Finite Group Databasebeta

← #10770301

Subgroups — #10770301

Order 768.

996 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3559C211yesmaximal, characteristic, complemented
2#83302C211yesmaximal, characteristic, complemented
3#83146C211yesmaximal, characteristic, complemented
4#83067C211yesmaximal, characteristic, complemented
5#71888C214yesmaximal, complemented
6#71878C214yesmaximal, complemented
7#71812C211yesmaximal, characteristic, complemented
8#71754C211yesmaximal, characteristic, complemented
9#71737C211yesmaximal, characteristic, complemented
10#18430—31nomaximal, Sylow
11C3×#563—21no—
12C3×#562—21no—
13C3×#554—21no—
14C3×#553—21no—
15C3×#553—21no—
16C3×#547—22no—
17C3×#547—21no—
18C3×#544—21no—
19C3×#540—21no—
20C3×#539—22no—
21C3×#539C2212yes—
22C3×#537C2212yes—
23C3×#532C2211yescharacteristic
24C3×#529C2211yescharacteristic
25C2×C3×#125C2211yescharacteristic
26C2×#874C2211yescharacteristic
27#5088—21no—
28#5073—21no—
29#5067—21no—
30#5058—21no—
31#5046C2212yes—
32#5036C2214yes—
33#5036C2212yes—
34#5027—21no—
35#5015—22no—
36#5015—21no—
37#4991—22no—
38#4991C2214yes—
39#4988—21no—
40#4988—21no—
41#4932C2211yescharacteristic
42#4926C2211yescharacteristic
43#4925C2211yescharacteristic
44#4538C2212yes—
45#4537C2212yes—
46#4537C2214yes—
47#4528C2214yes—
48#3559D311yescharacteristic, complemented
49#3217—31no—
50#3205—31no—
51#2641—31no—
52#1598—34no—
53#1597—34no—
54#1505—31no—
55#1500—31no—
56#1470—31no—
57C2×C3×#41—21no—
58C2×C3×#41—22no—
59C22×D12—21no—
60C22×D12—21no—
61C22×D12C2312yes—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8D411yescharacteristic
65C3×#127D411yescharacteristic
66C3×#126—22no—
67C3×#126—22no—
68C3×#126—21no—
69C3×#125—21no—
70C3×#125—22no—
71C3×#125—21no—
72C3×#125—21no—
73C3×#125—21no—
74C3×#125—21no—
75C3×#125D411yescharacteristic
76C3×#124—21no—
77C3×#124D411yescharacteristic
78C3×#123—22no—
79C3×#123—22no—
80C3×#123—22no—
81C3×#123—22no—
82C3×#123—22no—
83C3×#123—21no—
84C3×#123—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8C2311yescharacteristic
92C3×#121—22no—
93C3×#121—21no—
94C3×#121—21no—
95C3×#121—22no—
96C3×#121—22no—
97C3×#121—21no—
98C3×#121—22no—
99C3×#121C2311yescharacteristic
100C3×#121C2312yes—
101C3×C4×Q8D411yescharacteristic
102C3×C4×D4—22no—
103C3×C4×D4—21no—
104C3×C4×D4—21no—
105C3×C4×D4—21no—
106C3×C4×D4—22no—
107C3×C4×D4—21no—
108C3×C4×D4D411yescharacteristic
109C3×C4×D4D412yes—
110C3×#117—21no—
111C3×#117—21no—
112C2×C3×C4⋊C4—22no—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4D411yescharacteristic
115C2×C3×C4⋊C4D411yescharacteristic
116C2×C3×C4⋊C4D412yes—
117C2×C3×C4⋊C4C2312yes—
118C2×C3×C42C2311yescharacteristic
119C2×#215—22no—
120C2×#213—22no—
121C2×#213—24no—
122C2×(C2×D3)⋊C4—24no—
123C2×(C2×D3)⋊C4—24no—
124C2×C4×Dic3—21no—
125#874—21no—
126#874—21no—
127#874—21no—
128#874—21no—
129C3⋊#95—21no—
130C3⋊#95C2312yes—
131#830C2312yes—
132#830C2312yes—
133#809—24no—
134#809—22no—
135#809—22no—
136C4⋊(C3⋊C8)—21no—
137C4⋊(C3⋊C8)—21no—
138#803—21no—
139C4×C3⋊C8—21no—
140#563—21no—
141#562—21no—
142#554—21no—
143#553—21no—
144#553—21no—
145#547—22no—
146#547—21no—
147#544—21no—
148#540—21no—
149#539—22no—
150#539C2×D312yes—
151#537C2×D312yes—
152#532C2×D311yescharacteristic
153C2×#127—31no—
154#529C2×D311yescharacteristic
155C2×#125C2×D311yescharacteristic
156#496—61no—
157#493—61no—
158#490—61no—
159#486—61no—
160#460—61no—
161#459—62no—
162#459—61no—
163#450—32no—
164#447—34no—
165#447—32no—
166#432—31no—
167#431—31no—
168#394—34no—
169#394—62no—
170#390—61no—
171#390—61no—
172#376—31no—
173#328—32no—
174#327—34no—
175#327—32no—
176#324—34no—
177C23×D3—22no—
178C3×#41—82no—
179C3×#41—84no—
180C2×C3×Q8—22no—
181C2×C3×Q8—21no—
182C2×C3×Q8C2×D411yescharacteristic
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4C2×D411yescharacteristic
190C2×C3×D4C2×D412yes—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—44no—
205C2×D12—41no—
206C2×D12—41no—
207C2×D12—41no—
208C2×D12—41no—
209C2×D12—82no—
210C2×D12—82no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4C2×D412yes—
229C3×C4⋊C4C2×D411yescharacteristic
230C3×C4⋊C4C2×D411yescharacteristic
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—21no—
237C3×C22⋊C4—21no—
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—22no—
243C3×C42—22no—
244C3×C42—21no—
245C3×C42C2×D411yescharacteristic
246C3×C42C2×D411yescharacteristic
247C3×C42C2×D411yescharacteristic
248C3×C42C2×D411yescharacteristic
249#215—82no—
250#213—84no—
251#213—82no—
252(C2×D3)⋊C4—84no—
253(C2×D3)⋊C4—84no—
254C4×Dic3—42no—
255C2×C3⋊C8—22no—
256C2×C3⋊C8—22no—
257C2×#41—21no—
258C2×#41—22no—
259C22×D4—32no—
260C22×D4—61no—
261C22×D4—61no—
262C2×SD16—62no—
263C2×D8—64no—
264C2×D8—62no—
265C4⋊Q8—21no—
266C4⋊Q8—22no—
267C4⋊Q8#6711yescharacteristic
268#127—61no—
269#127—61no—
270#127—61no—
271#127—61no—
272#127#6711yescharacteristic
273#126—22no—
274#126—22no—
275#126—21no—
276#125—21no—
277#125—22no—
278#125—21no—
279#125—21no—
280#125—21no—
281#125—21no—
282#125#6711yescharacteristic
283#124—21no—
284#124#6711yescharacteristic
285#123—22no—
286#123—22no—
287#123—22no—
288#123—22no—
289#123—22no—
290#123—21no—
291#123—22no—
292C22⋊Q8—22no—
293C22⋊Q8—22no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8C22×D311yescharacteristic
299#121—21no—
300#121—22no—
301#121—21no—
302#121—22no—
303#121—22no—
304#121—21no—
305#121—22no—
306#121C22×D311yescharacteristic
307#121C22×D312yes—
308C4×Q8#6711yescharacteristic
309C4×D4—22no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—22no—
314C4×D4—21no—
315C4×D4#6711yescharacteristic
316C4×D4#6712yes—
317#117—21no—
318#117—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4#6711yescharacteristic
322C2×C4⋊C4#6712yes—
323C2×C4⋊C4#6711yescharacteristic
324C2×C4⋊C4C22×D312yes—
325C2×C22⋊C4—64no—
326C2×C22⋊C4—64no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329C4⋊C8—61no—
330C4⋊C8—61no—
331#102—64no—
332#102—62no—
333#102—62no—
334C22⋊C8—32no—
335C22⋊C8—32no—
336#97—61no—
337C4×C8—61no—
338#95—32no—
339#95—61no—
340C23×C3—21no—
341C23×C3—22no—
342C23×C3C22×D411yescharacteristic
343C22×D3—44no—
344C22×D3—44no—
345C22×D3—42no—
346C22×D3—44no—
347C3×Q8—41no—
348C3×Q8—81no—
349C3×Q8—82no—
350C3×D4—42no—
351C3×D4—41no—
352C3×D4—44no—
353C3×D4—42no—
354C3×D4—42no—
355C3×D4—81no—
356C3×D4—82no—
357C3×D4—82no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—41no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—82no—
371C2×C3×C4—82no—
372C2×C3×C4—82no—
373C2×C3×C4—82no—
374C2×C3×C4#12011yescharacteristic
375C2×C3×C4#12011yescharacteristic
376C2×C3×C4#12011yescharacteristic
377C2×C3×C4#12011yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383D12—44no—
384D12—42no—
385D12—42no—
386D12—82no—
387D12—82no—
388D12—82no—
389D12—82no—
390C3⋊C8—42no—
391C3⋊C8—42no—
392C24—62no—
393#41—82no—
394#41—84no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8C2×#6711yescharacteristic
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—124no—
407C2×D4—121no—
408C2×D4—121no—
409C2×D4—121no—
410C2×D4—121no—
411C2×D4—242no—
412C2×D4—242no—
413C2×D4C2×#6711yescharacteristic
414C2×D4C2×#6712yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—62no—
421C22×C4C2×#6712yes—
422C22×C4C2×#6711yescharacteristic
423C22×C4C2×#6711yescharacteristic
424C22×C4C2×#6712yes—
425C22×C4C23×D311yescharacteristic
426SD16—242no—
427D8—244no—
428D8—242no—
429C2×C8—62no—
430C2×C8—62no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4C2×#6712yes—
449C4⋊C4C2×#6711yescharacteristic
450C4⋊C4C2×#6711yescharacteristic
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—22no—
461C22⋊C4—21no—
462C22⋊C4—22no—
463C22⋊C4—244no—
464C22⋊C4—244no—
465C42—22no—
466C42—21no—
467C42—122no—
468C42C2×#6711yescharacteristic
469C42C2×#6711yescharacteristic
470C42C2×#6711yescharacteristic
471C42C2×#6711yescharacteristic
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—42no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—41no—
480C22×C3C2×#12011yescharacteristic
481C22×C3#57612yes—
482C2×D3—88no—
483C2×D3—82no—
484C2×D3—82no—
485C2×D3—82no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—42no—
496C3×C4—41no—
497C3×C4#45412yes—
498Dic3—82no—
499C23—21no—
500C23—22no—
501C23—124no—
502C23—124no—
503C23—124no—
504C23—122no—
505C23C22×#6711yescharacteristic
506Q8—41no—
507Q8—81no—
508Q8—82no—
509D4—42no—
510D4—41no—
511D4—44no—
512D4—42no—
513D4—42no—
514D4—81no—
515D4—82no—
516D4—82no—
517D4—124no—
518D4—122no—
519D4—122no—
520D4—242no—
521D4—242no—
522D4—242no—
523D4—242no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4#95311yescharacteristic
544C2×C4#95311yescharacteristic
545C2×C4#95311yescharacteristic
546C2×C4#95311yescharacteristic
547C2×C4C22×#6711yescharacteristic
548C2×C4C22×#6711yescharacteristic
549C8—122no—
550C8—122no—
551C2×C3—22no—
552C2×C3—81no—
553C2×C3—82no—
554C2×C3#300811yescharacteristic
555C2×C3C2×#45411yescharacteristic
556C2×C3#302711yescharacteristic
557D3—162no—
558C22—21no—
559C22—21no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22—42no—
565C22—41no—
566C22—248no—
567C22—242no—
568C22—242no—
569C22—242no—
570C22#583512yes—
571C22C2×#95311yescharacteristic
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—42no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—242no—
584C4#524512yes—
585C3#1843011yescharacteristic, complemented, Sylow
586C2—22no—
587C2—81no—
588C2—82no—
589C2—482no—
590C2#8410111yescharacteristic
591C2C2×#524511yescharacteristic
592C2#8412611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3559C2
socleC22×C3C2×#120
hypercenter#3559D3
hypocenterC3#18430
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3559

Complements

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

NH ≅ G/Ntype
1. C3×#3559C2semidirect
2. #83302C2semidirect
3. #83146C2semidirect
4. #83067C2semidirect
5. #71888C2semidirect
6. #71878C2semidirect
7. #71812C2semidirect
8. #71754C2semidirect
9. #71737C2semidirect
48. #3559D3semidirect
585. C3#18430semidirect

Maximal subgroups

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

subgroupindex
C3×#35592
#833022
#831462
#830672
#718882
#718782
#718122
#717542
#717372
#184303