Finite Group Databasebeta

← #11069493

Subgroups — #11069493

Order 768.

2078 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5603C212yesmaximal, complemented
2C3×#2535C211yesmaximal, characteristic
3#82643C212yesmaximal, complemented
4#82375C211yesmaximal, characteristic, complemented
5#82333C211yesmaximal, characteristic, complemented
6C4×#936C211yesmaximal, characteristic, complemented
7#82319C211yesmaximal, characteristic, complemented
8#77570C214yesmaximal, complemented
9#76726C214yesmaximal, complemented
10#74748C214yesmaximal, complemented
11#74739C214yesmaximal, complemented
12#74221C211yesmaximal, characteristic, complemented
13#74157C211yesmaximal, characteristic, complemented
14#73954C212yesmaximal, complemented
15#73892C212yesmaximal, complemented
16#39374—31nomaximal, Sylow
17C2×C3×#121C2212yes—
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#936C2214yes—
21C2×#936C2211yescharacteristic
22C2×#935C2214yes—
23C2×D4×Dic3C2212yes—
24C2×#926C2212yes—
25C2×#924C2212yes—
26#5603—216no—
27C2×#888C2212yes—
28C2×#887C2212yes—
29C2×#879C2214yes—
30C2×#878C2214yes—
31C2×C4×Dic6C2211yescharacteristic
32C3×#397C2212yes—
33C3×#392C2212yes—
34C3×#386C2214yes—
35C3×#382C2211yescharacteristic
36C3×#380C2212yes—
37#5222C2212yes—
38#5222C2214yes—
39#5221C2212yes—
40#5221C2211yescharacteristic
41#5220C2214yes—
42#5212C2212yes—
43#5212C2212yes—
44C4×#217C2212yes—
45C4×#217C2214yes—
46C2×C3⋊#95C2212yes—
47#4984C2212yes—
48#4984C2212yes—
49#4983C2212yes—
50#4977C2211yescharacteristic
51#4977C2212yes—
52#4973C2211yescharacteristic
53#4972C2212yes—
54#4952C2214yes—
55#4952C2214yes—
56#4951C2212yes—
57#4951C2212yes—
58#4951C2214yes—
59#4950C2214yes—
60#4949C2214yes—
61#4947C2214yes—
62#4945C2214yes—
63#4944C2214yes—
64#4943C2214yes—
65#4936C2211yescharacteristic
66C4×C4⋊Dic3C2211yescharacteristic
67C42×Dic3C2211yescharacteristic
68#4660C2214yes—
69#4658C2214yes—
70#4656C2214yes—
71#4654C2214yes—
72#4650C2212yes—
73#4649C2212yes—
74#4647C2214yes—
75#4646C2214yes—
76#4646C2214yes—
77C2×#542—32no—
78#2862—31no—
79#2710—34no—
80#2689—34no—
81#2628—32no—
82#2536—31no—
83#2535D311yescharacteristic
84#2521—34no—
85#2488—34no—
86#2421—32no—
87#2420—32no—
88#2351—31no—
89#2349—31no—
90C4×#124—31no—
91#2293—31no—
92C22×C3×D4C2311yescharacteristic
93C22×C3×D4C2314yes—
94C22×C3×D4C2311yescharacteristic
95C23×C3×C4C2312yes—
96C23×C3×C4C2312yes—
97C23×Dic3C2314yes—
98C22×Dic6C2311yescharacteristic
99C22×Dic6C2314yes—
100C3×#121—28no—
101C3×C4×D4—24no—
102C3×C4×D4—24no—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C22⋊C4C2312yes—
108C2×C3×C22⋊C4C2312yes—
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C42C2311yescharacteristic
111C2×#217C2314yes—
112C2×#217C2312yes—
113C2×#217C2312yes—
114C2×#217C2314yes—
115C2×#217C2314yes—
116C2×#217C2314yes—
117C2×#217C2312yes—
118C2×#217C2312yes—
119C2×#217C2314yes—
120#936—216no—
121#936D418yescomplemented
122#935—216no—
123D4×Dic3—28no—
124#926—28no—
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2312yes—
127C2×C4⋊Dic3C2312yes—
128C2×C4⋊Dic3C2311yescharacteristic
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2314yes—
131#924—28no—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2312yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2314yes—
137C2×C4×Dic3C2314yes—
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2312yes—
141C2×C4×Dic3C2312yes—
142C2×C4×Dic3C2312yes—
143C2×C4×Dic3C2311yescharacteristic
144#888—28no—
145#887—28no—
146#879—216no—
147#878—216no—
148C4×Dic6—24no—
149C3×#95C2312yes—
150C3×#95C2312yes—
151C3×#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2312yes—
155C3⋊#95C2312yes—
156C3⋊#95C2314yes—
157C3⋊#95C2311yescharacteristic
158C3⋊#95C2311yescharacteristic
159C3⋊#95C2312yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C3⋊#95C2314yes—
163C3⋊#95C2312yes—
164C3⋊#95C2314yes—
165#542—616no—
166C2×C4⋊Q8—32no—
167C2×#126—34no—
168C2×#124—31no—
169C2×#124—34no—
170C2×#123—34no—
171C2×C22⋊Q8—32no—
172C2×C22⋊Q8—34no—
173C2×#121C2×D312yes—
174C2×C4×Q8—32no—
175C2×C4×Q8—31no—
176C2×C4×D4—32no—
177C2×C4×D4C2×D311yescharacteristic
178C2×C4×D4C2×D311yescharacteristic
179C2×#117—32no—
180#400—34no—
181#400—34no—
182#399—34no—
183#398—32no—
184#397—34no—
185#397—34no—
186#397C2×D312yes—
187#395—34no—
188#394—34no—
189#394—32no—
190#393—34no—
191#392C2×D312yes—
192#391—31no—
193#391—34no—
194#391—34no—
195#390—31no—
196#390—32no—
197#389—32no—
198#389—32no—
199#389—32no—
200#389—34no—
201#388—34no—
202#388—34no—
203#388—34no—
204#388—32no—
205#388—32no—
206#387—34no—
207#386—34no—
208#386C2×D314yes—
209#385—34no—
210#382—32no—
211#382—31no—
212#382—31no—
213#382—32no—
214#382C2×D311yescharacteristic
215#381—34no—
216#380—32no—
217#380—32no—
218#380C2×D312yes—
219C4×C4⋊C4—31no—
220C4×C4⋊C4—32no—
221C4×C22⋊C4—32no—
222C4×C22⋊C4—34no—
223C2×#95—32no—
224C43—31no—
225C24×C3C2412yes—
226C24×C3C2412yes—
227C2×C3×D4—28no—
228C2×C3×D4—22no—
229C2×C3×D4—22no—
230C2×C3×D4—22no—
231C2×C3×D4—42no—
232C2×C3×D4—44no—
233C2×C3×D4—44no—
234C2×C3×D4C2×D414yes—
235C22×C3×C4—24no—
236C22×C3×C4—24no—
237C22×C3×C4—24no—
238C22×C3×C4—24no—
239C22×C3×C4—44no—
240C22×C3×C4C2412yes—
241C22×C3×C4C2412yes—
242C22×C3×C4C2414yes—
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2411yescharacteristic
245C22×C3×C4C2411yescharacteristic
246C22×Dic3—28no—
247C22×Dic3—44no—
248C22×Dic3—44no—
249C22×Dic3C2412yes—
250C22×Dic3C2412yes—
251C22×Dic3C2412yes—
252C22×Dic3C2414yes—
253C22×Dic3C2412yes—
254C22×Dic3C2412yes—
255C22×Dic3C2412yes—
256C2×Dic6—22no—
257C2×Dic6—22no—
258C2×Dic6—28no—
259C2×Dic6—44no—
260C2×Dic6—44no—
261C2×Dic6C2×D414yes—
262C3×C4⋊C4—24no—
263C3×C4⋊C4—22no—
264C3×C4⋊C4—22no—
265C3×C22⋊C4—28no—
266C3×C22⋊C4—24no—
267C3×C22⋊C4—24no—
268C3×C22⋊C4—44no—
269C3×C42#4114yes—
270#217—28no—
271#217—28no—
272#217—24no—
273#217—24no—
274#217—44no—
275#217—42no—
276#217—42no—
277#217—44no—
278#217C2×D4116yes—
279C4⋊Dic3—24no—
280C4⋊Dic3—22no—
281C4⋊Dic3—24no—
282C4⋊Dic3—24no—
283C4⋊Dic3—42no—
284C4⋊Dic3—44no—
285Dic3⋊C4—24no—
286Dic3⋊C4—28no—
287Dic3⋊C4—28no—
288Dic3⋊C4—28no—
289Dic3⋊C4—44no—
290C4×Dic3—24no—
291C4×Dic3—24no—
292C4×Dic3—22no—
293C4×Dic3—28no—
294C4×Dic3—42no—
295C4×Dic3—42no—
296C4×Dic3C2×D414yes—
297C22×Q8—31no—
298C22×Q8—34no—
299C22×D4C22×D311yescharacteristic
300C22×D4C22×D311yescharacteristic
301C22×D4C22×D314yes—
302C23×C4—34no—
303C23×C4C22×D312yes—
304C23×C4C22×D312yes—
305C4⋊Q8—68no—
306#126—616no—
307#124—38no—
308#124—616no—
309#123—616no—
310C22⋊Q8—616no—
311C22⋊Q8—68no—
312#121—28no—
313C4×Q8—68no—
314C4×Q8—64no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—68no—
318#117—68no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—32no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—34no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C4⋊C4C22×D312yes—
333C2×C22⋊C4—34no—
334C2×C22⋊C4—32no—
335C2×C22⋊C4—32no—
336C2×C22⋊C4—34no—
337C2×C22⋊C4—34no—
338C2×C22⋊C4—34no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4C22×D312yes—
343C2×C22⋊C4C22×D312yes—
344C2×C22⋊C4C22×D314yes—
345C2×C22⋊C4C22×D314yes—
346C2×C42—31no—
347C2×C42—32no—
348C2×C42—34no—
349C2×C42—32no—
350C2×C42—31no—
351C2×C42—32no—
352C2×C42—32no—
353C2×C42C22×D311yescharacteristic
354#95—31no—
355#95—32no—
356#95—34no—
357#95—32no—
358#95—34no—
359#95—32no—
360#95—34no—
361#95—31no—
362#95—34no—
363#95—34no—
364#95—34no—
365#95—32no—
366#95—34no—
367#95C22×D312yes—
368#95C22×D314yes—
369#95C22×D312yes—
370C23×C3—22no—
371C23×C3—22no—
372C23×C3—24no—
373C23×C3—24no—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—22no—
377C23×C3—22no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—22no—
381C23×C3C22×D414yes—
382C23×C3C2511yescharacteristic
383C3×D4—28no—
384C3×D4—44no—
385C3×D4—88no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—24no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—24no—
394C2×C3×C4—24no—
395C2×C3×C4—24no—
396C2×C3×C4—48no—
397C2×C3×C4—84no—
398C2×C3×C4C22×D412yes—
399C2×C3×C4C2×#4112yes—
400C2×C3×C4C2×#4112yes—
401C2×C3×C4C2×#4112yes—
402C2×C3×C42−1+412yes—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—24no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—24no—
416C2×Dic3—22no—
417C2×Dic3—24no—
418C2×Dic3—22no—
419C2×Dic3—24no—
420C2×Dic3—22no—
421C2×Dic3—88no—
422C2×Dic3C22×D414yes—
423C2×Dic3C22×D414yes—
424Dic6—28no—
425Dic6—88no—
426C24C23×D312yes—
427C24C23×D312yes—
428C2×Q8—34no—
429C2×Q8—68no—
430C2×Q8—62no—
431C2×Q8—62no—
432C2×Q8—124no—
433C2×Q8—124no—
434C2×D4—28no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4D3×D414yes—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—32no—
450C22×C4—32no—
451C22×C4—34no—
452C22×C4—32no—
453C22×C4—44no—
454C22×C4—68no—
455C22×C4—124no—
456C22×C4—124no—
457C22×C4C23×D312yes—
458C22×C4C23×D312yes—
459C22×C4C23×D314yes—
460C22×C4C23×D311yescharacteristic
461C22×C4C23×D311yescharacteristic
462C22×C4C23×D311yescharacteristic
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C4⋊C4—68no—
469C4⋊C4—68no—
470C4⋊C4—68no—
471C4⋊C4—64no—
472C4⋊C4—64no—
473C4⋊C4—62no—
474C4⋊C4—124no—
475C4⋊C4—122no—
476C4⋊C4—124no—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—316no—
481C22⋊C4—44no—
482C22⋊C4—64no—
483C22⋊C4—64no—
484C22⋊C4—68no—
485C22⋊C4—68no—
486C22⋊C4—124no—
487C22⋊C4—124no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C42—34no—
491C42—68no—
492C42—64no—
493C42—64no—
494C42—62no—
495C42—122no—
496C42—122no—
497C42#23714yes—
498C22×C3—24no—
499C22×C3—28no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—44no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3C23×D411yescharacteristic
511C22×C3C22×#4111yescharacteristic
512C22×C3C2×2−1+411yescharacteristic
513C22×C3#58512yes—
514C22×C3#58512yes—
515C3×C4—22no—
516C3×C4—22no—
517C3×C4—42no—
518C3×C4—44no—
519C3×C4—42no—
520C3×C4#54714yes—
521Dic3—24no—
522Dic3—24no—
523Dic3—42no—
524Dic3—44no—
525Dic3—42no—
526Dic3—42no—
527Dic3—42no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23C2×D3×D414yes—
540C23C24×D311yescharacteristic
541Q8—68no—
542Q8—248no—
543D4—28no—
544D4—44no—
545D4—88no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—34no—
557C2×C4—34no—
558C2×C4—48no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—64no—
577C2×C4—84no—
578C2×C4—248no—
579C2×C4C2×D3×D412yes—
580C2×C4C2×#23712yes—
581C2×C4C2×#23712yes—
582C2×C4C2×#23712yes—
583C2×C4#101012yes—
584C2×C3—44no—
585C2×C3—82no—
586C2×C3#345511yescharacteristic
587C2×C3C2×#54711yescharacteristic
588C2×C3#350012yes—
589C2×C3#351011yescharacteristic
590C2×C3#356512yes—
591C22—28no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22C22×D3×D411yescharacteristic
604C22C22×#23711yescharacteristic
605C22C2×#101011yescharacteristic
606C22#596412yes—
607C22#596612yes—
608C4—22no—
609C4—22no—
610C4—42no—
611C4—44no—
612C4—42no—
613C4—64no—
614C4—64no—
615C4—122no—
616C4—122no—
617C4—124no—
618C4—122no—
619C4—122no—
620C4#561114yes—
621C3#3937411yescharacteristic, complemented, Sylow
622C2—44no—
623C2—82no—
624C2C2×#561111yescharacteristic
625C2#8601911yescharacteristic
626C2#8632011yescharacteristic
627C2#8663212yes—
628C2#8690412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
Fitting, prime radicalC3×#2535C2
hypercenter#2535D3
hypocenterC3#39374
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. C2×#5603C2semidirect
3. #82643C2semidirect
4. #82375C2semidirect
5. #82333C2semidirect
6. C4×#936C2semidirect
7. #82319C2semidirect
8. #77570C2semidirect
9. #76726C2semidirect
10. #74748C2semidirect
11. #74739C2semidirect
12. #74221C2semidirect
13. #74157C2semidirect
14. #73954C2semidirect
15. #73892C2semidirect
121. #936D4semidirect
621. C3#39374semidirect

Maximal subgroups

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

subgroupindex
C2×#56032
C3×#25352
#826432
#823752
#823332
C4×#9362
#823192
#775702
#767262
#747482
#747392
#742212
#741572
#739542
#738922
#393743