Finite Group Databasebeta

← #10844613

Subgroups — #10844613

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82474C211yesmaximal, characteristic
2C2×#5020C211yesmaximal, characteristic, complemented
3C2×#4714C211yesmaximal, characteristic
4#73656C211yesmaximal, characteristic, complemented
5C3×#1763C211yesmaximal, characteristic, complemented
6#70284C211yesmaximal, characteristic, complemented
7#70279C211yesmaximal, characteristic
8#70230C211yesmaximal, characteristic
9#70229C211yesmaximal, characteristic, complemented
10#69836C211yesmaximal, characteristic, complemented
11#69488C211yesmaximal, characteristic
12#68912C211yesmaximal, characteristic, complemented
13#68900C211yesmaximal, characteristic
14#68793C211yesmaximal, characteristic, complemented
15#68792C211yesmaximal, characteristic
16#25287—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×#875C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34#5020—24no—
35#4994—21no—
36#4994—21no—
37#4988C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4973C2211yescharacteristic
41#4972C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#808C2211yescharacteristic
49#4940—21no—
50#4940—21no—
51C4×C4⋊Dic3—21no—
52C4×C4⋊Dic3C2211yescharacteristic
53#4928—21no—
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55#4714—24no—
56#4646—21no—
57#4646—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62C2×#489—31no—
63C2×#475—31no—
64#2545—31no—
65#2369—31no—
66#2066—31no—
67#2065—31no—
68#2043—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1981—31no—
73#1940—31no—
74#1927—31no—
75#1926—31no—
76#1763D311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×Dic6—21no—
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92#947—22no—
93#947—41no—
94#944—22no—
95#944—41no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#875—22no—
118#875—22no—
119#869—22no—
120#869D414yes—
121C3×#103—22no—
122C3×#103—22no—
123C3×#97—22no—
124C3×#95—21no—
125#835—22no—
126#835—22no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135#821—22no—
136#821—22no—
137#816—22no—
138#816—22no—
139Dic3⋊C8—41no—
140Dic3⋊C8—41no—
141#810—22no—
142#810—41no—
143#809—22no—
144#809—41no—
145#808—22no—
146#808—22no—
147C4⋊(C3⋊C8)—22no—
148C2×C4⋊Q8—31no—
149C2×C4⋊Q8—31no—
150C2×#124—31no—
151C2×#124—31no—
152C2×C4×Q8C2×D311yescharacteristic
153#489—64no—
154#475—64no—
155C2×#106—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#97C2×D311yescharacteristic
173#396—61no—
174#396—61no—
175#391—31no—
176#391—61no—
177#391—61no—
178#390—31no—
179#389—31no—
180#389—31no—
181#389—61no—
182#388—61no—
183#388—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—61no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—24no—
227C2×Dic6—81no—
228C2×Dic6—81no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4#4112yes—
242C3×C42—42no—
243C3×C42—42no—
244C3×C42C2×D412yes—
245C3×C42#4112yes—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252C4⋊Dic3—24no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—22no—
256C4⋊Dic3—41no—
257C4⋊Dic3C2×D412yes—
258C4⋊Dic3C2×D412yes—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—81no—
262Dic3⋊C4—81no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—81no—
268C4×Dic3—81no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C22×Q8—31no—
276C22×Q8—61no—
277C22×Q8C22×D311yescharacteristic
278C22×D4—31no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283C4⋊Q8—34no—
284C4⋊Q8—62no—
285C4⋊Q8—62no—
286C4⋊Q8—121no—
287#124—62no—
288#124—62no—
289#124—62no—
290#124—121no—
291C4×Q8—22no—
292C4×Q8—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#106—62no—
320#106—62no—
321C4⋊C8—62no—
322C4⋊C8—121no—
323C4⋊C8—121no—
324#103—22no—
325#103—22no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330#103—62no—
331#103—121no—
332#102—62no—
333#102—62no—
334#102—62no—
335#102—121no—
336#97—22no—
337#95—21no—
338#95—31no—
339#95—31no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3C22×D411yescharacteristic
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4#12112yes—
375C2×C3×C4#12112yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×C3×C42−1+411yescharacteristic
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3#13712yes—
403D12—41no—
404D12—41no—
405D12—41no—
406D12—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410Dic6—41no—
411Dic6—84no—
412C3×C8—41no—
413C3×C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C24—61no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—64no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—241no—
430C2×Q8—241no—
431C2×Q8#23512yes—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—32no—
470C4⋊C4—32no—
471C4⋊C4—42no—
472C4⋊C4—62no—
473C4⋊C4—64no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—121no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—241no—
480C4⋊C4—241no—
481C4⋊C4#23512yes—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C42—42no—
489C42—42no—
490C42—61no—
491C42—61no—
492C42—61no—
493C42—61no—
494C42—241no—
495C42—241no—
496C42#23512yes—
497C42D3×D412yes—
498C22×C3C2×#12111yescharacteristic
499C22×C3#54611yescharacteristic
500C22×C3#54811yescharacteristic
501C22×C3C2×#13711yescharacteristic
502C22×C3#57511yescharacteristic
503C22×C3#57711yescharacteristic
504C22×C3#57711yescharacteristic
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4—81no—
517C3×C4—81no—
518C3×C4#52911yescharacteristic
519C3×C4#54211yescharacteristic
520C3×C4#54811yescharacteristic
521C3×C4#55011yescharacteristic
522Dic3—22no—
523Dic3—42no—
524Dic3—42no—
525Dic3—41no—
526Dic3—41no—
527Dic3—41no—
528Dic3—81no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23C2×D3×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—121no—
542Q8—121no—
543Q8—121no—
544Q8—121no—
545Q8—244no—
546D4—121no—
547D4—121no—
548D4—121no—
549D4—121no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—32no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4#88412yes—
588C2×C4#88412yes—
589C2×C4C2×#23511yescharacteristic
590C2×C4C2×#23511yescharacteristic
591C2×C4C2×#23511yescharacteristic
592C2×C4C2×D3×D411yescharacteristic
593C2×C4#100911yescharacteristic
594C2×C4#101011yescharacteristic
595C8—41no—
596C8—41no—
597C8—121no—
598C8—121no—
599C2×C3#242611yescharacteristic
600C2×C3#303711yescharacteristic
601C2×C3#304211yescharacteristic
602C2×C3#331611yescharacteristic
603C2×C3#332111yescharacteristic
604C2×C3#336911yescharacteristic
605C2×C3#337611yescharacteristic
606D3—161no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22C2×#88411yescharacteristic
615C22#555011yescharacteristic
616C22#555211yescharacteristic
617C22#577311yescharacteristic
618C22D3×#13711yescharacteristic
619C22#577811yescharacteristic
620C22#577811yescharacteristic
621C4—22no—
622C4—22no—
623C4—42no—
624C4—42no—
625C4—62no—
626C4—81no—
627C4—81no—
628C4—122no—
629C4—122no—
630C4—121no—
631C4—121no—
632C4—121no—
633C4—241no—
634C4#547911yescharacteristic
635C4#553411yescharacteristic
636C4#555211yescharacteristic
637C4#555311yescharacteristic
638C3#2528711yescharacteristic, complemented, Sylow
639C2—481no—
640C2#7348511yescharacteristic
641C2#7891311yescharacteristic
642C2#7897611yescharacteristic
643C2#7975711yescharacteristic
644C2#7980511yescharacteristic
645C2#7981511yescharacteristic
646C2#7984311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1763C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1763D3
hypocenterC3#25287
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. C2×#5020C2semidirect
4. #73656C2semidirect
5. C3×#1763C2semidirect
6. #70284C2semidirect
9. #70229C2semidirect
10. #69836C2semidirect
12. #68912C2semidirect
14. #68793C2semidirect
76. #1763D3semidirect
638. C3#25287semidirect

Maximal subgroups

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

subgroupindex
#824742
C2×#50202
C2×#47142
#736562
C3×#17632
#702842
#702792
#702302
#702292
#698362
#694882
#689122
#689002
#687932
#687922
#252873