Finite Group Databasebeta

← #10945150

Subgroups — #10945150

Order 768.

864 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5185C211yesmaximal, characteristic
2C4×Q8×Dic3C211yesmaximal, characteristic, complemented
3C2×#4796C211yesmaximal, characteristic, complemented
4#76606C211yesmaximal, characteristic
5C3×#1878C211yesmaximal, characteristic, complemented
6#72065C211yesmaximal, characteristic, complemented
7#71633C211yesmaximal, characteristic
8#70240C211yesmaximal, characteristic
9#70230C211yesmaximal, characteristic
10#70224C211yesmaximal, characteristic, complemented
11#69666C211yesmaximal, characteristic, complemented
12#68801C211yesmaximal, characteristic, complemented
13#68792C211yesmaximal, characteristic
14#68625C211yesmaximal, characteristic
15#68605C211yesmaximal, characteristic, complemented
16#33029—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C22×#215C2211yescharacteristic
23C2×#890C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#391C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5185—24no—
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#4988C2211yescharacteristic
38Dic3×C4⋊C4—21no—
39Dic3×C4⋊C4—21no—
40Dic3×C4⋊C4C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#808C2211yescharacteristic
44#4939C2211yescharacteristic
45C4×C4⋊Dic3—21no—
46C4×C4⋊Dic3C2211yescharacteristic
47C42×Dic3—21no—
48C42×Dic3C2211yescharacteristic
49#4796—24no—
50#4670—21no—
51#4670—21no—
52#4669—21no—
53#4669—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#479—31no—
63C2×#464—31no—
64#2861—31no—
65C42×Q8—31no—
66#1989—31no—
67#1983—31no—
68#1928—31no—
69#1927—31no—
70#1926—31no—
71#1925—31no—
72#1880—31no—
73#1880—31no—
74#1878—31no—
75#1878D311yescharacteristic, complemented
76#1831—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C2×C3×Q16—22no—
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92#947—22no—
93#947—22no—
94Q8×Dic3—22no—
95Q8×Dic3—22no—
96Q8×Dic3—22no—
97Q8×Dic3D414yes—
98C2×#215—22no—
99C2×#215—22no—
100C2×#215—22no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3D411yescharacteristic
122C2×C4×Dic3D411yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127#890—22no—
128#890—22no—
129C3×#103—22no—
130C3×#103—22no—
131C3×#95—21no—
132C3×#95—21no—
133#835—22no—
134#835—22no—
135#835—22no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#821—22no—
142#821—22no—
143#818—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8—22no—
146#809—22no—
147#808—22no—
148C22×Q16C2×D311yescharacteristic
149C22×SD16—31no—
150C2×#125—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×Q8C2×D311yescharacteristic
155#479—64no—
156#464—64no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169#391—31no—
170#391C2×D311yescharacteristic
171#390—31no—
172#390—31no—
173#388—61no—
174#388—61no—
175#388—61no—
176#388—61no—
177C4×C4⋊C4—31no—
178C4×C4⋊C4—31no—
179C4×C4⋊C4—31no—
180C4×C4⋊C4—61no—
181C4×C4⋊C4—61no—
182C4×C4⋊C4—61no—
183C43—31no—
184C43—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#336—31no—
190#336—31no—
191#336—31no—
192#336C2×D311yescharacteristic
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—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8C2×D412yes—
204C2×C3×Q8#4112yes—
205C2×C3×Q8#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C3×Q16—44no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4#4112yes—
238C3×C42—24no—
239C3×C42—21no—
240C3×C42—21no—
241#215—44no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248C4⋊Dic3—24no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—24no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—42no—
255C4⋊Dic3C2×D412yes—
256C4⋊Dic3#4112yes—
257Dic3⋊C4—42no—
258C4×Dic3—24no—
259C4×Dic3—22no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—24no—
263C4×Dic3—22no—
264C4×Dic3—41no—
265C4×Dic3—41no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3C2×D412yes—
270C4×Dic3#4112yes—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C22×Q8C22×D311yescharacteristic
278C22×Q8C22×D311yescharacteristic
279C22×D4—31no—
280C2×Q16—22no—
281C2×Q16—22no—
282C2×Q16—22no—
283C2×SD16—62no—
284C2×SD16—62no—
285C2×SD16—62no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290#125—62no—
291#125—62no—
292#124—62no—
293#124—62no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×Q8—34no—
297C4×Q8—62no—
298C4×Q8—62no—
299C4×Q8—62no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C42—21no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329#107—62no—
330#106—62no—
331C4⋊C8—62no—
332C4⋊C8—62no—
333#103—22no—
334#103—22no—
335#103—62no—
336#103—62no—
337#103—62no—
338#102—62no—
339#102—62no—
340#102—62no—
341#95—21no—
342#95—21no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3C22×D411yescharacteristic
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—22no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8#12414yes—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4#13512yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3#12112yes—
407C2×Dic3#12112yes—
408C2×Dic3#13712yes—
409D12—41no—
410D12—41no—
411D12—41no—
412D12—41no—
413C3×C8—41no—
414C3×C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C24—61no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8#23512yes—
428C2×Q8D3×D412yes—
429C2×Q8#23712yes—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—44no—
451SD16—124no—
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—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—32no—
469C4⋊C4—32no—
470C4⋊C4—64no—
471C4⋊C4—64no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4#23712yes—
479C22⋊C4—122no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C42—24no—
486C42—21no—
487C42—21no—
488C42—32no—
489C42—32no—
490C42—62no—
491C42—64no—
492C42—64no—
493C42—62no—
494C42—61no—
495C42—61no—
496C42—121no—
497C42—121no—
498C42—122no—
499C42—122no—
500C42—122no—
501C22×C3C2×#12111yescharacteristic
502C22×C3#52511yescharacteristic
503C22×C3#52511yescharacteristic
504C22×C3C2×#13511yescharacteristic
505C22×C3C2×#13711yescharacteristic
506C22×C3#57511yescharacteristic
507C22×C3#57511yescharacteristic
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C3×C4—24no—
516C3×C4—22no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4C2×#12411yescharacteristic
522C3×C4#52911yescharacteristic
523C3×C4#52911yescharacteristic
524C3×C4#53211yescharacteristic
525Dic3—22no—
526Dic3—22no—
527Dic3—22no—
528Dic3—41no—
529Dic3—42no—
530Dic3—42no—
531Dic3—81no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23C2×D3×D411yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—22no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8#88514yes—
548D4—121no—
549D4—121no—
550D4—121no—
551D4—121no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—32no—
562C2×C4—32no—
563C2×C4—32no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
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—62no—
581C2×C4—61no—
582C2×C4—62no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4#91912yes—
591C2×C4C2×#23511yescharacteristic
592C2×C4C2×D3×D411yescharacteristic
593C2×C4C2×#23711yescharacteristic
594C2×C4C2×#23711yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C8—41no—
598C8—41no—
599C8—121no—
600C8—121no—
601C2×C3#285011yescharacteristic
602C2×C3#303511yescharacteristic
603C2×C3#303711yescharacteristic
604C2×C3#310211yescharacteristic
605C2×C3#310811yescharacteristic
606C2×C3#311111yescharacteristic
607C2×C3#311711yescharacteristic
608D3—161no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22#554711yescharacteristic
617C22#558611yescharacteristic
618C22D3×#12111yescharacteristic
619C22C2×#91911yescharacteristic
620C22#576811yescharacteristic
621C22D3×#13711yescharacteristic
622C22#577711yescharacteristic
623C4—22no—
624C4—24no—
625C4—41no—
626C4—42no—
627C4—62no—
628C4—62no—
629C4—62no—
630C4—81no—
631C4—81no—
632C4—122no—
633C4—121no—
634C4—122no—
635C4—241no—
636C4C2×#88511yescharacteristic
637C4#553811yescharacteristic
638C4#559611yescharacteristic
639C4#564911yescharacteristic
640C3#3302911yescharacteristic, complemented, Sylow
641C2—481no—
642C2#7486911yescharacteristic
643C2#7974611yescharacteristic
644C2#7983411yescharacteristic
645C2#8017311yescharacteristic
646C2#8021411yescharacteristic
647C2#8047511yescharacteristic
648C2#8059811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1878C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1878D3
hypocenterC3#33029
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1878

Complements

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

NH ≅ G/Ntype
2. C4×Q8×Dic3C2semidirect
3. C2×#4796C2semidirect
5. C3×#1878C2semidirect
6. #72065C2semidirect
10. #70224C2semidirect
11. #69666C2semidirect
12. #68801C2semidirect
15. #68605C2semidirect
75. #1878D3semidirect
640. C3#33029semidirect

Maximal subgroups

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

subgroupindex
C2×#51852
C4×Q8×Dic32
C2×#47962
#766062
C3×#18782
#720652
#716332
#702402
#702302
#702242
#696662
#688012
#687922
#686252
#686052
#330293