Finite Group Databasebeta

← #10765678

Subgroups — #10765678

Order 768.

590 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79591C211yesmaximal, characteristic
2#79526C211yesmaximal, characteristic, complemented
3#79523C211yesmaximal, characteristic, complemented
4C3×#1589C211yesmaximal, characteristic, complemented
5#70879C211yesmaximal, characteristic, complemented
6#70860C211yesmaximal, characteristic, complemented
7#70860C211yesmaximal, characteristic, complemented
8#68572C211yesmaximal, characteristic, complemented
9#68560C211yesmaximal, characteristic, complemented
10#68174C211yesmaximal, characteristic, complemented
11#68170C211yesmaximal, characteristic, complemented
12#68159C211yesmaximal, characteristic, complemented
13#68156C211yesmaximal, characteristic, complemented
14#68056C211yesmaximal, characteristic, complemented
15#68048C211yesmaximal, characteristic, complemented
16#17954—31nomaximal, Sylow
17#5557C2211yescharacteristic
18C4×#237C2211yescharacteristic
19C3×C4×D8C2211yescharacteristic
20C3×#436C2211yescharacteristic
21C3×#436C2211yescharacteristic
22#5158C411yescharacteristic
23#5153C411yescharacteristic
24#5128—21no—
25#5127—21no—
26#5119—21no—
27#5118—21no—
28C4×#214C2211yescharacteristic
29C4×#214C2211yescharacteristic
30#5012C2211yescharacteristic
31D4×C3⋊C8C2211yescharacteristic
32#4888C411yescharacteristic
33#4877C411yescharacteristic
34#4837C2211yescharacteristic
35D3×C4⋊C8C2211yescharacteristic
36#4831C2211yescharacteristic
37#4830C2211yescharacteristic
38#4784C411yescharacteristic
39#4783C411yescharacteristic
40#4767C411yescharacteristic
41#4766C411yescharacteristic
42#4730—21no—
43#4728—21no—
44#4720—21no—
45#4719—21no—
46C4×Dic12C2211yescharacteristic
47C4×ΓL(1,𝔽25)C2211yescharacteristic
48#4685C2211yescharacteristic
49#4685C2211yescharacteristic
50#4679C2211yescharacteristic
51#4679C2211yescharacteristic
52C3×#335C2211yescharacteristic
53C3×D4⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic
55C3×C82C2211yescharacteristic
56D4⋊(C3⋊C8)C2211yescharacteristic
57D4⋊(C3⋊C8)C2211yescharacteristic
58#4484C2211yescharacteristic
59#4484C2211yescharacteristic
60#4479C2211yescharacteristic
61#4479C2211yescharacteristic
62#4458C2211yescharacteristic
63#4456C2211yescharacteristic
64#4456C2211yescharacteristic
65#4455C2211yescharacteristic
66#4454C2211yescharacteristic
67#4454C2211yescharacteristic
68#3102—31no—
69#2993—31no—
70#2989—31no—
71#1592—31no—
72#1592—31no—
73#1591—31no—
74#1589D311yescharacteristic, complemented
75#1588—31no—
76#1585—31no—
77#1584—31no—
78#1581—31no—
79#1571—31no—
80#1489—31no—
81#1483—31no—
82#1453—31no—
83C2×#237C2×C411yescharacteristic
84C2×C3×D8C2×C411yescharacteristic
85C2×C3×M16—21no—
86C2×C3×M16—21no—
87C3×C4×D4C2311yescharacteristic
88C3×C4×D4C2311yescharacteristic
89#938C2×C411yescharacteristic
90#936C2×C411yescharacteristic
91D4×Dic3C2×C411yescharacteristic
92C2×#214C2×C411yescharacteristic
93C2×#214C2×C411yescharacteristic
94C4×#67—21no—
95C4×#67—21no—
96#926—21no—
97C2×C4×Dic3—21no—
98C2×#208—21no—
99C22×C3⋊C8—21no—
100C2×Dic12C2×C411yescharacteristic
101C2×ΓL(1,𝔽25)—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×D3×C8—21no—
104#897C2×C411yescharacteristic
105#892C2×C411yescharacteristic
106#890C2×C411yescharacteristic
107#887C2×C411yescharacteristic
108#883—21no—
109#67⋊C4—21no—
110#879—21no—
111#877—21no—
112C42×D3C2311yescharacteristic
113C4×Dic6C2311yescharacteristic
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(C2×D3)⋊C8C2×C411yescharacteristic
130(C2×D3)⋊C8C2×C411yescharacteristic
131(C2×D3)⋊C8D411yescharacteristic
132(C2×D3)⋊C8D411yescharacteristic
133#818C2×C411yescharacteristic
134#816C2×C411yescharacteristic
135#816C2×C411yescharacteristic
136#815—21no—
137#815—21no—
138#815—21no—
139#815—21no—
140Dic3⋊C8—21no—
141Dic3⋊C8—21no—
142Dic3⋊C8—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146Dic3⋊C8C2×C411yescharacteristic
147Dic3⋊C8C2×C411yescharacteristic
148Dic3⋊C8D411yescharacteristic
149Dic3⋊C8D411yescharacteristic
150C8×Dic3—21no—
151#810C2×C411yescharacteristic
152#810C2×C411yescharacteristic
153#808C2×C411yescharacteristic
154#808C2×C411yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#529—31no—
161C4×#41—31no—
162#485—31no—
163#482—31no—
164#479—31no—
165#478—31no—
166#464—31no—
167#463—31no—
168#452—31no—
169#449—31no—
170#446—31no—
171#446—31no—
172#446—31no—
173C8×Q8—31no—
174C4×Q16—31no—
175C4×SD16—31no—
176C4×SD16—31no—
177C4×D8C2×D311yescharacteristic
178#436—31no—
179#436—61no—
180#436C2×D311yescharacteristic
181#436C2×D311yescharacteristic
182#435—61no—
183#435—61no—
184#435—61no—
185C8×D4—31no—
186#433—61no—
187#432—61no—
188#431—31no—
189#431—31no—
190#424—61no—
191#423—31no—
192#423—61no—
193C2×C4⋊C8—31no—
194C4×M16—31no—
195#335—31no—
196#335—31no—
197#335—31no—
198#335C2×D311yescharacteristic
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201Q8⋊C8—31no—
202Q8⋊C8—31no—
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8C2×D311yescharacteristic
206D4⋊C8C2×D311yescharacteristic
207#322—31no—
208#322—31no—
209#322—31no—
210C82C2×D311yescharacteristic
211C2×C3×D4C22×C411yescharacteristic
212C2×C3×D4C22×C411yescharacteristic
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C2×#67—21no—
216C2×#67—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219#237—41no—
220#237—41no—
221C2×C4×D3C22×C411yescharacteristic
222C2×C4×D3C2×D411yescharacteristic
223C2×C4×D3#4111yescharacteristic
224C2×Dic6C22×C411yescharacteristic
225C2×Dic6C22×C411yescharacteristic
226C3×D8—41no—
227C3×M16—81no—
228C3×M16—81no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8C22×C411yescharacteristic
233C2×C3×C8C22×C411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8C2×D411yescharacteristic
236C3×C4⋊C4C22×C411yescharacteristic
237C3×C4⋊C4C22×C411yescharacteristic
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C42C2411yescharacteristic
241#217—21no—
242#217—21no—
243#217—21no—
244#217—21no—
245#214—41no—
246#214—41no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3C22×C411yescharacteristic
252C4⋊Dic3C22×C411yescharacteristic
253Dic3⋊C4—21no—
254Dic3⋊C4—21no—
255Dic3⋊C4—21no—
256Dic3⋊C4—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3C22×C411yescharacteristic
262C4×Dic3C2×D411yescharacteristic
263C4×Dic3#4111yescharacteristic
264#208—81no—
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
273Dic12—41no—
274ΓL(1,𝔽25)—42no—
275ΓL(1,𝔽25)—81no—
276D3×C8—81no—
277C2×#41—31no—
278C2×Q16—31no—
279C2×SD16—31no—
280C2×SD16—31no—
281C2×D8C4×D311yescharacteristic
282C2×M16—21no—
283C2×M16—21no—
284C2×M16—61no—
285C2×M16—61no—
286C2×M16—61no—
287C22×C8—61no—
288C22×C8—61no—
289#126—61no—
290#125—31no—
291#124—31no—
292#123—61no—
293C22⋊Q8—31no—
294#121—31no—
295C4×Q8—31no—
296C4×Q8—31no—
297C4×Q8—31no—
298C4×D4—31no—
299C4×D4—61no—
300C4×D4—61no—
301C4×D4—61no—
302C4×D4C22×D311yescharacteristic
303C4×D4C22×D311yescharacteristic
304#117—31no—
305#117—61no—
306#117—61no—
307C2×C42—31no—
308C2×C42—61no—
309#107—31no—
310#107C4×D311yescharacteristic
311#106—31no—
312#106—31no—
313C4⋊C8—31no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—61no—
320C4⋊C8—61no—
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#103—31no—
330#103—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—
343#97—61no—
344#97—61no—
345#97—61no—
346#97—61no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—61no—
351C4×C8C22×D311yescharacteristic
352C4×C8C22×D311yescharacteristic
353C4×C8C22×D311yescharacteristic
354C23×C3—21no—
355C23×C3—21no—
356C22×D3C4×D411yescharacteristic
357C3×D4—21no—
358C3×D4—21no—
359C3×D4—21no—
360C3×D4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4C23×C411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370#67—42no—
371#67—81no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—42no—
377C2×Dic3—81no—
378C2×Dic3C4×D411yescharacteristic
379C2×Dic3C4×D411yescharacteristic
380C2×Dic3C4×D411yescharacteristic
381C4×D3—21no—
382C4×D3—21no—
383C4×D3—21no—
384C4×D3—21no—
385C4×D3—41no—
386Dic6—21no—
387Dic6—21no—
388Dic6—21no—
389Dic6—21no—
390C3×C8—22no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—21no—
394C3×C8—21no—
395C3×C8—41no—
396C3×C8—41no—
397C3⋊C8—21no—
398C3⋊C8—21no—
399C3⋊C8—21no—
400C3⋊C8—21no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404#41—121no—
405#41—121no—
406C2×Q8—31no—
407C2×Q8—31no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4C2×C4×D311yescharacteristic
411C2×D4C2×C4×D311yescharacteristic
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—61no—
418C22×C4—61no—
419Q16—121no—
420SD16—121no—
421SD16—121no—
422D8—41no—
423M16—81no—
424M16—81no—
425M16—122no—
426M16—241no—
427M16—241no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—31no—
434C2×C8—31no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—241no—
439C2×C8—241no—
440C2×C8C2×C4×D311yescharacteristic
441C2×C8C2×C4×D311yescharacteristic
442C2×C8D3×D411yescharacteristic
443C2×C8D3×D411yescharacteristic
444C4⋊C4—31no—
445C4⋊C4—31no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4C2×C4×D311yescharacteristic
453C4⋊C4C2×C4×D311yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—61no—
457C22⋊C4—61no—
458C22⋊C4—61no—
459C22⋊C4—61no—
460C22⋊C4—61no—
461C22⋊C4—61no—
462C42—31no—
463C42—31no—
464C42—31no—
465C42—61no—
466C42—61no—
467C42—61no—
468C42—61no—
469C42C23×D311yescharacteristic
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3C2×C4×D411yescharacteristic
477C2×D3—21no—
478C2×D3—21no—
479C2×D3—21no—
480C3×C4—21no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4#56811yescharacteristic
484C3×C4#56811yescharacteristic
485C3×C4#57611yescharacteristic
486C3×C4#57811yescharacteristic
487Dic3—21no—
488Dic3—21no—
489Dic3—21no—
490Dic3—41no—
491Dic3—41no—
492Dic3—41no—
493Dic3—41no—
494C23—21no—
495C23—21no—
496C23—31no—
497Q8—61no—
498Q8—61no—
499Q8—61no—
500Q8—61no—
501D4—21no—
502D4—21no—
503D4—21no—
504D4—21no—
505D4—122no—
506D4—241no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—31no—
510C2×C4—31no—
511C2×C4—31no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—121no—
525C2×C4—122no—
526C2×C4—241no—
527C2×C4C22×C4×D311yescharacteristic
528C2×C4C2×D3×D411yescharacteristic
529C2×C4D3×#4111yescharacteristic
530C8—22no—
531C8—21no—
532C8—21no—
533C8—21no—
534C8—21no—
535C8—41no—
536C8—41no—
537C8—61no—
538C8—61no—
539C8—61no—
540C8—61no—
541C8—121no—
542C8—121no—
543C8—121no—
544C2×C3—81no—
545C2×C3—81no—
546C2×C3#293011yescharacteristic
547C2×C3#295111yescharacteristic
548C2×C3#295611yescharacteristic
549D3—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—61no—
557C22—61no—
558C22—61no—
559C22C4×D3×D411yescharacteristic
560C4—21no—
561C4—41no—
562C4—41no—
563C4—61no—
564C4—61no—
565C4—61no—
566C4—121no—
567C4—121no—
568C4—121no—
569C4—121no—
570C4#574511yescharacteristic
571C4#575011yescharacteristic
572C4#575711yescharacteristic
573C4#577111yescharacteristic
574C3#1795411yescharacteristic, complemented, Sylow
575C2—81no—
576C2—81no—
577C2—121no—
578C2#7953711yescharacteristic
579C2#7958911yescharacteristic
580C2#7988911yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1589C2
socleC22×C3C2×C4×D4
prime radical#5158C4
cosocleC3×C42C24
hypercenter#1589D3
hypocenterC3#17954
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1589

Complements

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

NH ≅ G/Ntype
2. #79526C2semidirect
3. #79523C2semidirect
4. C3×#1589C2semidirect
5. #70879C2semidirect
6. #70860C2semidirect
7. #70860C2semidirect
8. #68572C2semidirect
9. #68560C2semidirect
10. #68174C2semidirect
11. #68170C2semidirect
12. #68159C2semidirect
13. #68156C2semidirect
14. #68056C2semidirect
15. #68048C2semidirect
74. #1589D3semidirect
574. C3#17954semidirect

Maximal subgroups

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

subgroupindex
#795912
#795262
#795232
C3×#15892
#708792
#708602
#685722
#685602
#681742
#681702
#681592
#681562
#680562
#680482
#179543