Finite Group Databasebeta

← #10877208

Subgroups — #10877208

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#75964C211yesmaximal, characteristic
2#74148C211yesmaximal, characteristic, complemented
3C3×#2101C211yesmaximal, characteristic, complemented
4#69613C211yesmaximal, characteristic, complemented
5#69558C211yesmaximal, characteristic, complemented
6#68892C211yesmaximal, characteristic, complemented
7#68892C211yesmaximal, characteristic, complemented
8#68881C211yesmaximal, characteristic
9#68881C211yesmaximal, characteristic
10#68877C211yesmaximal, characteristic, complemented
11#68876C211yesmaximal, characteristic, complemented
12#68796C211yesmaximal, characteristic, complemented
13#68795C211yesmaximal, characteristic, complemented
14#68680C211yesmaximal, characteristic, complemented
15#68662C211yesmaximal, characteristic, complemented
16#27679—31nomaximal, Sylow
17C2×#895C2211yescharacteristic
18C2×#893—21no—
19C2×#893C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×C42×D3C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×#382C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4990—21no—
34#4987C2211yescharacteristic
35#4975C2211yescharacteristic
36#4975C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4939C2211yescharacteristic
43#4936C2211yescharacteristic
44#4667—21no—
45#4667—21no—
46#4666—21no—
47#4665—21no—
48#4649—21no—
49#4649—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60C4×#121—31no—
61#2285—31no—
62#2101—31no—
63#2101D311yescharacteristic, complemented
64#2092—31no—
65#2092—31no—
66#1948—31no—
67#1947—31no—
68#1875—31no—
69#1875—31no—
70#1854—31no—
71#1854—31no—
72#1802—31no—
73#1802—31no—
74#1797—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3D411yescharacteristic
79C22×C4×D3D411yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3D411yescharacteristic
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#895—22no—
112#895Q814yes—
113#893—22no—
114#893—22no—
115#893—44no—
116#892—41no—
117#892—41no—
118#892—41no—
119#892—41no—
120C42×D3—22no—
121C42×D3—22no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129#821—41no—
130#821—41no—
131(C2×D3)⋊C8—22no—
132(C2×D3)⋊C8—22no—
133#818—41no—
134#817—41no—
135Dic3⋊C8—22no—
136Dic3⋊C8—22no—
137#809—41no—
138#809—41no—
139#808—41no—
140#807—41no—
141C2×#121—31no—
142C2×C4×D4—31no—
143C2×C4×D4—61no—
144C2×#117—31no—
145C2×#117—31no—
146C22×C42—31no—
147C2×#107—31no—
148C2×#107—31no—
149C2×#106—31no—
150C2×#106—31no—
151C2×C4⋊C8—31no—
152C2×C4⋊C8—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159#390—31no—
160#389C2×D311yescharacteristic
161#388—61no—
162#386—61no—
163#384—31no—
164#384—31no—
165#382—31no—
166#382C2×D311yescharacteristic
167#380—31no—
168#380—61no—
169#380—61no—
170C4×C4⋊C4—31no—
171C4×C4⋊C4—61no—
172C4×C4⋊C4—61no—
173C4×C22⋊C4—61no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#336—31no—
183#336C2×D311yescharacteristic
184C23×D3—21no—
185C23×D3C2×D411yescharacteristic
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—24no—
204C2×D12—21no—
205C2×D12—42no—
206C2×D12—42no—
207C2×D12C2×Q812yes—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3—41no—
217C2×C4×D3—41no—
218C2×C4×D3—42no—
219C2×C4×D3—42no—
220C2×C4×D3—42no—
221C2×C4×D3C2×Q812yes—
222C2×C4×D3#4112yes—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4C2×Q812yes—
238C3×C42—21no—
239C3×C42—21no—
240(C2×D3)⋊C4—24no—
241(C2×D3)⋊C4—42no—
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—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257C4×Dic3—22no—
258C4×Dic3—22no—
259C4×Dic3—42no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C4×Dic3#4112yes—
263C4×Dic3#4112yes—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C22×D4—31no—
271C22×D4—61no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—31no—
275C23×C4—61no—
276C22×C8—31no—
277C22×C8—31no—
278C22×C8C22×D311yescharacteristic
279C22×C8C22×D311yescharacteristic
280#121—34no—
281#121—62no—
282C4×D4—62no—
283C4×D4—62no—
284C4×D4—124no—
285#117—121no—
286#117—121no—
287#117—121no—
288#117—121no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—62no—
312C2×C42—62no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—121no—
318#107—121no—
319#106—121no—
320#106—121no—
321C4⋊C8—62no—
322C4⋊C8—62no—
323#102—121no—
324#102—121no—
325#102—121no—
326#102—121no—
327C22⋊C8—62no—
328C22⋊C8—62no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336C23×C3C22×D411yescharacteristic
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—21no—
340C22×D3—21no—
341C22×D3—21no—
342C22×D3—21no—
343C22×D3—22no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3C22⋊Q812yes—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
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—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4C22×Q811yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3C22⋊Q812yes—
389C2×Dic3#12312yes—
390C2×Dic3#12312yes—
391C2×Dic3#13512yes—
392C2×Dic3AGL(1,ℤ/8)12yes—
393D12—22no—
394D12—22no—
395D12—48no—
396D12—41no—
397D12—41no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—22no—
401C4×D3—22no—
402C4×D3—44no—
403C4×D3—44no—
404C4×D3—84no—
405C3×C8—41no—
406C3×C8—41no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C24—31no—
410C24—61no—
411C2×D4—32no—
412C2×D4—64no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—122no—
419C2×D4—122no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—32no—
428C22×C4—32no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—121no—
441C22×C4—121no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4D3×D411yescharacteristic
446C22×C4D3×D411yescharacteristic
447C22×C4C23×D311yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4D3×Q812yes—
479C22⋊C4—64no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C42—21no—
487C42—21no—
488C42—32no—
489C42—32no—
490C42—62no—
491C42—62no—
492C42—122no—
493C42—122no—
494C42—122no—
495C22×C3C2×C22⋊Q811yescharacteristic
496C22×C3C2×#12311yescharacteristic
497C22×C3#53511yescharacteristic
498C22×C3C2×#13511yescharacteristic
499C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
500C22×C3#57511yescharacteristic
501C22×C3#57511yescharacteristic
502C2×D3—22no—
503C2×D3—22no—
504C2×D3—22no—
505C2×D3—41no—
506C2×D3—41no—
507C2×D3—41no—
508C2×D3—41no—
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C3×C4—42no—
517C3×C4—41no—
518C3×C4—41no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4#55311yescharacteristic
523C3×C4#55311yescharacteristic
524C3×C4#55411yescharacteristic
525C3×C4#55411yescharacteristic
526Dic3—22no—
527Dic3—22no—
528Dic3—81no—
529Dic3—81no—
530Dic3—81no—
531Dic3#48214yes—
532C23—32no—
533C23—61no—
534C23—61no—
535C23—61no—
536C23—61no—
537C23—61no—
538C23—61no—
539C23—62no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23—121no—
546C23C2×D3×D411yescharacteristic
547D4—62no—
548D4—62no—
549D4—128no—
550D4—121no—
551D4—121no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—32no—
560C2×C4—32no—
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—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—124no—
584C2×C4—124no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—244no—
595C2×C4C2×D3×Q811yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C2×C4D3×#4111yescharacteristic
598C2×C4D3×#4111yescharacteristic
599C2×C4#100911yescharacteristic
600C2×C4#100911yescharacteristic
601C8—41no—
602C8—41no—
603C8—121no—
604C8—121no—
605C2×C3#248711yescharacteristic
606C2×C3#307611yescharacteristic
607C2×C3#307711yescharacteristic
608C2×C3C2×#48211yescharacteristic
609C2×C3#309311yescharacteristic
610C2×C3#321211yescharacteristic
611C2×C3#321411yescharacteristic
612D3—42no—
613D3—82no—
614C22—62no—
615C22—62no—
616C22—62no—
617C22—121no—
618C22—121no—
619C22—121no—
620C22—121no—
621C22—122no—
622C22—122no—
623C22—122no—
624C22—241no—
625C22—241no—
626C22—241no—
627C22—241no—
628C22D3×C22⋊Q811yescharacteristic
629C22D3×#12311yescharacteristic
630C22#567411yescharacteristic
631C22D3×#13511yescharacteristic
632C22#576811yescharacteristic
633C22D3×AGL(1,ℤ/8)11yescharacteristic
634C22#577711yescharacteristic
635C4—34no—
636C4—42no—
637C4—41no—
638C4—41no—
639C4—62no—
640C4—62no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—241no—
645C4—241no—
646C4—241no—
647C4#569011yescharacteristic
648C4#569011yescharacteristic
649C4#570911yescharacteristic
650C4#570911yescharacteristic
651C3#2767911yescharacteristic, complemented, Sylow
652C2—122no—
653C2—242no—
654C2#7814611yescharacteristic
655C2#8116611yescharacteristic
656C2#8122911yescharacteristic
657C2D3×#48211yescharacteristic
658C2#8151611yescharacteristic
659C2#8164311yescharacteristic
660C2#8168511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2101C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2101D3
hypocenterC3#27679
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
2. #74148C2semidirect
3. C3×#2101C2semidirect
4. #69613C2semidirect
5. #69558C2semidirect
6. #68892C2semidirect
7. #68892C2semidirect
10. #68877C2semidirect
11. #68876C2semidirect
12. #68796C2semidirect
13. #68795C2semidirect
14. #68680C2semidirect
15. #68662C2semidirect
63. #2101D3semidirect
651. C3#27679semidirect

Maximal subgroups

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

subgroupindex
#759642
#741482
C3×#21012
#696132
#695582
#688922
#688812
#688772
#688762
#687962
#687952
#686802
#686622
#276793