Finite Group Databasebeta

← #10858993

Subgroups — #10858993

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82244C211yesmaximal, characteristic
2#82171C211yesmaximal, characteristic, complemented
3C3×#1736C211yesmaximal, characteristic, complemented
4#71995C211yesmaximal, characteristic, complemented
5#71990C211yesmaximal, characteristic
6#71576C211yesmaximal, characteristic, complemented
7#71560C211yesmaximal, characteristic
8#68901C211yesmaximal, characteristic, complemented
9#68898C211yesmaximal, characteristic, complemented
10#68895C211yesmaximal, characteristic, complemented
11#68892C211yesmaximal, characteristic, complemented
12#68811C211yesmaximal, characteristic, complemented
13#68808C211yesmaximal, characteristic, complemented
14#68800C211yesmaximal, characteristic, complemented
15#68788C211yesmaximal, characteristic, complemented
16#26425—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C2×#924—21no—
22C2×#893C2211yescharacteristic
23C2×#888C2211yescharacteristic
24#5212C2211yescharacteristic
25#5210—21no—
26C4×#217C2211yescharacteristic
27C2×#830C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4988C2211yescharacteristic
33#4984C2211yescharacteristic
34#4982—21no—
35#4979—21no—
36#4975C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4972C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4957—21no—
45#4956—21no—
46#4940—21no—
47C4×(C2×D3)⋊C4—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2440—31no—
61#2363—31no—
62#2100—31no—
63#2087—31no—
64#2067—31no—
65#2023—31no—
66#1988—31no—
67#1919—31no—
68#1895—31no—
69#1874—31no—
70#1864—31no—
71#1854—31no—
72#1851—31no—
73#1802—31no—
74#1736D311yescharacteristic, complemented
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6—21no—
81C2×C3×M16—22no—
82C2×C3×M16—41no—
83C2×C3×M16—41no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#117—22no—
87C3×#117—41no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—41no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99C2×#217D411yescharacteristic
100#930—22no—
101#930Q814yescomplemented
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110#924—44no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#893—22no—
122#893—41no—
123#888—22no—
124#888—41no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132#830—22no—
133#821—41no—
134#821—41no—
135Dic3⋊C8—41no—
136Dic3⋊C8—41no—
137#809—22no—
138#809—41no—
139#808—22no—
140#807—41no—
141C22×M16C2×D311yescharacteristic
142C2×C4⋊Q8—31no—
143C2×C22⋊Q8—61no—
144C2×#121—31no—
145C2×C4×D4—31no—
146C2×#117C2×D311yescharacteristic
147C22×C4⋊C4C2×D311yescharacteristic
148C2×#107—31no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×C22⋊C8—31no—
157#398—61no—
158#395—61no—
159#390—31no—
160#390—61no—
161#389—31no—
162#389—31no—
163#389—31no—
164#388—61no—
165#386—61no—
166#385—61no—
167#384—31no—
168#380—31no—
169C4×C4⋊C4—31no—
170C4×C22⋊C4—31no—
171C4×C22⋊C4—61no—
172#340—31no—
173#340—31no—
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#336—31no—
184C24×C3C2×D411yescharacteristic
185C23×D3—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4—81no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×Q812yes—
199C22×C3×C4#4112yes—
200C22×C3×C4C2411yescharacteristic
201C2×#67—24no—
202C2×#67—81no—
203C2×#67—81no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12C2×Q812yes—
217C2×C4×D3—42no—
218C2×C4×D3—42no—
219C2×C4×D3—81no—
220C2×Dic6—42no—
221C2×Dic6—42no—
222C2×Dic6—81no—
223C3×M16—44no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—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⋊C4#4112yes—
238C3×C4⋊C4#4112yes—
239C3×C22⋊C4—42no—
240C3×C42—42no—
241#217—22no—
242#217—22no—
243#217—42no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—81no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—42no—
254C4⋊Dic3C2×Q812yes—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258Dic3⋊C4—81no—
259Dic3⋊C4—81no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—81no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C22×Q8—61no—
272C22×D4—31no—
273C22×D4—61no—
274C23×C4—21no—
275C23×C4—61no—
276C23×C4C22×D311yescharacteristic
277C2×M16—22no—
278C2×M16—41no—
279C2×M16—41no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C4⋊Q8—62no—
285C4⋊Q8—121no—
286C22⋊Q8—124no—
287#121—34no—
288#121—62no—
289C4×D4—62no—
290C4×D4—121no—
291#117—22no—
292#117—41no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—41no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—31no—
311C2×C22⋊C4—31no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—21no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—61no—
321#107—121no—
322#106—62no—
323C4⋊C8—121no—
324C4⋊C8—121no—
325#102—62no—
326#102—121no—
327#102—121no—
328#102—121no—
329C22⋊C8—62no—
330#95—31no—
331#95—31no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3C22⋊Q812yes—
344C23×C3C22×D411yescharacteristic
345C22×D3—22no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
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×C4—41no—
370C2×C3×C4—84no—
371C2×C3×C4C22⋊Q812yes—
372C2×C3×C4C22×Q811yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378#67—84no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
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×Dic3AGL(1,ℤ/8)12yes—
399C2×Dic3#13712yes—
400D12—22no—
401D12—22no—
402D12—41no—
403D12—41no—
404C4×D3—84no—
405Dic6—84no—
406C3×C8—41no—
407C3×C8—41no—
408C3⋊C8—41no—
409C3⋊C8—41no—
410C24—61no—
411C24D3×D411yescharacteristic
412C2×Q8—122no—
413C2×Q8—122no—
414C2×Q8—241no—
415C2×D4—32no—
416C2×D4—64no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—61no—
422C2×D4—241no—
423C2×D4—241no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—81no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—241no—
445C22×C4#23512yes—
446C22×C4D3×D411yescharacteristic
447C22×C4D3×Q812yes—
448C22×C4C23×D311yescharacteristic
449M16—44no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—32no—
469C4⋊C4—42no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—241no—
480C4⋊C4—241no—
481C4⋊C4#23512yes—
482C4⋊C4#23512yes—
483C22⋊C4—42no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—64no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C22⋊C4—241no—
491C42—42no—
492C42—61no—
493C42—61no—
494C42—61no—
495C42—61no—
496C42—241no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3C2×C22⋊Q811yescharacteristic
505C22×C3#54611yescharacteristic
506C22×C3#54611yescharacteristic
507C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
508C22×C3C2×#13711yescharacteristic
509C22×C3#57611yescharacteristic
510C22×C3#57711yescharacteristic
511C2×D3—42no—
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—81no—
515C2×D3—81no—
516C2×D3—81no—
517C2×D3—81no—
518C3×C4—22no—
519C3×C4—42no—
520C3×C4—42no—
521C3×C4—81no—
522C3×C4—81no—
523C3×C4#52911yescharacteristic
524C3×C4#53811yescharacteristic
525C3×C4#55411yescharacteristic
526C3×C4#55411yescharacteristic
527Dic3—22no—
528Dic3—22no—
529Dic3—41no—
530Dic3—41no—
531Dic3—42no—
532Dic3—81no—
533Dic3—81no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—62no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23#89612yes—
548C23C2×D3×D411yescharacteristic
549Q8—244no—
550D4—62no—
551D4—62no—
552D4—121no—
553D4—121no—
554D4—244no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—32no—
564C2×C4—32no—
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—61no—
580C2×C4—61no—
581C2×C4—62no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—84no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—244no—
596C2×C4#89612yes—
597C2×C4C2×#23511yescharacteristic
598C2×C4C2×#23511yescharacteristic
599C2×C4C2×#23511yescharacteristic
600C2×C4#100411yescharacteristic
601C2×C4#100411yescharacteristic
602C2×C4C2×D3×Q811yescharacteristic
603C8—41no—
604C8—41no—
605C8—121no—
606C8—121no—
607C2×C3—42no—
608C2×C3#245011yescharacteristic
609C2×C3#307911yescharacteristic
610C2×C3#308111yescharacteristic
611C2×C3#331411yescharacteristic
612C2×C3#331811yescharacteristic
613C2×C3#332911yescharacteristic
614C2×C3#333211yescharacteristic
615D3—82no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—122no—
624C22—122no—
625C22—122no—
626C22—241no—
627C22—241no—
628C22—241no—
629C22—241no—
630C22#549011yescharacteristic
631C22#549011yescharacteristic
632C22C2×#89611yescharacteristic
633C22D3×AGL(1,ℤ/8)11yescharacteristic
634C22#577411yescharacteristic
635C22D3×#13711yescharacteristic
636C22#577811yescharacteristic
637C4—22no—
638C4—42no—
639C4—42no—
640C4—62no—
641C4—62no—
642C4—81no—
643C4—81no—
644C4—121no—
645C4—121no—
646C4—122no—
647C4—241no—
648C4—241no—
649C4#547911yescharacteristic
650C4#549411yescharacteristic
651C4#551111yescharacteristic
652C4#551111yescharacteristic
653C3#2642511yescharacteristic, complemented, Sylow
654C2—42no—
655C2—242no—
656C2#7335111yescharacteristic
657C2#7889811yescharacteristic
658C2#7890811yescharacteristic
659C2#7893611yescharacteristic
660C2#7897311yescharacteristic
661C2#7924011yescharacteristic
662C2#7924211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1736C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1736D3
hypocenterC3#26425
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. #82171C2semidirect
3. C3×#1736C2semidirect
4. #71995C2semidirect
6. #71576C2semidirect
8. #68901C2semidirect
9. #68898C2semidirect
10. #68895C2semidirect
11. #68892C2semidirect
12. #68811C2semidirect
13. #68808C2semidirect
14. #68800C2semidirect
15. #68788C2semidirect
74. #1736D3semidirect
101. #930Q8semidirect
653. C3#26425semidirect

Maximal subgroups

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

subgroupindex
#822442
#821712
C3×#17362
#719952
#719902
#715762
#715602
#689012
#688982
#688952
#688922
#688112
#688082
#688002
#687882
#264253