Finite Group Databasebeta

← #10815014

Subgroups — #10815014

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82338C211yesmaximal, characteristic, complemented
2C2×#5014C211yesmaximal, characteristic
3#74324C211yesmaximal, characteristic, complemented
4C3×#1802C211yesmaximal, characteristic, complemented
5#70480C211yesmaximal, characteristic, complemented
6#70467C211yesmaximal, characteristic, complemented
7#70191C211yesmaximal, characteristic
8#70190C211yesmaximal, characteristic
9#70174C211yesmaximal, characteristic, complemented
10#70173C211yesmaximal, characteristic, complemented
11#69578C211yesmaximal, characteristic, complemented
12#68792C211yesmaximal, characteristic, complemented
13#68791C211yesmaximal, characteristic, complemented
14#68786C211yesmaximal, characteristic, complemented
15#68784C211yesmaximal, characteristic, complemented
16#22817—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C22×#213C2211yescharacteristic
19C2×#892—21no—
20C2×C4×D12C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#384C2211yescharacteristic
25#5221C2211yescharacteristic
26#5221C2211yescharacteristic
27C2×#832C2211yescharacteristic
28#5211—21no—
29C4×#217—21no—
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#5014C418yescomplemented
35#4990—21no—
36#4990—21no—
37#4988C2211yescharacteristic
38#4988C2211yescharacteristic
39#4987—21no—
40Dic3×C4⋊C4C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4951—21no—
51#4951—21no—
52#4936C2211yescharacteristic
53C4×C4⋊Dic3C2211yescharacteristic
54C2×#803C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61C2×#442—31no—
62#2357—31no—
63#2354—31no—
64#1972—31no—
65#1970—31no—
66#1926—31no—
67#1926—31no—
68#1925—31no—
69#1925—31no—
70#1847—31no—
71#1804—31no—
72#1802D311yescharacteristic, complemented
73#1776—31no—
74C4×#106—31no—
75#1771—31no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×D4—22no—
83C3×C4×D4C2×C414yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#213—22no—
96C2×#213—22no—
97C2×#213C2×C414yes—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116#892—44no—
117C4×D12—22no—
118C4×D12C2×C414yes—
119C3×C4⋊C8—22no—
120C3×#102—41no—
121C3×#102—41no—
122#832C2×C414yes—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#821—41no—
132#821—41no—
133#815—22no—
134C8×Dic3—22no—
135#809C2×C414yes—
136#808—22no—
137#808—41no—
138#808C2×C414yes—
139#807—22no—
140#807—41no—
141#803C2×C414yes—
142C22×D8—31no—
143C2×C4×D4—31no—
144C2×C4×D4C2×D311yescharacteristic
145C2×#117—61no—
146#442—38no—
147C2×#107—31no—
148C2×#107—31no—
149C2×#106—31no—
150C2×#106—31no—
151C2×#106—31no—
152C2×C4⋊C8C2×D311yescharacteristic
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102C2×D311yescharacteristic
158C2×#102C2×D311yescharacteristic
159C2×#97—31no—
160C2×#97—31no—
161C2×C4×C8—31no—
162#390—31no—
163#390—31no—
164#390—31no—
165#390—31no—
166#388—61no—
167#388—61no—
168#388—61no—
169#388—61no—
170#384C2×D311yescharacteristic
171#382—31no—
172#381—61no—
173#380—61no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177C4×C4⋊C4—31no—
178C4×C4⋊C4—31no—
179C4×C22⋊C4—61no—
180#340—31no—
181#340—31no—
182#340C2×D311yescharacteristic
183#340C2×D311yescharacteristic
184#336—31no—
185#336—31no—
186C24×C3—21no—
187C23×D3—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4C22×C412yes—
194C22×C3×C4—21no—
195C22×C3×C4—24no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—81no—
199C22×C3×C4—81no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12C22×C412yes—
216C2×C4×D3—24no—
217C2×C4×D3—42no—
218C2×C4×D3—42no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—81no—
232C3×C4⋊C4C22×C412yes—
233C3×C22⋊C4—24no—
234C3×C42C22×C412yes—
235C3×C42#4112yes—
236#217—42no—
237#217—42no—
238#217—81no—
239#217—81no—
240#213—28no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—81no—
245(C2×D3)⋊C4—81no—
246C4⋊Dic3—21no—
247C4⋊Dic3—22no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—41no—
253C4⋊Dic3C22×C412yes—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—42no—
259C4×Dic3—42no—
260C4×Dic3—81no—
261C4×Dic3—81no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8C22×C412yes—
265C2×C3⋊C8C22×C412yes—
266C2×C3⋊C8#4112yes—
267C2×C3⋊C8#4112yes—
268C22×D4—31no—
269C22×D4C22×D311yescharacteristic
270C23×C4—21no—
271C23×C4—61no—
272C2×D8—34no—
273C2×D8—62no—
274C2×D8—62no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279C4×D4—22no—
280C4×D4—34no—
281C4×D4—62no—
282C4×D4C4×D314yes—
283#117—124no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C42—31no—
305C2×C42—31no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42—61no—
309C2×C42—61no—
310C2×C42C22×D311yescharacteristic
311#107—62no—
312#107—121no—
313#106—34no—
314#106—62no—
315#106—121no—
316C4⋊C8—22no—
317#102—34no—
318#102—34no—
319#102—41no—
320#102—41no—
321#102—121no—
322#102—121no—
323#97—34no—
324#97—62no—
325C4×C8—62no—
326#95—31no—
327#95—31no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334C23×C3—22no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3C22×D411yescharacteristic
342C22×D3—22no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C3×D4—22no—
350C3×D4—22no—
351C3×D4—41no—
352C3×D4—41no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—84no—
367C2×C3×C4C4×D412yes—
368C2×C3×C4C4×D412yes—
369C2×C3×C4C23×C411yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C42−1+411yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3AGL(1,ℤ/8)12yes—
396D12—22no—
397D12—22no—
398D12—41no—
399D12—41no—
400C4×D3—48no—
401C3×C8—41no—
402C3×C8—41no—
403C3⋊C8—22no—
404C3⋊C8#11714yescomplemented
405C24—21no—
406C24—61no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—32no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4C2×C4×D312yes—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—61no—
428C22×C4—64no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—122no—
434C22×C4—122no—
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4C23×D311yescharacteristic
438D8—68no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—32no—
446C2×C8—32no—
447C2×C8—32no—
448C2×C8—32no—
449C2×C8—61no—
450C2×C8—61no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—32no—
457C4⋊C4—42no—
458C4⋊C4—62no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—81no—
465C4⋊C4—121no—
466C4⋊C4C2×C4×D312yes—
467C22⋊C4—24no—
468C22⋊C4—64no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C22⋊C4—241no—
474C22⋊C4—241no—
475C22⋊C4—241no—
476C22⋊C4—241no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—61no—
481C42—122no—
482C42—122no—
483C42—241no—
484C42—241no—
485C42C2×C4×D312yes—
486C42#23712yes—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3C2×C4×D411yescharacteristic
495C22×C3#54711yescharacteristic
496C22×C3#55011yescharacteristic
497C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
498C22×C3#57511yescharacteristic
499C22×C3#57611yescharacteristic
500C22×C3#57711yescharacteristic
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—42no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4C2×#11711yescharacteristic
514C3×C4#52011yescharacteristic
515C3×C4#55511yescharacteristic
516C3×C4#55611yescharacteristic
517Dic3—22no—
518Dic3—41no—
519Dic3—41no—
520Dic3—41no—
521Dic3—42no—
522Dic3—81no—
523Dic3—81no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—62no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23C2×D3×D411yescharacteristic
539D4—22no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—62no—
544D4—62no—
545D4—121no—
546D4—121no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—32no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—62no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—84no—
574C2×C4—128no—
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×C4#67⋊C412yes—
584C2×C4#67⋊C412yes—
585C2×C4C22×C4×D311yescharacteristic
586C2×C4C2×#23711yescharacteristic
587C2×C4#100411yescharacteristic
588C2×C4#100711yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C8—34no—
592C8—41no—
593C8—41no—
594C8—62no—
595C2×C3—82no—
596C2×C3#235411yescharacteristic
597C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
598C2×C3#294611yescharacteristic
599C2×C3#334511yescharacteristic
600C2×C3#334711yescharacteristic
601C2×C3#339911yescharacteristic
602C2×C3#340111yescharacteristic
603D3—82no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22—241no—
615C22—241no—
616C22—241no—
617C22—241no—
618C22C2×#67⋊C411yescharacteristic
619C22#559911yescharacteristic
620C22#563111yescharacteristic
621C22#575711yescharacteristic
622C22#576211yescharacteristic
623C22D3×AGL(1,ℤ/8)11yescharacteristic
624C22#577711yescharacteristic
625C4—22no—
626C4—22no—
627C4—42no—
628C4—62no—
629C4—81no—
630C4—81no—
631C4—122no—
632C4—121no—
633C4—121no—
634C4—121no—
635C4—241no—
636C4—241no—
637C4#550411yescharacteristic
638C4D3×#11711yescharacteristic
639C4#561611yescharacteristic
640C4#564111yescharacteristic
641C3#2281711yescharacteristic, complemented, Sylow
642C2—82no—
643C2—242no—
644C2#7388511yescharacteristic
645C2#7872611yescharacteristic
646C2#7886611yescharacteristic
647C2#8032811yescharacteristic
648C2#8042411yescharacteristic
649C2#8098911yescharacteristic
650C2#8108311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC22×#213C22
cosocleC22×C3×C4C24
hypercenter#1802D3
hypocenterC3#22817
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. #82338C2semidirect
3. #74324C2semidirect
4. C3×#1802C2semidirect
5. #70480C2semidirect
6. #70467C2semidirect
9. #70174C2semidirect
10. #70173C2semidirect
11. #69578C2semidirect
12. #68792C2semidirect
13. #68791C2semidirect
14. #68786C2semidirect
15. #68784C2semidirect
34. #5014C4semidirect
72. #1802D3semidirect
404. C3⋊C8#117semidirect
641. C3#22817semidirect

Maximal subgroups

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

subgroupindex
#823382
C2×#50142
#743242
C3×#18022
#704802
#704672
#701912
#701902
#701742
#701732
#695782
#687922
#687912
#687862
#687842
#228173