Finite Group Databasebeta

← #10764800

Subgroups — #10764800

Order 768.

662 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
1D3×#446C211yesmaximal, characteristic, complemented
2C4×#915C211yesmaximal, characteristic
3C8×#237C211yesmaximal, characteristic
4C3×#1585C211yesmaximal, characteristic, complemented
5#70957C211yesmaximal, characteristic, complemented
6#70895C211yesmaximal, characteristic, complemented
7#70843C211yesmaximal, characteristic, complemented
8#68568C211yesmaximal, characteristic, complemented
9#68560C211yesmaximal, characteristic, complemented
10#68231C211yesmaximal, characteristic, complemented
11#68219C211yesmaximal, characteristic, complemented
12#68166C211yesmaximal, characteristic
13#68154C211yesmaximal, characteristic, complemented
14#68087C211yesmaximal, characteristic, complemented
15#68070C211yesmaximal, characteristic, complemented
16#17872—31nomaximal, Sylow
17C2×#915C411yescharacteristic
18C4×D3×Q8C2211yescharacteristic
19C4×#237C2211yescharacteristic
20C3×#446C2211yescharacteristic
21C3×C4×SD16C2211yescharacteristic
22C3×C8×D4C2211yescharacteristic
23#5164C411yescharacteristic
24C8×#67—21no—
25C2×C8×Dic3—21no—
26C4×#216C2211yescharacteristic
27#5024C2211yescharacteristic
28C4×#214C2211yescharacteristic
29D4×C3⋊C8C2211yescharacteristic
30#4861C411yescharacteristic
31#4853C411yescharacteristic
32#4833C2211yescharacteristic
33D3×C4⋊C8—21no—
34D3×C4⋊C8C2211yescharacteristic
35#4831C2211yescharacteristic
36#4830C2211yescharacteristic
37#4828—21no—
38#4802C411yescharacteristic
39#4789C411yescharacteristic
40#4771C411yescharacteristic
41#4757C411yescharacteristic
42#67⋊C8—21no—
43#4718—21no—
44D3×#97—21no—
45#4701—21no—
46C4×Dic12C2211yescharacteristic
47C4×ΓL(1,𝔽25)C2211yescharacteristic
48#4685C2211yescharacteristic
49C4×D3×C8C2211yescharacteristic
50#4679C2211yescharacteristic
51C8×Dic6C2211yescharacteristic
52C3×#335C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic
55C3×#322C2211yescharacteristic
56#4535C2211yescharacteristic
57D4⋊(C3⋊C8)C2211yescharacteristic
58#4484C2211yescharacteristic
59#4484C2211yescharacteristic
60#4480C2211yescharacteristic
61#4479C2211yescharacteristic
62#4457C2211yescharacteristic
63#4456C2211yescharacteristic
64#4456C2211yescharacteristic
65#4454C2211yescharacteristic
66#4454C2211yescharacteristic
67#4454C2211yescharacteristic
68C8×#41—31no—
69C2×#446—31no—
70C4×#137—31no—
71#1587—31no—
72#1587—31no—
73#1585—31no—
74#1585D311yescharacteristic, complemented
75#1584—31no—
76#1584—31no—
77#1582—31no—
78#1582—31no—
79#1565—31no—
80#1479—31no—
81#1477—31no—
82#1451—31no—
83C2×D3×Q8C2×C411yescharacteristic
84C2×#237C2×C411yescharacteristic
85C2×C3×SD16C2×C411yescharacteristic
86C22×C3×C8—21no—
87C3×C4×Q8C2311yescharacteristic
88C3×C4×D4C2311yescharacteristic
89Q8×Dic3C2×C411yescharacteristic
90C2×#216C2×C411yescharacteristic
91D4×Dic3C2×C411yescharacteristic
92C2×#214C2×C411yescharacteristic
93C4×#67—21no—
94C2×C4×Dic3—21no—
95C22×C3⋊C8—21no—
96#915—24no—
97C2×Dic12C2×C411yescharacteristic
98C2×ΓL(1,𝔽25)C2×C411yescharacteristic
99C2×ΓL(1,𝔽25)C2×C411yescharacteristic
100C2×D3×C8—21no—
101C2×D3×C8D411yescharacteristic
102C2×D3×C8D411yescharacteristic
103#892C2×C411yescharacteristic
104D3×C4⋊C4—21no—
105D3×C4⋊C4C2×C411yescharacteristic
106#887—21no—
107#887C2×C411yescharacteristic
108#887C2×C411yescharacteristic
109#67⋊C4—21no—
110#877—21no—
111C42×D3—21no—
112C42×D3C2311yescharacteristic
113C4×Dic6—21no—
114C4×Dic6C2311yescharacteristic
115C4×Dic6C2311yescharacteristic
116C3×#106C2×C411yescharacteristic
117C3×C4⋊C8—21no—
118C3×C4⋊C8C2311yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×#103C2×C411yescharacteristic
121C3×#102C2×C411yescharacteristic
122C3×C22⋊C8—21no—
123C3×#97—21no—
124C3×C4×C8C2311yescharacteristic
125C3×C4×C8C2311yescharacteristic
126C3×C4×C8C2311yescharacteristic
127#835C2×C411yescharacteristic
128#832C2×C411yescharacteristic
129#830—21no—
130(C2×D3)⋊C8—21no—
131(C2×D3)⋊C8—21no—
132#818C2×C411yescharacteristic
133#816C2×C411yescharacteristic
134#816C2×C411yescharacteristic
135#815—21no—
136#815—21no—
137#815C2×C411yescharacteristic
138#815C2×C411yescharacteristic
139Dic3⋊C8—21no—
140Dic3⋊C8—21no—
141Dic3⋊C8—21no—
142Dic3⋊C8—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145C8×Dic3—21no—
146C8×Dic3—21no—
147C8×Dic3—21no—
148C8×Dic3D411yescharacteristic
149C8×Dic3D411yescharacteristic
150#810C2×C411yescharacteristic
151#810C2×C411yescharacteristic
152#808C2×C411yescharacteristic
153#807C2×C411yescharacteristic
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157#803—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#137—31no—
162C4×#41—31no—
163C2×C4×Q8—31no—
164#446—31no—
165#446—31no—
166#446—31no—
167#446—61no—
168#446—61no—
169#446C2×D311yescharacteristic
170C8×Q8—31no—
171C8×Q8—31no—
172#441—31no—
173#441—31no—
174#440—31no—
175#440—31no—
176C4×Q16—31no—
177C4×Q16—31no—
178C4×SD16—31no—
179C4×SD16C2×D311yescharacteristic
180C8×D4—31no—
181C8×D4—61no—
182C8×D4—61no—
183C8×D4C2×D311yescharacteristic
184#431—31no—
185#431—31no—
186#431—61no—
187#428—31no—
188C2×C4⋊C8—31no—
189C2×C4⋊C8—61no—
190#419—31no—
191#417—31no—
192C4×M16—31no—
193C2×#97—61no—
194C2×C4×C8—31no—
195C2×C4×C8—61no—
196#335—31no—
197#335C2×D311yescharacteristic
198#334—31no—
199#334—31no—
200Q8⋊C8—31no—
201Q8⋊C8—31no—
202Q8⋊C8—31no—
203Q8⋊C8—31no—
204Q8⋊C8—31no—
205Q8⋊C8C2×D311yescharacteristic
206D4⋊C8—31no—
207D4⋊C8C2×D311yescharacteristic
208#322—31no—
209#322—31no—
210#322—31no—
211#322C2×D311yescharacteristic
212C2×C3×Q8C22×C411yescharacteristic
213C2×C3×D4C22×C411yescharacteristic
214C22×C3×C4—21no—
215C2×#67—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218D3×Q8—22no—
219D3×Q8—22no—
220#237—22no—
221#237M1614yes—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3C22×C411yescharacteristic
225C2×C4×D3C2×D411yescharacteristic
226C2×C4×D3#4111yescharacteristic
227C2×Dic6—21no—
228C2×Dic6C22×C411yescharacteristic
229C2×Dic6C22×C411yescharacteristic
230C3×SD16—22no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—42no—
235C2×C3×C8C22×C411yescharacteristic
236C2×C3×C8C22×C411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C3×C4⋊C4—21no—
240C3×C4⋊C4C22×C411yescharacteristic
241C3×C4⋊C4C22×C411yescharacteristic
242C3×C22⋊C4—21no—
243C3×C42—21no—
244C3×C42C2411yescharacteristic
245#217—21no—
246#216—22no—
247#214—22no—
248(C2×D3)⋊C4—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3C22×C411yescharacteristic
251C4⋊Dic3C22×C411yescharacteristic
252Dic3⋊C4—21no—
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×Dic3—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—42no—
270C2×C3⋊C8C22×C411yescharacteristic
271C2×C3⋊C8C22×C411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274Dic12—22no—
275ΓL(1,𝔽25)—22no—
276ΓL(1,𝔽25)—22no—
277D3×C8—22no—
278D3×C8—41no—
279D3×C8—42no—
280C2×#41—31no—
281C22×Q8—31no—
282#137—64no—
283C2×Q16—31no—
284C2×Q16—31no—
285C2×SD16—31no—
286C2×SD16C4×D311yescharacteristic
287C2×M16—31no—
288C2×M16—31no—
289C22×C8—21no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8—61no—
293C22×C8—61no—
294C4×Q8—31no—
295C4×Q8—31no—
296C4×Q8—31no—
297C4×Q8—31no—
298C4×Q8—31no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×Q8C22×D311yescharacteristic
302C4×D4—31no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4C22×D311yescharacteristic
306#117—31no—
307#117—61no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—61no—
310C2×C42—31no—
311C2×C42—61no—
312C2×C42—61no—
313#107—31no—
314#107—31no—
315#106—31no—
316#106C4×D311yescharacteristic
317C4⋊C8—21no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—61no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8—61no—
326C4⋊C8—61no—
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—31no—
330#103—31no—
331#103—31no—
332#103—31no—
333#103—31no—
334#103C4×D311yescharacteristic
335#102—31no—
336#102C4×D311yescharacteristic
337C22⋊C8—21no—
338C22⋊C8—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341#97—21no—
342#97—31no—
343#97—31no—
344#97—61no—
345#97—61no—
346#97—61no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—31no—
352C4×C8—61no—
353C4×C8—61no—
354C4×C8—61no—
355C4×C8C22×D311yescharacteristic
356C4×C8C22×D311yescharacteristic
357C4×C8C22×D311yescharacteristic
358C23×C3—21no—
359C22×D3C4×D411yescharacteristic
360C3×Q8—21no—
361C3×Q8—21no—
362C3×D4—21no—
363C3×D4C2×M1612yes—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4C23×C411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372#67—24no—
373C2×Dic3—21no—
374C2×Dic3—24no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—42no—
379C2×Dic3C4×D411yescharacteristic
380C2×Dic3C4×D411yescharacteristic
381C2×Dic3C4×D411yescharacteristic
382C4×D3—21no—
383C4×D3—21no—
384C4×D3—41no—
385C4×D3—42no—
386C4×D3—42no—
387C4×D3C2×M1612yes—
388C4×D3#13112yes—
389Dic6—21no—
390Dic6—21no—
391Dic6—21no—
392Dic6—42no—
393Dic6C2×M1612yes—
394C3×C8—21no—
395C3×C8—21no—
396C3×C8—21no—
397C3×C8—21no—
398C3×C8—41no—
399C3×C8—41no—
400C3×C8—41no—
401C3⋊C8—21no—
402C3⋊C8—21no—
403C3⋊C8—21no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C3⋊C8#13712yescomplemented
408#41—34no—
409#41—62no—
410C2×Q8—31no—
411C2×Q8—31no—
412C2×Q8—61no—
413C2×Q8—62no—
414C2×Q8—62no—
415C2×Q8C2×C4×D311yescharacteristic
416C2×D4—61no—
417C2×D4C2×C4×D311yescharacteristic
418C22×C4—21no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426Q16—62no—
427Q16—62no—
428SD16—22no—
429SD16—62no—
430M16—62no—
431M16—62no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—31no—
436C2×C8—31no—
437C2×C8—31no—
438C2×C8—31no—
439C2×C8—42no—
440C2×C8—62no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—121no—
445C2×C8—122no—
446C2×C8—122no—
447C2×C8C2×C4×D311yescharacteristic
448C2×C8C2×C4×D311yescharacteristic
449C2×C8D3×D411yescharacteristic
450C2×C8D3×D411yescharacteristic
451C4⋊C4—21no—
452C4⋊C4—31no—
453C4⋊C4—31no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4C2×C4×D311yescharacteristic
461C4⋊C4C2×C4×D311yescharacteristic
462C22⋊C4—21no—
463C22⋊C4—61no—
464C22⋊C4—61no—
465C42—21no—
466C42—31no—
467C42—31no—
468C42—31no—
469C42—61no—
470C42—61no—
471C42—61no—
472C42—61no—
473C42—61no—
474C42—61no—
475C42C23×D311yescharacteristic
476C22×C3—22no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3C2×C4×D411yescharacteristic
480C2×D3—21no—
481C2×D3—21no—
482C2×D3#43512yes—
483C3×C4—21no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4C22×M1611yescharacteristic
488C3×C4C2×#13111yescharacteristic
489C3×C4C2×#13711yescharacteristic
490C3×C4#57511yescharacteristic
491Dic3—22no—
492Dic3—21no—
493Dic3—21no—
494Dic3—41no—
495Dic3—41no—
496Dic3—41no—
497Dic3#43512yes—
498C23—21no—
499C23—31no—
500Q8—21no—
501Q8—21no—
502Q8—32no—
503Q8—61no—
504Q8—61no—
505Q8—61no—
506Q8—122no—
507D4—21no—
508D4—64no—
509D4D3×M1612yes—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—31no—
514C2×C4—32no—
515C2×C4—32no—
516C2×C4—31no—
517C2×C4—31no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—64no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—121no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4C22×C4×D311yescharacteristic
532C2×C4C2×D3×D411yescharacteristic
533C2×C4D3×#4111yescharacteristic
534C8—21no—
535C8—21no—
536C8—21no—
537C8—21no—
538C8—32no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—61no—
543C8—61no—
544C8—61no—
545C8—121no—
546C8—121no—
547C8—121no—
548C2×C3—42no—
549C2×C3C2×#43511yescharacteristic
550C2×C3C4×#13711yescharacteristic
551C2×C3#295511yescharacteristic
552D3—22no—
553C22—22no—
554C22—32no—
555C22—41no—
556C22—41no—
557C22—61no—
558C22—61no—
559C22C4×D3×D411yescharacteristic
560C4—21no—
561C4—32no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—61no—
566C4—62no—
567C4—61no—
568C4—121no—
569C4—121no—
570C4—121no—
571C4C2×D3×M1611yescharacteristic
572C4D3×#13111yescharacteristic
573C4#577311yescharacteristic
574C4D3×#13711yescharacteristic
575C3#1787211yescharacteristic, complemented, Sylow
576C2—42no—
577C2—62no—
578C2D3×#43511yescharacteristic
579C2#7977411yescharacteristic
580C2#7988811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1585C2
socleC22×C3C2×C4×D4
prime radicalC2×#237C2×C4
cosocleC3×C42C24
hypercenter#1585D3
hypocenterC3#17872
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1585

Complements

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

NH ≅ G/Ntype
1. D3×#446C2semidirect
4. C3×#1585C2semidirect
5. #70957C2semidirect
6. #70895C2semidirect
7. #70843C2semidirect
8. #68568C2semidirect
9. #68560C2semidirect
10. #68231C2semidirect
11. #68219C2semidirect
13. #68154C2semidirect
14. #68087C2semidirect
15. #68070C2semidirect
74. #1585D3semidirect
407. C3⋊C8#137semidirect
575. C3#17872semidirect

Maximal subgroups

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

subgroupindex
D3×#4462
C4×#9152
C8×#2372
C3×#15852
#709572
#708952
#708432
#685682
#685602
#682312
#682192
#681662
#681542
#680872
#680702
#178723