Finite Group Databasebeta

← #616436

Subgroups — #616436

Order 512.

2598 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2607C212yesmaximal
2#15253C212yesmaximal, complemented
3#15251C212yesmaximal, complemented
4#15161C214yesmaximal
5#15135C212yesmaximal
6#15125C212yesmaximal
7#14714C212yesmaximal, complemented
8#14699C212yesmaximal, complemented
9#14685C211yesmaximal, characteristic
10#14091C214yesmaximal, complemented
11#14053C212yesmaximal, complemented
12#14029C212yesmaximal, complemented
13#13953C212yesmaximal
14C4×#380C211yesmaximal, characteristic
15C42×C22⋊C4C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C23×C42C2211yescharacteristic
18#2607—216no—
19C2×#399C2214yes—
20C2×#393C2214yes—
21C2×#391C2214yes—
22C2×#391C2214yes—
23C2×#391C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#387C2214yes—
27C2×#387C2214yes—
28C2×#384C2212yes—
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C43C2211yescharacteristic
41C2×C43C2211yescharacteristic
42#1446C2214yes—
43#1446C2214yes—
44#1445C2214yes—
45#1445C2214yes—
46#1444C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56C4⋊#95C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65#1434C2212yes—
66#1434C2212yes—
67#1434C2214yes—
68#1434C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C4×#95C2212yes—
72C4×#95C2212yes—
73C4×#95C2211yescharacteristic
74C4×#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×Q8C2314yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—44no—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C42—42no—
98C22×C42—42no—
99C22×C42Q818yescomplemented
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114#399—216no—
115#393—216no—
116#391—216no—
117#391—216no—
118#391—216no—
119#389—28no—
120#389—44no—
121#387—216no—
122#387—216no—
123#384—28no—
124#381—28no—
125#380—28no—
126#380—42no—
127#380—42no—
128C4×C4⋊C4—42no—
129C4×C4⋊C4—42no—
130C4×C22⋊C4—28no—
131C4×C22⋊C4—28no—
132C4×C22⋊C4—28no—
133C4×C22⋊C4D418yes—
134C4×C22⋊C4Q818yescomplemented
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C43—42no—
163C43—42no—
164C25C2411yescharacteristic
165C22×Q8—28no—
166C22×Q8—28no—
167C22×Q8—28no—
168C22×Q8—28no—
169C22×Q8—44no—
170C22×Q8—44no—
171C22×Q8—44no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—41no—
178C23×C4—41no—
179C23×C4—42no—
180C23×C4—41no—
181C23×C4—41no—
182C23×C4—41no—
183C23×C4—41no—
184C23×C4C2×D414yes—
185C23×C4C2×Q814yes—
186C23×C4C2×Q814yes—
187C23×C4C2×Q814yes—
188C23×C4C2×Q814yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C4⋊Q8—432no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4#4118yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4C2×D418yes—
245C2×C22⋊C4C2×Q818yes—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—416no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42—41no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—41no—
266C2×C42—41no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42C2×D414yes—
280C2×C42C2×D414yes—
281C2×C42C2×Q814yes—
282C2×C42C2×Q814yes—
283C2×C42C2×Q814yes—
284C2×C42C2×Q814yes—
285C2×C42C2×Q814yes—
286C2×C42C2×Q814yes—
287C2×C42#4118yes—
288C2×C42#4118yes—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—42no—
313#95—44no—
314#95C2×D418yes—
315#95C2×Q818yes—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—21no—
323C24—21no—
324C24C22×D412yes—
325C24C22×Q812yes—
326C24C22×Q812yes—
327C24C2511yescharacteristic
328C2×Q8—416no—
329C2×Q8—416no—
330C2×Q8—416no—
331C2×Q8—416no—
332C2×Q8—88no—
333C2×Q8—88no—
334C2×Q8—88no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
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—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
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—24no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4C4⋊Q8116yescomplemented
418C22×C4C22×D414yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D414yes—
422C22×C4C22×Q814yes—
423C22×C4C22×Q812yes—
424C22×C4C22×Q812yes—
425C22×C4C22×Q812yes—
426C22×C4C22×Q812yes—
427C22×C4C22×Q812yes—
428C22×C4C22×Q812yes—
429C22×C4C22×Q814yes—
430C22×C4C22×Q812yes—
431C22×C4C22×Q812yes—
432C22×C4C2×#4118yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)12yes—
442C4⋊C4—216no—
443C4⋊C4—416no—
444C4⋊C4—416no—
445C4⋊C4—416no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—416no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C42—28no—
457C42—28no—
458C42—216no—
459C42—44no—
460C42—44no—
461C42—44no—
462C42—44no—
463C42—44no—
464C42—44no—
465C42—48no—
466C42—48no—
467C42—48no—
468C42—84no—
469C42—84no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C2×C4⋊Q814yes—
493C23C23×D411yescharacteristic
494C23C23×Q811yescharacteristic
495C23C23×Q811yescharacteristic
496C23C22×#4112yes—
497C23C22×#4112yes—
498C23C22×#4112yes—
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×H(5,𝔽2)11yescharacteristic
501C23#58512yes—
502C23#58511yescharacteristic
503C23#58511yescharacteristic
504Q8—832no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—416no—
549C2×C4—416no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—416no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
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×C4C2×C4⋊Q814yes—
589C2×C4C2×C4⋊Q814yes—
590C2×C4#53218yes—
591C2×C4#53318yes—
592C2×C4#53314yes—
593C2×C4#53314yes—
594C2×C4#53314yes—
595C2×C4#54314yes—
596C2×C4#55118yes—
597C2×C4#55918yes—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22C22×C4⋊Q811yescharacteristic
610C22C2×#53212yes—
611C22C2×#53311yescharacteristic
612C22C2×#53311yescharacteristic
613C22C2×#53312yes—
614C22C2×#53311yescharacteristic
615C22C2×#54311yescharacteristic
616C22#345811yescharacteristic
617C22#345811yescharacteristic
618C22D4×#4112yes—
619C22C2×#55112yes—
620C22#347512yes—
621C22Q8×#4112yes—
622C22C2×#55912yes—
623C22#348012yes—
624C22#349611yescharacteristic
625C22#349611yescharacteristic
626C22#350111yescharacteristic
627C22#350111yescharacteristic
628C22#350512yes—
629C22#350612yes—
630C22#351012yes—
631C22#351012yes—
632C4—28no—
633C4—28no—
634C4—24no—
635C4—24no—
636C4—24no—
637C4—42no—
638C4—44no—
639C4—44no—
640C4—48no—
641C4—42no—
642C4—84no—
643C4—84no—
644C4—84no—
645C4#260718yescomplemented
646C2—44no—
647C2C2×#260711yescharacteristic
648C2#3491211yescharacteristic
649C2#3493811yescharacteristic
650C2#3500211yescharacteristic
651C2#3503311yescharacteristic
652C2#3534411yescharacteristic
653C2#3626212yes—
654C2#3627712yes—
655C2#3646912yes—
656C2#3647612yes—
657C2#3664511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #15253C2semidirect
3. #15251C2semidirect
7. #14714C2semidirect
8. #14699C2semidirect
10. #14091C2semidirect
11. #14053C2semidirect
12. #14029C2semidirect
99. C22×C42Q8semidirect
134. C4×C22⋊C4Q8semidirect
417. C22×C4C4⋊Q8semidirect
645. C4#2607semidirect

Maximal subgroups

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

subgroupindex
C2×#26072
#152532
#152512
#151612
#151352
#151252
#147142
#146992
#146852
#140912
#140532
#140292
#139532
C4×#3802
C42×C22⋊C42