Finite Group Databasebeta

← #10824827

Subgroups — #10824827

Order 768.

820 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4840C211yesmaximal, characteristic
2C2×#4789C211yesmaximal, characteristic, complemented
3#76611C211yesmaximal, characteristic
4#76197C211yesmaximal, characteristic
5C3×#2068C211yesmaximal, characteristic, complemented
6#70635C211yesmaximal, characteristic
7#70605C211yesmaximal, characteristic, complemented
8#70488C211yesmaximal, characteristic, complemented
9#70273C211yesmaximal, characteristic
10#70243C211yesmaximal, characteristic
11Dic3×#103C211yesmaximal, characteristic, complemented
12#69681C211yesmaximal, characteristic, complemented
13#69624C211yesmaximal, characteristic, complemented
14#68833C211yesmaximal, characteristic
15#68781C211yesmaximal, characteristic, complemented
16#23713—31nomaximal, Sylow
17C2×#946C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C2×#897C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×#872C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#815C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4990—21no—
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4972—21no—
43C2×#810C2211yescharacteristic
44C2×#808C2211yescharacteristic
45#4929C2211yescharacteristic
46#4928C2211yescharacteristic
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48#4840—24no—
49#4789—24no—
50#4670—21no—
51#4669—21no—
52#4669—21no—
53#4654—21no—
54#4652—21no—
55#4647—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#441—31no—
63C2×#433—31no—
64#2323—31no—
65#2323—31no—
66#2088—31no—
67#2077—31no—
68#2068D311yescharacteristic, complemented
69#2066—31no—
70#2056—31no—
71#2036—31no—
72#2035—31no—
73#2001—31no—
74#1776—31no—
75C4×#103—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C4×D3C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87#946—22no—
88#946—22no—
89Q8×Dic3—22no—
90Q8×Dic3—22no—
91C2×#216—22no—
92C2×#216—22no—
93C2×#216—22no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4D411yescharacteristic
109C2×Dic3⋊C4D411yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#897—22no—
118#897—22no—
119#887—22no—
120#887—22no—
121#872—22no—
122#872—22no—
123C3×#107—22no—
124C3×C4⋊C8—22no—
125C3×#103—22no—
126C3×#103—22no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138(C2×D3)⋊C8—22no—
139(C2×D3)⋊C8—22no—
140#817—22no—
141#816—22no—
142#816—22no—
143#815D414yes—
144C8×Dic3—22no—
145#810—22no—
146#808—22no—
147C4⋊(C3⋊C8)—22no—
148C22×Q16—31no—
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×#117—31no—
154#441—64no—
155#433—64no—
156C2×#107C2×D311yescharacteristic
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8C2×D311yescharacteristic
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#103C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×#97—31no—
170C2×C4×C8—31no—
171#391—31no—
172#391C2×D311yescharacteristic
173#389—31no—
174#389—61no—
175#389—61no—
176#389C2×D311yescharacteristic
177#388—61no—
178#388—61no—
179#388—61no—
180#388—61no—
181#387—61no—
182#387—61no—
183#380—31no—
184#380—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193C23×D3C2×D411yescharacteristic
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×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6#4112yes—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8C2×D412yes—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—81no—
238C3×C4⋊C4#4112yes—
239C3×C42—21no—
240C3×C42—21no—
241#216—28no—
242(C2×D3)⋊C4—24no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4—22no—
246(C2×D3)⋊C4—81no—
247(C2×D3)⋊C4—81no—
248C4⋊Dic3—24no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—42no—
255C4⋊Dic3#4112yes—
256Dic3⋊C4—24no—
257Dic3⋊C4—22no—
258Dic3⋊C4—22no—
259Dic3⋊C4—42no—
260Dic3⋊C4—81no—
261C4×Dic3—42no—
262C4×Dic3—42no—
263C4×Dic3—81no—
264C4×Dic3—81no—
265C4×Dic3C2×D412yes—
266C4×Dic3#4112yes—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C2×D412yes—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C23×C4—31no—
276C2×Q16—62no—
277C2×Q16—62no—
278C2×Q16—62no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283C22⋊Q8—62no—
284C22⋊Q8—62no—
285C22⋊Q8—62no—
286C22⋊Q8—62no—
287C4×Q8—62no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×Q8—62no—
291#117—62no—
292#117—62no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—31no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#107—22no—
320#106—62no—
321#106—62no—
322C4⋊C8—22no—
323C4⋊C8—62no—
324#103—22no—
325#103—22no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330C22⋊C8—62no—
331C22⋊C8—62no—
332#97—34no—
333C4×C8—62no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3C22×D411yescharacteristic
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3—21no—
350C22×D3—21no—
351C22×D3—21no—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—41no—
355C3×Q8—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
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×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3#12112yes—
400C2×Dic3#12112yes—
401C4×D3—41no—
402C4×D3—41no—
403C4×D3—41no—
404C4×D3—41no—
405Dic6—22no—
406Dic6—22no—
407Dic6—41no—
408Dic6—41no—
409C3×C8—22no—
410C3×C8—41no—
411C3⋊C8—22no—
412C3⋊C8—41no—
413C24—31no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—32no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8#23912yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4D3×D411yescharacteristic
444C22×C4D3×D411yescharacteristic
445C22×C4C23×D311yescharacteristic
446Q16—68no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—32no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8D3×D412yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—32no—
465C4⋊C4—62no—
466C4⋊C4—62no—
467C4⋊C4—64no—
468C4⋊C4—64no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—81no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—241no—
478C4⋊C4#23712yes—
479C22⋊C4—62no—
480C22⋊C4—62no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—241no—
484C22⋊C4—241no—
485C42—21no—
486C42—21no—
487C42—32no—
488C42—32no—
489C42—122no—
490C42—122no—
491C42—241no—
492C42—241no—
493C22×C3C2×#12111yescharacteristic
494C22×C3#52511yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3#57711yescharacteristic
497C22×C3#57711yescharacteristic
498C22×C3#57811yescharacteristic
499C22×C3#57811yescharacteristic
500C2×D3—41no—
501C2×D3—41no—
502C2×D3—41no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—41no—
506C2×D3—41no—
507C3×C4—42no—
508C3×C4—42no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4#52511yescharacteristic
514C3×C4#52911yescharacteristic
515C3×C4#52911yescharacteristic
516C3×C4#53211yescharacteristic
517Dic3—22no—
518Dic3—22no—
519Dic3—42no—
520Dic3—42no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524C23—61no—
525C23—61no—
526C23—61no—
527C23—61no—
528C23—61no—
529C23—61no—
530C23—61no—
531C23C2×D3×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—62no—
537Q8—62no—
538Q8—121no—
539Q8—121no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—32no—
549C2×C4—32no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—62no—
566C2×C4—61no—
567C2×C4—62no—
568C2×C4—61no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4C2×D3×D411yescharacteristic
584C2×C4C2×#23711yescharacteristic
585C2×C4C2×#23911yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C8—22no—
590C8—41no—
591C8—62no—
592C8—121no—
593C2×C3#239211yescharacteristic
594C2×C3#306311yescharacteristic
595C2×C3#306311yescharacteristic
596C2×C3#312011yescharacteristic
597C2×C3#312111yescharacteristic
598C2×C3#312511yescharacteristic
599C2×C3#312711yescharacteristic
600D3—81no—
601C22—121no—
602C22—121no—
603C22—121no—
604C22—121no—
605C22—121no—
606C22—121no—
607C22—121no—
608C22D3×#12111yescharacteristic
609C22#560611yescharacteristic
610C22#562711yescharacteristic
611C22#576611yescharacteristic
612C22#577011yescharacteristic
613C22#577511yescharacteristic
614C22#577911yescharacteristic
615C4—41no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—62no—
620C4—62no—
621C4—81no—
622C4—81no—
623C4—122no—
624C4—121no—
625C4—122no—
626C4—241no—
627C4—241no—
628C4#559611yescharacteristic
629C4#562411yescharacteristic
630C4#562711yescharacteristic
631C4#570611yescharacteristic
632C3#2371311yescharacteristic, complemented, Sylow
633C2—241no—
634C2#7728111yescharacteristic
635C2#8046111yescharacteristic
636C2#8051111yescharacteristic
637C2#8076211yescharacteristic
638C2#8081011yescharacteristic
639C2#8109611yescharacteristic
640C2#8120911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2068C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2068D3
hypocenterC3#23713
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2068

Complements

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

NH ≅ G/Ntype
2. C2×#4789C2semidirect
5. C3×#2068C2semidirect
7. #70605C2semidirect
8. #70488C2semidirect
11. Dic3×#103C2semidirect
12. #69681C2semidirect
13. #69624C2semidirect
15. #68781C2semidirect
68. #2068D3semidirect
632. C3#23713semidirect

Maximal subgroups

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

subgroupindex
C2×#48402
C2×#47892
#766112
#761972
C3×#20682
#706352
#706052
#704882
#702732
#702432
Dic3×#1032
#696812
#696242
#688332
#687812
#237133