Finite Group Databasebeta

← #10836690

Subgroups — #10836690

Order 768.

690 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3080C211yesmaximal, characteristic, complemented
2C2×#5066C211yesmaximal, characteristic, complemented
3C4×#949C211yesmaximal, characteristic
4#82515C211yesmaximal, characteristic, complemented
5#71742C211yesmaximal, characteristic, complemented
6#70673C211yesmaximal, characteristic, complemented
7#70673C211yesmaximal, characteristic, complemented
8#70669C211yesmaximal, characteristic, complemented
9#70668C211yesmaximal, characteristic, complemented
10#70527C211yesmaximal, characteristic, complemented
11#70520C211yesmaximal, characteristic, complemented
12#70307C211yesmaximal, characteristic
13#70307C211yesmaximal, characteristic
14#70302C211yesmaximal, characteristic, complemented
15#70301C211yesmaximal, characteristic, complemented
16#24724—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19C2×#949C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C3×#479—21no—
22C3×#479—21no—
23C3×#478—21no—
24C3×#478—21no—
25C3×#428C2211yescharacteristic
26C3×#428C2211yescharacteristic
27C3×#423C2211yescharacteristic
28C3×#417C2211yescharacteristic
29C3×#417C2211yescharacteristic
30#5234C2211yescharacteristic
31#5234C2211yescharacteristic
32#5232C2211yescharacteristic
33C2×#823C2211yescharacteristic
34C2×#823C2211yescharacteristic
35C2×#822C2211yescharacteristic
36C2×#822C2211yescharacteristic
37#5066—21no—
38#5066—21no—
39#5066—21no—
40#5066—21no—
41#5026—21no—
42C4×#215—21no—
43#5014—21no—
44C4×#213—21no—
45#5001C2211yescharacteristic
46#5000—21no—
47#5000—21no—
48#4997C2211yescharacteristic
49#4997C2211yescharacteristic
50#4995C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#809C2211yescharacteristic
53C2×#808C2211yescharacteristic
54C2×#807C2211yescharacteristic
55#4926—21no—
56#4925—21no—
57#4924C2211yescharacteristic
58C2×C4⋊(C3⋊C8)C2211yescharacteristic
59C4×#208C2211yescharacteristic
60#4558C2211yescharacteristic
61#4558C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64#4530C2211yescharacteristic
65#4530C2211yescharacteristic
66#4530C2211yescharacteristic
67#4530C2211yescharacteristic
68#3080D311yescharacteristic, complemented
69C2×#478—31no—
70C4×AGL(1,ℤ/8)—31no—
71#2925—31no—
72#2079—31no—
73#2078—31no—
74#2051—31no—
75#2051—31no—
76#1998—31no—
77#1952—31no—
78#1951—31no—
79#1950—31no—
80#1931—31no—
81#1931—31no—
82#1800—31no—
83C2×C3×#41C2311yescharacteristic
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C3×#125—21no—
89C3×#125—21no—
90C3×#125—21no—
91C3×#125—21no—
92C3×C4×Q8D411yescharacteristic
93C3×C4×Q8D411yescharacteristic
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×C4×D4D411yescharacteristic
97C3×C4×D4D411yescharacteristic
98C3×#117—21no—
99C3×#117C2311yescharacteristic
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C42—21no—
105C2×C3×C42C2311yescharacteristic
106#949—22no—
107#949—22no—
108C2×#215—21no—
109C2×#213—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C2×#208C2311yescharacteristic
115C2×#208C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C4×D12—21no—
119C4×D12—21no—
120C4×D12—21no—
121C4×D12—21no—
122C3×#107—21no—
123C3×#107—21no—
124C3×#106—21no—
125C3×#106—21no—
126C3×C4⋊C8—21no—
127C3×C4⋊C8—21no—
128C3×#103—21no—
129C3×#103—21no—
130C3×#102—21no—
131C3×#102—21no—
132#835—21no—
133#832—21no—
134#830—21no—
135#830—21no—
136#830—21no—
137#830—21no—
138#823—22no—
139#823—22no—
140#822—22no—
141#822—22no—
142#809—21no—
143#809—21no—
144#809—21no—
145#809—21no—
146#809—21no—
147#809—21no—
148#808—21no—
149#808—21no—
150#808—21no—
151#807—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)—21no—
156C4⋊(C3⋊C8)D411yescharacteristic
157C4⋊(C3⋊C8)D411yescharacteristic
158C4⋊(C3⋊C8)D411yescharacteristic
159C4⋊(C3⋊C8)D411yescharacteristic
160#803—21no—
161#803—21no—
162#803—21no—
163C4×C3⋊C8—21no—
164C4×C3⋊C8—21no—
165C4×C3⋊C8—21no—
166C2×AGL(1,ℤ/8)—31no—
167C2×#125C2×D311yescharacteristic
168C4×#41C2×D311yescharacteristic
169C2×C4×D4—31no—
170#479—21no—
171#479—21no—
172#478—21no—
173#478—21no—
174#478—61no—
175#478—61no—
176#478—61no—
177#478—61no—
178#442—61no—
179#440—61no—
180C4×SD16—61no—
181C4×D8—61no—
182#433—61no—
183#433—61no—
184#432—61no—
185#431—61no—
186C2×#108—31no—
187C2×#108—31no—
188#428—31no—
189#428C2×D311yescharacteristic
190#428C2×D311yescharacteristic
191C2×#107—31no—
192C2×#106—31no—
193#423—31no—
194#423C2×D311yescharacteristic
195C2×C4⋊C8—31no—
196#418—31no—
197#417—31no—
198#417C2×D311yescharacteristic
199#417C2×D311yescharacteristic
200C2×#102—31no—
201C2×#102—31no—
202C2×#100—31no—
203C2×#100—31no—
204#408—31no—
205#408—31no—
206#405—31no—
207#405—31no—
208C4×M16—31no—
209#343—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#343—31no—
214#343—31no—
215#341—31no—
216#341—31no—
217C23×D3—21no—
218C3×#41—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41—21no—
222C2×C3×Q8C2×D411yescharacteristic
223C2×C3×D4—21no—
224C2×C3×D4C2×D411yescharacteristic
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4#4111yescharacteristic
231C22×C3×C4#4111yescharacteristic
232C22×C3×C4C2411yescharacteristic
233C22×Dic3—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12#4112yes—
240C2×C4×D3—41no—
241C2×C4×D3—41no—
242C2×C4×D3—82no—
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—41no—
256C3×C4⋊C4—41no—
257C3×C4⋊C4—41no—
258C3×C4⋊C4—41no—
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C22⋊C4—21no—
262C3×C22⋊C4—21no—
263C3×C22⋊C4—21no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42C2×D411yescharacteristic
268C3×C42C2×D411yescharacteristic
269C3×C42C2×D411yescharacteristic
270C3×C42C2×D411yescharacteristic
271#215—24no—
272#213—24no—
273(C2×D3)⋊C4—41no—
274(C2×D3)⋊C4—41no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277#208—21no—
278#208—21no—
279#208—21no—
280#208—21no—
281#208—21no—
282#208—21no—
283#208—21no—
284#208#4112yes—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8C2×D411yescharacteristic
294C2×C3⋊C8C2×D411yescharacteristic
295C2×C3⋊C8C2×D411yescharacteristic
296C2×C3⋊C8C2×D411yescharacteristic
297C2×#41C22×D311yescharacteristic
298C22×D4—31no—
299C23×C4—61no—
300AGL(1,ℤ/8)—62no—
301AGL(1,ℤ/8)—62no—
302C2×SD16—61no—
303C2×D8—61no—
304C2×M16—31no—
305C2×M16—31no—
306C2×M16—31no—
307C2×M16—31no—
308C2×M16C22×D311yescharacteristic
309C2×M16C22×D311yescharacteristic
310C22×C8—31no—
311C22×C8—31no—
312#125—21no—
313#125—21no—
314#125—21no—
315#125—21no—
316C4×Q8#6711yescharacteristic
317C4×Q8#6711yescharacteristic
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—61no—
321C4×D4—61no—
322C4×D4—61no—
323C4×D4—61no—
324C4×D4#6711yescharacteristic
325C4×D4#6711yescharacteristic
326#117—21no—
327#117C22×D311yescharacteristic
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C22⋊C4—61no—
334C2×C42—21no—
335C2×C42C22×D311yescharacteristic
336#108—62no—
337#108—62no—
338#107—21no—
339#107—21no—
340#107—61no—
341#107—61no—
342#107—61no—
343#106—21no—
344#106—21no—
345#106—61no—
346#106—61no—
347#106—61no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C4⋊C8—31no—
351C4⋊C8—31no—
352C4⋊C8—31no—
353C4⋊C8—31no—
354C4⋊C8—61no—
355C4⋊C8—61no—
356#103—21no—
357#103—21no—
358#103—61no—
359#102—21no—
360#102—21no—
361#102—61no—
362#102—61no—
363#102—61no—
364#102—61no—
365#102—61no—
366#102—61no—
367#102—61no—
368#100—62no—
369#100—62no—
370C22⋊C8—61no—
371C22⋊C8—61no—
372C22⋊C8—61no—
373C22⋊C8—61no—
374#97—61no—
375#97—61no—
376#97—61no—
377C4×C8—61no—
378C4×C8—61no—
379C4×C8—61no—
380C23×C3—21no—
381C23×C3C2×#4111yescharacteristic
382C22×D3—22no—
383C22×D3—41no—
384C22×D3—41no—
385C22×D3—81no—
386C22×D3—81no—
387C3×Q8—21no—
388C3×Q8—21no—
389C3×D4—21no—
390C3×D4—21no—
391C3×D4—41no—
392C3×D4—41no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—42no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—81no—
404C2×C3×C4—81no—
405C2×C3×C4—81no—
406C2×C3×C4#12111yescharacteristic
407C2×C3×C4#12111yescharacteristic
408C2×C3×C4#12111yescharacteristic
409C2×C3×C4#12111yescharacteristic
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C42−1+411yescharacteristic
415C2×C3×C42−1+411yescharacteristic
416C2×Dic3—41no—
417C2×Dic3—81no—
418D12—22no—
419D12—22no—
420D12—41no—
421D12—41no—
422C4×D3—82no—
423C4×D3—82no—
424C3×C8—41no—
425C3×C8—41no—
426C3⋊C8—22no—
427C3⋊C8—21no—
428C3⋊C8—21no—
429C3⋊C8—21no—
430C3⋊C8—21no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C3⋊C8—41no—
434C24—61no—
435#41—21no—
436#41—21no—
437#41—21no—
438#41—21no—
439C2×Q8C2×#6711yescharacteristic
440C2×D4—21no—
441C2×D4—32no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4C2×#6711yescharacteristic
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—61no—
454C22×C4—121no—
455C22×C4—121no—
456C22×C4—242no—
457C22×C4#23711yescharacteristic
458C22×C4#23711yescharacteristic
459C22×C4C23×D311yescharacteristic
460SD16—64no—
461D8—64no—
462M16—21no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—32no—
467M16—61no—
468M16—61no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16—61no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—31no—
477C2×C8—31no—
478C2×C8—31no—
479C2×C8—31no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4C2×#6711yescharacteristic
501C4⋊C4C2×#6711yescharacteristic
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—121no—
506C22⋊C4—121no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42D3×D411yescharacteristic
511C42D3×D411yescharacteristic
512C42C2×#6711yescharacteristic
513C42C2×#6711yescharacteristic
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3C2×#12111yescharacteristic
520C22×C3#55211yescharacteristic
521C22×C3#55611yescharacteristic
522C2×D3—42no—
523C2×D3—82no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—82no—
527C3×C4—21no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—41no—
532C3×C4—41no—
533C3×C4—41no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4#52511yescharacteristic
537C3×C4#53611yescharacteristic
538C3×C4#54711yescharacteristic
539C3×C4#54811yescharacteristic
540Dic3—81no—
541C23—21no—
542C23—62no—
543C23—121no—
544C23—121no—
545C23—241no—
546C23—241no—
547C23C2×#23711yescharacteristic
548Q8—21no—
549Q8—21no—
550D4—21no—
551D4—21no—
552D4—41no—
553D4—41no—
554D4—62no—
555D4—62no—
556D4—121no—
557D4—121no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—121no—
572C2×C4—242no—
573C2×C4—242no—
574C2×C4—241no—
575C2×C4#93911yescharacteristic
576C2×C4#93911yescharacteristic
577C2×C4#93911yescharacteristic
578C2×C4#93911yescharacteristic
579C2×C4C2×D3×D411yescharacteristic
580C2×C4#100711yescharacteristic
581C2×C4#100711yescharacteristic
582C2×C4D3×#4111yescharacteristic
583C2×C4D3×#4111yescharacteristic
584C2×C4C22×#6711yescharacteristic
585C8—41no—
586C8—41no—
587C8—62no—
588C8—61no—
589C8—61no—
590C8—61no—
591C8—61no—
592C8—121no—
593C8—121no—
594C8—121no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—81no—
598C2×C3#241511yescharacteristic
599C2×C3#306711yescharacteristic
600C2×C3#306811yescharacteristic
601D3—82no—
602C22—21no—
603C22—21no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—122no—
608C22—242no—
609C22—242no—
610C22—241no—
611C22—241no—
612C22#563911yescharacteristic
613C22#564111yescharacteristic
614C22C2×#93911yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—241no—
625C4#563111yescharacteristic
626C4#563311yescharacteristic
627C4#581611yescharacteristic
628C4#582811yescharacteristic
629C3#2472411yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—81no—
633C2—242no—
634C2#8281411yescharacteristic
635C2#8400111yescharacteristic
636C2#8400211yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3080C2
socleC22×C3C2×#121
prime radicalC2×#949C22
hypercenter#3080D3
hypocenterC3#24724
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3080

Complements

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

NH ≅ G/Ntype
1. C3×#3080C2semidirect
2. C2×#5066C2semidirect
4. #82515C2semidirect
5. #71742C2semidirect
6. #70673C2semidirect
7. #70673C2semidirect
8. #70669C2semidirect
9. #70668C2semidirect
10. #70527C2semidirect
11. #70520C2semidirect
14. #70302C2semidirect
15. #70301C2semidirect
68. #3080D3semidirect
629. C3#24724semidirect

Maximal subgroups

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

subgroupindex
C3×#30802
C2×#50662
C4×#9492
#825152
#717422
#706732
#706692
#706682
#705272
#705202
#703072
#703022
#703012
#247243