Finite Group Databasebeta

← #10848880

Subgroups — #10848880

Order 768.

1240 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic, complemented
2#83287C211yesmaximal, characteristic, complemented
3#83251C211yesmaximal, characteristic, complemented
4#82526C211yesmaximal, characteristic, complemented
5#71879C214yesmaximal, complemented
6#71815C211yesmaximal, characteristic, complemented
7#71790C211yesmaximal, characteristic, complemented
8#71667C214yesmaximal, complemented
9C4×#837C211yesmaximal, characteristic, complemented
10#25616—31nomaximal, Sylow
11C3×#556—21no—
12C3×#555—21no—
13C3×#554—21no—
14C3×#553—22no—
15C3×#548—22no—
16C3×#547—21no—
17C3×#542—22no—
18C3×#537C2212yes—
19C3×#533—21no—
20C3×#532—21no—
21C3×#532C2211yescharacteristic
22C2×C3×C4⋊Q8C2211yescharacteristic
23C3×#520C2212yes—
24C3×C4×#41C2211yescharacteristic
25C2×#949C2214yes—
26C2×#949C2212yes—
27C2×#947—22no—
28C2×#874C2211yescharacteristic
29C2×#837C2212yes—
30C2×#837C2214yes—
31#5085—21no—
32#5073—21no—
33#5058—22no—
34#5027—21no—
35#5027—21no—
36#5015—21no—
37#5015—21no—
38#5001C2212yes—
39#4988—22no—
40#4988C2214yes—
41C42×Dic3C2211yescharacteristic
42#4924C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4532C2212yes—
45#4530C2214yes—
46#3511D311yescharacteristic, complemented
47#3151—31no—
48#3038—31no—
49#2858—31no—
50#2006—34no—
51#1997—31no—
52#1962—31no—
53#1883—34no—
54C4×#104—31no—
55C2×C3×#41—22no—
56C2×C3×#41C2311yescharacteristic
57C2×C3×#41C2312yes—
58C22×C3×Q8D412yes—
59C22×D12—21no—
60C22×D12—21no—
61C22×D12C2312yes—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8D411yescharacteristic
67C3×#127D411yescharacteristic
68C3×#126—24no—
69C3×#125—22no—
70C3×#124—22no—
71C3×#124—24no—
72C3×#124D412yes—
73C3×#123—24no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—24no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#121—21no—
82C3×#121—21no—
83C3×#121—24no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8D411yescharacteristic
87C3×C4×Q8D411yescharacteristic
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C3×C4×D4—24no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×C4×D4D411yescharacteristic
96C3×C4×D4D411yescharacteristic
97C3×#117—22no—
98C3×#117—22no—
99C3×#117—21no—
100C3×#117C2311yescharacteristic
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4D412yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C42—21no—
105C2×C3×C42C2311yescharacteristic
106#949—28no—
107#949—44no—
108#949—44no—
109#947—48no—
110C2×#215—22no—
111C2×#215—24no—
112C2×#213—22no—
113C2×#213—24no—
114C2×(C2×D3)⋊C4—28no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2312yes—
118C2×#208C2312yes—
119C2×#208C2312yes—
120#874—21no—
121#874—21no—
122#874—21no—
123#874—21no—
124#837—24no—
125#837—44no—
126#809—22no—
127#809—22no—
128C4⋊(C3⋊C8)—21no—
129C4⋊(C3⋊C8)—21no—
130#803—21no—
131C4×C3⋊C8—21no—
132C2×AGL(1,ℤ/8)—34no—
133C2×AGL(1,ℤ/8)—32no—
134#556—21no—
135#555—21no—
136#554—21no—
137#553—22no—
138#548—22no—
139#547—21no—
140#542—22no—
141#537C2×D312yes—
142#533—21no—
143#532—21no—
144#532C2×D311yescharacteristic
145C2×C4⋊Q8C2×D311yescharacteristic
146C2×#127—31no—
147C2×#124—62no—
148#520C2×D312yes—
149C4×#41C2×D311yescharacteristic
150#496—62no—
151#493—61no—
152#492—61no—
153#460—61no—
154#460—61no—
155#459—61no—
156#459—61no—
157#423—31no—
158C2×#104—32no—
159C2×#104—34no—
160#418—32no—
161C4×M16—31no—
162#390—34no—
163#390—62no—
164C43—31no—
165#343—34no—
166#339—32no—
167C23×D3—22no—
168C3×#41—22no—
169C3×#41—24no—
170C3×#41—24no—
171C3×#41—22no—
172C3×#41—84no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—44no—
176C2×C3×Q8C2×D412yes—
177C2×C3×Q8C2×D411yescharacteristic
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4—24no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4C2×D412yes—
184C2×C3×D4C2×D411yescharacteristic
185C22×C3×C4—24no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D414yes—
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—22no—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—24no—
198C2×D12—24no—
199C2×D12—44no—
200C2×D12—44no—
201C2×D12—41no—
202C2×D12—41no—
203C2×D12—41no—
204C2×D12—41no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4C2×D411yescharacteristic
216C3×C4⋊C4C2×D411yescharacteristic
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—21no—
221C3×C22⋊C4—21no—
222C3×C22⋊C4—21no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C42—24no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D411yescharacteristic
231C3×C42C2×D411yescharacteristic
232C3×C42C2×D411yescharacteristic
233#215—48no—
234#215—44no—
235#213—48no—
236#213—44no—
237(C2×D3)⋊C4—416no—
238C4×Dic3—24no—
239C4×Dic3—22no—
240C4×Dic3—22no—
241C4×Dic3—42no—
242#208—22no—
243#208—24no—
244#208#4114yescomplemented
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×#41—22no—
248C2×#41C22×D311yescharacteristic
249C2×#41C22×D312yes—
250C22×Q8#6712yes—
251C22×D4—32no—
252C22×D4—61no—
253C22×D4—61no—
254AGL(1,ℤ/8)—68no—
255AGL(1,ℤ/8)—124no—
256AGL(1,ℤ/8)—124no—
257C2×SD16—62no—
258C2×SD16—64no—
259C2×D8—62no—
260C2×D8—64no—
261C2×M16—32no—
262C2×M16—32no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8#6711yescharacteristic
268#127—61no—
269#127—61no—
270#127—61no—
271#127—61no—
272#127#6711yescharacteristic
273#126—24no—
274#125—22no—
275#124—22no—
276#124—24no—
277#124—128no—
278#124#6712yes—
279#123—24no—
280C22⋊Q8—22no—
281C22⋊Q8—24no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284#121—22no—
285#121—22no—
286#121—22no—
287#121—21no—
288#121—21no—
289#121—24no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8#6711yescharacteristic
293C4×Q8#6711yescharacteristic
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—24no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4#6711yescharacteristic
302C4×D4#6711yescharacteristic
303#117—22no—
304#117—22no—
305#117—21no—
306#117C22×D311yescharacteristic
307C2×C4⋊C4—22no—
308C2×C4⋊C4#6712yes—
309C2×C4⋊C4C22×D312yes—
310C2×C22⋊C4—68no—
311C2×C42—21no—
312C2×C42—32no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42C22×D311yescharacteristic
316C4⋊C8—61no—
317C4⋊C8—61no—
318#104—64no—
319#104—124no—
320#102—62no—
321#102—62no—
322#97—61no—
323C4×C8—61no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3C22×D411yescharacteristic
327C22×D3—44no—
328C22×D3—44no—
329C22×D3—42no—
330C22×D3—44no—
331C3×Q8—21no—
332C3×Q8—21no—
333C3×Q8—24no—
334C3×Q8—44no—
335C3×Q8—82no—
336C3×D4—21no—
337C3×D4—21no—
338C3×D4—24no—
339C3×D4—42no—
340C3×D4—44no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—84no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—24no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—44no—
357C2×C3×C4—84no—
358C2×C3×C4#12011yescharacteristic
359C2×C3×C4#12011yescharacteristic
360C2×C3×C4#12011yescharacteristic
361C2×C3×C4#12011yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C2×#4112yes—
365C2×C3×C4H(5,𝔽2)12yes—
366C2×Dic3—24no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369D12—42no—
370D12—42no—
371D12—44no—
372D12—82no—
373D12—82no—
374D12—82no—
375D12—82no—
376C3⋊C8—24no—
377C3⋊C8—42no—
378C24—62no—
379#41—22no—
380#41—24no—
381#41—24no—
382#41—22no—
383#41—84no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—44no—
387C2×Q8C2×#6712yes—
388C2×Q8C2×#6711yescharacteristic
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—64no—
395C2×D4—64no—
396C2×D4—62no—
397C2×D4—62no—
398C2×D4—124no—
399C2×D4—124no—
400C2×D4—121no—
401C2×D4—121no—
402C2×D4—121no—
403C2×D4—121no—
404C2×D4C2×#6712yes—
405C2×D4C2×#6711yescharacteristic
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—62no—
412C22×C4C2×#6711yescharacteristic
413C22×C4C2×#6711yescharacteristic
414C22×C4C2×#6714yes—
415C22×C4C23×D311yescharacteristic
416SD16—128no—
417SD16—124no—
418D8—128no—
419D8—124no—
420M16—34no—
421M16—62no—
422M16—64no—
423C2×C8—62no—
424C2×C8—62no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4C2×#6711yescharacteristic
436C4⋊C4C2×#6711yescharacteristic
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—1216no—
446C42—24no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—64no—
451C42—62no—
452C42—62no—
453C42—122no—
454C42C2×#6711yescharacteristic
455C42C2×#6711yescharacteristic
456C42C2×#6711yescharacteristic
457C42C2×#6711yescharacteristic
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—42no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—41no—
465C22×C3—42no—
466C22×C3C2×#12011yescharacteristic
467C22×C3#55012yes—
468C2×D3—88no—
469C2×D3—82no—
470C2×D3—82no—
471C2×D3—82no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—42no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—44no—
478C3×C4—41no—
479C3×C4#45312yes—
480C3×C4#45412yes—
481C3×C4#53211yescharacteristic
482C3×C4#53511yescharacteristic
483C3×C4#54612yes—
484Dic3—44no—
485C23—21no—
486C23—22no—
487C23—124no—
488C23—124no—
489C23—124no—
490C23—122no—
491C23C22×#6711yescharacteristic
492Q8—24no—
493Q8—21no—
494Q8—21no—
495Q8—44no—
496Q8—82no—
497D4—24no—
498D4—21no—
499D4—21no—
500D4—42no—
501D4—44no—
502D4—42no—
503D4—42no—
504D4—84no—
505D4—124no—
506D4—122no—
507D4—122no—
508D4—242no—
509D4—242no—
510D4—242no—
511D4—242no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—64no—
526C2×C4—84no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4#95311yescharacteristic
530C2×C4#95311yescharacteristic
531C2×C4#95311yescharacteristic
532C2×C4#95311yescharacteristic
533C2×C4D3×#4112yes—
534C2×C4#100912yes—
535C2×C4C22×#6711yescharacteristic
536C2×C4C22×#6711yescharacteristic
537C8—64no—
538C8—122no—
539C2×C3—22no—
540C2×C3—81no—
541C2×C3—82no—
542C2×C3#243111yescharacteristic
543C2×C3C2×#45311yescharacteristic
544C2×C3C2×#45411yescharacteristic
545D3—162no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—42no—
550C22—41no—
551C22—42no—
552C22—41no—
553C22—42no—
554C22—248no—
555C22—242no—
556C22—242no—
557C22—242no—
558C22#583412yes—
559C22C2×#95311yescharacteristic
560C4—21no—
561C4—21no—
562C4—41no—
563C4—42no—
564C4—41no—
565C4—44no—
566C4—41no—
567C4—124no—
568C4#524112yes—
569C4#524512yes—
570C4#582912yes—
571C4#583011yescharacteristic
572C4#583211yescharacteristic
573C3#2561611yescharacteristic, complemented, Sylow
574C2—22no—
575C2—81no—
576C2—82no—
577C2—482no—
578C2#8409211yescharacteristic
579C2C2×#524111yescharacteristic
580C2C2×#524511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3511C2
socleC22×C3C2×#120
hypercenter#3511D3
hypocenterC3#25616
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. C3×#3511C2semidirect
2. #83287C2semidirect
3. #83251C2semidirect
4. #82526C2semidirect
5. #71879C2semidirect
6. #71815C2semidirect
7. #71790C2semidirect
8. #71667C2semidirect
9. C4×#837C2semidirect
46. #3511D3semidirect
244. #208#41semidirect
573. C3#25616semidirect

Maximal subgroups

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

subgroupindex
C3×#35112
#832872
#832512
#825262
#718792
#718152
#717902
#716672
C4×#8372
#256163