Finite Group Databasebeta

← #10824904

Subgroups — #10824904

Order 768.

808 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#932C211yesmaximal, characteristic, complemented
2C2×#5014C211yesmaximal, characteristic, complemented
3C4×#908C211yesmaximal, characteristic, complemented
4#78520C211yesmaximal, characteristic, complemented
5C3×C4×#100C211yesmaximal, characteristic, complemented
6#70092C211yesmaximal, characteristic, complemented
7#70090C211yesmaximal, characteristic
8#70083C211yesmaximal, characteristic, complemented
9#70075C211yesmaximal, characteristic, complemented
10#69873C211yesmaximal, characteristic, complemented
11C4×#823C211yesmaximal, characteristic, complemented
12#69011C211yesmaximal, characteristic, complemented
13#68996C211yesmaximal, characteristic, complemented
14#68986C211yesmaximal, characteristic, complemented
15#68981C211yesmaximal, characteristic, complemented
16#23719—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#932C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C2×#908C2211yescharacteristic
21C4×#235C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C3×#100C2211yescharacteristic
24C2×C3×#100C2211yescharacteristic
25C3×C4×M16C2211yescharacteristic
26C3×C4×M16C2211yescharacteristic
27#5215C2211yescharacteristic
28C2×#832C2211yescharacteristic
29#5133C2211yescharacteristic
30#5131C2211yescharacteristic
31C2×#823C2211yescharacteristic
32C2×#823C2211yescharacteristic
33#5129C2211yescharacteristic
34#5129C2211yescharacteristic
35C2×#822C2211yescharacteristic
36C2×#822C2211yescharacteristic
37#5121C2211yescharacteristic
38#5119C2211yescharacteristic
39#5119C2211yescharacteristic
40#5018—21no—
41C4×#214—21no—
42#5014—21no—
43#5014—22no—
44#5014—22no—
45C4×#213—21no—
46#4966C2211yescharacteristic
47C2×#809C2211yescharacteristic
48#4964C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C4×#208C2211yescharacteristic
51C2×#803C2211yescharacteristic
52#4711—21no—
53#4710—21no—
54#4705—22no—
55C4×D24—21no—
56C4×#204—21no—
57#4690—22no—
58C3×#343C2211yescharacteristic
59C3×#343C2211yescharacteristic
60#4558C2211yescharacteristic
61#4558C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64C4×AGL(1,ℤ/8)—31no—
65C4×AGL(1,ℤ/8)—31no—
66C2×#442—31no—
67#2923—31no—
68#2090—31no—
69#2079—31no—
70#2063—31no—
71#2063—31no—
72#2051—31no—
73#2051—31no—
74#2045—31no—
75#2018—31no—
76#1778—31no—
77C4×#100—31no—
78C4×#100D311yescharacteristic, complemented
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C2×#235C2311yescharacteristic
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C2×C3×M16C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C3×C4×D4—21no—
89C3×C4×D4—21no—
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#214—21no—
96#932—22no—
97#932—22no—
98C2×#213—22no—
99C2×#213—21no—
100C2×#213—22no—
101C2×#213—22no—
102C4×#67—21no—
103C4×#67—21no—
104C2×(C2×D3)⋊C4—21no—
105#926C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C2×#208C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#908—22no—
112#908—22no—
113C2×D24—21no—
114C2×#204—21no—
115C4×D12—21no—
116C4×D12—21no—
117C4×D12—21no—
118C4×D12—21no—
119C4×D12D411yescharacteristic
120C4×D12D411yescharacteristic
121#872—21no—
122C42×D3—21no—
123C4×Dic6D411yescharacteristic
124C4×Dic6D411yescharacteristic
125C3×#100—22no—
126C3×#100—22no—
127C3×#97—21no—
128C3×#97—21no—
129C3×C4×C8—21no—
130C3×C4×C8—21no—
131#832—21no—
132#832—21no—
133#832—22no—
134#823—22no—
135#823—22no—
136#822—22no—
137#822—22no—
138#821—21no—
139#821—21no—
140#821—21no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143#818—21no—
144#817—21no—
145#816—21no—
146Dic3⋊C8—22no—
147Dic3⋊C8—22no—
148#810—21no—
149#809—21no—
150#809—22no—
151#808—21no—
152#808—22no—
153#807—21no—
154#803—21no—
155#803D412yes—
156#803D412yes—
157C4×C3⋊C8—21no—
158C2×AGL(1,ℤ/8)—31no—
159C2×AGL(1,ℤ/8)—31no—
160C22×D8—31no—
161C4×#41—31no—
162C2×C4×D4—31no—
163C2×C4×D4C2×D311yescharacteristic
164#442—61no—
165#442—62no—
166#442—62no—
167#442—61no—
168#440—61no—
169#440—61no—
170C4×SD16—61no—
171C4×SD16—61no—
172C4×D8—61no—
173C4×D8—61no—
174#433—62no—
175#432—62no—
176C2×#108—31no—
177C2×#108—31no—
178#428—31no—
179#428—31no—
180C2×#106—31no—
181#424—31no—
182#424—31no—
183#418—31no—
184#418—31no—
185#417—31no—
186#417—31no—
187C2×#102—31no—
188C2×#102—31no—
189C2×#100—31no—
190C2×#100—31no—
191C2×#100C2×D311yescharacteristic
192C2×#100C2×D311yescharacteristic
193#408—31no—
194#408—31no—
195C4×M16—31no—
196C4×M16C2×D311yescharacteristic
197C4×M16C2×D311yescharacteristic
198C2×#97—31no—
199#343—31no—
200#343—31no—
201#343C2×D311yescharacteristic
202#343C2×D311yescharacteristic
203#341—31no—
204#341—31no—
205C24×C3—21no—
206C23×D3—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4#4112yes—
213C22×C3×C4—21no—
214C22×C3×C4—41no—
215C22×C3×C4—41no—
216C22×C3×C4—82no—
217C22×C3×C4#4111yescharacteristic
218C22×C3×C4#4111yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C2×#67—21no—
221C22×Dic3—21no—
222#235—21no—
223#235—21no—
224#235—21no—
225#235—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12C2×D411yescharacteristic
232C2×D12#4112yes—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—41no—
237C2×C4×D3—41no—
238C2×C4×D3—82no—
239C2×Dic6C2×D411yescharacteristic
240C3×M16—21no—
241C3×M16—21no—
242C3×M16—21no—
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16—21no—
247C3×M16#4112yes—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C22⋊C4—41no—
255C3×C22⋊C4—41no—
256C3×C42C2×D411yescharacteristic
257C3×C42C2×D411yescharacteristic
258C3×C42C2×D411yescharacteristic
259C3×C42C2×D411yescharacteristic
260#217—21no—
261#214—24no—
262#213—28no—
263#213—24no—
264(C2×D3)⋊C4—21no—
265(C2×D3)⋊C4—21no—
266(C2×D3)⋊C4—41no—
267(C2×D3)⋊C4—41no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3C2×D411yescharacteristic
271C4⋊Dic3C2×D411yescharacteristic
272Dic3⋊C4—21no—
273Dic3⋊C4—21no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277#208—21no—
278#208—21no—
279#208—21no—
280#208#4112yes—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8C2×D412yes—
288C2×C3⋊C8C2×D412yes—
289D24—24no—
290#204—24no—
291C2×#41—31no—
292C22×D4—31no—
293C22×D4C22×D311yescharacteristic
294C23×C4—21no—
295C23×C4—61no—
296AGL(1,ℤ/8)—62no—
297AGL(1,ℤ/8)—62no—
298AGL(1,ℤ/8)—62no—
299AGL(1,ℤ/8)—62no—
300C2×SD16—61no—
301C2×SD16—61no—
302C2×D8—61no—
303C2×D8—62no—
304C2×D8—62no—
305C2×D8—61no—
306C2×D8—62no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C2×M16C22×D311yescharacteristic
312C2×M16C22×D311yescharacteristic
313C22×C8—31no—
314C22×C8—31no—
315C4×Q8—31no—
316C4×Q8—31no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—31no—
322C4×D4—31no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—61no—
329#117—31no—
330#117—61no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—61no—
335C2×C42—61no—
336C2×C42C22×D311yescharacteristic
337#108—62no—
338#108—62no—
339#107—61no—
340#107—61no—
341#106—61no—
342#106—62no—
343#106—61no—
344C4⋊C8—62no—
345C4⋊C8—62no—
346#103—61no—
347#103—61no—
348#102—62no—
349#102—61no—
350#102—61no—
351#102—61no—
352#102—61no—
353#102—61no—
354#102—61no—
355#102—62no—
356#100—22no—
357#100—22no—
358#100—62no—
359#100—62no—
360C22⋊C8—62no—
361C22⋊C8—62no—
362#97—21no—
363#97—21no—
364#97—32no—
365#97—32no—
366#97—61no—
367C4×C8—21no—
368C4×C8—21no—
369C4×C8—61no—
370C23×C3—22no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—81no—
374C23×C3—81no—
375C23×C3C2×#4111yescharacteristic
376C22×D3—21no—
377C22×D3—22no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—81no—
381C22×D3—81no—
382C3×D4—22no—
383C3×D4—22no—
384C3×D4—41no—
385C3×D4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—82no—
390C2×C3×C4—82no—
391C2×C3×C4—81no—
392C2×C3×C4#12111yescharacteristic
393C2×C3×C4#12111yescharacteristic
394C2×C3×C4#12111yescharacteristic
395C2×C3×C4#12111yescharacteristic
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402#67—41no—
403#67—41no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—41no—
408C2×Dic3—81no—
409D12—22no—
410D12—21no—
411D12—22no—
412D12—21no—
413D12—41no—
414D12—41no—
415C4×D3—41no—
416C4×D3—41no—
417C4×D3—81no—
418C4×D3—81no—
419C4×D3—82no—
420C4×D3—82no—
421Dic6—21no—
422Dic6—21no—
423C3×C8—22no—
424C3×C8—41no—
425C3×C8—41no—
426C3×C8—41no—
427C3⋊C8—22no—
428C3⋊C8—22no—
429C3⋊C8—22no—
430C3⋊C8—41no—
431C24—21no—
432C24—61no—
433#41—61no—
434#41—61no—
435#41—61no—
436#41—61no—
437C2×Q8—31no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—31no—
444C2×D4—32no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4#23512yes—
452C22×C4—21no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—82no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4—242no—
463C22×C4#23511yescharacteristic
464C22×C4#23511yescharacteristic
465C22×C4C23×D311yescharacteristic
466SD16—64no—
467SD16—64no—
468D8—64no—
469D8—68no—
470D8—64no—
471M16—21no—
472M16—21no—
473M16—21no—
474M16—21no—
475M16—21no—
476M16—21no—
477M16—21no—
478M16—32no—
479M16—61no—
480M16—61no—
481M16—61no—
482M16#23512yes—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—32no—
488C2×C8—32no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—31no—
498C4⋊C4—31no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—61no—
506C22⋊C4—61no—
507C22⋊C4—61no—
508C22⋊C4—121no—
509C22⋊C4—121no—
510C42—61no—
511C42—61no—
512C42—61no—
513C42D3×D411yescharacteristic
514C42D3×D411yescharacteristic
515C42D3×D411yescharacteristic
516C42D3×D411yescharacteristic
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—42no—
520C22×C3—82no—
521C22×C3—82no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3C2×#12111yescharacteristic
525C22×C3#52911yescharacteristic
526C22×C3#52911yescharacteristic
527C2×D3—41no—
528C2×D3—42no—
529C2×D3—41no—
530C2×D3—41no—
531C2×D3—82no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—82no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—81no—
540C3×C4#52511yescharacteristic
541C3×C4#52511yescharacteristic
542C3×C4#52511yescharacteristic
543C3×C4#53211yescharacteristic
544Dic3—41no—
545Dic3—41no—
546Dic3—41no—
547Dic3—81no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—61no—
552C23—62no—
553C23—81no—
554C23—81no—
555C23—121no—
556C23—121no—
557C23—241no—
558C23—241no—
559C23C2×#23511yescharacteristic
560Q8—61no—
561Q8—61no—
562D4—22no—
563D4—22no—
564D4—41no—
565D4—41no—
566D4—62no—
567D4—61no—
568D4—61no—
569D4—62no—
570D4—121no—
571D4—121no—
572D4—121no—
573D4—121no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—81no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—241no—
587C2×C4—241no—
588C2×C4—242no—
589C2×C4—242no—
590C2×C4—241no—
591C2×C4#88411yescharacteristic
592C2×C4#88411yescharacteristic
593C2×C4#88411yescharacteristic
594C2×C4#88411yescharacteristic
595C2×C4C2×#23511yescharacteristic
596C2×C4C2×#23511yescharacteristic
597C2×C4C2×D3×D411yescharacteristic
598C2×C4C2×D3×D411yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C8—22no—
602C8—41no—
603C8—41no—
604C8—41no—
605C8—62no—
606C8—62no—
607C8—62no—
608C8—121no—
609C2×C3—21no—
610C2×C3—21no—
611C2×C3—82no—
612C2×C3#239211yescharacteristic
613C2×C3#306811yescharacteristic
614C2×C3#306811yescharacteristic
615D3—81no—
616D3—82no—
617C22—21no—
618C22—21no—
619C22—42no—
620C22—82no—
621C22—82no—
622C22—81no—
623C22—81no—
624C22—121no—
625C22—122no—
626C22—121no—
627C22—121no—
628C22—242no—
629C22—242no—
630C22—241no—
631C22—241no—
632C22#547911yescharacteristic
633C22C2×#88411yescharacteristic
634C22#552811yescharacteristic
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—81no—
640C4—121no—
641C4—121no—
642C4—121no—
643C4—241no—
644C4#554711yescharacteristic
645C4#554711yescharacteristic
646C4#554711yescharacteristic
647C4#554811yescharacteristic
648C3#2371911yescharacteristic, complemented, Sylow
649C2—21no—
650C2—21no—
651C2—82no—
652C2—241no—
653C2—242no—
654C2#7358111yescharacteristic
655C2#7870311yescharacteristic
656C2#7872011yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#100C2
socleC22×C3C2×#121
prime radical#70090C2
hypercenterC4×#100D3
hypocenterC3#23719
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
1. C4×#932C2semidirect
2. C2×#5014C2semidirect
3. C4×#908C2semidirect
4. #78520C2semidirect
5. C3×C4×#100C2semidirect
6. #70092C2semidirect
8. #70083C2semidirect
9. #70075C2semidirect
10. #69873C2semidirect
11. C4×#823C2semidirect
12. #69011C2semidirect
13. #68996C2semidirect
14. #68986C2semidirect
15. #68981C2semidirect
78. C4×#100D3semidirect
648. C3#23719semidirect

Maximal subgroups

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

subgroupindex
C4×#9322
C2×#50142
C4×#9082
#785202
C3×C4×#1002
#700922
#700902
#700832
#700752
#698732
C4×#8232
#690112
#689962
#689862
#689812
#237193