Finite Group Databasebeta

← #535079

Subgroups — #535079

Order 512.

3306 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2733C211yesmaximal, characteristic
2C2×#2372C212yesmaximal, complemented
3#14467C211yesmaximal, characteristic, complemented
4#14445C212yesmaximal, complemented
5#14228C214yesmaximal, complemented
6#14227C212yesmaximal, complemented
7#14224C212yesmaximal, complemented
8#14187C214yesmaximal, complemented
9#14079C214yesmaximal, complemented
10#14026C212yesmaximal, complemented
11#13816C214yesmaximal, complemented
12C4×#391C211yesmaximal, characteristic, complemented
13C4×#390C212yesmaximal, complemented
14C22×#124C2212yes—
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2212yes—
17#2733C4116yescomplemented
18C2×#394C2212yes—
19C2×#392C2214yes—
20#2372—216no—
21C2×#391C2212yes—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#386C2214yes—
35C2×#386C2214yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C2×C4×C22⋊C4C2214yes—
40C2×C4×C22⋊C4C2214yes—
41C2×#376C2214yes—
42C2×C43C2211yescharacteristic
43#1444C2214yes—
44#1442C2212yes—
45#1441C2212yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1438C2212yes—
50#1438C2214yes—
51#1437C2214yes—
52#1437C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2212yes—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64#1434C2214yes—
65C22⋊#95C2214yes—
66C4×#95C2212yes—
67C4×#95C2214yes—
68C4×#95C2212yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C24×C4C2314yes—
74C2×#124—28no—
75C2×#124—28no—
76C2×#124C2×C4116yescomplemented
77C2×C4×Q8—24no—
78C2×C4×Q8—42no—
79C2×C4×Q8—42no—
80C2×C4×D4—28no—
81C2×C4×D4—44no—
82C2×C4×D4—44no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2314yes—
106C22×C42C2312yes—
107C22×C42C2314yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114#394C2×C4116yescomplemented
115#392C2×C4132yescomplemented
116#391—28no—
117#391C2×C418yescomplemented
118#391D418yescomplemented
119#390—28no—
120#390—44no—
121#390—44no—
122#390—44no—
123#390C2×C4116yescomplemented
124#388—216no—
125#388—216no—
126#388—216no—
127#388—48no—
128#388C2×C4132yescomplemented
129#386—216no—
130#386—48no—
131C4×C4⋊C4—44no—
132C4×C4⋊C4—42no—
133C4×C22⋊C4—216no—
134C4×C22⋊C4—216no—
135C4×C22⋊C4—216no—
136#376—216no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C43—42no—
155C25C2414yes—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8C22×C414yes—
161C22×Q8C2×D414yes—
162C22×Q8#4114yes—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4C22×C418yes—
180C22×D4C22×C418yes—
181C22×D4C22×C418yes—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201#124—264no—
202C4×Q8—416no—
203C4×D4—432no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C22×C418yes—
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4#4118yes—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4C22×C4116yes—
260C2×C22⋊C4C22×C4116yes—
261C2×C22⋊C4C22×C4116yes—
262C2×C22⋊C4C22×C4116yes—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—24no—
274C2×C42—28no—
275C2×C42—28no—
276C2×C42—24no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—42no—
281C2×C42—41no—
282C2×C42—41no—
283C2×C42—42no—
284C2×C42—41no—
285C2×C42—41no—
286C2×C42—42no—
287C2×C42—44no—
288C2×C42—42no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42C22×C418yes—
292C2×C42C22×C414yes—
293C2×C42C2×D414yes—
294C2×C42#4118yes—
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—42no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95C22×C418yes—
310#95C22×C4116yes—
311#95C2×D4116yes—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24C23×C418yes—
327C24C2511yescharacteristic
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8C4×D4116yescomplemented
335C2×D4—216no—
336C2×D4—216no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—416no—
346C2×D4—48no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—88no—
353C2×D4—84no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4C23×C414yes—
417C22×C4C23×C418yes—
418C22×C4C23×C414yes—
419C22×C4C23×C412yes—
420C22×C4C23×C414yes—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C22×D414yes—
424C22×C4C22×D414yes—
425C22×C4C2×#4118yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4114yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)14yes—
434C22×C4H(5,𝔽2)12yes—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—44no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C22⋊C4—232no—
445C22⋊C4—232no—
446C22⋊C4—232no—
447C22⋊C4—416no—
448C22⋊C4—416no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C42—216no—
456C42—28no—
457C42—216no—
458C42—44no—
459C42—44no—
460C42—48no—
461C42—416no—
462C42—48no—
463C42—416no—
464C42—84no—
465C42—84no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23C24×C411yescharacteristic
502C23C23×D411yescharacteristic
503C23C22×#4111yescharacteristic
504C23C22×#4112yes—
505C23C22×#4112yes—
506C23C2×H(5,𝔽2)12yes—
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×H(5,𝔽2)11yescharacteristic
509C23#58512yes—
510C23#58511yescharacteristic
511C23#58511yescharacteristic
512Q8—416no—
513D4—432no—
514D4—816no—
515D4—816no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—832no—
586C2×C4C2×C4×D418yes—
587C2×C4C2×C4×D414yes—
588C2×C4C4×#4118yes—
589C2×C4#55018yes—
590C2×C4#55514yes—
591C22—48no—
592C22—48no—
593C22—48no—
594C22—48no—
595C22—48no—
596C22—48no—
597C22—48no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22C22×C4×D411yescharacteristic
607C22C2×C4×#4112yes—
608C22C4×H(5,𝔽2)12yes—
609C22C4×H(5,𝔽2)11yescharacteristic
610C22C4×H(5,𝔽2)11yescharacteristic
611C22C2×#55012yes—
612C22D4×#4112yes—
613C22C2×#55511yescharacteristic
614C22#347512yes—
615C22#348012yes—
616C22#348312yes—
617C22#348311yescharacteristic
618C22#349012yes—
619C22#349011yescharacteristic
620C22#349111yescharacteristic
621C22#349312yes—
622C22#349611yescharacteristic
623C22#349611yescharacteristic
624C22#350512yes—
625C22#350612yes—
626C22#354512yes—
627C22#354511yescharacteristic
628C22#354911yescharacteristic
629C4—28no—
630C4—28no—
631C4—24no—
632C4—28no—
633C4—24no—
634C4—48no—
635C4—42no—
636C4—48no—
637C4—44no—
638C4—44no—
639C4—82no—
640C4—84no—
641C2—88no—
642C2C4×#55012yes—
643C2C4×#55511yescharacteristic
644C2#3427311yescharacteristic
645C2#3427712yes—
646C2#3452411yescharacteristic
647C2#5981912yes—
648C2#5982211yescharacteristic
649C2#5983411yescharacteristic
650C2#5987812yes—
651C2#5988511yescharacteristic
652C2#6004811yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC2×#2733C2
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×#2372C2semidirect
3. #14467C2semidirect
4. #14445C2semidirect
5. #14228C2semidirect
6. #14227C2semidirect
7. #14224C2semidirect
8. #14187C2semidirect
9. #14079C2semidirect
10. #14026C2semidirect
11. #13816C2semidirect
12. C4×#391C2semidirect
13. C4×#390C2semidirect
17. #2733C4semidirect
76. C2×#124C2×C4semidirect
114. #394C2×C4semidirect
115. #392C2×C4semidirect
117. #391C2×C4semidirect
118. #391D4semidirect
123. #390C2×C4semidirect
128. #388C2×C4semidirect
334. C2×Q8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27332
C2×#23722
#144672
#144452
#142282
#142272
#142242
#141872
#140792
#140262
#138162
C4×#3912
C4×#3902