Finite Group Databasebeta

← #10773142

Subgroups — #10773142

Order 768.

878 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3552C211yesmaximal, characteristic
2C2×#5078C211yesmaximal, characteristic, complemented
3#83154C211yesmaximal, characteristic, complemented
4#83124C211yesmaximal, characteristic, complemented
5#71937C212yesmaximal, complemented
6#71933C212yesmaximal, complemented
7#71916C212yesmaximal, complemented
8#71912C212yesmaximal, complemented
9#71776C211yesmaximal, characteristic, complemented
10#71752C211yesmaximal, characteristic, complemented
11#71752C211yesmaximal, characteristic, complemented
12#18699—31nomaximal, Sylow
13C3×#563—22no—
14C3×#557C2211yescharacteristic
15C3×#556C2211yescharacteristic
16C3×#554—21no—
17C3×#553—21no—
18C3×D4×Q8—21no—
19C3×D4×Q8C2211yescharacteristic
20C3×#548—21no—
21C3×#547C2211yescharacteristic
22C3×#544—21no—
23C3×#542—21no—
24C3×#541—21no—
25C3×#539—22no—
26C3×#536—21no—
27C2×C3×C4⋊Q8C2211yescharacteristic
28C3×#529C2211yescharacteristic
29C3×#529C2211yescharacteristic
30C2×#869C2211yescharacteristic
31#5078—22no—
32#5078C2212yes—
33#5078C2212yes—
34#5069—21no—
35#5064—21no—
36#5060—21no—
37#5055—21no—
38#5050—21no—
39#5044—21no—
40#5040—21no—
41#5034—21no—
42#5031C2212yes—
43#5020C2212yes—
44#5019C2212yes—
45#5007C2212yes—
46C2×#807—22no—
47#4926C2211yescharacteristic
48#4926C2211yescharacteristic
49C2×C4×C3⋊C8C2211yescharacteristic
50#4547C2212yes—
51#4547C2212yes—
52#4535C2212yes—
53#4535C2212yes—
54D4⋊(C3⋊C8)C2212yes—
55D4⋊(C3⋊C8)C2212yes—
56#3552D311yescharacteristic
57#3239—31no—
58#3218—31no—
59C2×#500—31no—
60#1650—32no—
61#1649—32no—
62#1648—32no—
63#1647—32no—
64#1556—31no—
65#1494—31no—
66#1494—31no—
67C2×C3×#41—21no—
68C22×C3×Q8—21no—
69C22×C3×Q8—21no—
70C22×Dic6—21no—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8—21no—
76C3×C4⋊Q8D412yes—
77C3×C4⋊Q8C2311yescharacteristic
78C3×C4⋊Q8C2311yescharacteristic
79C3×#126—22no—
80C3×#126—22no—
81C3×#126—22no—
82C3×#126—21no—
83C3×#126—21no—
84C3×#125—22no—
85C3×#125—22no—
86C3×#125—22no—
87C3×#125—21no—
88C3×#125D411yescharacteristic
89C3×#125D411yescharacteristic
90C3×#124—22no—
91C3×#124—21no—
92C3×#124D411yescharacteristic
93C3×#124D411yescharacteristic
94C3×#123—22no—
95C3×#123—22no—
96C3×#123—21no—
97C3×#123—21no—
98C3×#123—22no—
99C3×C22⋊Q8—22no—
100C3×C22⋊Q8—22no—
101C3×C22⋊Q8—22no—
102C3×C22⋊Q8—22no—
103C3×C22⋊Q8—22no—
104C3×C22⋊Q8—21no—
105C3×C22⋊Q8—22no—
106C3×C22⋊Q8—21no—
107C3×C22⋊Q8—21no—
108C3×C22⋊Q8D411yescharacteristic
109C3×C22⋊Q8D411yescharacteristic
110C3×#121—22no—
111C3×#121—21no—
112C3×#121D411yescharacteristic
113C3×#121D411yescharacteristic
114C3×C4×Q8—21no—
115C3×C4×Q8—21no—
116C3×C4×Q8—21no—
117C3×C4×Q8C2311yescharacteristic
118C3×C4×Q8C2311yescharacteristic
119C3×C4×D4—22no—
120C3×C4×D4—22no—
121C3×C4×D4—21no—
122C3×C4×D4—21no—
123C3×C4×D4—21no—
124C3×C4×D4C2311yescharacteristic
125C3×C4×D4C2311yescharacteristic
126C3×#117—21no—
127C3×#117—21no—
128C3×#117—21no—
129C2×C3×C4⋊C4—21no—
130C2×C3×C4⋊C4—21no—
131C2×C3×C4⋊C4D412yes—
132C2×C3×C42C2311yescharacteristic
133C2×#216—22no—
134C2×#214—22no—
135C2×C4⋊Dic3—21no—
136C2×C4⋊Dic3—21no—
137C22×C3⋊C8—21no—
138#869—21no—
139#869—21no—
140#869—21no—
141#869C2312yes—
142#835—22no—
143#832—22no—
144#830—21no—
145#830—21no—
146#810—22no—
147#810—22no—
148#808—22no—
149#807—24no—
150#807—24no—
151#807—22no—
152C4⋊(C3⋊C8)C2312yes—
153C4⋊(C3⋊C8)C2312yes—
154#803—21no—
155#803—21no—
156C4×C3⋊C8—21no—
157C4×C3⋊C8C2312yes—
158#563—22no—
159#557C2×D311yescharacteristic
160#556C2×D311yescharacteristic
161#554—21no—
162#553—21no—
163D4×Q8—21no—
164D4×Q8C2×D311yescharacteristic
165#548—21no—
166#547C2×D311yescharacteristic
167#544—21no—
168#542—21no—
169#541—21no—
170#539—22no—
171#536—21no—
172C2×C4⋊Q8—31no—
173C2×C4⋊Q8C2×D311yescharacteristic
174#529C2×D311yescharacteristic
175#529C2×D311yescharacteristic
176#501—61no—
177#500—32no—
178#500—32no—
179#500—62no—
180#497—61no—
181#491—61no—
182#489—61no—
183#484—61no—
184#481—61no—
185#475—32no—
186#474—32no—
187#462—32no—
188#461—32no—
189#451—61no—
190#448—61no—
191#432—31no—
192#432—31no—
193C2×#107—62no—
194C2×C4×C8—31no—
195#332—32no—
196#332—32no—
197Q8⋊C8—32no—
198Q8⋊C8—32no—
199D4⋊C8—32no—
200D4⋊C8—32no—
201C3×#41—82no—
202C2×C3×Q8—22no—
203C2×C3×Q8—22no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—82no—
207C2×C3×Q8—82no—
208C2×C3×Q8C2×D411yescharacteristic
209C2×C3×Q8C2×D411yescharacteristic
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—22no—
213C2×C3×D4C2×D411yescharacteristic
214C2×C3×D4C2×D411yescharacteristic
215C22×C3×C4—22no—
216C22×C3×C4—22no—
217C22×C3×C4—22no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×Dic3—41no—
224C2×Dic6—22no—
225C2×Dic6—41no—
226C2×Dic6—82no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4C2×D411yescharacteristic
246C3×C4⋊C4C2×D412yes—
247C3×C4⋊C4C2×D412yes—
248C3×C4⋊C4C2×D411yescharacteristic
249C3×C4⋊C4C2×D411yescharacteristic
250C3×C4⋊C4C2×D411yescharacteristic
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—21no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—21no—
258C3×C22⋊C4—21no—
259C3×C22⋊C4—21no—
260C3×C22⋊C4—21no—
261C3×C22⋊C4—21no—
262C3×C22⋊C4—21no—
263C3×C22⋊C4—21no—
264C3×C22⋊C4—21no—
265C3×C42—22no—
266C3×C42—22no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42C2×D411yescharacteristic
270C3×C42C2×D411yescharacteristic
271C3×C42C2×D411yescharacteristic
272C3×C42C2411yescharacteristic
273#216—82no—
274#214—82no—
275C4⋊Dic3—22no—
276C4⋊Dic3—22no—
277C4⋊Dic3—41no—
278C4⋊Dic3—41no—
279C2×C3⋊C8—24no—
280C2×C3⋊C8—22no—
281C2×C3⋊C8—22no—
282C2×C3⋊C8—22no—
283C2×#41—21no—
284C22×Q8—21no—
285C22×Q8—21no—
286C22×Q8—61no—
287C2×Q16—62no—
288C2×SD16—62no—
289C22×C8—61no—
290C4⋊Q8—22no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—32no—
296C4⋊Q8—61no—
297C4⋊Q8—61no—
298C4⋊Q8—61no—
299C4⋊Q8#6712yes—
300C4⋊Q8C22×D311yescharacteristic
301C4⋊Q8C22×D311yescharacteristic
302#126—22no—
303#126—22no—
304#126—22no—
305#126—21no—
306#126—21no—
307#125—22no—
308#125—22no—
309#125—22no—
310#125—21no—
311#125#6711yescharacteristic
312#125#6711yescharacteristic
313#124—22no—
314#124—21no—
315#124#6711yescharacteristic
316#124#6711yescharacteristic
317#123—22no—
318#123—22no—
319#123—21no—
320#123—21no—
321#123—22no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—21no—
328C22⋊Q8—22no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8#6711yescharacteristic
332C22⋊Q8#6711yescharacteristic
333#121—22no—
334#121—21no—
335#121#6711yescharacteristic
336#121#6711yescharacteristic
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—21no—
340C4×Q8C22×D311yescharacteristic
341C4×Q8C22×D311yescharacteristic
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4C22×D311yescharacteristic
348C4×D4C22×D311yescharacteristic
349#117—21no—
350#117—21no—
351#117—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—61no—
355C2×C4⋊C4—61no—
356C2×C4⋊C4#6712yes—
357C2×C42C22×D311yescharacteristic
358#107—64no—
359#107—64no—
360#107—62no—
361#106—62no—
362C4⋊C8—32no—
363C4⋊C8—32no—
364#103—62no—
365#103—62no—
366#103—62no—
367#102—62no—
368C22⋊C8—61no—
369C22⋊C8—61no—
370#97—61no—
371#97—61no—
372C4×C8—32no—
373C4×C8—61no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3#12011yescharacteristic
377C3×Q8—22no—
378C3×Q8—22no—
379C3×Q8—42no—
380C3×Q8—42no—
381C3×Q8—82no—
382C3×Q8—82no—
383C3×D4—22no—
384C3×D4—22no—
385C3×D4—42no—
386C3×D4—42no—
387C3×D4—82no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—42no—
400C2×C3×C4—42no—
401C2×C3×C4—82no—
402C2×C3×C4—82no—
403C2×C3×C4—82no—
404C2×C3×C4#12011yescharacteristic
405C2×C3×C4#12011yescharacteristic
406C2×C3×C4#12011yescharacteristic
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C22×D411yescharacteristic
410C2×Dic3—42no—
411C2×Dic3—82no—
412Dic6—42no—
413Dic6—42no—
414Dic6—82no—
415C3⋊C8—24no—
416C3⋊C8—42no—
417C3⋊C8—42no—
418#41—82no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—62no—
424C2×Q8—82no—
425C2×Q8—82no—
426C2×Q8—121no—
427C2×Q8—242no—
428C2×Q8C2×#6711yescharacteristic
429C2×Q8C2×#6711yescharacteristic
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4C2×#6711yescharacteristic
434C2×D4C2×#6711yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—121no—
441C22×C4C2×#6711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443C22×C4C2×#6711yescharacteristic
444Q16—242no—
445SD16—242no—
446C2×C8—64no—
447C2×C8—62no—
448C2×C8—62no—
449C2×C8—62no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—121no—
471C4⋊C4—121no—
472C4⋊C4C2×#6711yescharacteristic
473C4⋊C4C2×#6711yescharacteristic
474C4⋊C4C2×#6711yescharacteristic
475C4⋊C4C2×#6711yescharacteristic
476C4⋊C4C2×#6712yes—
477C4⋊C4C2×#6712yes—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—21no—
496C42C2×#6711yescharacteristic
497C42C2×#6711yescharacteristic
498C42C2×#6711yescharacteristic
499C42C23×D311yescharacteristic
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3#45612yes—
509C22×C3C2×#12011yescharacteristic
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—41no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—42no—
518C3×C4—41no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4#57612yes—
523C3×C4#57812yes—
524Dic3—82no—
525C23—21no—
526C23—21no—
527C23#95311yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—42no—
531Q8—42no—
532Q8—82no—
533Q8—82no—
534Q8—122no—
535Q8—122no—
536Q8—242no—
537D4—22no—
538D4—22no—
539D4—42no—
540D4—42no—
541D4—82no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—122no—
559C2×C4—242no—
560C2×C4#95311yescharacteristic
561C2×C4#95311yescharacteristic
562C2×C4#95311yescharacteristic
563C2×C4C22×#6711yescharacteristic
564C2×C4C22×#6711yescharacteristic
565C2×C4C22×#6711yescharacteristic
566C8—64no—
567C8—122no—
568C8—122no—
569C2×C3—22no—
570C2×C3—81no—
571C2×C3—81no—
572C2×C3#300811yescharacteristic
573C2×C3#301111yescharacteristic
574C2×C3C2×#45611yescharacteristic
575C22—21no—
576C22—21no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22#524612yes—
584C22C2×#95311yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—242no—
598C4#583512yes—
599C4#583812yes—
600C3#1869911yescharacteristic, complemented, Sylow
601C2—22no—
602C2—81no—
603C2—81no—
604C2#8410111yescharacteristic
605C2#8411711yescharacteristic
606C2C2×#524611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
Fitting, prime radicalC3×#3552C2
socleC22×C3C2×#120
hypercenter#3552D3
hypocenterC3#18699
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3552

Complements

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

NH ≅ G/Ntype
2. C2×#5078C2semidirect
3. #83154C2semidirect
4. #83124C2semidirect
5. #71937C2semidirect
6. #71933C2semidirect
7. #71916C2semidirect
8. #71912C2semidirect
9. #71776C2semidirect
10. #71752C2semidirect
11. #71752C2semidirect
600. C3#18699semidirect

Maximal subgroups

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

subgroupindex
C3×#35522
C2×#50782
#831542
#831242
#719372
#719332
#719162
#719122
#717762
#717522
#186993