Finite Group Databasebeta

← #773265

Subgroups — #773265

Order 512.

3206 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2451C212yesmaximal, complemented
2C2×Q8×C22⋊C4C212yesmaximal
3#16303C212yesmaximal
4#15965C212yesmaximal, complemented
5#15511C212yesmaximal, complemented
6#15332C214yesmaximal, complemented
7#15220C212yesmaximal
8#15125C211yesmaximal, characteristic
9#14775C214yesmaximal
10#14213C212yesmaximal
11#14017C212yesmaximal, complemented
12#13978C212yesmaximal, complemented
13#13942C212yesmaximal, complemented
14#13922C211yesmaximal, characteristic
15C4×#399C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2212yes—
20#2451—216no—
21C2×#399C2211yescharacteristic
22C2×#398C2214yes—
23C2×#395C2214yes—
24C2×#393C2212yes—
25Q8×C22⋊C4—216no—
26C2×#391C2214yes—
27C2×#391C2212yes—
28C2×#391C2214yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2214yes—
32C2×#391C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#387C2214yes—
36C2×#382C2212yes—
37C2×#380C2212yes—
38C2×#380C2214yes—
39C2×#380C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2214yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45#1446C2212yes—
46#1446C2212yes—
47#1446C2212yes—
48#1445C2211yescharacteristic
49#1445C2212yes—
50#1444C2211yescharacteristic
51#1444C2214yes—
52#1443C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1439C2212yes—
61#1439C2211yescharacteristic
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C4×#95C2211yescharacteristic
77C23×Q8—28no—
78C23×Q8—28no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8Q818yescomplemented
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—42no—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—44no—
94C2×C4×Q8—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2314yes—
117C22×C42C2314yes—
118C22×C42C2312yes—
119#399—24no—
120#398—216no—
121#395—216no—
122#393—28no—
123#391—28no—
124#391—28no—
125#391—28no—
126#391—216no—
127#391—216no—
128#391—216no—
129#391D4116yes—
130#389—28no—
131#389—28no—
132#387—216no—
133#382—44no—
134#380—28no—
135#380—216no—
136#380—42no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—42no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—216no—
141C4×C22⋊C4—42no—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2311yescharacteristic
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2411yescharacteristic
173C22×Q8—264no—
174C22×Q8—24no—
175C22×Q8—28no—
176C22×Q8—28no—
177C22×Q8—24no—
178C22×Q8—28no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—44no—
186C22×Q8—42no—
187C22×Q8—416no—
188C22×Q8—416no—
189C22×Q8—432no—
190C22×Q8—416no—
191C22×Q8C2×D418yes—
192C22×Q8C2×Q818yes—
193C22×Q8C2×Q818yes—
194C22×Q8#4118yes—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4C2×Q814yes—
204C23×C4C2×Q818yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C4⋊Q8—416no—
220C4×Q8—432no—
221C4×Q8—432no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D418yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4#4114yes—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—28no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42C2×D418yes—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—22no—
306#95—24no—
307#95—28no—
308#95—41no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—41no—
313#95—41no—
314#95—42no—
315#95—42no—
316#95—44no—
317#95—44no—
318#95C2×D4116yes—
319#95C2×D4116yes—
320C24—22no—
321C24—22no—
322C24—21no—
323C24—21no—
324C24—22no—
325C24—21no—
326C24—22no—
327C24—21no—
328C24C22×Q812yes—
329C24C2×#4112yes—
330C242−1+412yes—
331C24C2511yescharacteristic
332C2×Q8—216no—
333C2×Q8—232no—
334C2×Q8—216no—
335C2×Q8—216no—
336C2×Q8—216no—
337C2×Q8—216no—
338C2×Q8—216no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—416no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—416no—
345C2×Q8—4128no—
346C2×Q8—88no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×Q8C22⋊Q8132yes—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—232no—
395C22×C4—216no—
396C22×C4—28no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—416no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D418yes—
433C22×C4C22×D418yes—
434C22×C4C22×D414yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q818yes—
437C22×C4C22×Q812yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4118yes—
441C22×C4C2×#4114yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C42−1+414yes—
445C22×C42−1+412yes—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—88no—
457C4⋊C4—88no—
458C22⋊C4—216no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—416no—
462C22⋊C4—44no—
463C22⋊C4—84no—
464C42—216no—
465C42—416no—
466C42—416no—
467C42—48no—
468C42—44no—
469C42—44no—
470C23—24no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—41no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23#55714yes—
493C23C23×D412yes—
494C23C23×Q811yescharacteristic
495C23C22×#4112yes—
496C23C22×#4111yescharacteristic
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×2−1+411yescharacteristic
500C23C2×2−1+412yes—
501C23C2×2−1+411yescharacteristic
502C23#58512yes—
503C23#58511yescharacteristic
504Q8—264no—
505Q8—432no—
506Q8—816no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—432no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—416no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4C2×C22⋊Q8116yes—
590C2×C4C2×C22⋊Q818yes—
591C2×C4#54314yes—
592C2×C4D4214yes—
593C2×C4#54618yes—
594C2×C4#55114yes—
595C22—24no—
596C22—28no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—44no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22C22×C22⋊Q812yes—
607C22C2×#54311yescharacteristic
608C22C2×D4211yescharacteristic
609C22C2×#54612yes—
610C22D4×#4112yes—
611C22C2×#55111yescharacteristic
612C22C2×#55711yescharacteristic
613C22#348212yes—
614C22#348612yes—
615C22#348612yes—
616C22#350012yes—
617C22#350212yes—
618C22#350211yescharacteristic
619C22#350511yescharacteristic
620C22#351111yescharacteristic
621C22#351612yes—
622C22#351812yes—
623C22#351812yes—
624C22#353811yescharacteristic
625C22#353811yescharacteristic
626C22#355111yescharacteristic
627C22#355412yes—
628C22#356111yescharacteristic
629C4—216no—
630C4—42no—
631C4—44no—
632C4—42no—
633C4—48no—
634C4—42no—
635C4—44no—
636C4—44no—
637C4—42no—
638C4—82no—
639C4—84no—
640C4—84no—
641C4—81no—
642C4#243718yes—
643C2—44no—
644C2C2×#243711yescharacteristic
645C2#3571112yes—
646C2#3643512yes—
647C2#3975211yescharacteristic
648C2#4281811yescharacteristic
649C2#4528912yes—
650C2#4530811yescharacteristic
651C2#4541811yescharacteristic
652C2#5910012yes—
653C2#5937911yescharacteristic
654C2#5981811yescharacteristic
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
1. C2×#2451C2semidirect
4. #15965C2semidirect
5. #15511C2semidirect
6. #15332C2semidirect
11. #14017C2semidirect
12. #13978C2semidirect
13. #13942C2semidirect
81. C23×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24512
C2×Q8×C22⋊C42
#163032
#159652
#155112
#153322
#152202
#151252
#147752
#142132
#140172
#139782
#139422
#139222
C4×#3992