Finite Group Databasebeta

← #517285

Subgroups — #517285

Order 512.

3074 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic, complemented
3C22×#376C211yesmaximal, characteristic
4#14697C212yesmaximal
5#14686C211yesmaximal, characteristic
6#14283C211yesmaximal, characteristic, complemented
7#14277C211yesmaximal, characteristic, complemented
8#14266C212yesmaximal, complemented
9#14088C214yesmaximal, complemented
10#14087C214yesmaximal, complemented
11#14004C214yesmaximal
12#13980C214yesmaximal
13#13783C212yesmaximal, complemented
14#13779C211yesmaximal, characteristic, complemented
15C4×#380C212yesmaximal
16C22×#125C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2851C4116yescomplemented
21#2330—28no—
22C2×#389C2214yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#387C2214yes—
27C2×#387C2214yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2214yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C2×#376—28no—
40C2×#376C2212yes—
41C2×#376C2212yes—
42C2×#376C2211yescharacteristic
43C2×#376C2211yescharacteristic
44C2×#376C2211yescharacteristic
45C2×#376C2211yescharacteristic
46C22×#95C2212yes—
47C22×#95C2212yes—
48#1445C2214yes—
49#1444C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C4×#95C2214yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C2×#125—24no—
80C2×#125—24no—
81C2×#125C2×C418yescomplemented
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C42—24no—
95C22×C42—24no—
96C22×C42—24no—
97C22×C42—24no—
98C22×C42—24no—
99C22×C42—24no—
100C22×C42—24no—
101C22×C42C2×C418yes—
102C22×C42Q818yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2314yes—
112C22×C42C2314yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#389—28no—
116#389—28no—
117#389—216no—
118#389C2×C4116yescomplemented
119#387—216no—
120#387C2×C4132yescomplemented
121#385—48no—
122#385C2×C4132yes—
123#384—24no—
124#384—24no—
125#382C2×C418yescomplemented
126#380—28no—
127#380—48no—
128#380C2×C4116yes—
129C4×C4⋊C4—28no—
130C4×C4⋊C4—28no—
131C4×C22⋊C4—48no—
132#376—24no—
133#376—24no—
134#376—24no—
135#376—28no—
136#376—28no—
137#376D418yes—
138C2×#95—28no—
139C2×#95—28no—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C25C2411yescharacteristic
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4C22×C418yes—
194C23×C4C22×C414yes—
195C23×C4C2×Q818yes—
196C23×C4C2×Q814yes—
197C23×C4#4114yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212#125—232no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C22×C418yes—
240C2×C4⋊C4C22×C4116yes—
241C2×C4⋊C4C22×C418yes—
242C2×C4⋊C4C22×C4116yes—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4C22×C4116yes—
249C2×C42—232no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—232no—
269C2×C42—232no—
270C2×C42—28no—
271C2×C42—28no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42C22×C418yes—
277C2×C42C22×C414yes—
278C2×C42C22×C414yes—
279C2×C42C2×D414yes—
280C2×C42C2×D414yes—
281C2×C42C2×D414yes—
282C2×C42C2×Q814yes—
283C2×C42C2×Q818yes—
284C2×C42C2×Q814yes—
285C2×C42#4114yes—
286C2×C42#4114yes—
287C2×C42#4114yes—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95C22×C4116yes—
310#95C22×C4116yes—
311#95C22×C4116yes—
312#95C2×D418yes—
313#95C2×D418yes—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24C23×C412yes—
321C24C22×Q812yes—
322C24C2×#4112yes—
323C24H(5,𝔽2)14yes—
324C242−1+414yes—
325C24C2511yescharacteristic
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
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—24no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4C4×Q8116yes—
423C22×C4C23×C418yes—
424C22×C4C23×C414yes—
425C22×C4C23×C418yes—
426C22×C4C23×C414yes—
427C22×C4C23×C412yes—
428C22×C4C23×C412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q814yes—
437C22×C4C22×Q812yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C4⋊C4—232no—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—88no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C4⋊C4—88no—
457C22⋊C4—232no—
458C22⋊C4—416no—
459C42—28no—
460C42—28no—
461C42—216no—
462C42—28no—
463C42—216no—
464C42—28no—
465C42—28no—
466C42—28no—
467C42—88no—
468C42—88no—
469C42C4×D4116yes—
470C42C22⋊Q8116yes—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23C2×C4×Q814yes—
490C23#52018yes—
491C23#54014yes—
492C23#54114yes—
493C23#54418yes—
494C23C24×C411yescharacteristic
495C23C23×D411yescharacteristic
496C23C23×Q811yescharacteristic
497C23C22×#4111yescharacteristic
498C23C22×#4111yescharacteristic
499C23C22×#4111yescharacteristic
500C23C22×#4111yescharacteristic
501C23C2×H(5,𝔽2)12yes—
502C23C2×2−1+412yes—
503C23#58512yes—
504C23#58512yes—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—232no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—432no—
545C2×C4—432no—
546C2×C4—416no—
547C2×C4—416no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
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—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4C2×C4×D418yes—
591C2×C4C2×C4×D414yes—
592C2×C4C2×C4×Q814yes—
593C2×C4C2×C4×Q814yes—
594C2×C4C4×#4114yes—
595C2×C4C2×C22⋊Q818yes—
596C2×C4C2×C22⋊Q814yes—
597C2×C4#52514yes—
598C2×C4#52914yes—
599C2×C4#53314yes—
600C22—24no—
601C22—28no—
602C22—28no—
603C22—28no—
604C22—24no—
605C22—24no—
606C22—28no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22#233018yescomplemented
611C22C22×C4×D411yescharacteristic
612C22C22×C4×Q811yescharacteristic
613C22C2×C4×#4111yescharacteristic
614C22C2×#52012yes—
615C22#342812yes—
616C22C22×C22⋊Q811yescharacteristic
617C22C2×#52511yescharacteristic
618C22C2×#52911yescharacteristic
619C22C2×#53311yescharacteristic
620C22C2×#54011yescharacteristic
621C22C2×#54111yescharacteristic
622C22#345511yescharacteristic
623C22#345511yescharacteristic
624C22C2×#54412yes—
625C22#345812yes—
626C22#348812yes—
627C22#348912yes—
628C22#349412yes—
629C22#349412yes—
630C22#349412yes—
631C22#349512yes—
632C22#349512yes—
633C22#349512yes—
634C4—28no—
635C4—28no—
636C4—24no—
637C4—48no—
638C4—44no—
639C4—48no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—84no—
646C4—84no—
647C4C4×C22⋊Q818yes—
648C2—28no—
649C2C2×C4×C22⋊Q811yescharacteristic
650C2C2×#233011yescharacteristic
651C2#3407411yescharacteristic
652C2#3420212yes—
653C2#3422012yes—
654C2#3525312yes—
655C2#3526111yescharacteristic
656C2#3527111yescharacteristic
657C2#3530412yes—
658C2#3531011yescharacteristic
659C2#3531511yescharacteristic
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. C2×#2330C2semidirect
6. #14283C2semidirect
7. #14277C2semidirect
8. #14266C2semidirect
9. #14088C2semidirect
10. #14087C2semidirect
13. #13783C2semidirect
14. #13779C2semidirect
20. #2851C4semidirect
81. C2×#125C2×C4semidirect
118. #389C2×C4semidirect
120. #387C2×C4semidirect
125. #382C2×C4semidirect
610. C22#2330semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#23302
C22×#3762
#146972
#146862
#142832
#142772
#142662
#140882
#140872
#140042
#139802
#137832
#137792
C4×#3802