Finite Group Databasebeta

← #10844986

Subgroups — #10844986

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5163C211yesmaximal, characteristic
2#78949C211yesmaximal, characteristic, complemented
3#78921C211yesmaximal, characteristic
4#78544C211yesmaximal, characteristic, complemented
5C3×#1977C211yesmaximal, characteristic, complemented
6#72053C211yesmaximal, characteristic, complemented
7#70715C211yesmaximal, characteristic, complemented
8#70695C211yesmaximal, characteristic, complemented
9#70339C211yesmaximal, characteristic, complemented
10#70317C211yesmaximal, characteristic, complemented
11#70107C211yesmaximal, characteristic, complemented
12Dic3×#101C211yesmaximal, characteristic, complemented
13#70072C211yesmaximal, characteristic, complemented
14#70064C211yesmaximal, characteristic
15#69892C211yesmaximal, characteristic, complemented
16#25315—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#940C2211yescharacteristic
21#5517C2211yescharacteristic
22C2×C3×#108C2211yescharacteristic
23C3×#419C2211yescharacteristic
24C3×#418C2211yescharacteristic
25C2×C3×#101C2211yescharacteristic
26C2×C3×#100C2211yescharacteristic
27C3×#405C2211yescharacteristic
28C2×#836C2211yescharacteristic
29#5225C2211yescharacteristic
30C2×#833C2211yescharacteristic
31#5215C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5163—21no—
34#5163—21no—
35#5163—21no—
36#5163—21no—
37#5131C2211yescharacteristic
38C2×#823C2211yescharacteristic
39C2×#823C2211yescharacteristic
40C2×#822C2211yescharacteristic
41#5122C2211yescharacteristic
42#5121C2211yescharacteristic
43#5118C2211yescharacteristic
44Dic3×M16C2211yescharacteristic
45Dic3×M16C2211yescharacteristic
46C2×#821C2211yescharacteristic
47#5102C2211yescharacteristic
48C2×Dic3⋊C8C2211yescharacteristic
49#5001C2211yescharacteristic
50#4999C2211yescharacteristic
51#4995C2211yescharacteristic
52#4799—21no—
53#4796—21no—
54#4793—21no—
55#4791—21no—
56#4764—21no—
57#4764—21no—
58#4760—21no—
59#4760—21no—
60#4701—21no—
61#4701—21no—
62#4679—21no—
63C8×Dic6—21no—
64#4558C2211yescharacteristic
65#4554C2211yescharacteristic
66#4554C2211yescharacteristic
67#4552C2211yescharacteristic
68#3138—31no—
69#3078—31no—
70C2×#460—31no—
71#2962—31no—
72#2063—31no—
73#2063—31no—
74#2046—31no—
75#2046—31no—
76#1977—31no—
77#1977D311yescharacteristic, complemented
78#1965—31no—
79#1950—31no—
80#1931—31no—
81#1931—31no—
82C4×#101—31no—
83C22×C3×Q8C2311yescharacteristic
84C22×C3×D4C2311yescharacteristic
85C22×#67—21no—
86C22×#67—21no—
87C22×D12C2311yescharacteristic
88C2×C3×SD16—22no—
89C2×C3×SD16—21no—
90C2×C3×SD16—21no—
91C2×C3×SD16—41no—
92C2×C3×M16C2311yescharacteristic
93C2×C3×M16C2311yescharacteristic
94C2×C3×M16C2311yescharacteristic
95C22×C3×C8C2311yescharacteristic
96C3×#117C2311yescharacteristic
97Q8×Dic3—21no—
98Q8×Dic3—21no—
99Q8×Dic3—21no—
100Q8×Dic3—21no—
101C2×#215—22no—
102C2×#215—21no—
103C2×#215—21no—
104C2×#215—41no—
105#940—21no—
106#940—21no—
107#940—21no—
108#940—21no—
109#926C2311yescharacteristic
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×#208C2311yescharacteristic
115C2×#208C2311yescharacteristic
116C2×#208C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#890D411yescharacteristic
119#890D411yescharacteristic
120#888D411yescharacteristic
121#888D411yescharacteristic
122#878—21no—
123#878—21no—
124C4×Dic6—21no—
125C4×Dic6—21no—
126C3×#108—41no—
127C3×#103—21no—
128C3×#103—21no—
129C3×#102—21no—
130C3×#102—21no—
131C3×#101—22no—
132C3×#100—22no—
133C3×#97—21no—
134C3×C4×C8—21no—
135#836—22no—
136#835—21no—
137#835—21no—
138#833—41no—
139#832—21no—
140#832—21no—
141#832—21no—
142#832—21no—
143#823—22no—
144#823—41no—
145#822—41no—
146#821—21no—
147#821—21no—
148#821—21no—
149#821—21no—
150#818—21no—
151#817—21no—
152#815—21no—
153#815—21no—
154#815—21no—
155Dic3⋊C8—21no—
156Dic3⋊C8—21no—
157Dic3⋊C8—21no—
158Dic3⋊C8—21no—
159Dic3⋊C8D411yescharacteristic
160Dic3⋊C8D411yescharacteristic
161Dic3⋊C8D411yescharacteristic
162Dic3⋊C8D411yescharacteristic
163C8×Dic3—21no—
164C8×Dic3—21no—
165C8×Dic3—21no—
166#809—21no—
167#809—21no—
168#808—21no—
169#807—21no—
170C4⋊(C3⋊C8)—21no—
171C4⋊(C3⋊C8)—21no—
172C22×SD16—31no—
173C22×SD16C2×D311yescharacteristic
174#533—31no—
175C2×#127—31no—
176C2×C4×Q8—31no—
177#490—61no—
178#490—61no—
179#486—61no—
180#486—61no—
181#479—61no—
182#479—61no—
183#477—61no—
184#475—61no—
185#460—61no—
186#460—61no—
187#460—61no—
188#460—61no—
189#446—61no—
190#446—61no—
191#446—61no—
192C8×Q8—61no—
193C2×#108—31no—
194C2×#108C2×D311yescharacteristic
195#428—31no—
196#428—31no—
197#424—31no—
198#424—31no—
199#423—31no—
200C2×C4⋊C8—31no—
201#419—31no—
202#419C2×D311yescharacteristic
203#418—31no—
204#418—31no—
205#418—31no—
206#418C2×D311yescharacteristic
207C2×#102—31no—
208C2×#102—31no—
209C2×#101—31no—
210C2×#101C2×D311yescharacteristic
211C2×#100—31no—
212C2×#100—31no—
213C2×#100—31no—
214C2×#100C2×D311yescharacteristic
215#405—31no—
216#405C2×D311yescharacteristic
217C4×M16—31no—
218C4×M16—31no—
219#343—31no—
220#343—31no—
221#341—31no—
222#341—31no—
223C24×C3—21no—
224C23×D3—21no—
225C2×C3×Q8—21no—
226C2×C3×Q8—21no—
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8#4112yes—
231C2×C3×D4—21no—
232C2×C3×D4—21no—
233C2×C3×D4—21no—
234C2×C3×D4—21no—
235C2×C3×D4—21no—
236C2×C3×D4—21no—
237C22×C3×C4—21no—
238C22×C3×C4C2411yescharacteristic
239C2×#67—41no—
240C2×#67—41no—
241C2×#67—41no—
242C2×#67—41no—
243C2×#67—82no—
244C2×#67—82no—
245C22×Dic3—21no—
246C22×Dic3#4111yescharacteristic
247C22×Dic3#4111yescharacteristic
248C2×D12—21no—
249C2×D12—21no—
250C2×D12—21no—
251C2×D12—21no—
252C2×D12—21no—
253C2×D12—21no—
254C2×Dic6—21no—
255C2×Dic6—21no—
256C3×SD16—42no—
257C3×SD16—42no—
258C3×M16—21no—
259C3×M16—21no—
260C3×M16—21no—
261C3×M16—21no—
262C3×M16—21no—
263C3×M16#4112yes—
264C2×C3×C8—21no—
265C2×C3×C8—21no—
266C2×C3×C8—21no—
267C2×C3×C8—21no—
268C2×C3×C8—21no—
269C2×C3×C8C2×D411yescharacteristic
270C2×C3×C8C2×D411yescharacteristic
271C3×C4⋊C4C2×D411yescharacteristic
272C3×C4⋊C4C2×D411yescharacteristic
273C3×C22⋊C4—21no—
274C3×C42—21no—
275#217—21no—
276#215—42no—
277#215—42no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—41no—
283C4⋊Dic3—41no—
284C4⋊Dic3C2×D411yescharacteristic
285C4⋊Dic3C2×D411yescharacteristic
286Dic3⋊C4—21no—
287Dic3⋊C4—21no—
288Dic3⋊C4—21no—
289Dic3⋊C4—21no—
290C4×Dic3—21no—
291C4×Dic3—41no—
292C4×Dic3—41no—
293C4×Dic3C2×D411yescharacteristic
294C4×Dic3C2×D411yescharacteristic
295C4×Dic3C2×D411yescharacteristic
296C4×Dic3C2×D411yescharacteristic
297#208—21no—
298#208—21no—
299#208—21no—
300#208—21no—
301#208—21no—
302#208—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—21no—
307C2×C3⋊C8—21no—
308C2×C3⋊C8C2×D411yescharacteristic
309C2×C3⋊C8C2×D411yescharacteristic
310C22×Q8C22×D311yescharacteristic
311C22×D4—31no—
312C22×D4—61no—
313C22×D4—61no—
314C22×D4C22×D311yescharacteristic
315C2×SD16—22no—
316C2×SD16—21no—
317C2×SD16—21no—
318C2×SD16—41no—
319C2×SD16—61no—
320C2×SD16—61no—
321C2×SD16—62no—
322C2×SD16—121no—
323C2×M16—31no—
324C2×M16—31no—
325C2×M16—31no—
326C2×M16C22×D311yescharacteristic
327C2×M16C22×D311yescharacteristic
328C2×M16C22×D311yescharacteristic
329C22×C8—31no—
330C22×C8C22×D311yescharacteristic
331C4⋊Q8—31no—
332C4⋊Q8—31no—
333#127—61no—
334#127—61no—
335#127—61no—
336#127—61no—
337#125—31no—
338#125—31no—
339C22⋊Q8—61no—
340C22⋊Q8—61no—
341C4×Q8—61no—
342C4×Q8—61no—
343C4×Q8—61no—
344C4×Q8—61no—
345C4×Q8—61no—
346C4×Q8—61no—
347#117—31no—
348#117C22×D311yescharacteristic
349C2×C4⋊C4—31no—
350C2×C4⋊C4—61no—
351C2×C42—31no—
352C2×C42—61no—
353#108—41no—
354#108—121no—
355#107—61no—
356#107—61no—
357#106—61no—
358#106—61no—
359C4⋊C8—31no—
360C4⋊C8—31no—
361C4⋊C8—31no—
362C4⋊C8—31no—
363C4⋊C8—61no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366C4⋊C8—61no—
367C4⋊C8—61no—
368C4⋊C8—61no—
369#103—21no—
370#103—21no—
371#103—61no—
372#103—61no—
373#102—21no—
374#102—21no—
375#102—61no—
376#102—61no—
377#102—61no—
378#102—61no—
379#102—61no—
380#102—61no—
381#102—61no—
382#102—61no—
383#102—61no—
384#102—61no—
385#101—22no—
386#101—62no—
387#100—22no—
388#100—62no—
389#100—121no—
390#100—121no—
391#97—21no—
392#97—61no—
393#97—61no—
394#97—61no—
395C4×C8—21no—
396C4×C8—61no—
397C4×C8—61no—
398C4×C8—61no—
399C23×C3—41no—
400C23×C3—41no—
401C23×C3—41no—
402C23×C3—81no—
403C23×C3—81no—
404C23×C3C2×#4111yescharacteristic
405C22×D3—41no—
406C22×D3—41no—
407C22×D3—41no—
408C22×D3—81no—
409C22×D3—81no—
410C3×Q8—22no—
411C3×Q8—22no—
412C3×Q8—41no—
413C3×Q8—41no—
414C3×D4—41no—
415C3×D4—41no—
416C3×D4—41no—
417C3×D4—41no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—22no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C22×D411yescharacteristic
425C2×C3×C4C2×#4111yescharacteristic
426C2×C3×C4C2×#4111yescharacteristic
427C2×C3×C4H(5,𝔽2)11yescharacteristic
428C2×C3×C42−1+411yescharacteristic
429#67—82no—
430#67—82no—
431#67—82no—
432#67—82no—
433C2×Dic3—22no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—81no—
439C2×Dic3#12111yescharacteristic
440C2×Dic3#12111yescharacteristic
441C2×Dic3#12111yescharacteristic
442C2×Dic3#12111yescharacteristic
443D12—41no—
444D12—41no—
445D12—41no—
446D12—41no—
447Dic6—42no—
448Dic6—81no—
449C3×C8—21no—
450C3×C8—21no—
451C3×C8—22no—
452C3×C8—41no—
453C3×C8—41no—
454C3⋊C8—21no—
455C3⋊C8—21no—
456C3⋊C8—41no—
457C3⋊C8—41no—
458C3⋊C8—41no—
459C24—21no—
460C24—61no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—61no—
467C2×Q8—61no—
468C2×Q8#23512yes—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—61no—
476C2×D4—61no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—61no—
481C2×D4—121no—
482C2×D4—121no—
483C2×D4—121no—
484C2×D4—121no—
485C2×D4—242no—
486C2×D4—242no—
487C22×C4—21no—
488C22×C4—31no—
489C22×C4—31no—
490C22×C4—61no—
491C22×C4C23×D311yescharacteristic
492SD16—42no—
493SD16—42no—
494SD16—122no—
495SD16—122no—
496M16—21no—
497M16—21no—
498M16—21no—
499M16—21no—
500M16—21no—
501M16—61no—
502M16—61no—
503M16—61no—
504M16—61no—
505M16—61no—
506M16—61no—
507M16#23712yes—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—31no—
514C2×C8—31no—
515C2×C8—61no—
516C2×C8—61no—
517C2×C8—61no—
518C2×C8—61no—
519C2×C8—61no—
520C2×C8D3×D411yescharacteristic
521C2×C8D3×D411yescharacteristic
522C4⋊C4—31no—
523C4⋊C4—31no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—61no—
531C4⋊C4—61no—
532C4⋊C4—121no—
533C4⋊C4—121no—
534C4⋊C4D3×D411yescharacteristic
535C4⋊C4D3×D411yescharacteristic
536C22⋊C4—21no—
537C22⋊C4—61no—
538C42—21no—
539C42—31no—
540C42—31no—
541C42—31no—
542C42—31no—
543C42—61no—
544C42—121no—
545C42—121no—
546C22×C3—21no—
547C22×C3—21no—
548C22×C3—81no—
549C22×C3—81no—
550C22×C3—81no—
551C22×C3—161no—
552C22×C3—161no—
553C22×C3C2×#12111yescharacteristic
554C22×C3#52911yescharacteristic
555C22×C3#54211yescharacteristic
556C2×D3—81no—
557C2×D3—81no—
558C2×D3—81no—
559C2×D3—161no—
560C2×D3—161no—
561C3×C4—42no—
562C3×C4—41no—
563C3×C4—41no—
564C3×C4#54611yescharacteristic
565C3×C4#54811yescharacteristic
566C3×C4#54811yescharacteristic
567C3×C4#55011yescharacteristic
568Dic3—21no—
569Dic3—21no—
570Dic3—21no—
571Dic3—41no—
572Dic3—41no—
573Dic3—81no—
574Dic3#47112yes—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—81no—
579C23—81no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23—241no—
584C23—241no—
585C23D3×#4111yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—41no—
589Q8—41no—
590Q8—122no—
591Q8—241no—
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—121no—
597D4—121no—
598D4—121no—
599D4—121no—
600D4—242no—
601D4—242no—
602D4—242no—
603D4—242no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—31no—
608C2×C4—31no—
609C2×C4—31no—
610C2×C4—31no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—62no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—241no—
619C2×C4C2×#23511yescharacteristic
620C2×C4C2×D3×D411yescharacteristic
621C2×C4C2×D3×D411yescharacteristic
622C2×C4C2×#23711yescharacteristic
623C2×C4#100411yescharacteristic
624C2×C4#101011yescharacteristic
625C8—21no—
626C8—21no—
627C8—22no—
628C8—41no—
629C8—41no—
630C8—61no—
631C8—61no—
632C8—121no—
633C8—121no—
634C8—121no—
635C2×C3—21no—
636C2×C3—21no—
637C2×C3—161no—
638C2×C3#242611yescharacteristic
639C2×C3C2×#47111yescharacteristic
640C2×C3#306811yescharacteristic
641D3—161no—
642C22—21no—
643C22—21no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—161no—
648C22—161no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22—481no—
653C22—481no—
654C22#553811yescharacteristic
655C22D3×#12111yescharacteristic
656C22#566911yescharacteristic
657C4—32no—
658C4—42no—
659C4—41no—
660C4—41no—
661C4—61no—
662C4—61no—
663C4—61no—
664C4—121no—
665C4—121no—
666C4—241no—
667C4#549011yescharacteristic
668C4#555211yescharacteristic
669C4#559811yescharacteristic
670C4#571811yescharacteristic
671C3#2531511yescharacteristic, complemented, Sylow
672C2—21no—
673C2—21no—
674C2—161no—
675C2—481no—
676C2#7631211yescharacteristic
677C2D3×#47111yescharacteristic
678C2#8093011yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1977C2
socleC22×C3C2×#121
prime radicalC2×#940C22
hypercenter#1977D3
hypocenterC3#25315
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1977

Complements

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

NH ≅ G/Ntype
2. #78949C2semidirect
4. #78544C2semidirect
5. C3×#1977C2semidirect
6. #72053C2semidirect
7. #70715C2semidirect
8. #70695C2semidirect
9. #70339C2semidirect
10. #70317C2semidirect
11. #70107C2semidirect
12. Dic3×#101C2semidirect
13. #70072C2semidirect
15. #69892C2semidirect
77. #1977D3semidirect
671. C3#25315semidirect

Maximal subgroups

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

subgroupindex
C2×#51632
#789492
#789212
#785442
C3×#19772
#720532
#707152
#706952
#703392
#703172
#701072
Dic3×#1012
#700722
#700642
#698922
#253153