Finite Group Databasebeta

← #10951663

Subgroups — #10951663

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83920C211yesmaximal, characteristic, complemented
2C2×#5193C211yesmaximal, characteristic, complemented
3C2×D3×SD32C211yesmaximal, characteristic, complemented
4#80729C211yesmaximal, characteristic, complemented
5C3×#2186C211yesmaximal, characteristic, complemented
6#71475C211yesmaximal, characteristic, complemented
7#71467C211yesmaximal, characteristic, complemented
8#71466C211yesmaximal, characteristic, complemented
9#71074C211yesmaximal, characteristic, complemented
10#69223C211yesmaximal, characteristic, complemented
11#69208C211yesmaximal, characteristic, complemented
12#69203C211yesmaximal, characteristic, complemented
13#69195C211yesmaximal, characteristic, complemented
14#69110C211yesmaximal, characteristic, complemented
15#69103C211yesmaximal, characteristic, complemented
16#33831—31nomaximal, Sylow
17C2×C3×#135C2211yescharacteristic, complemented
18#5831—21no—
19C2×D3×Q16C2211yescharacteristic, complemented
20C2×D3×D8C2211yescharacteristic
21C22×D24C2211yescharacteristic
22#5609—21no—
23C2×C3×SD32C2211yescharacteristic, complemented
24C2×C3×D16C2211yescharacteristic, complemented
25C3×#466C2211yescharacteristic
26#5239—21no—
27#5193—24no—
28Dic3×Q16C2211yescharacteristic, complemented
29C2×#828C2211yescharacteristic
30C2×#828C2211yescharacteristic, complemented
31#5170—21no—
32#5166—21no—
33#5157C2211yescharacteristic, complemented
34#5155C2211yescharacteristic, complemented
35#5151C2211yescharacteristic
36C2×#827C2211yescharacteristic, complemented
37C2×#826C2211yescharacteristic
38C2×#826C2211yescharacteristic, complemented
39#5116C2211yescharacteristic, complemented
40C8×#67C2211yescharacteristic
41C2×#812C2211yescharacteristic, complemented
42#5036—21no—
43D3×SD32—24no—
44C2×#800C2211yescharacteristic
45C2×#800C2211yescharacteristic, complemented
46C2×D3×C16C2211yescharacteristic, complemented
47#4883C2211yescharacteristic
48#4872C2211yescharacteristic
49#4786—21no—
50#4773—21no—
51C3×#358C2211yescharacteristic, complemented
52C3×#357C2211yescharacteristic, complemented
53C3×C22⋊C16C2211yescharacteristic, complemented
54#4562C2211yescharacteristic, complemented
55#4509C2211yescharacteristic
56#4509C2211yescharacteristic, complemented
57(C2×D3)⋊C16C2211yescharacteristic, complemented
58#4502C2211yescharacteristic, complemented
59Dic3⋊C16C2211yescharacteristic, complemented
60#4492C2211yescharacteristic, complemented
61#4489C2211yescharacteristic, complemented
62C2×#509—31no—
63C22×SD32—31no—
64#3285—31no—
65D4×D8—31no—
66#2250—31no—
67#2220—31no—
68#2215—31no—
69#2210—31no—
70#2207—31no—
71#2187—31no—
72#2186D311yescharacteristic, complemented
73#2185—31no—
74#2184—31no—
75#2175—31no—
76#2168—31no—
77C2×C3×#41—21no—
78C2×D3×Q8—21no—
79C2×D3×D4—21no—
80C2×D3×D4—41no—
81C22×D12—21no—
82C3×#135—22no—
83C3×#135—41no—
84C2×C3×Q16C2311yescharacteristic
85C2×C3×SD16—21no—
86C2×C3×D8C2311yescharacteristic
87C2×C3×D8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#121—21no—
90Q8×Dic3—21no—
91C2×#216—21no—
92#940—21no—
93#938—21no—
94#938—41no—
95#937—41no—
96#936—21no—
97#936—41no—
98C2×#214—21no—
99C2×#213—21no—
100C2×#213—21no—
101C4×#67—41no—
102C4×#67D411yescharacteristic
103#924—21no—
104#924—41no—
105D3×Q16—22no—
106D3×Q16—41no—
107D3×D8—22no—
108D3×D8—41no—
109C2×Dic12C2311yescharacteristic
110C2×D24—22no—
111C2×D24—21no—
112C2×D24—41no—
113C2×D24C2311yescharacteristic
114C2×D24C2311yescharacteristic
115C2×D3×C8C2311yescharacteristic
116#895—21no—
117#893—21no—
118#884—41no—
119#882—41no—
120C3×SD32—22no—
121C3×D16—41no—
122C2×C3×C16C2311yescharacteristic
123C2×C3×C16C2311yescharacteristic
124C3×#107C2311yescharacteristic
125C3×#102—21no—
126#835—21no—
127#832—21no—
128#830D411yescharacteristic
129#828—22no—
130#828D414yescomplemented
131#827—22no—
132#826—22no—
133#826—22no—
134(C2×D3)⋊C8D411yescharacteristic
135#818C2311yescharacteristic
136#816—21no—
137Dic3⋊C8D411yescharacteristic
138C8×Dic3C2311yescharacteristic
139#812—22no—
140C2×C3⋊C16C2311yescharacteristic
141C2×C3⋊C16C2311yescharacteristic
142#809—21no—
143#809—21no—
144#800—22no—
145#800—41no—
146D3×C16—22no—
147C2×#135C2×D311yescharacteristic, complemented
148C22×Q16—31no—
149C22×D8—31no—
150C22×D8—31no—
151#548—61no—
152D42—61no—
153#509—64no—
154C2×SD32—31no—
155C2×SD32—31no—
156C2×SD32—31no—
157C2×SD32—31no—
158C2×SD32—31no—
159C2×SD32—64no—
160C2×SD32C2×D311yescharacteristic, complemented
161C2×D16—31no—
162C2×D16—31no—
163C2×D16C2×D311yescharacteristic, complemented
164C2×M32—31no—
165C22×C16—31no—
166#495—31no—
167#493—31no—
168#467—31no—
169#466—31no—
170#466—31no—
171#466C2×D311yescharacteristic
172#464—61no—
173#459—61no—
174#452—61no—
175#448—61no—
176#447—61no—
177#447—61no—
178C4×Q16—31no—
179C4×D8—31no—
180C8×D4—31no—
181#366—31no—
182C4⋊C16—31no—
183#358—31no—
184#358C2×D311yescharacteristic, complemented
185#357—31no—
186#357—31no—
187#357—31no—
188#357C2×D311yescharacteristic, complemented
189C22⋊C16—31no—
190C22⋊C16C2×D311yescharacteristic, complemented
191#346—31no—
192C23×D3—41no—
193C23×D3—41no—
194C3×#41—42no—
195C3×#41—81no—
196C2×C3×Q8—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—41no—
200C2×C3×D4—41no—
201C22×C3×C4—41no—
202C22×C3×C4C2×D411yescharacteristic
203C2×#67—41no—
204C2×#67—41no—
205C2×#67—41no—
206C2×#67—41no—
207C2×#67D811yescharacteristic
208D3×Q8—42no—
209D3×Q8—81no—
210D3×D4—42no—
211D3×D4—81no—
212D3×D4—84no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—42no—
216C2×D12—41no—
217C2×D12—41no—
218C2×D12—81no—
219C2×C4×D3—41no—
220C2×C4×D3—41no—
221C2×C4×D3C2×D411yescharacteristic
222C2×Dic6—21no—
223C2×Dic6—41no—
224C3×Q16—21no—
225C3×Q16C2×D412yescomplemented
226C3×SD16—42no—
227C3×D8—21no—
228C3×D8—21no—
229C3×D8—21no—
230C3×D8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2411yescharacteristic
235C3×C4⋊C4—21no—
236C3×C22⋊C4—41no—
237#217—41no—
238#217—41no—
239#217—41no—
240#217—41no—
241#217D811yescharacteristic
242#216—42no—
243#214—81no—
244#213—42no—
245#213—81no—
246(C2×D3)⋊C4—41no—
247(C2×D3)⋊C4—41no—
248(C2×D3)⋊C4D811yescharacteristic
249C4⋊Dic3—21no—
250C4⋊Dic3—41no—
251Dic3⋊C4—41no—
252Dic3⋊C4D811yescharacteristic
253C4×Dic3—41no—
254C4×Dic3C2×D411yescharacteristic
255C2×C3⋊C8C2×D411yescharacteristic
256C2×C3⋊C8C2×D411yescharacteristic
257Dic12—21no—
258Dic12—21no—
259D24—21no—
260D24—21no—
261D24—21no—
262D24—42no—
263D24C2×D412yes—
264D3×C8—21no—
265D3×C8—21no—
266C3×C16—21no—
267C3×C16—21no—
268C3⋊C16—21no—
269C3⋊C16C2×D412yescomplemented
270C2×#41—21no—
271C22×Q8—61no—
272C22×D4—61no—
273C22×D4—61no—
274C22×D4—121no—
275#135—22no—
276#135—41no—
277C2×Q16—31no—
278C2×Q16—61no—
279C2×Q16—62no—
280C2×Q16—121no—
281C2×Q16C22×D311yescharacteristic
282C2×SD16—21no—
283C2×SD16—61no—
284C2×D8—31no—
285C2×D8—31no—
286C2×D8—62no—
287C2×D8—61no—
288C2×D8—61no—
289C2×D8—62no—
290C2×D8—61no—
291C2×D8—121no—
292C2×D8—121no—
293C2×D8C22×D311yescharacteristic
294C2×D8C22×D311yescharacteristic
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297#127—61no—
298#124—61no—
299#124—121no—
300C22⋊Q8—61no—
301C22⋊Q8—121no—
302#121—21no—
303#121—61no—
304#121—61no—
305#121—121no—
306#121—121no—
307#120—121no—
308#120—121no—
309C4×Q8—61no—
310C4×D4—31no—
311C4×D4—61no—
312C4×D4—121no—
313SD32—22no—
314SD32—34no—
315SD32—62no—
316SD32—62no—
317SD32—62no—
318SD32—121no—
319D16—41no—
320D16—62no—
321D16—62no—
322M32—62no—
323C2×C16—31no—
324C2×C16—31no—
325C2×C16—62no—
326C2×C16C22×D311yescharacteristic
327C2×C16C22×D311yescharacteristic
328#107—31no—
329#107C22×D311yescharacteristic
330C4⋊C8—31no—
331#103—61no—
332#103—61no—
333#102—21no—
334#102—61no—
335#102—61no—
336#102—61no—
337C22⋊C8—31no—
338C22⋊C8—31no—
339C4×C8—31no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3C2×D811yescharacteristic
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—82no—
346C22×D3—82no—
347C22×D3—82no—
348C22×D3—82no—
349C22×D3—161no—
350C22×D3—161no—
351C22×D3C2×D811yescharacteristic
352C3×Q8—22no—
353C3×Q8—41no—
354C3×D4—41no—
355C3×D4—41no—
356C3×D4—41no—
357C3×D4—41no—
358C3×D4—82no—
359C3×D4—82no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—82no—
365C2×C3×C4C2×D811yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367#67—22no—
368#67—82no—
369#67—82no—
370#67—82no—
371#67—161no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3C2×D811yescharacteristic
375C2×Dic3C2×D811yescharacteristic
376C2×Dic3C2×D811yescharacteristic
377D12—22no—
378D12—41no—
379D12—41no—
380D12—41no—
381D12—82no—
382D12—82no—
383C4×D3—21no—
384C4×D3—21no—
385C4×D3—82no—
386C4×D3—82no—
387Dic6—41no—
388Dic6—41no—
389Dic6—82no—
390C3×C8—21no—
391C3×C8C22×D411yescharacteristic
392C3×C8H(5,𝔽2)11yescharacteristic
393C3⋊C8—21no—
394C3⋊C8—21no—
395C24—121no—
396C24—121no—
397#41—42no—
398#41—81no—
399C2×Q8—21no—
400C2×Q8—61no—
401C2×Q8—121no—
402C2×Q8—122no—
403C2×Q8—241no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—31no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—122no—
412C2×D4—121no—
413C2×D4—121no—
414C2×D4—121no—
415C2×D4—121no—
416C2×D4—122no—
417C2×D4—121no—
418C2×D4—121no—
419C2×D4—241no—
420C2×D4—241no—
421C2×D4—244no—
422C22×C4—31no—
423C22×C4—41no—
424C22×C4—121no—
425C22×C4—121no—
426C22×C4D3×D411yescharacteristic
427Q16—21no—
428Q16—61no—
429Q16—61no—
430Q16—122no—
431Q16D3×D412yescomplemented
432SD16—42no—
433SD16—241no—
434D8—21no—
435D8—21no—
436D8—21no—
437D8—21no—
438D8—32no—
439D8—61no—
440D8—61no—
441D8—61no—
442D8—122no—
443D8—122no—
444D8—241no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—31no—
448C2×C8—31no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8D3×D411yescharacteristic
452C2×C8C23×D311yescharacteristic
453C4⋊C4—21no—
454C4⋊C4—31no—
455C4⋊C4—61no—
456C4⋊C4—121no—
457C4⋊C4—121no—
458C22⋊C4—31no—
459C22⋊C4—31no—
460C22⋊C4—41no—
461C22⋊C4—121no—
462C22⋊C4—121no—
463C22⋊C4—121no—
464C22⋊C4—121no—
465C22⋊C4—121no—
466C22⋊C4—121no—
467C42—31no—
468C42—121no—
469C16—21no—
470C16—21no—
471C16—32no—
472C16—61no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3C22×D811yescharacteristic
483C2×D3—21no—
484C2×D3—21no—
485C2×D3—42no—
486C2×D3—81no—
487C2×D3—81no—
488C2×D3—81no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—164no—
492C2×D3—164no—
493C2×D3#50912yescomplemented
494C3×C4—21no—
495C3×C4—42no—
496C3×C4—81no—
497C3×C4D4211yescharacteristic
498C3×C4#57611yescharacteristic
499Dic3—21no—
500Dic3—21no—
501Dic3—21no—
502Dic3—81no—
503Dic3—81no—
504C23—31no—
505C23—41no—
506C23—41no—
507C23—121no—
508C23—121no—
509C23—242no—
510C23—242no—
511C23—242no—
512C23—242no—
513C23—481no—
514C23—481no—
515C23D3×D811yescharacteristic
516Q8—22no—
517Q8—41no—
518Q8—121no—
519Q8—121no—
520Q8—242no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—41no—
525D4—62no—
526D4—62no—
527D4—82no—
528D4—82no—
529D4—121no—
530D4—121no—
531D4—121no—
532D4—242no—
533D4—242no—
534D4—242no—
535D4—242no—
536D4—242no—
537D4—481no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—31no—
541C2×C4—31no—
542C2×C4—31no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—82no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—242no—
551C2×C4—242no—
552C2×C4D3×D811yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C8—21no—
555C8—61no—
556C8—61no—
557C8C2×D3×D411yescharacteristic
558C8#100411yescharacteristic
559C2×C3—41no—
560C2×C3—161no—
561C2×C3—161no—
562C2×C3D4×D811yescharacteristic
563C2×C3C2×#50911yescharacteristic
564C2×C3#341611yescharacteristic
565D3—22no—
566D3—82no—
567D3—161no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—32no—
572C22—61no—
573C22—61no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—122no—
581C22—241no—
582C22—241no—
583C22—241no—
584C22—241no—
585C22—241no—
586C22—484no—
587C22—484no—
588C22C2×D3×D811yescharacteristic
589C4—21no—
590C4—42no—
591C4—61no—
592C4—61no—
593C4—61no—
594C4—81no—
595C4—241no—
596C4—241no—
597C4#559011yescharacteristic
598C4#577411yescharacteristic
599C3#3383111yescharacteristic, complemented, Sylow
600C2—41no—
601C2—62no—
602C2—161no—
603C2—161no—
604C2—242no—
605C2—481no—
606C2#8002411yescharacteristic
607C2#8203211yescharacteristic
608C2D3×#50911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2186C2
socleC22×C3C22×D8
hypercenter#2186D3
hypocenterC3#33831
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2186

Complements

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

NH ≅ G/Ntype
1. #83920C2semidirect
2. C2×#5193C2semidirect
3. C2×D3×SD32C2semidirect
4. #80729C2semidirect
5. C3×#2186C2semidirect
6. #71475C2semidirect
7. #71467C2semidirect
8. #71466C2semidirect
9. #71074C2semidirect
10. #69223C2semidirect
11. #69208C2semidirect
12. #69203C2semidirect
13. #69195C2semidirect
14. #69110C2semidirect
15. #69103C2semidirect
17. C2×C3×#135C22semidirect
19. C2×D3×Q16C22semidirect
23. C2×C3×SD32C22semidirect
24. C2×C3×D16C22semidirect
28. Dic3×Q16C22semidirect
30. C2×#828C22semidirect
33. #5157C22semidirect
34. #5155C22semidirect
36. C2×#827C22semidirect
38. C2×#826C22semidirect
39. #5116C22semidirect
41. C2×#812C22semidirect
45. C2×#800C22semidirect
46. C2×D3×C16C22semidirect
51. C3×#358C22semidirect
52. C3×#357C22semidirect
53. C3×C22⋊C16C22semidirect
54. #4562C22semidirect
56. #4509C22semidirect
57. (C2×D3)⋊C16C22semidirect
58. #4502C22semidirect
59. Dic3⋊C16C22semidirect
60. #4492C22semidirect
61. #4489C22semidirect
72. #2186D3semidirect
130. #828D4semidirect
147. C2×#135C2×D3semidirect
160. C2×SD32C2×D3semidirect
163. C2×D16C2×D3semidirect
184. #358C2×D3semidirect
188. #357C2×D3semidirect
190. C22⋊C16C2×D3semidirect
225. C3×Q16C2×D4semidirect
269. C3⋊C16C2×D4semidirect
431. Q16D3×D4semidirect
493. C2×D3#509semidirect
599. C3#33831semidirect

Maximal subgroups

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

subgroupindex
#839202
C2×#51932
C2×D3×SD322
#807292
C3×#21862
#714752
#714672
#714662
#710742
#692232
#692082
#692032
#691952
#691102
#691032
#338313