Finite Group Databasebeta

← #732405

Subgroups — #732405

Order 512.

2418 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#16575C214yesmaximal, complemented
4#16076C214yesmaximal, complemented
5#16071C214yesmaximal, complemented
6#15226C212yesmaximal
7#15186C212yesmaximal, complemented
8#14859C214yesmaximal, complemented
9#14274C211yesmaximal, characteristic, complemented
10#14150C211yesmaximal, characteristic, complemented
11#14026C212yesmaximal
12#13941C211yesmaximal, characteristic
13#13931C212yesmaximal, complemented
14#13854C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×C4×Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2547—28no—
21C2×#397C2214yes—
22#2365—28no—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#387C2214yes—
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×#376C2211yescharacteristic
41#1445C2214yes—
42#1445C2214yes—
43#1445C2214yes—
44#1445C2214yes—
45#1445C2214yes—
46#1444C2212yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2212yes—
50#1442C2212yes—
51#1442C2212yes—
52#1442C2212yes—
53#1442C2212yes—
54#1442C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71#1434C2214yes—
72#1434C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8D418yes—
83C2×#124—28no—
84C2×#124—28no—
85C2×#124—44no—
86C2×C4×Q8—28no—
87C2×C4×Q8—28no—
88C2×C4×Q8—44no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118#397—216no—
119#391—28no—
120#391—28no—
121#391—44no—
122#391—44no—
123#390—24no—
124#390—24no—
125#390—44no—
126#389—28no—
127#389—44no—
128#387—216no—
129#382—24no—
130#380—28no—
131#380—28no—
132C4×C4⋊C4—28no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136#376—24no—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
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—
167C25C2412yes—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8C2×D418yes—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4#4114yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C4⋊Q8—232no—
202#124—432no—
203C4×Q8—432no—
204C4×D4—416no—
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—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C2×D418yes—
235C2×C4⋊C4C2×D418yes—
236C2×C4⋊C4#4114yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
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×C42—28no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—28no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42C2×D414yes—
276C2×C42#4114yes—
277C2×C42#4114yes—
278#95—24no—
279#95—24no—
280#95—28no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—42no—
297#95—44no—
298#95—42no—
299#95—42no—
300#95—42no—
301#95—44no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24C2×#4114yes—
319C24C2511yescharacteristic
320C2×Q8—216no—
321C2×Q8—216no—
322C2×Q8—216no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4C22×D418yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+414yes—
427C4⋊C4—28no—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—28no—
431C4⋊C4—416no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—88no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—48no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C22⋊C4—88no—
446C42—416no—
447C42—48no—
448C42—48no—
449C42—48no—
450C42—44no—
451C42—44no—
452C42#123116yes—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—28no—
457C23—28no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23C23×D411yescharacteristic
477C23C22×#4111yescharacteristic
478C23C22×#4111yescharacteristic
479C23C22×#4111yescharacteristic
480C23C22×#4111yescharacteristic
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×H(5,𝔽2)11yescharacteristic
483C23C2×2−1+412yes—
484C23#58512yes—
485C23#58512yes—
486C23#58512yes—
487Q8—432no—
488D4—416no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
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—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—816no—
569C2×C4C2×#12314yes—
570C2×C4C2×#12314yes—
571C2×C4C2×#12314yes—
572C2×C4#52914yes—
573C2×C4#54014yes—
574C2×C4#54614yes—
575C2×C4#54614yes—
576C2×C4#55314yes—
577C2×C4#55514yes—
578C22—44no—
579C22—48no—
580C22—48no—
581C22—44no—
582C22—44no—
583C22—82no—
584C22—82no—
585C22—84no—
586C22—84no—
587C22—82no—
588C22—82no—
589C22C22×#12311yescharacteristic
590C22C2×#52911yescharacteristic
591C22C2×#54011yescharacteristic
592C22#346112yes—
593C22C2×#54611yescharacteristic
594C22C2×#54611yescharacteristic
595C22C2×#55311yescharacteristic
596C22C2×#55511yescharacteristic
597C22#348611yescharacteristic
598C22#348711yescharacteristic
599C22#348711yescharacteristic
600C22#348711yescharacteristic
601C22#350712yes—
602C22#351012yes—
603C22#351012yes—
604C22#351512yes—
605C22#351612yes—
606C22#352112yes—
607C22#352112yes—
608C22#355412yes—
609C22#356112yes—
610C22#356212yes—
611C22#356212yes—
612C4—24no—
613C4—24no—
614C4—48no—
615C4—44no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—84no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4#242018yes—
626C2—84no—
627C2C2×#242011yescharacteristic
628C2#3518211yescharacteristic
629C2#3567911yescharacteristic
630C2#3597312yes—
631C2#3829112yes—
632C2#3928612yes—
633C2#4270512yes—
634C2#4526811yescharacteristic
635C2#4526911yescharacteristic
636C2#4526911yescharacteristic
637C2#4533511yescharacteristic
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
2. C2×#2365C2semidirect
3. #16575C2semidirect
4. #16076C2semidirect
5. #16071C2semidirect
7. #15186C2semidirect
8. #14859C2semidirect
9. #14274C2semidirect
10. #14150C2semidirect
13. #13931C2semidirect
14. #13854C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#23652
#165752
#160762
#160712
#152262
#151862
#148592
#142742
#141502
#140262
#139412
#139312
#138542
#137852