Finite Group Databasebeta

← #10876060

Subgroups — #10876060

Order 768.

724 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×#2512C211yesmaximal, characteristic
2#73637C211yesmaximal, characteristic, complemented
3#69660C211yesmaximal, characteristic
4#69660C211yesmaximal, characteristic
5#69660C211yesmaximal, characteristic
6#69653C211yesmaximal, characteristic
7#69645C211yesmaximal, characteristic
8#69641C211yesmaximal, characteristic
9#69631C211yesmaximal, characteristic, complemented
10#69613C211yesmaximal, characteristic, complemented
11#69588C211yesmaximal, characteristic, complemented
12#69571C211yesmaximal, characteristic, complemented
13#69483C211yesmaximal, characteristic, complemented
14#69475C211yesmaximal, characteristic, complemented
15#69461C211yesmaximal, characteristic, complemented
16#27578—31nomaximal, Sylow
17C2×C3×#126—21no—
18C22×#214C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#397—21no—
23C3×#396—21no—
24C3×#390C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#388—21no—
29C3×#384C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35#4976—21no—
36#4976—21no—
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#807C2211yescharacteristic
41#4936C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4650—21no—
47#4649—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512D311yescharacteristic
61#2351—31no—
62#2101—31no—
63#2099—31no—
64#2073—31no—
65#2067—31no—
66#1936—31no—
67#1921—31no—
68#1917—31no—
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1863—31no—
73#1803—31no—
74#1763—31no—
75C22×C3×D4C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C3×#126—44no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×#214—22no—
97C2×#214—22no—
98C2×#214—22no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C4×Dic6—22no—
115C4×Dic6—22no—
116C4×Dic6—22no—
117C4×Dic6—22no—
118C4×Dic6—41no—
119C4×Dic6—41no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95D411yescharacteristic
123C3×#95D411yescharacteristic
124#832—22no—
125#832—41no—
126#832—41no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#810—22no—
132#810—22no—
133#810—22no—
134#807—22no—
135C4⋊(C3⋊C8)—41no—
136#803—41no—
137C22×SD16—31no—
138C2×#126—21no—
139C2×C4×Q8—31no—
140C2×C4×Q8—31no—
141C2×C4×Q8—31no—
142C2×#107—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#97—31no—
151#397—21no—
152#396—21no—
153#390C2×D311yescharacteristic
154#390C2×D311yescharacteristic
155#390C2×D311yescharacteristic
156#389—31no—
157#389—31no—
158#389—61no—
159#389—61no—
160#389C2×D311yescharacteristic
161#388—21no—
162#384C2×D311yescharacteristic
163#382—31no—
164#382—61no—
165C4×C4⋊C4—61no—
166C4×C4⋊C4C2×D311yescharacteristic
167C4×C4⋊C4C2×D311yescharacteristic
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179#336—31no—
180C24×C3—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6#4112yes—
205C2×Dic6#4112yes—
206C2×Dic6#4112yes—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—41no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—81no—
218C3×C22⋊C4—42no—
219C3×C22⋊C4—42no—
220C3×C22⋊C4—42no—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—81no—
223C3×C22⋊C4—81no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—42no—
231#214—44no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3#4112yes—
238Dic3⋊C4—24no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243C4×Dic3—24no—
244C4×Dic3—42no—
245C4×Dic3—42no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C22×Q8—31no—
259C22×D4C22×D311yescharacteristic
260C2×SD16—62no—
261C2×SD16—62no—
262C2×SD16—62no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—31no—
267#126—44no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×Q8—62no—
271C4×Q8—62no—
272C4×Q8—121no—
273C4×Q8—121no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4#6711yescharacteristic
286C2×C4⋊C4#6711yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C42—21no—
297C2×C42—61no—
298C2×C42—61no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#107—62no—
304C4⋊C8—121no—
305#103—62no—
306#103—62no—
307#103—62no—
308#102—62no—
309#102—121no—
310#102—121no—
311#97—121no—
312#95—21no—
313#95—21no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95#6711yescharacteristic
319#95#6711yescharacteristic
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3C22×D411yescharacteristic
328C3×D4—41no—
329C3×D4—41no—
330C3×D4—41no—
331C3×D4—41no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4AGL(1,ℤ/8)12yes—
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×C3×C42−1+411yescharacteristic
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377Dic6—22no—
378Dic6—22no—
379Dic6—22no—
380Dic6#12614yescomplemented
381C3⋊C8—41no—
382C3⋊C8—41no—
383C3⋊C8—41no—
384C3⋊C8—41no—
385C24—21no—
386C2×Q8—32no—
387C2×Q8—32no—
388C2×Q8—32no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4C2×#6711yescharacteristic
406C22×C4C2×#6711yescharacteristic
407C22×C4C2×#6711yescharacteristic
408C22×C4C2×#6711yescharacteristic
409C22×C4C2×#6711yescharacteristic
410C22×C4C2×#6711yescharacteristic
411C22×C4C23×D311yescharacteristic
412SD16—124no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—32no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—64no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—81no—
443C4⋊C4—122no—
444C4⋊C4—122no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—81no—
452C22⋊C4—81no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—42no—
460C42—64no—
461C42—122no—
462C42—122no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3—81no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3#52511yescharacteristic
471C22×C3#54611yescharacteristic
472C22×C3#54811yescharacteristic
473C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
474C22×C3#57511yescharacteristic
475C22×C3#57711yescharacteristic
476C22×C3#57811yescharacteristic
477C3×C4—22no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—81no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4C2×#12611yescharacteristic
487C3×C4#53911yescharacteristic
488C3×C4#55211yescharacteristic
489C3×C4#55311yescharacteristic
490Dic3—24no—
491Dic3—42no—
492Dic3—81no—
493Dic3—81no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C22×#6711yescharacteristic
502Q8—34no—
503Q8—62no—
504Q8—62no—
505Q8—62no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4#93212yes—
549C2×C4D3×#4111yescharacteristic
550C2×C4D3×#4111yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4D3×#4111yescharacteristic
553C2×C4#100911yescharacteristic
554C2×C4#101011yescharacteristic
555C8—121no—
556C8—121no—
557C8—121no—
558C8—121no—
559C2×C3—161no—
560C2×C3#248111yescharacteristic
561C2×C3#311511yescharacteristic
562C2×C3#312111yescharacteristic
563C2×C3#330811yescharacteristic
564C2×C3#332611yescharacteristic
565C2×C3#336811yescharacteristic
566C2×C3#337711yescharacteristic
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22C2×#93211yescharacteristic
575C22#582211yescharacteristic
576C22#582811yescharacteristic
577C22#582911yescharacteristic
578C22#583111yescharacteristic
579C22#583611yescharacteristic
580C22#583811yescharacteristic
581C4—22no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—64no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—122no—
592C4—241no—
593C4—241no—
594C4D3×#12611yescharacteristic
595C4#570711yescharacteristic
596C4#570811yescharacteristic
597C4#570911yescharacteristic
598C3#2757811yescharacteristic, complemented, Sylow
599C2—161no—
600C2#8308211yescharacteristic
601C2#8330111yescharacteristic
602C2#8346711yescharacteristic
603C2#8353211yescharacteristic
604C2#8358711yescharacteristic
605C2#8362111yescharacteristic
606C2#8367811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2512C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2512D3
hypocenterC3#27578
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2512

Complements

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

NH ≅ G/Ntype
2. #73637C2semidirect
9. #69631C2semidirect
10. #69613C2semidirect
11. #69588C2semidirect
12. #69571C2semidirect
13. #69483C2semidirect
14. #69475C2semidirect
15. #69461C2semidirect
380. Dic6#126semidirect
598. C3#27578semidirect

Maximal subgroups

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

subgroupindex
C3×#25122
#736372
#696602
#696532
#696452
#696412
#696312
#696132
#695882
#695712
#694832
#694752
#694612
#275783