Finite Group Databasebeta

← #1008470

Subgroups — #1008470

Order 512.

2338 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C212yesmaximal, complemented
2#17121C212yesmaximal
3#16652C212yesmaximal
4#16640C214yesmaximal, complemented
5#16115C212yesmaximal, complemented
6#15504C214yesmaximal, complemented
7#15232C212yesmaximal, complemented
8#15196C211yesmaximal, characteristic, complemented
9#15132C212yesmaximal, complemented
10#14501C212yesmaximal, complemented
11#14155C211yesmaximal, characteristic, complemented
12#13973C212yesmaximal, complemented
13#13879C212yesmaximal, complemented
14#13871C212yesmaximal
15C4×#394C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2212yes—
19#2439—216no—
20C2×#397C2214yes—
21C2×#396C2214yes—
22C2×#394C2211yescharacteristic
23C2×#393C2214yes—
24C2×#391C2212yes—
25C2×#391C2214yes—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#387C2214yes—
30C2×#384C2211yescharacteristic
31C2×#382C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39#1445C2212yes—
40#1445C2214yes—
41#1445C2212yes—
42#1444C2212yes—
43#1444C2212yes—
44#1443C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2211yescharacteristic
50#1442C2214yes—
51#1441C2212yes—
52#1441C2211yescharacteristic
53#1441C2212yes—
54#1440C2212yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1438C2214yes—
62#1438C2211yescharacteristic
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C4×#95C2211yescharacteristic
76C23×Q8C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8D418yescomplemented
82C2×C4×D4—28no—
83C2×C4×D4—42no—
84C2×#117—28no—
85C2×#117—44no—
86C2×#117—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113C22×C42C2312yes—
114#397—48no—
115#396—216no—
116#394—24no—
117#393—216no—
118#391—216no—
119#391—28no—
120#391—44no—
121#389—28no—
122#389—28no—
123#387—216no—
124#384—42no—
125#382—28no—
126#380—28no—
127#380—28no—
128#380—44no—
129C4×C4⋊C4—42no—
130C4×C4⋊C4—42no—
131C4×C22⋊C4—28no—
132C4×C22⋊C4—28no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2311yescharacteristic
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C25C2412yes—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8C2×D418yes—
170C22×D4—24no—
171C22×D4—22no—
172C22×D4—41no—
173C22×D4—42no—
174C22×D4—41no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4#4118yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C4⋊Q8—416no—
197C4×D4—416no—
198#117—432no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C2×D418yes—
230C2×C4⋊C4C2×D418yes—
231C2×C4⋊C4#4114yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
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×C42—22no—
260C2×C42—22no—
261C2×C42—28no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42C2×D414yes—
275#95—24no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—22no—
288#95—28no—
289#95—24no—
290#95—42no—
291#95—44no—
292#95—41no—
293#95—42no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—41no—
299#95—41no—
300#95—42no—
301#95—44no—
302#95—44no—
303#95—44no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C2×#4114yes—
319C24C2511yescharacteristic
320C2×Q8—216no—
321C2×Q8—48no—
322C2×Q8—48no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8—84no—
327C2×D4—48no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—48no—
331C2×D4—44no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4C22×D418yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4114yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+412yes—
424C22×C42−1+414yes—
425C4⋊C4—216no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—416no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—88no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—416no—
442C22⋊C4—48no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C22⋊C4—88no—
446C42—28no—
447C42—416no—
448C42—48no—
449C42—44no—
450C42—44no—
451C42—84no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23C23×D411yescharacteristic
488C23C22×#4112yes—
489C23C22×#4111yescharacteristic
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×H(5,𝔽2)11yescharacteristic
493C23C2×2−1+412yes—
494C23C2×2−1+411yescharacteristic
495C23#58512yes—
496C23#58512yes—
497C23#58511yescharacteristic
498Q8—816no—
499D4—416no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—432no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4#53414yes—
582C2×C4#53918yes—
583C2×C4#54714yes—
584C2×C4#54818yes—
585C2×C4#55114yes—
586C2×C4#56314yes—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22C2×#53411yescharacteristic
603C22C2×#53912yes—
604C22C2×#54711yescharacteristic
605C22C2×#54812yes—
606C22C2×#55111yescharacteristic
607C22#347512yes—
608C22#348712yes—
609C22#349312yes—
610C22#349412yes—
611C22#349611yescharacteristic
612C22#350511yescharacteristic
613C22#350712yes—
614C22#350811yescharacteristic
615C22#351812yes—
616C22C2×#56311yescharacteristic
617C22#353611yescharacteristic
618C22#354011yescharacteristic
619C22#354712yes—
620C22#355111yescharacteristic
621C22#355512yes—
622C22#356111yescharacteristic
623C22#356212yes—
624C22#356512yes—
625C4—24no—
626C4—42no—
627C4—48no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—81no—
633C4—82no—
634C4—84no—
635C4—84no—
636C4—82no—
637C4—84no—
638C4#272818yes—
639C2—84no—
640C2#3948611yescharacteristic
641C2#3991512yes—
642C2#4265512yes—
643C2#4288311yescharacteristic
644C2#4489312yes—
645C2#4511511yescharacteristic
646C2#4619611yescharacteristic
647C2C2×#272811yescharacteristic
648C2#5308111yescharacteristic
649C2#5906512yes—
650C2#5920211yescharacteristic
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×#2439C2semidirect
4. #16640C2semidirect
5. #16115C2semidirect
6. #15504C2semidirect
7. #15232C2semidirect
8. #15196C2semidirect
9. #15132C2semidirect
10. #14501C2semidirect
11. #14155C2semidirect
12. #13973C2semidirect
13. #13879C2semidirect
81. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24392
#171212
#166522
#166402
#161152
#155042
#152322
#151962
#151322
#145012
#141552
#139732
#138792
#138712
C4×#3942