Finite Group Databasebeta

← #642700

Subgroups — #642700

Order 512.

2990 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic
3#16599C214yesmaximal, complemented
4#16555C214yesmaximal, complemented
5#15148C212yesmaximal, complemented
6#15141C212yesmaximal, complemented
7#14825C214yesmaximal, complemented
8#14733C214yesmaximal, complemented
9#14707C211yesmaximal, characteristic, complemented
10#14276C211yesmaximal, characteristic, complemented
11#14272C211yesmaximal, characteristic
12#14272C211yesmaximal, characteristic
13#14018C212yesmaximal, complemented
14#13920C212yesmaximal, complemented
15#13759C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×C4×D4C2214yes—
19#2647—28no—
20C2×#400C2214yes—
21C2×#399C2214yes—
22C2×#398C2214yes—
23C2×#396C2214yes—
24C4×#127—28no—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2214yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#385C2214yes—
36C2×#384C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#380C2214yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×#376C2211yescharacteristic
43C2×#376C2211yescharacteristic
44C2×C43C2211yescharacteristic
45#1441C2214yes—
46#1441C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2212yes—
65#1436C2212yes—
66#1436C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C22⋊#95C2214yes—
70C22⋊#95C2214yes—
71C4×#95C2211yescharacteristic
72C4×#95C2211yescharacteristic
73C23×D4C2312yes—
74C23×D4C2314yes—
75C24×C4C2314yes—
76C2×#127—24no—
77C2×#127—24no—
78C2×#127Q818yes—
79C2×C4×D4—28no—
80C2×C4×D4—216no—
81C2×C4×D4—28no—
82C2×C4×D4—48no—
83C2×C4×D4—44no—
84C2×C4×D4—48no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116#400—216no—
117#399—48no—
118#398—216no—
119#396—48no—
120#390—28no—
121#390—216no—
122#390—28no—
123#390—44no—
124#390—44no—
125#390—44no—
126#389—28no—
127#389—28no—
128#389—44no—
129#389—44no—
130#385—48no—
131#384D418yes—
132#382—24no—
133#382—24no—
134#380—48no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—24no—
137#376—24no—
138#376—24no—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C43—24no—
161C25C2414yes—
162C22×D4—28no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—28no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4C2×Q818yes—
175C22×D4C2×Q8116yes—
176C23×C4—28no—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—28no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4C2411yescharacteristic
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197#127—232no—
198C4×D4—432no—
199C4×D4—464no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C2×D418yes—
230C2×C4⋊C4C2×D4116yes—
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—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
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⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42C2×D414yes—
287C2×C42C2×Q814yes—
288C2×C42#4114yes—
289C2×C42#4114yes—
290C2×C42#4114yes—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—28no—
302#95—24no—
303#95—28no—
304#95—24no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
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—
326C24C22×Q818yes—
327C24C2511yescharacteristic
328C2×D4—232no—
329C2×D4—216no—
330C2×D4—216no—
331C2×D4—216no—
332C2×D4—416no—
333C2×D4—416no—
334C2×D4—416no—
335C2×D4—416no—
336C2×D4—416no—
337C2×D4—416no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4C22×D418yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×Q814yes—
422C22×C4C22×Q812yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4H(5,𝔽2)14yes—
432C22×C4H(5,𝔽2)14yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C4H(5,𝔽2)12yes—
435C4⋊C4—216no—
436C4⋊C4—416no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C22⋊C4—416no—
447C22⋊C4—416no—
448C22⋊C4—416no—
449C22⋊C4—416no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C22⋊C4—88no—
456C22⋊C4—88no—
457C42—216no—
458C42—28no—
459C42—28no—
460C42—48no—
461C42—44no—
462C42—44no—
463C42—44no—
464C42—44no—
465C42C22⋊Q8116yes—
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—
501C23C23×D411yescharacteristic
502C23C23×Q811yescharacteristic
503C23C22×#4111yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4111yescharacteristic
506C23C2×H(5,𝔽2)12yes—
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×H(5,𝔽2)12yes—
510C23#58512yes—
511C23#58512yes—
512D4—464no—
513D4—432no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—832no—
592C2×C4C2×C22⋊Q818yes—
593C2×C4C2×C22⋊Q814yes—
594C2×C4#52914yes—
595C2×C4#54014yes—
596C2×C4#54014yes—
597C2×C4#54614yes—
598C2×C4#55014yes—
599C2×C4#55414yes—
600C2×C4#55414yes—
601C22—48no—
602C22—48no—
603C22—48no—
604C22—48no—
605C22—48no—
606C22—48no—
607C22—48no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22C22×C22⋊Q811yescharacteristic
617C22C2×#52911yescharacteristic
618C22C2×#54011yescharacteristic
619C22C2×#54011yescharacteristic
620C22#345511yescharacteristic
621C22#346012yes—
622C22#346012yes—
623C22#346012yes—
624C22C2×#54611yescharacteristic
625C22C2×#55011yescharacteristic
626C22C2×#55411yescharacteristic
627C22C2×#55411yescharacteristic
628C22#348412yes—
629C22#348712yes—
630C22#349412yes—
631C22#350112yes—
632C22#350112yes—
633C22#350811yescharacteristic
634C22#351211yescharacteristic
635C22#354312yes—
636C22#354612yes—
637C22#355812yes—
638C22#356212yes—
639C4—24no—
640C4—28no—
641C4—42no—
642C4—48no—
643C4—42no—
644C4—44no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—84no—
649C4—82no—
650C4—82no—
651C4—84no—
652C4#245418yes—
653C2—88no—
654C2#3462111yescharacteristic
655C2C2×#245411yescharacteristic
656C2#3517412yes—
657C2#3528711yescharacteristic
658C2#3758712yes—
659C2#3769411yescharacteristic
660C2#3770111yescharacteristic
661C2#3777212yes—
662C2#3962811yescharacteristic
663C2#3963411yescharacteristic
664C2#3967212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2647C2semidirect
3. #16599C2semidirect
4. #16555C2semidirect
5. #15148C2semidirect
6. #15141C2semidirect
7. #14825C2semidirect
8. #14733C2semidirect
9. #14707C2semidirect
10. #14276C2semidirect
13. #14018C2semidirect
14. #13920C2semidirect
15. #13759C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
C2×C4×#1272
#165992
#165552
#151482
#151412
#148252
#147332
#147072
#142762
#142722
#140182
#139202
#137592