Finite Group Databasebeta

← #10858279

Subgroups — #10858279

Order 768.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×SD16C211yesmaximal, characteristic
2#82460C211yesmaximal, characteristic, complemented
3C2×#4793C211yesmaximal, characteristic, complemented
4#76323C211yesmaximal, characteristic
5C3×#1880C211yesmaximal, characteristic
6#72048C211yesmaximal, characteristic
7#71628C211yesmaximal, characteristic, complemented
8#70237C211yesmaximal, characteristic, complemented
9#70194C211yesmaximal, characteristic
10#70168C211yesmaximal, characteristic
11#69654C211yesmaximal, characteristic
12#68859C211yesmaximal, characteristic, complemented
13#68827C211yesmaximal, characteristic, complemented
14#68641C211yesmaximal, characteristic, complemented
15#68610C211yesmaximal, characteristic, complemented
16#26377—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×#878—21no—
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32Dic3×SD16—24no—
33C2×#818C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4984—21no—
38#4984—21no—
39#4984C2211yescharacteristic
40#4984C2211yescharacteristic
41#4972C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4948—21no—
46#4947—21no—
47#4946—21no—
48#4936—21no—
49#4936C2211yescharacteristic
50C4×C4⋊Dic3—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52#4928C2211yescharacteristic
53#4793—24no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#477—31no—
63C2×C4×SD16—31no—
64#2452—31no—
65#2351—31no—
66#2092—31no—
67#2085—31no—
68#2067—31no—
69#2058—31no—
70#1942—31no—
71#1916—31no—
72#1880—31no—
73#1880D311yescharacteristic
74#1865—31no—
75#1863—31no—
76#1862—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×Dic3—21no—
80C22×Dic6—21no—
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C2×C3×SD16Q814yescomplemented
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98Q8×Dic3—22no—
99Q8×Dic3—22no—
100Q8×Dic3—41no—
101Q8×Dic3—41no—
102D4×Dic3—22no—
103D4×Dic3—41no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3D411yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#888—22no—
124#888—41no—
125#878—44no—
126C3×#103—22no—
127C3×#102—22no—
128#835—22no—
129#835D414yes—
130#832—22no—
131#832—41no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#818—22no—
141#817—22no—
142Dic3⋊C8—22no—
143C8×Dic3—22no—
144#808—41no—
145#807—22no—
146#807—22no—
147C22×SD16C2×D311yescharacteristic
148C2×C4⋊Q8—31no—
149C2×C22⋊Q8—61no—
150C2×C4×Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4—31no—
154#477—64no—
155C4×SD16—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×C4×C8—31no—
169#396—61no—
170#393—61no—
171#390—31no—
172#390C2×D311yescharacteristic
173#389—31no—
174#389—31no—
175#389—31no—
176#389—31no—
177#389—61no—
178#389—61no—
179#385—61no—
180#382—31no—
181#382—61no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—61no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340C2×D311yescharacteristic
188#336—31no—
189#336—31no—
190#336—31no—
191#336C2×D311yescharacteristic
192C24×C3—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8C2×D412yes—
197C2×C3×Q8C2×Q812yes—
198C2×C3×Q8#4112yes—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4C2×Q812yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—42no—
216C22×Dic3—42no—
217C22×Dic3—81no—
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C2×Dic6—42no—
223C2×Dic6—42no—
224C2×Dic6—81no—
225C3×SD16—28no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8C2×Q812yes—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4#4112yes—
237C3×C22⋊C4—42no—
238C3×C22⋊C4—81no—
239C3×C42—24no—
240C3×C42—21no—
241C3×C42—21no—
242#217—42no—
243#217—42no—
244#217—42no—
245C4⋊Dic3—21no—
246C4⋊Dic3—22no—
247C4⋊Dic3—22no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—42no—
255C4⋊Dic3—42no—
256C4⋊Dic3—42no—
257C4⋊Dic3—42no—
258C4⋊Dic3C2×D412yes—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8C2×D412yes—
274C22×Q8—61no—
275C22×Q8C22×D311yescharacteristic
276C22×D4C22×D311yescharacteristic
277C23×C4—61no—
278C2×SD16—22no—
279C2×SD16—22no—
280C2×SD16Dic614yescomplemented
281C22×C8—31no—
282C22×C8—31no—
283C22×C8C22×D311yescharacteristic
284C22×C8C22×D311yescharacteristic
285C4⋊Q8—62no—
286C4⋊Q8—121no—
287C22⋊Q8—124no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—121no—
293C4×Q8—121no—
294C4×D4—62no—
295C4×D4—121no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—21no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—62no—
324#107—62no—
325#107—62no—
326#106—62no—
327#106—121no—
328C4⋊C8—62no—
329#103—22no—
330#103—34no—
331#103—62no—
332#102—22no—
333#102—62no—
334#102—121no—
335C4×C8—62no—
336#95—31no—
337#95—31no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3—22no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3C22×D411yescharacteristic
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8C22⋊Q814yescomplemented
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—41no—
359C3×D4—41no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×Q811yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C42−1+411yescharacteristic
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—22no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—84no—
404C2×Dic3#13512yes—
405C2×Dic3AGL(1,ℤ/8)12yes—
406Dic6—84no—
407C3×C8—22no—
408C3×C8—41no—
409C3⋊C8—22no—
410C3⋊C8—41no—
411C24—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—122no—
416C2×Q8—122no—
417C2×Q8—241no—
418C2×Q8C2×Dic612yes—
419C2×Q8D3×D412yes—
420C2×Q8#23712yes—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C2×Dic612yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—122no—
439C22×C4—122no—
440C22×C4—241no—
441C22×C4D3×D411yescharacteristic
442C22×C4D3×D411yescharacteristic
443C22×C4C23×D311yescharacteristic
444SD16—28no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—32no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8C2×Dic612yes—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—32no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4#23712yes—
479C22⋊C4—42no—
480C22⋊C4—81no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C42—24no—
485C42—21no—
486C42—21no—
487C42—61no—
488C42—61no—
489C42—61no—
490C42—61no—
491C42—122no—
492C42—122no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3#52511yescharacteristic
501C22×C3D4211yescharacteristic
502C22×C3D4×Q811yescharacteristic
503C22×C3C2×#13511yescharacteristic
504C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
505C22×C3#57711yescharacteristic
506C22×C3#57811yescharacteristic
507C3×C4—24no—
508C3×C4—41no—
509C3×C4—42no—
510C3×C4—41no—
511C3×C4—81no—
512C3×C4C2×C22⋊Q811yescharacteristic
513C3×C4#53911yescharacteristic
514C3×C4#54711yescharacteristic
515C3×C4#55411yescharacteristic
516Dic3—22no—
517Dic3—22no—
518Dic3—41no—
519Dic3—42no—
520Dic3—41no—
521Dic3—81no—
522Dic3—81no—
523Dic3—81no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D3×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—22no—
535Q8—244no—
536Q8#87814yescomplemented
537D4—22no—
538D4—22no—
539D4—41no—
540D4—41no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—62no—
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—244no—
581C2×C4C22×Dic611yescharacteristic
582C2×C4C2×D3×D411yescharacteristic
583C2×C4C2×#23711yescharacteristic
584C2×C4C2×#23711yescharacteristic
585C2×C4#100911yescharacteristic
586C2×C4#101011yescharacteristic
587C8—22no—
588C8—41no—
589C8—62no—
590C8—121no—
591C2×C3—82no—
592C2×C3#244911yescharacteristic
593C2×C3#310611yescharacteristic
594C2×C3#312111yescharacteristic
595C2×C3#326511yescharacteristic
596C2×C3#328511yescharacteristic
597C2×C3#338111yescharacteristic
598C2×C3#338611yescharacteristic
599C22—42no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22D4×Dic611yescharacteristic
607C22#558611yescharacteristic
608C22#560911yescharacteristic
609C22#576211yescharacteristic
610C22D3×#13511yescharacteristic
611C22D3×AGL(1,ℤ/8)11yescharacteristic
612C22#577911yescharacteristic
613C4—24no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—62no—
618C4—62no—
619C4—81no—
620C4—121no—
621C4—122no—
622C4—121no—
623C4—241no—
624C4—241no—
625C4—241no—
626C4C2×#87811yescharacteristic
627C4#554311yescharacteristic
628C4#561111yescharacteristic
629C4#566511yescharacteristic
630C3#2637711yescharacteristic, complemented, Sylow
631C2—82no—
632C2#7471011yescharacteristic
633C2#7963311yescharacteristic
634C2#7976011yescharacteristic
635C2#8003211yescharacteristic
636C2#8006911yescharacteristic
637C2#8040411yescharacteristic
638C2#8061411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1880C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1880D3
hypocenterC3#26377
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
2. #82460C2semidirect
3. C2×#4793C2semidirect
7. #71628C2semidirect
8. #70237C2semidirect
12. #68859C2semidirect
13. #68827C2semidirect
14. #68641C2semidirect
15. #68610C2semidirect
83. C2×C3×SD16Q8semidirect
280. C2×SD16Dic6semidirect
355. C3×Q8C22⋊Q8semidirect
536. Q8#878semidirect
630. C3#26377semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SD162
#824602
C2×#47932
#763232
C3×#18802
#720482
#716282
#702372
#701942
#701682
#696542
#688592
#688272
#686412
#686102
#263773