Finite Group Databasebeta

← #10859163

Subgroups — #10859163

Order 768.

832 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82399C211yesmaximal, characteristic
2#76000C211yesmaximal, characteristic, complemented
3C3×#1864C211yesmaximal, characteristic, complemented
4#71604C211yesmaximal, characteristic, complemented
5#70579C211yesmaximal, characteristic, complemented
6#70557C211yesmaximal, characteristic, complemented
7#70203C211yesmaximal, characteristic
8#70201C211yesmaximal, characteristic
9#70193C211yesmaximal, characteristic, complemented
10#70164C211yesmaximal, characteristic, complemented
11#69610C211yesmaximal, characteristic, complemented
12#68867C211yesmaximal, characteristic, complemented
13#68802C211yesmaximal, characteristic, complemented
14#68667C211yesmaximal, characteristic, complemented
15#68642C211yesmaximal, characteristic, complemented
16#26436—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#938C2211yescharacteristic
19C2×C4×#67—21no—
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×#890C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#389C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31#5210—21no—
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34#4994—21no—
35#4987C2211yescharacteristic
36#4984—21no—
37#4984C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4956—21no—
41#4945—21no—
42C4×(C2×D3)⋊C4C2211yescharacteristic
43#4935C2211yescharacteristic
44#4928C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4677—21no—
47#4653—21no—
48C3×#340C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2441—31no—
61#2361—31no—
62#2099—31no—
63#2088—31no—
64#2065—31no—
65#2025—31no—
66#1989—31no—
67#1919—31no—
68#1895—31no—
69#1875—31no—
70#1864D311yescharacteristic, complemented
71#1853—31no—
72#1851—31no—
73#1802—31no—
74#1736—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×#67—21no—
78C22×C4×D3—21no—
79C22×C4×D3C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×D4—22no—
83C3×C4×D4—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93#938—22no—
94#938Q814yes—
95C4×#67—44no—
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⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×ΓL(1,𝔽25)—22no—
117C2×ΓL(1,𝔽25)—41no—
118C2×ΓL(1,𝔽25)—41no—
119#892—22no—
120#892—41no—
121D3×C4⋊C4—22no—
122D3×C4⋊C4—41no—
123#890—22no—
124#890—41no—
125C3×#106—22no—
126C3×#102—22no—
127C3×#102—41no—
128C3×#95—21no—
129#832—41no—
130#832—41no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137(C2×D3)⋊C8—22no—
138Dic3⋊C8—41no—
139#808—41no—
140C4⋊(C3⋊C8)—41no—
141C22×M16—31no—
142C2×#125—31no—
143C2×#121—31no—
144C2×C4×D4—61no—
145C2×C4×D4C2×D311yescharacteristic
146C2×#117—31no—
147C22×C4⋊C4—31no—
148C2×#106—31no—
149C2×#106C2×D311yescharacteristic
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102C2×D311yescharacteristic
155C2×#102C2×D311yescharacteristic
156C2×C22⋊C8—31no—
157#399—61no—
158#396—61no—
159#390—31no—
160#389—31no—
161#389—31no—
162#389—61no—
163#389C2×D311yescharacteristic
164#388—61no—
165#387—61no—
166#386—61no—
167#385—61no—
168#384—31no—
169#380—31no—
170C4×C4⋊C4—31no—
171C4×C22⋊C4—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#336—31no—
183#336C2×D311yescharacteristic
184C24×C3—21no—
185C23×D3C2×D411yescharacteristic
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4C2×Q812yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—81no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C2×#67—24no—
202C2×#67—42no—
203C2×#67—42no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—42no—
216C2×C4×D3—42no—
217C2×C4×D3—81no—
218C2×C4×D3C2×Q812yes—
219C2×C4×D3#4112yes—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4#4112yes—
232C3×C4⋊C4#4112yes—
233C3×C22⋊C4—42no—
234C3×C42—21no—
235C3×C42—21no—
236#217—24no—
237#217—42no—
238#217—81no—
239(C2×D3)⋊C4—22no—
240(C2×D3)⋊C4—41no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—81no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—42no—
251C4⋊Dic3—42no—
252C4⋊Dic3—81no—
253C4⋊Dic3—81no—
254C4⋊Dic3C2×Q812yes—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268ΓL(1,𝔽25)—44no—
269C22×D4—61no—
270C22×D4C22×D311yescharacteristic
271C23×C4—21no—
272C23×C4—31no—
273C23×C4—61no—
274C2×M16—62no—
275C2×M16—121no—
276C2×M16—121no—
277C22×C8—31no—
278C22×C8—31no—
279C22×C8C22×D311yescharacteristic
280C22×C8C22×D311yescharacteristic
281#125—62no—
282#125—121no—
283#121—34no—
284#121—62no—
285C4×D4—22no—
286C4×D4—41no—
287C4×D4—124no—
288#117—62no—
289#117—121no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—62no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—121no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—31no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C42—31no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#106—22no—
320#106—121no—
321C4⋊C8—121no—
322C4⋊C8—121no—
323#102—22no—
324#102—41no—
325#102—121no—
326#102—121no—
327C22⋊C8—62no—
328#95—21no—
329#95—31no—
330#95—31no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335C23×C3—22no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3C22×D411yescharacteristic
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3C22⋊Q812yes—
350C3×D4—22no—
351C3×D4—22no—
352C3×D4—41no—
353C3×D4—41no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
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—84no—
369C2×C3×C4AGL(1,ℤ/8)12yes—
370C2×C3×C4C22×Q811yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376#67—48no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—41no—
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×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3C22⋊Q812yes—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—22no—
401C4×D3—22no—
402C4×D3—84no—
403C3×C8—41no—
404C3×C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C24—21no—
408C24—31no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—64no—
415C2×D4—122no—
416C2×D4—122no—
417C2×D4D3×Q812yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—31no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—81no—
437C22×C4—122no—
438C22×C4—122no—
439C22×C4—241no—
440C22×C4D3×D411yescharacteristic
441C22×C4D3×D411yescharacteristic
442C22×C4C23×D311yescharacteristic
443M16—124no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—32no—
461C4⋊C4—42no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C4⋊C4—241no—
475C4⋊C4—241no—
476C4⋊C4#23512yes—
477C4⋊C4#23712yes—
478C22⋊C4—42no—
479C22⋊C4—64no—
480C22⋊C4—62no—
481C22⋊C4—121no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—241no—
485C22⋊C4—241no—
486C42—21no—
487C42—21no—
488C42—61no—
489C42—61no—
490C42—122no—
491C42—122no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3C2×C22⋊Q811yescharacteristic
500C22×C3#54611yescharacteristic
501C22×C3#54611yescharacteristic
502C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
503C22×C3#57511yescharacteristic
504C22×C3#57711yescharacteristic
505C22×C3#57711yescharacteristic
506C2×D3—22no—
507C2×D3—22no—
508C2×D3—22no—
509C2×D3—41no—
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—41no—
513C3×C4—22no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—81no—
518C3×C4#52911yescharacteristic
519C3×C4#53811yescharacteristic
520C3×C4#55411yescharacteristic
521C3×C4#55411yescharacteristic
522Dic3—22no—
523Dic3—41no—
524Dic3—42no—
525Dic3—41no—
526Dic3—81no—
527Dic3—81no—
528Dic3—81no—
529C23—22no—
530C23—32no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—61no—
538C23—61no—
539C23—61no—
540C23—61no—
541C23—61no—
542C23—61no—
543C23C2×D3×D411yescharacteristic
544D4—22no—
545D4—22no—
546D4—41no—
547D4—41no—
548D4—128no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—32no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—62no—
577C2×C4—84no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—244no—
590C2×C4#91412yes—
591C2×C4C2×#23511yescharacteristic
592C2×C4C2×#23711yescharacteristic
593C2×C4#100411yescharacteristic
594C2×C4C2×D3×Q811yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4#100911yescharacteristic
597C8—41no—
598C8—41no—
599C8—121no—
600C8—121no—
601C2×C3—82no—
602C2×C3#245011yescharacteristic
603C2×C3#307711yescharacteristic
604C2×C3#308411yescharacteristic
605C2×C3#330711yescharacteristic
606C2×C3#331011yescharacteristic
607C2×C3#332611yescharacteristic
608C2×C3#333211yescharacteristic
609D3—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—62no—
614C22—62no—
615C22—62no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—121no—
621C22—121no—
622C22—121no—
623C22—121no—
624C22#555011yescharacteristic
625C22#559111yescharacteristic
626C22D3×C22⋊Q811yescharacteristic
627C22C2×#91411yescharacteristic
628C22#577011yescharacteristic
629C22#577311yescharacteristic
630C22#577511yescharacteristic
631C4—22no—
632C4—42no—
633C4—42no—
634C4—41no—
635C4—62no—
636C4—81no—
637C4—121no—
638C4—121no—
639C4—122no—
640C4—241no—
641C4—241no—
642C4—241no—
643C4#551111yescharacteristic
644C4#555711yescharacteristic
645C4#563511yescharacteristic
646C4#565711yescharacteristic
647C3#2643611yescharacteristic, complemented, Sylow
648C2—82no—
649C2—122no—
650C2#7461011yescharacteristic
651C2#7968711yescharacteristic
652C2#7988111yescharacteristic
653C2#8011511yescharacteristic
654C2#8016011yescharacteristic
655C2#8106011yescharacteristic
656C2#8117111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1864C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#1864D3
hypocenterC3#26436
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #76000C2semidirect
3. C3×#1864C2semidirect
4. #71604C2semidirect
5. #70579C2semidirect
6. #70557C2semidirect
9. #70193C2semidirect
10. #70164C2semidirect
11. #69610C2semidirect
12. #68867C2semidirect
13. #68802C2semidirect
14. #68667C2semidirect
15. #68642C2semidirect
70. #1864D3semidirect
647. C3#26436semidirect

Maximal subgroups

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

subgroupindex
#823992
#760002
C3×#18642
#716042
#705792
#705572
#702032
#702012
#701932
#701642
#696102
#688672
#688022
#686672
#686422
#264363