Finite Group Databasebeta

← #799356

Subgroups — #799356

Order 512.

2626 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2593C212yesmaximal, complemented
2#16599C212yesmaximal, complemented
3#15978C212yesmaximal, complemented
4#15831C214yesmaximal, complemented
5#15780C212yesmaximal, complemented
6#15152C211yesmaximal, characteristic, complemented
7#15127C212yesmaximal
8#14799C214yesmaximal, complemented
9#14154C211yesmaximal, characteristic
10#14022C212yesmaximal, complemented
11#14001C212yesmaximal, complemented
12#14001C212yesmaximal, complemented
13#14001C212yesmaximal, complemented
14C4×#398C211yesmaximal, characteristic, complemented
15C4⋊C42C212yesmaximal, complemented
16C22×#127C2211yescharacteristic
17#2593—216no—
18C2×#400C2212yes—
19C2×#398C2211yescharacteristic
20C2×#394C2214yes—
21C2×#394C2214yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2214yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46#1445C2211yescharacteristic
47#1445C2212yes—
48#1444C2211yescharacteristic
49#1442C2212yes—
50#1442C2212yes—
51#1441C2212yes—
52#1441C2211yescharacteristic
53#1441C2212yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2214yes—
63#1438C2214yes—
64#1438C2212yes—
65#1438C2212yes—
66#1437C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2214yes—
73C4⋊#95C2212yes—
74C4×#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C2×#127—24no—
80C2×#127—42no—
81C2×#127—42no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2314yes—
115#400—28no—
116#398D418yescomplemented
117#394—216no—
118#394—48no—
119#390—28no—
120#390—28no—
121#390—28no—
122#390—28no—
123#390—28no—
124#390—44no—
125#390—44no—
126#390—44no—
127#388—216no—
128#388—216no—
129#388—216no—
130#388—216no—
131#388—48no—
132#388—48no—
133#388—48no—
134#384—28no—
135#384—42no—
136C4×C4⋊C4—216no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—42no—
140C4×C4⋊C4—42no—
141C4×C4⋊C4—42no—
142C4×C4⋊C4—44no—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C25C2414yes—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—44no—
185C22×D4—44no—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200#127—416no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C2×D418yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4#4118yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—42no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—42no—
294C2×C42#4114yes—
295C2×C42#4118yes—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—41no—
308#95—42no—
309#95—42no—
310#95—41no—
311#95—41no—
312#95—44no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95C2×D414yes—
318#95C2×D418yes—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24C2511yescharacteristic
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—88no—
351C2×D4—84no—
352C2×D4—88no—
353C2×D4—84no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D414yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4118yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C4H(5,𝔽2)14yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C42−1+412yes—
440C4⋊C4—216no—
441C4⋊C4—216no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—416no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—84no—
451C4⋊C4—88no—
452C4⋊C4—84no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—416no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—416no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C42—216no—
466C42—28no—
467C42—48no—
468C42—416no—
469C42—44no—
470C42—44no—
471C42—84no—
472C42—88no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23C23×D411yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4112yes—
511C23C22×#4112yes—
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)12yes—
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×2−1+411yescharacteristic
517C23#58512yes—
518C23#58511yescharacteristic
519D4—816no—
520D4—816no—
521D4—816no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4#52918yes—
602C2×C4#53814yes—
603C2×C4#54714yes—
604C2×C4#55018yes—
605C2×C4#55318yes—
606C2×C4#55514yes—
607C2×C4#55914yes—
608C22—84no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22C2×#52912yes—
624C22C2×#53811yescharacteristic
625C22C2×#54711yescharacteristic
626C22C2×#55012yes—
627C22D4×#4112yes—
628C22C2×#55312yes—
629C22C2×#55511yescharacteristic
630C22C2×#55911yescharacteristic
631C22#349412yes—
632C22#349611yescharacteristic
633C22#349811yescharacteristic
634C22#350612yes—
635C22#350612yes—
636C22#350611yescharacteristic
637C22#350712yes—
638C22#351912yes—
639C22#352212yes—
640C22#352412yes—
641C22#353611yescharacteristic
642C22#354011yescharacteristic
643C22#354612yes—
644C22#354911yescharacteristic
645C22#356111yescharacteristic
646C4—28no—
647C4—24no—
648C4—48no—
649C4—44no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—84no—
656C4—84no—
657C4—82no—
658C4—81no—
659C4#259318yescomplemented
660C2—164no—
661C2C2×#259311yescharacteristic
662C2#3578512yes—
663C2#3611712yes—
664C2#3937511yescharacteristic
665C2#4297511yescharacteristic
666C2#4496912yes—
667C2#4504711yescharacteristic
668C2#4536911yescharacteristic
669C2#5962811yescharacteristic
670C2#5968012yes—
671C2#5982211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2593C2semidirect
2. #16599C2semidirect
3. #15978C2semidirect
4. #15831C2semidirect
5. #15780C2semidirect
6. #15152C2semidirect
8. #14799C2semidirect
10. #14022C2semidirect
11. #14001C2semidirect
12. #14001C2semidirect
13. #14001C2semidirect
14. C4×#398C2semidirect
15. C4⋊C42C2semidirect
116. #398D4semidirect
659. C4#2593semidirect

Maximal subgroups

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

subgroupindex
C2×#25932
#165992
#159782
#158312
#157802
#151522
#151272
#147992
#141542
#140222
#140012
C4×#3982
C4⋊C422