Finite Group Databasebeta

← #608730

Subgroups — #608730

Order 512.

4456 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#120C211yesmaximal, characteristic
2C22×C4×C22⋊C4C211yesmaximal, characteristic, complemented
3#15205C214yesmaximal, complemented
4#15162C214yesmaximal, complemented
5#15026C211yesmaximal, characteristic, complemented
6#14955C211yesmaximal, characteristic, complemented
7#14946C211yesmaximal, characteristic, complemented
8#14945C211yesmaximal, characteristic, complemented
9#14727C212yesmaximal, complemented
10#14718C212yesmaximal
11C4×#397C214yesmaximal, complemented
12C4×#394C214yesmaximal, complemented
13C4×#379C211yesmaximal, characteristic, complemented
14C4×#385C212yesmaximal, complemented
15C4×#386C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C42C2212yes—
22C2×#397C2214yes—
23C2×#394C2214yes—
24C4×#120—28no—
25C2×#386C2212yes—
26C2×#386C2211yescharacteristic
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2211yescharacteristic
33C2×#379C2212yes—
34C2×#379C2211yescharacteristic
35C2×C4×C4⋊C4C2214yescomplemented
36C2×C4×C22⋊C4—28no—
37C2×C4×C22⋊C4C2214yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2214yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2214yes—
43C2×C4×C22⋊C4C2214yescomplemented
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1445C2214yescomplemented
47#1443C2214yescomplemented
48#1443C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2214yescomplemented
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yescomplemented
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yescomplemented
59#1438C2214yes—
60#1437C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2211yescharacteristic
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C4×#95C2214yescomplemented
72C4×#95C2214yescomplemented
73C26C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C23×D4C2312yes—
76C24×C4—28no—
77C24×C4—48no—
78C24×C4D418yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C2×#120—24no—
87C2×#120—24no—
88C2×#120—24no—
89C2×C4×D4—28no—
90C2×C4×D4—24no—
91C2×C4×D4—28no—
92C2×C4×D4—48no—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—216no—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C42—28no—
116C22×C42—48no—
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2314yes—
121C22×C42C2314yes—
122#397—216no—
123#394—216no—
124#386—28no—
125#386—24no—
126#386—28no—
127#386—28no—
128#385—24no—
129#385—28no—
130#385—28no—
131#385—28no—
132#379—24no—
133#379—28no—
134C4×C4⋊C4—48no—
135C4×C22⋊C4—216no—
136C4×C22⋊C4—216no—
137C4×C22⋊C4—28no—
138C4×C22⋊C4—24no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—48no—
141C4×C22⋊C4—48no—
142C2×#95—24no—
143C2×#95—24no—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C25—28no—
165C25—24no—
166C25—22no—
167C25—22no—
168C25—24no—
169C25—44no—
170C25C2×D414yes—
171C25C2411yescharacteristic
172C25C2412yes—
173C25C2411yescharacteristic
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—44no—
183C22×D4#4114yes—
184C23×C4—28no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—232no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—432no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—464no—
211C23×C4C2×D418yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D418yes—
215C23×C4#4118yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228#120—232no—
229C4×D4—432no—
230C4×D4—416no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C22⋊C4—264no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4#41116yes—
282C2×C22⋊C4#4114yes—
283C2×C22⋊C4#4118yes—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—432no—
298#95—28no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318C24—24no—
319C24—24no—
320C24—232no—
321C24—28no—
322C24—216no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—28no—
333C24—28no—
334C24—21no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—21no—
339C24—24no—
340C24—48no—
341C24—416no—
342C24—416no—
343C24—416no—
344C24—416no—
345C24—416no—
346C24—48no—
347C24—48no—
348C24C22×D414yes—
349C24C22×D412yes—
350C24C2×#4112yes—
351C24C2×#4118yes—
352C24C2×#4114yes—
353C24C2×#4112yes—
354C24C2×#4112yes—
355C24C2×#4114yes—
356C24C2511yescharacteristic
357C2×D4—216no—
358C2×D4—28no—
359C2×D4—216no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—44no—
367C22×C4—24no—
368C22×C4—216no—
369C22×C4—232no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—4128no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4#123116yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4118yes—
452C22×C4C2×#4118yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4H(5,𝔽2)18yes—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C22⋊C4—232no—
462C22⋊C4—216no—
463C22⋊C4—232no—
464C22⋊C4—28no—
465C22⋊C4—216no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C42—416no—
478C42—44no—
479C42—416no—
480C42—48no—
481C42—416no—
482C23—24no—
483C23—28no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—232no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—22no—
504C23—24no—
505C23—28no—
506C23—432no—
507C23—432no—
508C23—416no—
509C23—464no—
510C23—464no—
511C23—416no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—44no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23C2×#12314yes—
533C23#52518yes—
534C23#52514yes—
535C23#52918yes—
536C23#52918yes—
537C23#53214yes—
538C23C23×D411yescharacteristic
539C23C22×#4112yes—
540C23C22×#4111yescharacteristic
541C23C22×#4111yescharacteristic
542C23C22×#4114yes—
543C23C22×#4112yes—
544C23C2×H(5,𝔽2)14yes—
545D4—432no—
546D4—416no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—232no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—432no—
567C2×C4—432no—
568C2×C4—432no—
569C2×C4—416no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—432no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—88no—
610C2×C4C2×#12314yes—
611C2×C4C2×#12314yes—
612C2×C4#53418yes—
613C2×C4#538116yes—
614C22—24no—
615C22—28no—
616C22—28no—
617C22—216no—
618C22—216no—
619C22—24no—
620C22—28no—
621C22—28no—
622C22—28no—
623C22—464no—
624C22—44no—
625C22—44no—
626C22—48no—
627C22—48no—
628C22—48no—
629C22—48no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—48no—
634C22—42no—
635C22—81no—
636C22—82no—
637C22—84no—
638C22—81no—
639C22#239518yes—
640C22C22×#12311yescharacteristic
641C22C2×#52512yes—
642C22C2×#52511yescharacteristic
643C22C2×#52912yes—
644C22C2×#52912yes—
645C22C2×#53211yescharacteristic
646C22#344414yes—
647C22C2×#53412yes—
648C22C2×#53814yes—
649C22#348114yes—
650C22#348114yes—
651C22#348114yes—
652C22#348114yes—
653C4—24no—
654C4—48no—
655C4—44no—
656C4—48no—
657C4—48no—
658C4—42no—
659C4—44no—
660C4—81no—
661C4—82no—
662C4—84no—
663C4—81no—
664C4#262218yes—
665C2—28no—
666C2—48no—
667C2—82no—
668C2C2×#239511yescharacteristic
669C2C2×#262211yescharacteristic
670C2#3495612yes—
671C2#3495712yes—
672C2#3495914yes—
673C2#3495914yes—
674C2#3662611yescharacteristic
subgroupG/H
centerC23×C4C24
commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C22⋊C4C2semidirect
3. #15205C2semidirect
4. #15162C2semidirect
5. #15026C2semidirect
6. #14955C2semidirect
7. #14946C2semidirect
8. #14945C2semidirect
9. #14727C2semidirect
11. C4×#397C2semidirect
12. C4×#394C2semidirect
13. C4×#379C2semidirect
14. C4×#385C2semidirect
15. C4×#386C2semidirect
35. C2×C4×C4⋊C4C22semidirect
43. C2×C4×C22⋊C4C22semidirect
46. #1445C22semidirect
47. #1443C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
55. #1440C22semidirect
58. #1439C22semidirect
60. #1437C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
71. C4×#95C22semidirect
72. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1202
C22×C4×C22⋊C42
#152052
#151622
#150262
#149552
#149462
#149452
#147272
#147182
C4×#3972
C4×#3942
C4×#3792
C4×#3852
C4×#3862