Finite Group Databasebeta

← #10828166

Subgroups — #10828166

Order 768.

1364 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5552C211yesmaximal, characteristic
2C3×#2486C211yesmaximal, characteristic, complemented
3C22×#810C211yesmaximal, characteristic, complemented
4C2×#4967C211yesmaximal, characteristic, complemented
5#71610C211yesmaximal, characteristic, complemented
6#69631C212yesmaximal, complemented
7#69628C212yesmaximal, complemented
8#69527C212yesmaximal, complemented
9#69525C211yesmaximal, characteristic, complemented
10#69522C212yesmaximal, complemented
11#69515C211yesmaximal, characteristic, complemented
12#24035—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×C3×#123—21no—
16C2×C3×C4×D4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#939—21no—
19C2×#930C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22#5552—24no—
23#5552—24no—
24C2×#896—21no—
25C2×#885—21no—
26C2×#67⋊C4—21no—
27C2×#875C2211yescharacteristic
28C2×C4×Dic6C2211yescharacteristic
29C3×#398—21no—
30C3×#397—21no—
31C3×#389C2212yes—
32C3×#381—21no—
33C3×#380C2211yescharacteristic
34C3×#380C2212yes—
35C2×#830C2211yescharacteristic
36C2×#830C2212yes—
37C2×#810—24no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41C2×#810C2212yes—
42#4967—24no—
43C2×#809C2212yes—
44C2×#809C2212yes—
45C2×#807C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#548—31no—
49C2×#480—31no—
50C22×#103—31no—
51#2486D311yescharacteristic, complemented
52#2042—32no—
53#2036—32no—
54#1893—31no—
55#1869—31no—
56#1863—32no—
57#1788—32no—
58#1787—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×#67—21no—
65C23×Dic3—21no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—24no—
69C2×#235—42no—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×Dic6—24no—
73C22×Dic6—21no—
74C22×Dic6—22no—
75C22×Dic6—42no—
76C22×Dic6D414yes—
77C22×Dic6C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C3×#123—44no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4D411yescharacteristic
92C2×C3×C22⋊C4D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95#939—28no—
96#930—22no—
97#930D414yes—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102#924—22no—
103#924—22no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C22×C3⋊C8—22no—
108C22×C3⋊C8—42no—
109C22×C3⋊C8C2312yes—
110C22×C3⋊C8C2312yes—
111#896—28no—
112#885—28no—
113#67⋊C4—28no—
114#875—22no—
115#875—22no—
116C4×Dic6—22no—
117C4×Dic6—22no—
118C3×#95—22no—
119C3×#95—21no—
120C3×#95—22no—
121#830—22no—
122#830—42no—
123#810—24no—
124#810—24no—
125#810—24no—
126#810—42no—
127#809—24no—
128#809—42no—
129#807—24no—
130C22×#41—31no—
131C23×Q8—31no—
132C23×C8—31no—
133#548—64no—
134#548—64no—
135C2×#124—31no—
136C2×#124—61no—
137C2×#123—21no—
138C2×C22⋊Q8—31no—
139C2×C22⋊Q8—61no—
140C2×#121—31no—
141C2×#121—61no—
142C2×C4×Q8—31no—
143C2×C4×D4—61no—
144C2×C4×D4C2×D311yescharacteristic
145C22×C4⋊C4C2×D311yescharacteristic
146#480—64no—
147C2×#107—32no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#103—32no—
152C2×#103—64no—
153C2×#102—32no—
154C2×#102—32no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—32no—
157#398—21no—
158#397—21no—
159#389C2×D312yes—
160#381—21no—
161#380C2×D311yescharacteristic
162#380C2×D312yes—
163#340—32no—
164#336—32no—
165C24×C3C2×D411yescharacteristic
166C24×C3C2×D411yescharacteristic
167C23×D3—21no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4—42no—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C2×#67—24no—
190C2×#67—24no—
191C2×#67—44no—
192C2×#67—42no—
193C2×#67—42no—
194C22×Dic3—21no—
195C22×Dic3—24no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—44no—
199#235—48no—
200#235—44no—
201#235—44no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—21no—
205C2×D12C2×D412yes—
206C2×C4×D3—24no—
207C2×C4×D3—44no—
208C2×Dic6—21no—
209C2×Dic6—22no—
210C2×Dic6—22no—
211C2×Dic6—22no—
212C2×Dic6—28no—
213C2×Dic6—24no—
214C2×Dic6—21no—
215C2×Dic6—44no—
216C2×Dic6—48no—
217C2×Dic6—44no—
218C2×Dic6C2×D412yes—
219C2×Dic6C2×D412yes—
220C2×Dic6#4114yes—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—44no—
227C3×C4⋊C4—42no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—44no—
231C3×C22⋊C4—42no—
232C3×C42—21no—
233C3×C42—21no—
234#217—24no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—24no—
238C4⋊Dic3—21no—
239C4⋊Dic3C2×D412yes—
240Dic3⋊C4—24no—
241Dic3⋊C4—24no—
242C4×Dic3—24no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—44no—
250C2×#41—62no—
251C2×#41—62no—
252C2×#41—64no—
253C2×#41—122no—
254C22×Q8—34no—
255C22×Q8—31no—
256C22×Q8—31no—
257C22×Q8—64no—
258C22×Q8—62no—
259C22×Q8—61no—
260C22×Q8—122no—
261C22×D4—31no—
262C22×D4—61no—
263C22×D4—61no—
264C22×D4#6711yescharacteristic
265C23×C4—21no—
266C23×C4—61no—
267C23×C4—61no—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D311yescharacteristic
270C22×C8—32no—
271C22×C8—32no—
272C22×C8—62no—
273C22×C8—122no—
274#124—62no—
275#124—62no—
276#124—68no—
277#123—44no—
278C22⋊Q8—68no—
279C22⋊Q8—62no—
280C22⋊Q8—62no—
281#121—34no—
282#121—68no—
283#121—62no—
284C4×Q8—62no—
285C4×Q8—62no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—68no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4#6711yescharacteristic
297C2×C4⋊C4C22×D312yes—
298C2×C4⋊C4C22×D312yes—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4#6711yescharacteristic
306C2×C22⋊C4#6711yescharacteristic
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#107—64no—
310#103—64no—
311#103—64no—
312#103—64no—
313#103—122no—
314#102—64no—
315#102—122no—
316C22⋊C8—62no—
317C22⋊C8—122no—
318#95—22no—
319#95—21no—
320#95—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—21no—
324C23×C3—22no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—22no—
328C23×C3#12012yes—
329C23×C3#13512yes—
330C23×C3AGL(1,ℤ/8)12yes—
331C23×C3C22×D411yescharacteristic
332C22×D3—22no—
333C22×D3—42no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—42no—
337C3×D4—44no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—84no—
359C2×C3×C4#12012yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4112yes—
363C2×C3×C4H(5,𝔽2)12yes—
364#67—48no—
365#67—48no—
366C2×Dic3—22no—
367C2×Dic3—24no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—48no—
371C2×Dic3—42no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374D12—24no—
375D12—41no—
376D12—41no—
377C4×D3—48no—
378Dic6—22no—
379Dic6—24no—
380Dic6—22no—
381Dic6—22no—
382Dic6—48no—
383Dic6—42no—
384Dic6#12314yescomplemented
385C3⋊C8—42no—
386C3⋊C8—42no—
387C24—61no—
388C24C2×#6711yescharacteristic
389C24C2×#6711yescharacteristic
390#41—128no—
391#41—124no—
392#41—124no—
393C2×Q8—32no—
394C2×Q8—32no—
395C2×Q8—34no—
396C2×Q8—68no—
397C2×Q8—64no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—62no—
401C2×Q8—62no—
402C2×Q8—62no—
403C2×Q8—124no—
404C2×Q8—124no—
405C2×Q8—128no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—32no—
410C2×D4—64no—
411C2×D4—64no—
412C2×D4—61no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—124no—
416C2×D4—122no—
417C2×D4—122no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—64no—
430C22×C4—61no—
431C22×C4—64no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—124no—
435C22×C4—124no—
436C22×C4D3×D412yes—
437C22×C4D3×D412yes—
438C22×C4C2×#6711yescharacteristic
439C22×C4C2×#6711yescharacteristic
440C22×C4C2×#6711yescharacteristic
441C22×C4C2×#6711yescharacteristic
442C22×C4C23×D311yescharacteristic
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—62no—
449C2×C8—124no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—32no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—64no—
459C4⋊C4—64no—
460C4⋊C4—61no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—44no—
464C22⋊C4—42no—
465C22⋊C4—64no—
466C22⋊C4—64no—
467C22⋊C4—64no—
468C22⋊C4—64no—
469C42—21no—
470C42—21no—
471C42—64no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3#44914yes—
482C22×C3C2×#12011yescharacteristic
483C22×C3#54612yes—
484C22×C3C2×#13511yescharacteristic
485C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
486C22×C3#57812yes—
487C2×D3—42no—
488C2×D3—42no—
489C2×D3—42no—
490C2×D3—82no—
491C2×D3—82no—
492C3×C4—22no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—82no—
497C3×C4—82no—
498C3×C4C2×#12311yescharacteristic
499C3×C4#53411yescharacteristic
500C3×C4#54612yes—
501Dic3—24no—
502Dic3—42no—
503Dic3—42no—
504C23—22no—
505C23—22no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—62no—
512C23—121no—
513C23—121no—
514C23—122no—
515C23—122no—
516C23#93212yes—
517C23#93712yes—
518C23#95012yes—
519C23C22×#6711yescharacteristic
520Q8—34no—
521Q8—62no—
522Q8—64no—
523Q8—62no—
524Q8—62no—
525Q8—128no—
526Q8—122no—
527D4—44no—
528D4—64no—
529D4—128no—
530D4—128no—
531D4—121no—
532D4—121no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—62no—
554C2×C4—64no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—84no—
558C2×C4—128no—
559C2×C4—128no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—122no—
563C2×C4#93712yes—
564C2×C4C2×D3×D411yescharacteristic
565C2×C4C2×D3×D411yescharacteristic
566C2×C4D3×#4112yes—
567C2×C4#100912yes—
568C8—122no—
569C8—122no—
570C2×C3—24no—
571C2×C3—81no—
572C2×C3#240511yescharacteristic
573C2×C3C2×#44911yescharacteristic
574C2×C3#301111yescharacteristic
575C2×C3#330012yes—
576C2×C3#330812yes—
577D3—82no—
578C22—24no—
579C22—24no—
580C22—22no—
581C22—22no—
582C22—42no—
583C22—42no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—122no—
588C22—122no—
589C22—122no—
590C22—242no—
591C22—242no—
592C22#503714yes—
593C22C2×#93211yescharacteristic
594C22C2×#93711yescharacteristic
595C22C2×#95011yescharacteristic
596C22#582912yes—
597C22#583812yes—
598C4—22no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—64no—
603C4—82no—
604C4—82no—
605C4—122no—
606C4—122no—
607C4D3×#12311yescharacteristic
608C4#565311yescharacteristic
609C4#565412yes—
610C3#2403511yescharacteristic, complemented, Sylow
611C2—24no—
612C2—81no—
613C2—242no—
614C2C2×#503711yescharacteristic
615C2#8289311yescharacteristic
616C2#8341111yescharacteristic
617C2#8353212yes—
618C2#8360212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2486C2
socleC23×C3C22×D4
prime radicalC2×#5552C2
hypercenter#2486D3
hypocenterC3#24035
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. C3×#2486C2semidirect
3. C22×#810C2semidirect
4. C2×#4967C2semidirect
5. #71610C2semidirect
6. #69631C2semidirect
7. #69628C2semidirect
8. #69527C2semidirect
9. #69525C2semidirect
10. #69522C2semidirect
11. #69515C2semidirect
51. #2486D3semidirect
384. Dic6#123semidirect
610. C3#24035semidirect

Maximal subgroups

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

subgroupindex
C2×#55522
C3×#24862
C22×#8102
C2×#49672
#716102
#696312
#696282
#695272
#695252
#695222
#695152
#240353