Finite Group Databasebeta

← #10765252

Subgroups — #10765252

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79908C211yesmaximal, characteristic, complemented
2#79654C211yesmaximal, characteristic
3#79426C211yesmaximal, characteristic, complemented
4C3×C8×SD16C211yesmaximal, characteristic, complemented
5#70937C211yesmaximal, characteristic, complemented
6#70897C211yesmaximal, characteristic, complemented
7#70840C211yesmaximal, characteristic, complemented
8#68535C211yesmaximal, characteristic, complemented
9#68521C211yesmaximal, characteristic, complemented
10#68233C211yesmaximal, characteristic, complemented
11#68216C211yesmaximal, characteristic, complemented
12#68172C211yesmaximal, characteristic, complemented
13#68152C211yesmaximal, characteristic, complemented
14C8×D24C211yesmaximal, characteristic, complemented
15#68049C211yesmaximal, characteristic, complemented
16#17918—31nomaximal, Sylow
17#5580C2211yescharacteristic
18C4×D3×D4C2211yescharacteristic
19C3×C8×Q8C2211yescharacteristic, complemented
20C3×C4×SD16C2211yescharacteristic
21C3×C8×D4C2211yescharacteristic
22#5169C411yescharacteristic
23#5165C411yescharacteristic
24C8×#67—21no—
25C2×(C2×D3)⋊C8—21no—
26C4×#215C2211yescharacteristic
27#5024C2211yescharacteristic, complemented
28C4×#213C2211yescharacteristic
29D4×C3⋊C8C2211yescharacteristic
30#4871C411yescharacteristic
31#4863C411yescharacteristic
32#4839—21no—
33#4838C2211yescharacteristic, complemented
34#4837C2211yescharacteristic
35#4836—21no—
36#4834C2211yescharacteristic
37#4833C2211yescharacteristic
38#4819C411yescharacteristic
39#4811C411yescharacteristic
40#4773C411yescharacteristic
41#4766C411yescharacteristic
42#67⋊C8—21no—
43D3×C22⋊C8—21no—
44C4×D24C2211yescharacteristic
45#4690—21no—
46#4688—21no—
47C8×D12C2211yescharacteristic
48C8×D12C2211yescharacteristic, complemented
49#4685C2211yescharacteristic
50#4685C2211yescharacteristic
51#4685C2211yescharacteristic
52C3×#334C2211yescharacteristic, complemented
53C3×Q8⋊C8C2211yescharacteristic, complemented
54C3×D4⋊C8C2211yescharacteristic
55C3×C82C2211yescharacteristic, complemented
56#4535C2211yescharacteristic, complemented
57D4⋊(C3⋊C8)C2211yescharacteristic
58#4483C2211yescharacteristic
59#4483C2211yescharacteristic, complemented
60#4480C2211yescharacteristic, complemented
61#4479C2211yescharacteristic, complemented
62#4459C2211yescharacteristic
63#4459C2211yescharacteristic, complemented
64#4458C2211yescharacteristic, complemented
65#4454C2211yescharacteristic, complemented
66#4454C2211yescharacteristic, complemented
67#4454C2211yescharacteristic, complemented
68#3105—31no—
69#2993—31no—
70#2972—31no—
71#1586—31no—
72#1585—31no—
73#1582—31no—
74#1581—31no—
75#1580—31no—
76SD16⋊C8—31no—
77C8×SD16D311yescharacteristic, complemented
78C8×D8—31no—
79#1574—31no—
80#1486—31no—
81#1482—31no—
82#1454—31no—
83C2×D3×D4C2×C411yescharacteristic
84C22×C4×D3—21no—
85C2×C3×SD16C2×C411yescharacteristic
86C22×C3×C8—21no—
87C3×C4×Q8C2311yescharacteristic
88C3×C4×D4C2311yescharacteristic
89#947C2×C411yescharacteristic
90#946C2×C411yescharacteristic
91C2×#215C2×C411yescharacteristic
92D4×Dic3C2×C411yescharacteristic
93C2×#213C2×C411yescharacteristic
94C4×#67—21no—
95C22×C3⋊C8—21no—
96C2×D24C2×C411yescharacteristic
97C2×ΓL(1,𝔽25)—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×D3×C8—21no—
100C2×D3×C8—21no—
101C2×D3×C8—21no—
102#898—21no—
103#895C2×C411yescharacteristic
104#894—21no—
105#893C2×C411yescharacteristic
106D3×C4⋊C4C2×C411yescharacteristic
107#890C2×C411yescharacteristic
108#67⋊C4—21no—
109D3×C22⋊C4—21no—
110C4×D12—21no—
111C4×D12C2311yescharacteristic
112C4×D12C2311yescharacteristic
113#872—21no—
114C42×D3C2311yescharacteristic
115C3×#106C2×C411yescharacteristic
116C3×C4⋊C8—21no—
117C3×C4⋊C8C2311yescharacteristic
118C3×C4⋊C8C2311yescharacteristic
119C3×#103C2×C411yescharacteristic
120C3×#102C2×C411yescharacteristic
121C3×C22⋊C8—21no—
122C3×C4×C8—21no—
123C3×C4×C8C2311yescharacteristic
124C3×C4×C8C2311yescharacteristic
125C3×C4×C8C2311yescharacteristic
126#835C2×C411yescharacteristic
127#832C2×C411yescharacteristic
128#830—21no—
129#821C2×C411yescharacteristic
130#821C2×C411yescharacteristic
131(C2×D3)⋊C8—21no—
132(C2×D3)⋊C8—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8C2×C411yescharacteristic
138(C2×D3)⋊C8C2×C411yescharacteristic
139(C2×D3)⋊C8D411yescharacteristic
140(C2×D3)⋊C8D411yescharacteristic
141#818C2×C411yescharacteristic
142Dic3⋊C8—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8C2×C411yescharacteristic
145Dic3⋊C8C2×C411yescharacteristic
146Dic3⋊C8D411yescharacteristic
147Dic3⋊C8D411yescharacteristic
148C8×Dic3—21no—
149#809C2×C411yescharacteristic
150#809C2×C411yescharacteristic
151#808C2×C411yescharacteristic
152#807C2×C411yescharacteristic
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156#803—21no—
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#529—31no—
161C2×C4×D4—31no—
162#484—31no—
163#481—31no—
164#478—31no—
165#478—31no—
166#463—31no—
167#463—31no—
168#450—31no—
169#447—31no—
170#446—31no—
171C8×Q8C2×D311yescharacteristic, complemented
172C4×SD16—31no—
173C4×SD16C2×D311yescharacteristic
174C4×D8—31no—
175C4×D8—31no—
176#436—31no—
177#436—61no—
178#435—61no—
179C8×D4—31no—
180C8×D4—31no—
181C8×D4—31no—
182C8×D4—31no—
183C8×D4—61no—
184C8×D4—61no—
185C8×D4C2×D311yescharacteristic
186#433—61no—
187#431—31no—
188#431—31no—
189#431—31no—
190#431—31no—
191#424—61no—
192#423—31no—
193C2×C22⋊C8—61no—
194C2×C22⋊C8—61no—
195#335—31no—
196#335—31no—
197#334—31no—
198#334C2×D311yescharacteristic, complemented
199Q8⋊C8—31no—
200Q8⋊C8C2×D311yescharacteristic, complemented
201D4⋊C8—31no—
202D4⋊C8—31no—
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8—31no—
206D4⋊C8C2×D311yescharacteristic
207#322—31no—
208#322—31no—
209#322—31no—
210C82C2×D311yescharacteristic, complemented
211C23×D3—21no—
212C2×C3×Q8C22×C411yescharacteristic
213C2×C3×D4C22×C411yescharacteristic
214C22×C3×C4—21no—
215C2×#67—21no—
216C22×Dic3—21no—
217D3×D4—22no—
218D3×D4—22no—
219C2×D12—21no—
220C2×D12C22×C411yescharacteristic
221C2×D12C22×C411yescharacteristic
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—81no—
227C2×C4×D3—81no—
228C2×C4×D3C22×C411yescharacteristic
229C2×C4×D3C2×D411yescharacteristic
230C2×C4×D3#4111yescharacteristic
231C3×SD16—22no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—81no—
236C2×C3×C8C22×C411yescharacteristic
237C2×C3×C8C22×C411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C3×C4⋊C4—21no—
241C3×C4⋊C4C22×C411yescharacteristic
242C3×C4⋊C4C22×C411yescharacteristic
243C3×C22⋊C4—21no—
244C3×C42—21no—
245C3×C42C2411yescharacteristic
246#217—21no—
247#215—22no—
248#213—22no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—21no—
255(C2×D3)⋊C4—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3C22×C411yescharacteristic
258C4⋊Dic3C22×C411yescharacteristic
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262C4×Dic3C22×C411yescharacteristic
263C4×Dic3C2×D411yescharacteristic
264C4×Dic3#4111yescharacteristic
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—81no—
269C2×C3⋊C8C22×C411yescharacteristic
270C2×C3⋊C8C22×C411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273D24—22no—
274ΓL(1,𝔽25)—81no—
275ΓL(1,𝔽25)—81no—
276D3×C8—81no—
277D3×C8—81no—
278D3×C8—81no—
279C22×D4—31no—
280C23×C4—61no—
281C2×SD16—31no—
282C2×SD16C4×D311yescharacteristic
283C2×D8—31no—
284C2×D8—31no—
285C2×M16—61no—
286C2×M16—61no—
287C22×C8—21no—
288C22×C8—61no—
289C22×C8—61no—
290C22×C8—61no—
291C22×C8—61no—
292#126—61no—
293#125—31no—
294#124—31no—
295#123—61no—
296C22⋊Q8—31no—
297#121—31no—
298C4×Q8C22×D311yescharacteristic
299C4×D4—31no—
300C4×D4—31no—
301C4×D4—31no—
302C4×D4—31no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4—61no—
306C4×D4C22×D311yescharacteristic
307#117—61no—
308C2×C4⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C42—31no—
311#107—31no—
312#107—31no—
313#106—31no—
314#106C4×D311yescharacteristic
315C4⋊C8—21no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—31no—
328#103C4×D311yescharacteristic
329#102—31no—
330#102—31no—
331#102—31no—
332#102—31no—
333#102—31no—
334#102C4×D311yescharacteristic
335C22⋊C8—21no—
336C22⋊C8—31no—
337C22⋊C8—31no—
338C22⋊C8—31no—
339C22⋊C8—31no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C22⋊C8—61no—
347#97—61no—
348C4×C8—21no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—31no—
352C4×C8—61no—
353C4×C8C22×D311yescharacteristic
354C4×C8C22×D311yescharacteristic
355C4×C8C22×D311yescharacteristic
356C23×C3—21no—
357C22×D3—21no—
358C22×D3—24no—
359C22×D3—21no—
360C22×D3—81no—
361C22×D3—81no—
362C22×D3C4×D411yescharacteristic
363C3×Q8—21no—
364C3×Q8—21no—
365C3×D4—21no—
366C3×D4#13112yes—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4C23×C411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375#67—24no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—81no—
379C2×Dic3C4×D411yescharacteristic
380C2×Dic3C4×D411yescharacteristic
381C2×Dic3C4×D411yescharacteristic
382D12—21no—
383D12—21no—
384D12—21no—
385D12—81no—
386D12#13112yes—
387C4×D3—21no—
388C4×D3—21no—
389C4×D3—21no—
390C4×D3—21no—
391C4×D3—41no—
392C4×D3—41no—
393C4×D3—41no—
394C4×D3—41no—
395C4×D3—41no—
396C4×D3—81no—
397C4×D3—81no—
398C3×C8—21no—
399C3×C8—21no—
400C3×C8—21no—
401C3×C8—21no—
402C3×C8—41no—
403C3×C8—41no—
404C3×C8—41no—
405C3⋊C8—21no—
406C3⋊C8—21no—
407C3⋊C8—21no—
408C3⋊C8—21no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412C24—61no—
413C2×Q8C2×C4×D311yescharacteristic
414C2×D4—31no—
415C2×D4—31no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—62no—
419C2×D4—62no—
420C2×D4C2×C4×D311yescharacteristic
421C22×C4—21no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—241no—
431C22×C4—241no—
432SD16—22no—
433SD16—62no—
434D8—62no—
435D8—62no—
436M16—241no—
437M16—241no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—31no—
442C2×C8—31no—
443C2×C8—31no—
444C2×C8—31no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—81no—
449C2×C8—241no—
450C2×C8—241no—
451C2×C8—241no—
452C2×C8—241no—
453C2×C8C2×C4×D311yescharacteristic
454C2×C8C2×C4×D311yescharacteristic
455C2×C8D3×D411yescharacteristic
456C2×C8D3×D411yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—31no—
459C4⋊C4—31no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4C2×C4×D311yescharacteristic
465C4⋊C4C2×C4×D311yescharacteristic
466C22⋊C4—21no—
467C22⋊C4—61no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—61no—
472C22⋊C4—61no—
473C22⋊C4—61no—
474C22⋊C4—61no—
475C42—21no—
476C42—31no—
477C42—31no—
478C42—31no—
479C42C23×D311yescharacteristic
480C22×C3—22no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3C2×C4×D411yescharacteristic
484C2×D3—21no—
485C2×D3—22no—
486C2×D3—21no—
487C2×D3—21no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—84no—
494C3×C4—21no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4C2×#13111yescharacteristic
499C3×C4C2×#13111yescharacteristic
500C3×C4#57611yescharacteristic
501C3×C4#57711yescharacteristic
502Dic3—21no—
503Dic3—21no—
504Dic3—21no—
505Dic3—41no—
506Dic3—41no—
507C23—21no—
508C23—31no—
509C23—64no—
510C23—61no—
511C23—61no—
512C23—241no—
513C23—241no—
514Q8—21no—
515Q8—21no—
516D4—21no—
517D4—32no—
518D4—64no—
519D4—61no—
520D4—61no—
521D4—61no—
522D4—241no—
523D4#90112yes—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—31no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—241no—
544C2×C4—241no—
545C2×C4—241no—
546C2×C4C22×C4×D311yescharacteristic
547C2×C4C2×D3×D411yescharacteristic
548C2×C4D3×#4111yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—21no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—61no—
557C8—61no—
558C8—61no—
559C8—61no—
560C8—121no—
561C8—121no—
562C8—121no—
563C2×C3—42no—
564C2×C3#292911yescharacteristic
565C2×C3#294611yescharacteristic
566C2×C3#295711yescharacteristic
567D3—41no—
568D3—42no—
569D3—81no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—61no—
574C22—61no—
575C22—61no—
576C22—62no—
577C22—121no—
578C22—121no—
579C22—121no—
580C22—121no—
581C22—121no—
582C22—244no—
583C22C4×D3×D411yescharacteristic
584C4—21no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—61no—
589C4—61no—
590C4—61no—
591C4—121no—
592C4—121no—
593C4C2×#90111yescharacteristic
594C4D3×#13111yescharacteristic
595C4#576211yescharacteristic
596C4#576911yescharacteristic
597C3#1791811yescharacteristic, complemented, Sylow
598C2—42no—
599C2—121no—
600C2—122no—
601C2—241no—
602C2#7942111yescharacteristic
603C2#7964711yescharacteristic
604C2#7989011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×C8×SD16C2
socleC22×C3C2×C4×D4
prime radical#4773C4
cosocleC3×C42C24
hypercenterC8×SD16D3
hypocenterC3#17918
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
1. #79908C2semidirect
3. #79426C2semidirect
4. C3×C8×SD16C2semidirect
5. #70937C2semidirect
6. #70897C2semidirect
7. #70840C2semidirect
8. #68535C2semidirect
9. #68521C2semidirect
10. #68233C2semidirect
11. #68216C2semidirect
12. #68172C2semidirect
13. #68152C2semidirect
14. C8×D24C2semidirect
15. #68049C2semidirect
19. C3×C8×Q8C22semidirect
27. #5024C22semidirect
33. #4838C22semidirect
48. C8×D12C22semidirect
52. C3×#334C22semidirect
53. C3×Q8⋊C8C22semidirect
55. C3×C82C22semidirect
56. #4535C22semidirect
59. #4483C22semidirect
60. #4480C22semidirect
61. #4479C22semidirect
63. #4459C22semidirect
64. #4458C22semidirect
65. #4454C22semidirect
66. #4454C22semidirect
67. #4454C22semidirect
77. C8×SD16D3semidirect
171. C8×Q8C2×D3semidirect
198. #334C2×D3semidirect
200. Q8⋊C8C2×D3semidirect
210. C82C2×D3semidirect
597. C3#17918semidirect

Maximal subgroups

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

subgroupindex
#799082
#796542
#794262
C3×C8×SD162
#709372
#708972
#708402
#685352
#685212
#682332
#682162
#681722
#681522
C8×D242
#680492
#179183