Finite Group Databasebeta

← #732105

Subgroups — #732105

Order 512.

2622 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2402C211yesmaximal, characteristic, complemented
2C2×#2349C211yesmaximal, characteristic
3#16304C214yesmaximal, complemented
4#15679C214yesmaximal, complemented
5#15306C214yesmaximal, complemented
6#15200C212yesmaximal, complemented
7#15133C212yesmaximal, complemented
8#14766C214yesmaximal, complemented
9#14276C211yesmaximal, characteristic, complemented
10#14018C212yesmaximal
11#13955C211yesmaximal, characteristic, complemented
12#13910C211yesmaximal, characteristic, complemented
13#13906C212yesmaximal
14#13906C211yesmaximal, characteristic
15#13785C211yesmaximal, characteristic
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18#2402—28no—
19C2×#400C2212yes—
20C2×#400C2212yes—
21C2×#398C2212yes—
22C2×#398C2212yes—
23C2×#398C2214yes—
24#2349—28no—
25C2×#390C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#384C2211yescharacteristic
34C2×#384C2212yes—
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2214yes—
38C2×#380C2214yes—
39C2×#380C2214yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×#376C2211yescharacteristic
46C22×#95C2212yes—
47#1445C2214yes—
48#1444C2212yes—
49#1444C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1437C2214yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73#1434C2212yes—
74#1434C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×D4C2311yescharacteristic
78C24×C4C2314yes—
79C24×C4C2312yes—
80C2×C4×D4—28no—
81C2×C4×D4—24no—
82C2×C4×D4—28no—
83C2×C4×D4—24no—
84C2×C4×D4—44no—
85C2×C4×D4Q818yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115#400—28no—
116#400—28no—
117#398—28no—
118#398—28no—
119#398—48no—
120#390—28no—
121#390—24no—
122#390—24no—
123#389—28no—
124#389—48no—
125#389—44no—
126#385—216no—
127#385—48no—
128#384—24no—
129#384—44no—
130#382—24no—
131#380—28no—
132#380—48no—
133#380—48no—
134#380—48no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4D418yes—
140#376—24no—
141C2×#95—28no—
142C2×#95C2311yescharacteristic
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C25C2412yes—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4C2×Q814yes—
175C22×D4#4114yes—
176C22×D4#4114yes—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—28no—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4C2×Q818yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C4×D4—232no—
204C4×D4—432no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
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—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×Q814yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×Q818yes—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—44no—
285C2×C42C2×D418yes—
286C2×C42C2×D414yes—
287C2×C42C2×Q814yes—
288C2×C42#4114yes—
289#95—24no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—22no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95C2×D414yes—
315#95C2×D414yes—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24—22no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24C22×Q814yes—
325C24C2×#4114yes—
326C24C2×#4114yes—
327C24C2511yescharacteristic
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4#125116yes—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—216no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
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—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×Q814yes—
422C22×C4C22×Q812yes—
423C22×C4C22×Q812yes—
424C22×C4C22×Q812yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)14yes—
431C22×C4H(5,𝔽2)14yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C42−1+414yes—
434C22×C42−1+412yes—
435C4⋊C4—28no—
436C4⋊C4—216no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C4⋊C4—88no—
452C22⋊C4—216no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—88no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C42—216no—
459C42—44no—
460C42—48no—
461C42—44no—
462C42—84no—
463C42C22⋊Q8116yes—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23C2×#12518yes—
487C23C23×D411yescharacteristic
488C23C23×Q811yescharacteristic
489C23C22×#4111yescharacteristic
490C23C22×#4111yescharacteristic
491C23C22×#4111yescharacteristic
492C23C2×H(5,𝔽2)12yes—
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×H(5,𝔽2)12yes—
495C23C2×2−1+412yes—
496C23C2×2−1+411yescharacteristic
497C23#58512yes—
498D4—232no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—816no—
578C2×C4—832no—
579C2×C4C2×C22⋊Q814yes—
580C2×C4C2×C22⋊Q814yes—
581C2×C4C2×C22⋊Q814yes—
582C2×C4C2×#12514yes—
583C2×C4#53718yes—
584C2×C4#54414yes—
585C2×C4#54614yes—
586C2×C4#54814yes—
587C2×C4#55214yes—
588C2×C4#55314yes—
589C22—216no—
590C22—28no—
591C22—44no—
592C22—48no—
593C22—44no—
594C22—48no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—48no—
599C22—44no—
600C22C22×C22⋊Q811yescharacteristic
601C22C22×#12511yescharacteristic
602C22C2×#53712yes—
603C22C2×#54411yescharacteristic
604C22C2×#54611yescharacteristic
605C22C2×#54811yescharacteristic
606C22C2×#55211yescharacteristic
607C22C2×#55311yescharacteristic
608C22#348812yes—
609C22#350112yes—
610C22#350312yes—
611C22#350811yescharacteristic
612C22#350811yescharacteristic
613C22#350911yescharacteristic
614C22#351011yescharacteristic
615C22#351312yes—
616C22#351512yes—
617C22#351912yes—
618C22#352012yes—
619C22#354312yes—
620C22#354412yes—
621C22#354712yes—
622C22#356112yes—
623C4—28no—
624C4—24no—
625C4—24no—
626C4—44no—
627C4—44no—
628C4—44no—
629C4—82no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—84no—
636C4#244218yes—
637C2—48no—
638C2C2×#244211yescharacteristic
639C2#3528011yescharacteristic
640C2#3551511yescharacteristic
641C2#3585812yes—
642C2#3802412yes—
643C2#3912612yes—
644C2#4249712yes—
645C2#4504111yescharacteristic
646C2#4505611yescharacteristic
647C2#4508311yescharacteristic
648C2#4522111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2402C2semidirect
3. #16304C2semidirect
4. #15679C2semidirect
5. #15306C2semidirect
6. #15200C2semidirect
7. #15133C2semidirect
8. #14766C2semidirect
9. #14276C2semidirect
11. #13955C2semidirect
12. #13910C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24022
C2×#23492
#163042
#156792
#153062
#152002
#151332
#147662
#142762
#140182
#139552
#139102
#139062
#137852