Finite Group Databasebeta

← #517328

Subgroups — #517328

Order 512.

2686 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#14703C212yesmaximal, complemented
3#14685C211yesmaximal, characteristic
4#14297C212yesmaximal
5#14286C212yesmaximal, complemented
6#14107C214yesmaximal
7#14057C214yesmaximal, complemented
8#14029C212yesmaximal, complemented
9#14024C212yesmaximal, complemented
10#13997C212yesmaximal
11#13969C212yesmaximal
12#13792C212yesmaximal, complemented
13#13780C212yesmaximal
14C4×#391C212yesmaximal, complemented
15C42×C22⋊C4C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2212yes—
19C23×C42C2211yescharacteristic
20#2857C4116yescomplemented
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#387C2214yes—
29C2×#384C2211yescharacteristic
30C2×#381C2212yes—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C43C2211yescharacteristic
41C2×C43C2211yescharacteristic
42#1445C2214yes—
43#1445C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1437C2214yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65#1434C2214yes—
66#1434C2212yes—
67#1434C2212yes—
68#1434C2214yes—
69C22⋊#95C2212yes—
70C4×#95C2214yes—
71C4×#95C2212yes—
72C4×#95C2211yescharacteristic
73C4×#95C2211yescharacteristic
74C4×#95C2212yes—
75C4×#95C2214yes—
76C23×Q8C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8C2×C418yes—
83C2×C22⋊Q8—44no—
84C2×C22⋊Q8—44no—
85C2×C22⋊Q8C2×C4116yes—
86C2×C4×Q8—28no—
87C2×C4×Q8—44no—
88C2×C4×Q8—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C42—42no—
105C22×C42—42no—
106C22×C42C2×C418yes—
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2314yes—
114C22×C42C2314yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#391—28no—
122#391—44no—
123#391—44no—
124#391C2×C4116yes—
125#389—44no—
126#389C2×C4116yes—
127#389C2×C4116yes—
128#387—216no—
129#384C2×C418yes—
130#381—28no—
131#380—42no—
132#380—42no—
133#380C2×C4116yes—
134#380C2×C4116yes—
135C4×C4⋊C4—44no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—28no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4Q818yescomplemented
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C43—42no—
167C43—42no—
168C25C2411yescharacteristic
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8C22×C418yes—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—41no—
183C23×C4—41no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4C22×C414yes—
189C23×C4C22×C414yes—
190C23×C4C22×C414yes—
191C23×C4C2×Q814yes—
192C23×C4#4114yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C4⋊Q8—416no—
208C22⋊Q8—432no—
209C4×Q8—432no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C22×C418yes—
234C2×C4⋊C4C22×C418yes—
235C2×C4⋊C4C22×C4116yes—
236C2×C4⋊C4C22×C418yes—
237C2×C4⋊C4C22×C418yes—
238C2×C4⋊C4C22×C418yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4C22×C4116yes—
253C2×C22⋊C4C2×Q818yes—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—28no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—416no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42—41no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—41no—
285C2×C42—41no—
286C2×C42—41no—
287C2×C42—41no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42C22×C414yes—
291C2×C42C22×C414yes—
292C2×C42C22×C414yes—
293C2×C42C22×C414yes—
294C2×C42C2×Q814yes—
295C2×C42C2×Q814yes—
296C2×C42#4118yes—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—42no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95C22×C4116yes—
320#95C22×C4116yes—
321#95C2×Q818yes—
322#95#4118yes—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24C23×C412yes—
332C24C22×Q812yes—
333C24C2×#4112yes—
334C24C2511yescharacteristic
335C2×Q8—216no—
336C2×Q8—216no—
337C2×Q8—216no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4C4×Q8116yescomplemented
425C22×C4C23×C418yes—
426C22×C4C23×C418yes—
427C22×C4C23×C412yes—
428C22×C4C23×C412yes—
429C22×C4C23×C412yes—
430C22×C4C23×C412yes—
431C22×C4C23×C412yes—
432C22×C4C23×C412yes—
433C22×C4C22×Q814yes—
434C22×C4C22×Q812yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q814yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+414yes—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—48no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—416no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—216no—
461C22⋊C4—416no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C42—28no—
465C42—28no—
466C42—216no—
467C42—44no—
468C42—44no—
469C42—44no—
470C42—44no—
471C42—48no—
472C42—48no—
473C42—44no—
474C42—44no—
475C42—416no—
476C42—48no—
477C42—88no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23C2×C4×Q814yes—
501C23C24×C411yescharacteristic
502C23C23×Q811yescharacteristic
503C23C22×#4111yescharacteristic
504C23C22×#4112yes—
505C23C22×#4112yes—
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×H(5,𝔽2)12yes—
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×2−1+412yes—
510C23#58511yescharacteristic
511C23#58511yescharacteristic
512Q8—432no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—416no—
558C2×C4—416no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—416no—
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—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4C2×C4×Q814yes—
597C2×C4C2×C4×Q814yes—
598C2×C4C4×#4114yes—
599C2×C4C4×#4114yes—
600C2×C4#51814yes—
601C2×C4#51814yes—
602C2×C4#51914yes—
603C2×C4#53318yes—
604C2×C4#53818yes—
605C2×C4#54018yes—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22C22×C4×Q811yescharacteristic
618C22C2×C4×#4111yescharacteristic
619C22C2×C4×#4111yescharacteristic
620C22C2×#51811yescharacteristic
621C22C2×#51811yescharacteristic
622C22C2×#51911yescharacteristic
623C22#342811yescharacteristic
624C22C2×#53312yes—
625C22#344412yes—
626C22C2×#53812yes—
627C22C2×#54012yes—
628C22#346112yes—
629C22#346112yes—
630C22#350111yescharacteristic
631C22#350111yescharacteristic
632C22#350312yes—
633C22#350511yescharacteristic
634C22#350511yescharacteristic
635C22#351112yes—
636C22#351412yes—
637C22#351712yes—
638C22#352012yes—
639C22#352012yes—
640C4—24no—
641C4—24no—
642C4—24no—
643C4—24no—
644C4—28no—
645C4—42no—
646C4—48no—
647C4—44no—
648C4—44no—
649C4—48no—
650C4—44no—
651C4—84no—
652C4—84no—
653C4#233318yes—
654C2—44no—
655C2C4×#53311yescharacteristic
656C2C2×#233311yescharacteristic
657C2#3408511yescharacteristic
658C2#3421211yescharacteristic
659C2#3421611yescharacteristic
660C2#3423211yescharacteristic
661C2#3424511yescharacteristic
662C2#3501612yes—
663C2#3503612yes—
664C2#3510812yes—
665C2#3517612yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14703C2semidirect
5. #14286C2semidirect
7. #14057C2semidirect
8. #14029C2semidirect
9. #14024C2semidirect
12. #13792C2semidirect
14. C4×#391C2semidirect
20. #2857C4semidirect
139. C4×C22⋊C4Q8semidirect
424. C22×C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#147032
#146852
#142972
#142862
#141072
#140572
#140292
#140242
#139972
#139692
#137922
#137802
C4×#3912
C42×C22⋊C42