Finite Group Databasebeta

← #10827244

Subgroups — #10827244

Order 768.

1326 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3493C211yesmaximal, characteristic, complemented
2#82927C212yesmaximal, complemented
3#82269C211yesmaximal, characteristic, complemented
4#71903C212yesmaximal, complemented
5#71879C212yesmaximal, complemented
6#71712C212yesmaximal, complemented
7#71667C214yesmaximal, complemented
8#71589C211yesmaximal, characteristic, complemented
9#23952—31nomaximal, Sylow
10C23×D12C2211yescharacteristic
11C3×#556—22no—
12C3×#555—22no—
13C3×#553—22no—
14C3×#552—22no—
15C3×#544C2211yescharacteristic, complemented
16C3×#538—22no—
17C3×#537C2211yescharacteristic, complemented
18C3×#529—22no—
19C3×#525C2212yes—
20C3×#520C2212yescomplemented
21C2×C3×#117C2211yescharacteristic
22C2×#949C2214yes—
23C2×#926C2211yescharacteristic
24C2×#837C2214yescomplemented
25#5207C2212yes—
26#5046—22no—
27#5046—22no—
28#5036—22no—
29#5036—22no—
30#5001C2214yes—
31#4988C2212yes—
32#4988C2212yes—
33#4955—22no—
34#4939—22no—
35#4532C2214yescomplemented
36#4530C2212yescomplemented
37#4530C2212yescomplemented
38#3493D311yescharacteristic, complemented
39#3003—32no—
40#2309—31no—
41#2027—32no—
42#2006—32no—
43#1897—32no—
44#1883—34no—
45#1822—31no—
46C24×D3—21no—
47C2×C3×#41C2312yes—
48C23×C3×C4C2311yescharacteristic
49C22×D12—21no—
50C22×D12—24no—
51C22×D12—22no—
52C22×D12—22no—
53C22×D12—42no—
54C22×D12C2312yes—
55C3×C4⋊Q8—22no—
56C3×#126—24no—
57C3×#126—24no—
58C3×#125—22no—
59C3×#125—22no—
60C3×#124—22no—
61C3×#124—24no—
62C3×#123—24no—
63C3×#123—22no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8D412yes—
67C3×C22⋊Q8D412yes—
68C3×#121—24no—
69C3×#121—22no—
70C3×#121—22no—
71C3×#121D412yes—
72C3×#121D412yes—
73C3×#120—22no—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C3×C4×D4—24no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C3×#117—24no—
82C3×#117—22no—
83C3×#117—22no—
84C3×#117—22no—
85C3×#117C2312yes—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4D412yes—
90C2×C3×C42D412yes—
91C2×#217—21no—
92#949—28no—
93#949—44no—
94C2×#215—24no—
95C2×#213—24no—
96C2×(C2×D3)⋊C4—24no—
97C2×(C2×D3)⋊C4—24no—
98#926—42no—
99C2×C4⋊Dic3—21no—
100C2×C4×Dic3C2312yes—
101C2×#208C2314yes—
102#837—44no—
103#830—22no—
104#830—22no—
105#809—24no—
106#809—24no—
107C23×D4—31no—
108C2×AGL(1,ℤ/8)—34no—
109#556—22no—
110#555—22no—
111#553—22no—
112#552—22no—
113#544C2×D311yescharacteristic, complemented
114#538—22no—
115#537C2×D311yescharacteristic, complemented
116#529—22no—
117#525C2×D312yes—
118#520C2×D312yescomplemented
119C2×#117—31no—
120C2×#117C2×D311yescharacteristic
121#450—62no—
122#450—62no—
123#447—62no—
124#447—62no—
125C2×#104—34no—
126#418—34no—
127#407—32no—
128#390—32no—
129#390—32no—
130#390—62no—
131#379—62no—
132#343—32no—
133#343—32no—
134#339—34no—
135C24×C3C2×D411yescharacteristic
136C23×D3—22no—
137C23×D3—24no—
138C23×D3—42no—
139C23×D3—42no—
140C23×D3—82no—
141C23×D3—82no—
142C3×#41—24no—
143C3×#41—24no—
144C2×C3×Q8—22no—
145C2×C3×Q8C2×D412yes—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4C2×D412yes—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
154C22×C3×C4—22no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2411yescharacteristic
161C22×Dic3—22no—
162C2×D12—24no—
163C2×D12—24no—
164C2×D12—24no—
165C2×D12—22no—
166C2×D12—48no—
167C2×D12—48no—
168C2×D12—42no—
169C2×D12—41no—
170C2×D12—41no—
171C2×D12#4114yes—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4C2×D412yes—
181C3×C4⋊C4C2×D412yes—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—22no—
188C3×C22⋊C4—22no—
189C3×C22⋊C4—22no—
190C3×C42—24no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—22no—
194#217—42no—
195#215—48no—
196#213—48no—
197(C2×D3)⋊C4—84no—
198(C2×D3)⋊C4—84no—
199C4⋊Dic3—42no—
200C4×Dic3—24no—
201#208—24no—
202#208—24no—
203C2×C3⋊C8—24no—
204C25—61no—
205C2×#41C22×D312yes—
206C22×D4—32no—
207C22×D4—64no—
208C22×D4—61no—
209C22×D4—62no—
210C22×D4—62no—
211C22×D4—122no—
212C23×C4C22×D311yescharacteristic
213AGL(1,ℤ/8)—68no—
214AGL(1,ℤ/8)—124no—
215C2×SD16—64no—
216C2×D8—64no—
217C2×M16—34no—
218C4⋊Q8—22no—
219#126—24no—
220#126—24no—
221#125—22no—
222#125—22no—
223#124—22no—
224#124—24no—
225#123—24no—
226#123—22no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229C22⋊Q8#6712yes—
230C22⋊Q8#6712yes—
231#121—24no—
232#121—22no—
233#121—22no—
234#121#6712yes—
235#121#6712yes—
236#120—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—24no—
242C4×D4—22no—
243C4×D4—22no—
244#117—24no—
245#117—22no—
246#117—22no—
247#117—22no—
248#117—122no—
249#117C22×D312yes—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4C22×D311yescharacteristic
253C2×C4⋊C4C22×D311yescharacteristic
254C2×C22⋊C4—61no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4#6712yes—
258C2×C42—32no—
259C2×C42#6712yes—
260#104—124no—
261#102—64no—
262#102—64no—
263C22⋊C8—62no—
264C22⋊C8—62no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—22no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3#12012yes—
271C23×C3C22×D411yescharacteristic
272C22×D3—24no—
273C22×D3—44no—
274C22×D3—44no—
275C22×D3—48no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—88no—
279C22×D3—84no—
280C22×D3—84no—
281C22×D3—88no—
282C3×Q8—24no—
283C3×Q8—82no—
284C3×D4—24no—
285C3×D4—42no—
286C3×D4—42no—
287C3×D4—82no—
288C3×D4—82no—
289C3×D4—82no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—44no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—84no—
303C2×C3×C4#12011yescharacteristic
304C2×C3×C4#12011yescharacteristic
305C2×C3×C4C22×D412yes—
306C2×C3×C4C2×#4112yes—
307C2×C3×C4H(5,𝔽2)12yes—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311D12—28no—
312D12—44no—
313D12—44no—
314D12—44no—
315C3⋊C8—44no—
316C24—64no—
317C24—62no—
318C24—122no—
319C24—122no—
320C24—242no—
321C24—242no—
322C24C2×#6711yescharacteristic
323#41—24no—
324#41—24no—
325C2×Q8—22no—
326C2×Q8C2×#6712yes—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—34no—
332C2×D4—64no—
333C2×D4—62no—
334C2×D4—64no—
335C2×D4—64no—
336C2×D4—128no—
337C2×D4—128no—
338C2×D4—122no—
339C2×D4—121no—
340C2×D4—121no—
341C2×D4C2×#6712yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—62no—
347C22×C4C2×#6711yescharacteristic
348C22×C4C2×#6712yes—
349C22×C4C2×#6712yes—
350C22×C4C2×#6712yes—
351C22×C4C2×#6712yes—
352C22×C4C23×D311yescharacteristic
353SD16—128no—
354D8—128no—
355M16—64no—
356M16—64no—
357C2×C8—64no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—122no—
367C4⋊C4C2×#6712yes—
368C4⋊C4C2×#6712yes—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—122no—
378C22⋊C4—244no—
379C22⋊C4—244no—
380C42—24no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—64no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3C2×#12011yescharacteristic
396C22×C3#52511yescharacteristic
397C22×C3#53511yescharacteristic
398C2×D3—48no—
399C2×D3—44no—
400C2×D3—84no—
401C2×D3—816no—
402C2×D3—82no—
403C2×D3—82no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4#54614yes—
412Dic3—82no—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—41no—
417C23—41no—
418C23—64no—
419C23—124no—
420C23—124no—
421C23—128no—
422C23—122no—
423C23—122no—
424C23—244no—
425C23—244no—
426C23—248no—
427C23—248no—
428C23#95312yes—
429C23C22×#6711yescharacteristic
430Q8—24no—
431Q8—82no—
432D4—24no—
433D4—42no—
434D4—42no—
435D4—68no—
436D4—82no—
437D4—82no—
438D4—82no—
439D4—124no—
440D4—124no—
441D4—124no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—84no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4#95311yescharacteristic
459C2×C4#95311yescharacteristic
460C2×C4D3×#4112yes—
461C2×C4#100912yes—
462C2×C4C22×#6712yes—
463C8—124no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—42no—
467C2×C3—82no—
468C2×C3#240411yescharacteristic
469C2×C3#302111yescharacteristic
470C2×C3#302111yescharacteristic
471D3—84no—
472C22—21no—
473C22—21no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—128no—
483C22—124no—
484C22—244no—
485C22—2416no—
486C22—242no—
487C22—242no—
488C22#582811yescharacteristic
489C22#583011yescharacteristic
490C22C2×#95311yescharacteristic
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—242no—
499C4#582914yes—
500C3#2395211yescharacteristic, complemented, Sylow
501C2—21no—
502C2—21no—
503C2—42no—
504C2—82no—
505C2—244no—
506C2#8409611yescharacteristic
507C2#8412211yescharacteristic
508C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3493C2
socleC22×C3C2×#120
hypercenter#3493D3
hypocenterC3#23952
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3493

Complements

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

NH ≅ G/Ntype
1. C3×#3493C2semidirect
2. #82927C2semidirect
3. #82269C2semidirect
4. #71903C2semidirect
5. #71879C2semidirect
6. #71712C2semidirect
7. #71667C2semidirect
8. #71589C2semidirect
15. C3×#544C22semidirect
17. C3×#537C22semidirect
20. C3×#520C22semidirect
24. C2×#837C22semidirect
35. #4532C22semidirect
36. #4530C22semidirect
37. #4530C22semidirect
38. #3493D3semidirect
113. #544C2×D3semidirect
115. #537C2×D3semidirect
118. #520C2×D3semidirect
500. C3#23952semidirect

Maximal subgroups

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

subgroupindex
C3×#34932
#829272
#822692
#719032
#718792
#717122
#716672
#715892
#239523