Finite Group Databasebeta

← #10859156

Subgroups — #10859156

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#73494C211yesmaximal, characteristic, complemented
2#73467C211yesmaximal, characteristic
3C3×#1851C211yesmaximal, characteristic, complemented
4#71990C211yesmaximal, characteristic
5#71985C211yesmaximal, characteristic, complemented
6#70483C211yesmaximal, characteristic, complemented
7#70479C211yesmaximal, characteristic, complemented
8#70464C211yesmaximal, characteristic, complemented
9#70461C211yesmaximal, characteristic, complemented
10#69933C211yesmaximal, characteristic
11#69928C211yesmaximal, characteristic, complemented
12#68679C211yesmaximal, characteristic, complemented
13#68671C211yesmaximal, characteristic, complemented
14#68642C211yesmaximal, characteristic, complemented
15#68622C211yesmaximal, characteristic, complemented
16#26436—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#67⋊C4—21no—
22C2×C4×D12C2211yescharacteristic
23C2×#870C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×C4×C22⋊C4C2211yescharacteristic
28#5212C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#817C2211yescharacteristic
35#4958—21no—
36#4949—21no—
37#4948—21no—
38#4939C2211yescharacteristic
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4670—21no—
45#4668—21no—
46#4654—21no—
47#4647—21no—
48C3×#336C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
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#1864—31no—
71#1853—31no—
72#1851D311yescharacteristic, complemented
73#1802—31no—
74#1736—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3—21no—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C2×C3×M16—22no—
81C2×C3×M16—41no—
82C2×C3×M16—41no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93#930—22no—
94#930Q814yes—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99#926—22no—
100#926—41no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—41no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120#67⋊C4—44no—
121C4×D12—22no—
122C4×D12—41no—
123#870—22no—
124#870—41no—
125C3×C4⋊C8—41no—
126C3×C4⋊C8—41no—
127C3×C22⋊C8—22no—
128C3×#95D411yescharacteristic
129C3×#95D411yescharacteristic
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#821—22no—
136#821—41no—
137#821—41no—
138#821—41no—
139#817—22no—
140#817—41no—
141C22×M16C2×D311yescharacteristic
142C2×#125—31no—
143C2×#121—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×#117—31no—
147C22×C4⋊C4—31no—
148C2×#106—31no—
149C2×#106—31no—
150C2×C4⋊C8C2×D311yescharacteristic
151C2×C4⋊C8C2×D311yescharacteristic
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×C22⋊C8C2×D311yescharacteristic
157#399—61no—
158#396—61no—
159#390—31no—
160#389—31no—
161#389—31no—
162#389—31no—
163#389—61no—
164#388—61no—
165#387—61no—
166#386—61no—
167#385—61no—
168#384—31no—
169#380—31no—
170C4×C4⋊C4—31no—
171C4×C22⋊C4C2×D311yescharacteristic
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#340—31no—
182#336C2×D311yescharacteristic
183#336C2×D311yescharacteristic
184C24×C3C2×D411yescharacteristic
185C23×D3—21no—
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×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×Q812yes—
196C22×C3×C4#4112yes—
197C22×C3×C4C2411yescharacteristic
198C2×#67—24no—
199C2×#67—42no—
200C2×#67—42no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—42no—
209C22×Dic3—42no—
210C22×Dic3—81no—
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—
220C3×M16—44no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—41no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239#217—42no—
240#217—81no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—81no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—42no—
257C4⋊Dic3—42no—
258C4⋊Dic3—81no—
259C4⋊Dic3C2×Q812yes—
260C4⋊Dic3#4112yes—
261C4⋊Dic3#4112yes—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265Dic3⋊C4—42no—
266Dic3⋊C4—81no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C22×D4—31no—
270C22×D4—61no—
271C23×C4—61no—
272C23×C4—61no—
273C23×C4C22×D311yescharacteristic
274C2×M16—22no—
275C2×M16—41no—
276C2×M16—41no—
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279C22×C8C22×D311yescharacteristic
280C22×C8C22×D311yescharacteristic
281#125—62no—
282#125—121no—
283#121—34no—
284#121—62no—
285C4×D4—62no—
286C4×D4—121no—
287C4×D4—124no—
288#117—62no—
289#117—121no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—62no—
298C2×C4⋊C4—61no—
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⋊C4—61no—
306C2×C4⋊C4—121no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4D1211yescharacteristic
314C2×C22⋊C4D1211yescharacteristic
315C2×C42—61no—
316C2×C42—61no—
317C2×C42C22×D311yescharacteristic
318C2×C42C22×D311yescharacteristic
319#106—62no—
320#106—121no—
321C4⋊C8—41no—
322C4⋊C8—41no—
323#102—62no—
324#102—121no—
325#102—121no—
326#102—121no—
327C22⋊C8—22no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95D1211yescharacteristic
334#95D1211yescharacteristic
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3C22⋊Q812yes—
342C23×C3C22×D411yescharacteristic
343C22×D3—22no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
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—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4C22⋊Q812yes—
368C2×C3×C4AGL(1,ℤ/8)12yes—
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
375#67—48no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
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×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×Dic3—84no—
398D12—22no—
399D12—22no—
400D12—41no—
401D12—41no—
402C4×D3—84no—
403C3×C8—41no—
404C3×C8—41no—
405C3×C8—41no—
406C3×C8—41no—
407C24—61no—
408C24C2×D1211yescharacteristic
409C2×D4—32no—
410C2×D4—64no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—122no—
417C2×D4—122no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4—122no—
433C22×C4—241no—
434C22×C4—241no—
435C22×C4C2×D1211yescharacteristic
436C22×C4C2×D1211yescharacteristic
437C22×C4C2×D1211yescharacteristic
438C22×C4C2×D1211yescharacteristic
439C22×C4C2×D1211yescharacteristic
440C22×C4#23712yes—
441C22×C4D3×Q812yes—
442C22×C4C23×D311yescharacteristic
443M16—44no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C4⋊C4—32no—
457C4⋊C4—32no—
458C4⋊C4—32no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—241no—
477C4⋊C4—241no—
478C22⋊C4—22no—
479C22⋊C4—41no—
480C22⋊C4—64no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—241no—
485C22⋊C4—241no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—122no—
491C42—122no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
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—42no—
507C2×D3—42no—
508C2×D3—42no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—41no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4#52911yescharacteristic
519C3×C4#53811yescharacteristic
520C3×C4#55411yescharacteristic
521C3×C4#55411yescharacteristic
522Dic3—42no—
523Dic3—42no—
524Dic3—42no—
525Dic3—81no—
526Dic3—81no—
527Dic3—81no—
528Dic3—81no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—62no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23#89712yes—
543C23C22×D1211yescharacteristic
544D4—62no—
545D4—62no—
546D4—128no—
547D4—121no—
548D4—121no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—62no—
567C2×C4—62no—
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—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—244no—
588C2×C4—244no—
589C2×C4#89712yes—
590C2×C4#90812yes—
591C2×C4C2×#23711yescharacteristic
592C2×C4#100411yescharacteristic
593C2×C4#100411yescharacteristic
594C2×C4C2×D3×Q811yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C8—41no—
598C8—41no—
599C8—41no—
600C8—41no—
601C2×C3—42no—
602C2×C3#245011yescharacteristic
603C2×C3#307711yescharacteristic
604C2×C3#308411yescharacteristic
605C2×C3#330711yescharacteristic
606C2×C3#331011yescharacteristic
607C2×C3#332611yescharacteristic
608C2×C3#333211yescharacteristic
609D3—82no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—122no—
618C22—122no—
619C22—122no—
620C22—241no—
621C22—241no—
622C22—241no—
623C22—241no—
624C22C2×#89711yescharacteristic
625C22#555411yescharacteristic
626C22#555411yescharacteristic
627C22C2×#90811yescharacteristic
628C22#574811yescharacteristic
629C22#575111yescharacteristic
630C22#575111yescharacteristic
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—122no—
637C4—122no—
638C4—122no—
639C4—241no—
640C4—241no—
641C4—241no—
642C4—241no—
643C4#559311yescharacteristic
644C4#563611yescharacteristic
645C4#563611yescharacteristic
646C4#564911yescharacteristic
647C3#2643611yescharacteristic, complemented, Sylow
648C2—42no—
649C2—242no—
650C2#7442411yescharacteristic
651C2#7905911yescharacteristic
652C2#7909211yescharacteristic
653C2#7950011yescharacteristic
654C2#7950111yescharacteristic
655C2#7956111yescharacteristic
656C2#7956511yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1851C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1851D3
hypocenterC3#26436
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
1. #73494C2semidirect
3. C3×#1851C2semidirect
5. #71985C2semidirect
6. #70483C2semidirect
7. #70479C2semidirect
8. #70464C2semidirect
9. #70461C2semidirect
11. #69928C2semidirect
12. #68679C2semidirect
13. #68671C2semidirect
14. #68642C2semidirect
15. #68622C2semidirect
72. #1851D3semidirect
647. C3#26436semidirect

Maximal subgroups

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

subgroupindex
#734942
#734672
C3×#18512
#719902
#719852
#704832
#704792
#704642
#704612
#699332
#699282
#686792
#686712
#686422
#686222
#264363