Finite Group Databasebeta

← #409889

Subgroups — #409889

Order 512.

1160 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2101C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic
4C2×#1935C211yesmaximal, characteristic
5#15240C211yesmaximal, characteristic
6#15143C211yesmaximal, characteristic
7#11373C211yesmaximal, characteristic
8#11372C211yesmaximal, characteristic
9#11050C211yesmaximal, characteristic
10#11050C211yesmaximal, characteristic
11#10928C211yesmaximal, characteristic
12#10926C211yesmaximal, characteristic
13#10843C211yesmaximal, characteristic
14#10841C211yesmaximal, characteristic
15#10696C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#395—22no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34#2101—24no—
35#2024—24no—
36#1993—24no—
37#1935—24no—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53#1445—22no—
54#1445—22no—
55#1444—22no—
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C23×Q8—21no—
60C23×Q8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—41no—
70C2×C4⋊Q8—41no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—41no—
94C2×#103—41no—
95C2×#103—41no—
96C2×#103—41no—
97C2×#103Q814yescomplemented
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—41no—
101C2×#97—41no—
102C2×#97—41no—
103C2×#97D414yes—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—41no—
107#395—48no—
108#391—22no—
109#391—22no—
110#391—22no—
111#391—22no—
112#391—22no—
113#391—22no—
114#389—22no—
115#389—22no—
116#384—22no—
117#384—22no—
118#384—22no—
119#384—22no—
120#384—41no—
121#384—41no—
122#382—22no—
123#382—22no—
124#382—22no—
125#382—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95D412yes—
137C2×#95D412yes—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#336—22no—
151#336—22no—
152#336D414yes—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—44no—
169C22×Q8—42no—
170C22×Q8—82no—
171C22×Q8C2×Q812yes—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2411yescharacteristic
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×Q812yes—
212C22×C8#4112yes—
213C4⋊Q8—28no—
214C4⋊Q8—44no—
215C4⋊Q8—44no—
216C4⋊Q8—44no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4C2×Q812yes—
271C2×C4⋊C4#4112yes—
272C2×C4⋊C4#4112yes—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42#4112yes—
288C2×C42#4112yes—
289C2×C42#4112yes—
290C2×C42#4112yes—
291#103—28no—
292#103—44no—
293#103—44no—
294#103—44no—
295#97—44no—
296#97—44no—
297C4×C8—44no—
298#95—24no—
299#95—24no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—42no—
308#95—42no—
309#95—44no—
310C24C22×D411yescharacteristic
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—48no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—84no—
341C2×Q8—84no—
342C2×Q8—88no—
343C2×Q8—88no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4#12414yes—
405C22×C4#12414yes—
406C22×C4#12712yes—
407C22×C4#12712yes—
408C22×C4AGL(1,ℤ/8)14yes—
409C22×C4#13714yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×Q811yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C42−1+411yescharacteristic
423C22×C42−1+411yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8C22⋊Q814yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—48no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—88no—
482C4⋊C4—84no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C23C2×#12411yescharacteristic
498C23C2×#12411yescharacteristic
499C23C2×#12711yescharacteristic
500C23#54611yescharacteristic
501C23#54611yescharacteristic
502C23#54711yescharacteristic
503C23D4×Q811yescharacteristic
504C23C2×AGL(1,ℤ/8)12yes—
505C23C2×#13712yes—
506C23#57712yes—
507C23#57812yes—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—816no—
513Q8—81no—
514Q8—81no—
515Q8—81no—
516Q8—81no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4#48918yes—
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4#52511yescharacteristic
585C2×C4#52911yescharacteristic
586C2×C4#52911yescharacteristic
587C2×C4#53211yescharacteristic
588C2×C4#53211yescharacteristic
589C2×C4#53211yescharacteristic
590C2×C4#53211yescharacteristic
591C2×C4#53311yescharacteristic
592C2×C4#53311yescharacteristic
593C2×C4#53411yescharacteristic
594C2×C4#53611yescharacteristic
595C2×C4#53911yescharacteristic
596C2×C4#53911yescharacteristic
597C2×C4#54211yescharacteristic
598C2×C4#54211yescharacteristic
599C2×C4#54611yescharacteristic
600C2×C4#54611yescharacteristic
601C2×C4#54711yescharacteristic
602C2×C4D4×Q811yescharacteristic
603C2×C4#55111yescharacteristic
604C2×C4#55111yescharacteristic
605C2×C4#55211yescharacteristic
606C2×C4#55211yescharacteristic
607C2×C4#55411yescharacteristic
608C2×C4#55411yescharacteristic
609C2×C4#55611yescharacteristic
610C2×C4#55611yescharacteristic
611C8—24no—
612C8—42no—
613C8—42no—
614C8—42no—
615C22#248811yescharacteristic
616C22#248811yescharacteristic
617C22#249211yescharacteristic
618C22#249211yescharacteristic
619C22#253911yescharacteristic
620C22#254211yescharacteristic
621C22#285811yescharacteristic
622C22C2×#48912yes—
623C22#313712yes—
624C22#313712yes—
625C22#314112yes—
626C22#315111yescharacteristic
627C22#315211yescharacteristic
628C22#315411yescharacteristic
629C22#315611yescharacteristic
630C22#331012yes—
631C22#331212yes—
632C22#332412yes—
633C22#332812yes—
634C22#334312yes—
635C22#334412yes—
636C22#338612yes—
637C22#338712yes—
638C4—28no—
639C4—44no—
640C4—42no—
641C4—42no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—161no—
647C4—161no—
648C4—161no—
649C4—161no—
650C4#239611yescharacteristic
651C4#252111yescharacteristic
652C4#252111yescharacteristic
653C4#255611yescharacteristic
654C4#255611yescharacteristic
655C4#256411yescharacteristic
656C4#260411yescharacteristic
657C4#266311yescharacteristic
658C2#1506511yescharacteristic
659C2#2770612yes—
660C2#2771312yes—
661C2#2782312yes—
662C2#2782712yes—
663C2#2835511yescharacteristic
664C2#2836511yescharacteristic
665C2#2854511yescharacteristic
666C2#2855011yescharacteristic
667C2#3322911yescharacteristic
668C2#3323011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
97. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#21012
C2×#20242
C2×#19932
C2×#19352
#152402
#151432
#113732
#113722
#110502
#109282
#109262
#108432
#108412
#106962