111yescharacteristic
397C2×C3#311311yescharacteristic
398C2×C3#322911yescharacteristic
399C2×C3#323611yescharacteristic
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22#522714yes—
407C22C22×#21511yescharacteristic
408C22C2×#94211yescharacteristic
409C22#582211yescharacteristic
410C22C2×#95111yescharacteristic
411C22#582511yescharacteristic
412C22#582811yescharacteristic
413C22C2×#95311yescharacteristic
414C4—22no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—44no—
420C4—44no—
421C4—124no—
422C4—242no—
423C4#582414yes—
424C3#2406311yescharacteristic, complemented, Sylow
425C2—24no—
426C2#8332311yescharacteristic
427C2#8332611yescharacteristic
428C2#8335111yescharacteristic
429C2#8336911yescharacteristic
430C2C2×#522711yescharacteristic
431C2#8408411yescharacteristic
432C2#8409011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#533D3
hypocenterC3#24063
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
4. #71783C2semidirect
5. #71781C2semidirect
7. #71629C2semidirect
424. C3#24063semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#50432
C2×#49282
#717832
#717812
#716922
#716292
#715312
#240633