Finite Group Databasebeta

← #10765254

Subgroups — #10765254

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#79592C211yesmaximal, characteristic
2#79526C211yesmaximal, characteristic, complemented
3#79426C211yesmaximal, characteristic, complemented
4C3×#1581C211yesmaximal, characteristic, complemented
5#70878C211yesmaximal, characteristic, complemented
6#70863C211yesmaximal, characteristic, complemented
7#70843C211yesmaximal, characteristic, complemented
8#68575C211yesmaximal, characteristic, complemented
9#68558C211yesmaximal, characteristic, complemented
10#68172C211yesmaximal, characteristic, complemented
11#68169C211yesmaximal, characteristic, complemented
12#68149C211yesmaximal, characteristic, complemented
13#68148C211yesmaximal, characteristic, complemented
14#68079C211yesmaximal, characteristic, complemented
15#68074C211yesmaximal, characteristic, complemented
16#17918—31nomaximal, Sylow
17#5557C2211yescharacteristic
18C4×D3×D4C2211yescharacteristic
19C3×C4×D8C2211yescharacteristic
20C3×#436C2211yescharacteristic, complemented
21C3×C8×D4C2211yescharacteristic
22#5156C411yescharacteristic
23#5153C411yescharacteristic
24#5128—21no—
25#5119—21no—
26C8×#67—21no—
27C2×(C2×D3)⋊C8—21no—
28C4×#214C2211yescharacteristic
29C4×#213C2211yescharacteristic
30D4×C3⋊C8C2211yescharacteristic
31D4×C3⋊C8C2211yescharacteristic, complemented
32#4891C411yescharacteristic
33#4884C411yescharacteristic
34#4837C2211yescharacteristic
35#4834C2211yescharacteristic
36#4833C2211yescharacteristic
37#4830C2211yescharacteristic, complemented
38#4787C411yescharacteristic
39#4777C411yescharacteristic
40#4776C411yescharacteristic
41#4763C411yescharacteristic
42#4730—21no—
43#67⋊C8—21no—
44D3×C22⋊C8—21no—
45#4720—21no—
46C4×#204C2211yescharacteristic
47C8×D12C2211yescharacteristic
48#4685C2211yescharacteristic
49#4685C2211yescharacteristic
50#4685C2211yescharacteristic
51#4679C2211yescharacteristic, complemented
52C3×#335C2211yescharacteristic, complemented
53C3×D4⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic, complemented
55C3×#322C2211yescharacteristic, complemented
56D4⋊(C3⋊C8)C2211yescharacteristic
57D4⋊(C3⋊C8)C2211yescharacteristic, complemented
58#4484C2211yescharacteristic, complemented
59#4483C2211yescharacteristic
60#4480C2211yescharacteristic, complemented
61#4479C2211yescharacteristic, complemented
62#4459C2211yescharacteristic
63#4457C2211yescharacteristic, complemented
64#4456C2211yescharacteristic, complemented
65#4455C2211yescharacteristic, complemented
66#4454C2211yescharacteristic, complemented
67C8×C3⋊C8C2211yescharacteristic, complemented
68#3105—31no—
69#2993—31no—
70#2972—31no—
71#1586—31no—
72#1585—31no—
73#1582—31no—
74#1581D311yescharacteristic, complemented
75#1580—31no—
76SD16⋊C8—31no—
77C8×SD16—31no—
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×D8C2×C411yescharacteristic
86C2×C3×M16—21no—
87C22×C3×C8—21no—
88C3×C4×D4C2311yescharacteristic
89C3×C4×D4C2311yescharacteristic
90#938C2×C411yescharacteristic
91#936C2×C411yescharacteristic
92D4×Dic3C2×C411yescharacteristic
93C2×#214C2×C411yescharacteristic
94C2×#213C2×C411yescharacteristic
95C4×#67—21no—
96C4×#67—21no—
97#926—21no—
98C22×C3⋊C8—21no—
99C22×C3⋊C8—21no—
100C2×#204C2×C411yescharacteristic
101C2×ΓL(1,𝔽25)—21no—
102C2×D3×C8—21no—
103C2×D3×C8—21no—
104#897C2×C411yescharacteristic
105#893C2×C411yescharacteristic
106D3×C4⋊C4C2×C411yescharacteristic
107#890C2×C411yescharacteristic
108#883—21no—
109#67⋊C4—21no—
110D3×C22⋊C4—21no—
111#879—21no—
112C4×D12C2311yescharacteristic
113C42×D3C2311yescharacteristic
114C4×Dic6C2311yescharacteristic
115C3×#107C2×C411yescharacteristic
116C3×C4⋊C8C2311yescharacteristic
117C3×C4⋊C8C2311yescharacteristic
118C3×#102C2×C411yescharacteristic
119C3×#102C2×C411yescharacteristic
120C3×C22⋊C8—21no—
121C3×C22⋊C8—21no—
122C3×C4×C8C2311yescharacteristic
123C3×C4×C8C2311yescharacteristic
124C3×C4×C8C2311yescharacteristic
125#832C2×C411yescharacteristic
126#832C2×C411yescharacteristic
127#830—21no—
128#830—21no—
129#821C2×C411yescharacteristic
130(C2×D3)⋊C8—21no—
131(C2×D3)⋊C8—21no—
132(C2×D3)⋊C8—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8C2×C411yescharacteristic
135(C2×D3)⋊C8C2×C411yescharacteristic
136(C2×D3)⋊C8D411yescharacteristic
137(C2×D3)⋊C8D411yescharacteristic
138#817C2×C411yescharacteristic
139#816C2×C411yescharacteristic
140#815—21no—
141Dic3⋊C8—21no—
142Dic3⋊C8—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8C2×C411yescharacteristic
146Dic3⋊C8C2×C411yescharacteristic
147Dic3⋊C8D411yescharacteristic
148Dic3⋊C8D411yescharacteristic
149C8×Dic3—21no—
150C8×Dic3—21no—
151#810C2×C411yescharacteristic
152#809C2×C411yescharacteristic
153#808C2×C411yescharacteristic
154#807C2×C411yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
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×Q8—31no—
172C4×SD16—31no—
173C4×SD16—31no—
174C4×D8—31no—
175C4×D8C2×D311yescharacteristic
176#436—61no—
177#436C2×D311yescharacteristic, complemented
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#335C2×D311yescharacteristic, complemented
197#334—31no—
198#334—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201D4⋊C8—31no—
202D4⋊C8—31no—
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8C2×D311yescharacteristic
206D4⋊C8C2×D311yescharacteristic, complemented
207#322—31no—
208#322—31no—
209#322C2×D311yescharacteristic, complemented
210C82—31no—
211C23×D3—21no—
212C2×C3×D4C22×C411yescharacteristic
213C2×C3×D4C22×C411yescharacteristic
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C2×#67—21no—
217C2×#67—21no—
218C22×Dic3—21no—
219D3×D4—22no—
220D3×D4—22no—
221C2×D12C22×C411yescharacteristic
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—81no—
226C2×C4×D3—81no—
227C2×C4×D3C22×C411yescharacteristic
228C2×C4×D3C2×D411yescharacteristic
229C2×C4×D3#4111yescharacteristic
230C2×Dic6C22×C411yescharacteristic
231C3×D8—22no—
232C3×M16—81no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—81no—
237C2×C3×C8C22×C411yescharacteristic
238C2×C3×C8C22×C411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C3×C4⋊C4C22×C411yescharacteristic
242C3×C4⋊C4C22×C411yescharacteristic
243C3×C22⋊C4—21no—
244C3×C22⋊C4—21no—
245C3×C42C2411yescharacteristic
246#217—21no—
247#217—21no—
248#217—21no—
249#217—21no—
250#214—22no—
251#213—22no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3C22×C411yescharacteristic
258C4⋊Dic3C22×C411yescharacteristic
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262C4×Dic3—21no—
263C4×Dic3C22×C411yescharacteristic
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3#4111yescharacteristic
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—81no—
270C2×C3⋊C8—81no—
271C2×C3⋊C8C22×C411yescharacteristic
272C2×C3⋊C8C22×C411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275#204—22no—
276ΓL(1,𝔽25)—81no—
277D3×C8—81no—
278D3×C8—81no—
279C22×D4—31no—
280C23×C4—61no—
281C2×SD16—31no—
282C2×SD16—31no—
283C2×D8—31no—
284C2×D8C4×D311yescharacteristic
285C2×M16—21no—
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×Q8—31no—
299C4×D4—31no—
300C4×D4—31no—
301C4×D4—31no—
302C4×D4—61no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4C22×D311yescharacteristic
306C4×D4C22×D311yescharacteristic
307#117—61no—
308C2×C4⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C42—31no—
311#107—31no—
312#107C4×D311yescharacteristic
313#106—31no—
314#106—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—31no—
328#103—31no—
329#102—31no—
330#102—31no—
331#102—31no—
332#102—31no—
333#102C4×D311yescharacteristic
334#102C4×D311yescharacteristic
335C22⋊C8—21no—
336C22⋊C8—21no—
337C22⋊C8—31no—
338C22⋊C8—31no—
339C22⋊C8—31no—
340C22⋊C8—31no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C22⋊C8—61no—
347#97—61no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—61no—
352C4×C8—61no—
353C4×C8C22×D311yescharacteristic
354C4×C8C22×D311yescharacteristic
355C4×C8C22×D311yescharacteristic
356C23×C3—21no—
357C23×C3—21no—
358C22×D3—21no—
359C22×D3—24no—
360C22×D3—81no—
361C22×D3—81no—
362C22×D3C4×D411yescharacteristic
363C3×D4—21no—
364C3×D4—21no—
365C3×D4—21no—
366C3×D4#13112yes—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4C23×C411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376#67—24no—
377#67—81no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—81no—
382C2×Dic3C4×D411yescharacteristic
383C2×Dic3C4×D411yescharacteristic
384C2×Dic3C4×D411yescharacteristic
385D12—21no—
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—81no—
395C4×D3—81no—
396Dic6—21no—
397Dic6—21no—
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×Q8—31no—
414C2×D4—31no—
415C2×D4—61no—
416C2×D4—62no—
417C2×D4—62no—
418C2×D4—61no—
419C2×D4C2×C4×D311yescharacteristic
420C2×D4C2×C4×D311yescharacteristic
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—241no—
431C22×C4—241no—
432SD16—62no—
433SD16—62no—
434D8—22no—
435D8—62no—
436M16—81no—
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—31no—
458C4⋊C4—31no—
459C4⋊C4—61no—
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—21no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—61no—
472C22⋊C4—61no—
473C22⋊C4—61no—
474C22⋊C4—61no—
475C42—31no—
476C42—31no—
477C42—31no—
478C42—61no—
479C42C23×D311yescharacteristic
480C22×C3—22no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3C2×C4×D411yescharacteristic
487C2×D3—21no—
488C2×D3—22no—
489C2×D3—21no—
490C2×D3—21no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—84no—
494C3×C4—21no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4C2×#13111yescharacteristic
498C3×C4C2×#13111yescharacteristic
499C3×C4#57611yescharacteristic
500C3×C4#57711yescharacteristic
501Dic3—21no—
502Dic3—21no—
503Dic3—21no—
504Dic3—41no—
505Dic3—41no—
506Dic3—41no—
507C23—21no—
508C23—21no—
509C23—31no—
510C23—64no—
511C23—61no—
512C23—241no—
513C23—241no—
514Q8—61no—
515Q8—61no—
516D4—21no—
517D4—21no—
518D4—21no—
519D4—32no—
520D4—64no—
521D4—61no—
522D4—241no—
523D4#90712yes—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—31no—
527C2×C4—31no—
528C2×C4—31no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
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—81no—
565C2×C3#292911yescharacteristic
566C2×C3#294611yescharacteristic
567C2×C3#295711yescharacteristic
568D3—41no—
569D3—42no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—61no—
577C22—61no—
578C22—62no—
579C22—61no—
580C22—121no—
581C22—121no—
582C22—244no—
583C22C4×D3×D411yescharacteristic
584C4—21no—
585C4—41no—
586C4—41no—
587C4—61no—
588C4—61no—
589C4—61no—
590C4—121no—
591C4—121no—
592C4—121no—
593C4C2×#90711yescharacteristic
594C4D3×#13111yescharacteristic
595C4#575711yescharacteristic
596C4#577011yescharacteristic
597C3#1791811yescharacteristic, complemented, Sylow
598C2—42no—
599C2—81no—
600C2—121no—
601C2—122no—
602C2#7947811yescharacteristic
603C2#7958711yescharacteristic
604C2#7989911yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1581C2
socleC22×C3C2×C4×D4
prime radical#5156C4
cosocleC3×C42C24
hypercenter#1581D3
hypocenterC3#17918
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. #79526C2semidirect
3. #79426C2semidirect
4. C3×#1581C2semidirect
5. #70878C2semidirect
6. #70863C2semidirect
7. #70843C2semidirect
8. #68575C2semidirect
9. #68558C2semidirect
10. #68172C2semidirect
11. #68169C2semidirect
12. #68149C2semidirect
13. #68148C2semidirect
14. #68079C2semidirect
15. #68074C2semidirect
20. C3×#436C22semidirect
31. D4×C3⋊C8C22semidirect
37. #4830C22semidirect
51. #4679C22semidirect
52. C3×#335C22semidirect
54. C3×D4⋊C8C22semidirect
55. C3×#322C22semidirect
57. D4⋊(C3⋊C8)C22semidirect
58. #4484C22semidirect
60. #4480C22semidirect
61. #4479C22semidirect
63. #4457C22semidirect
64. #4456C22semidirect
65. #4455C22semidirect
66. #4454C22semidirect
67. C8×C3⋊C8C22semidirect
74. #1581D3semidirect
177. #436C2×D3semidirect
196. #335C2×D3semidirect
206. D4⋊C8C2×D3semidirect
209. #322C2×D3semidirect
597. C3#17918semidirect

Maximal subgroups

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

subgroupindex
#795922
#795262
#794262
C3×#15812
#708782
#708632
#708432
#685752
#685582
#681722
#681692
#681492
#681482
#680792
#680742
#179183