Finite Group Databasebeta

← #800727

Subgroups — #800727

Order 512.

2842 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2357C212yesmaximal, complemented
2#17015C212yesmaximal, complemented
3#15952C212yesmaximal, complemented
4#15417C212yesmaximal, complemented
5#15413C214yesmaximal, complemented
6#15134C212yesmaximal, complemented
7#15127C211yesmaximal, characteristic
8#14922C214yesmaximal, complemented
9#14224C212yesmaximal, complemented
10#14025C212yesmaximal
11#14022C212yesmaximal, complemented
12#13982C212yesmaximal, complemented
13#13979C212yesmaximal, complemented
14#13911C211yesmaximal, characteristic, complemented
15C4×#398C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4C2212yes—
18C2×#398C2211yescharacteristic
19C2×#396C2214yes—
20C2×#396C2214yes—
21C2×#394C2212yes—
22C2×#394C2212yes—
23#2357—216no—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#384C2211yescharacteristic
36C2×#382C2212yes—
37C2×#382C2212yes—
38C2×#381C2214yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2214yes—
45#1445C2211yescharacteristic
46#1445C2214yes—
47#1444C2211yescharacteristic
48#1443C2214yes—
49#1442C2212yes—
50#1442C2212yes—
51#1442C2212yes—
52#1441C2212yes—
53#1441C2211yescharacteristic
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2212yes—
61#1438C2212yes—
62#1438C2214yes—
63#1438C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C4×#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C24×C4C2314yes—
80C2×#127—28no—
81C2×#127—42no—
82C2×C4×D4—28no—
83C2×C4×D4—28no—
84C2×C4×D4—28no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
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×C22⋊C4C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2314yes—
114C22×C42C2312yes—
115#398—24no—
116#396—48no—
117#396—48no—
118#394—28no—
119#394—44no—
120#390—28no—
121#390—28no—
122#390—28no—
123#390—28no—
124#390—44no—
125#389—28no—
126#388—216no—
127#388—216no—
128#388—48no—
129#386—48no—
130#386—48no—
131#384—42no—
132#382—28no—
133#382—44no—
134#381—216no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—42no—
138C4×C4⋊C4—42no—
139C4×C4⋊C4D418yescomplemented
140C4×C22⋊C4—216no—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C25C2414yes—
167C22×D4—28no—
168C22×D4—28no—
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—42no—
178C22×D4—42no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4#4118yes—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—28no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205#127—416no—
206C4×D4—264no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D418yes—
240C2×C4⋊C4#4114yes—
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—28no—
256C2×C22⋊C4—28no—
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—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42C2×D414yes—
291C2×C42C2×D418yes—
292C2×C42#4118yes—
293#95—24no—
294#95—28no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—22no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—42no—
307#95—42no—
308#95—41no—
309#95—41no—
310#95—41no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—44no—
317#95C2×D418yes—
318C24—24no—
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—
332C24C2×#4118yes—
333C24C2511yescharacteristic
334C2×D4—216no—
335C2×D4—216no—
336C2×D4—216no—
337C2×D4—416no—
338C2×D4—416no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—88no—
348C2×D4—88no—
349C2×D4—84no—
350C2×D4—84no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
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—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—232no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4C22×D418yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+412yes—
444C4⋊C4—216no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—84no—
454C4⋊C4—88no—
455C4⋊C4—84no—
456C22⋊C4—232no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—416no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C42—216no—
468C42—28no—
469C42—48no—
470C42—48no—
471C42—44no—
472C42—44no—
473C42—88no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
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—
508C23—44no—
509C23C23×D411yescharacteristic
510C23C22×#4112yes—
511C23C22×#4112yes—
512C23C22×#4111yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×H(5,𝔽2)12yes—
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×2−1+411yescharacteristic
518C23#58512yes—
519C23#58511yescharacteristic
520D4—432no—
521D4—816no—
522D4—816no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—832no—
604C2×C4#52518yes—
605C2×C4#53714yes—
606C2×C4#55014yes—
607C2×C4#55114yes—
608C2×C4#55218yes—
609C2×C4#55514yes—
610C22—48no—
611C22—48no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—48no—
616C22—48no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22C2×#52512yes—
626C22C2×#53711yescharacteristic
627C22C2×#55011yescharacteristic
628C22C2×#55111yescharacteristic
629C22C2×#55212yes—
630C22C2×#55511yescharacteristic
631C22#348012yes—
632C22#348112yes—
633C22#348812yes—
634C22#349611yescharacteristic
635C22#350511yescharacteristic
636C22#350812yes—
637C22#350812yes—
638C22#351012yes—
639C22#351111yescharacteristic
640C22#351312yes—
641C22#351412yes—
642C22#351412yes—
643C22#353711yescharacteristic
644C22#353711yescharacteristic
645C22#355111yescharacteristic
646C22#355912yes—
647C22#355911yescharacteristic
648C4—24no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—48no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4—84no—
659C4—84no—
660C4—81no—
661C4#260918yes—
662C2—88no—
663C2C2×#260911yescharacteristic
664C2#3580812yes—
665C2#3649312yes—
666C2#3974211yescharacteristic
667C2#4295211yescharacteristic
668C2#4521112yes—
669C2#4521411yescharacteristic
670C2#4543711yescharacteristic
671C2#5937811yescharacteristic
672C2#5944612yes—
673C2#5990811yescharacteristic
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×#2357C2semidirect
2. #17015C2semidirect
3. #15952C2semidirect
4. #15417C2semidirect
5. #15413C2semidirect
6. #15134C2semidirect
8. #14922C2semidirect
9. #14224C2semidirect
11. #14022C2semidirect
12. #13982C2semidirect
13. #13979C2semidirect
14. #13911C2semidirect
15. C4×#398C2semidirect
139. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23572
#170152
#159522
#154172
#154132
#151342
#151272
#149222
#142242
#140252
#140222
#139822
#139792
#139112
C4×#3982