Finite Group Databasebeta

← #674788

Subgroups — #674788

Order 512.

2518 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C212yesmaximal, complemented
2#15913C214yesmaximal
3#15714C214yesmaximal, complemented
4#15183C212yesmaximal, complemented
5#15131C212yesmaximal, complemented
6#14853C214yesmaximal, complemented
7#14707C211yesmaximal, characteristic, complemented
8#14101C214yesmaximal, complemented
9#13973C212yesmaximal, complemented
10#13954C211yesmaximal, characteristic, complemented
11#13905C211yesmaximal, characteristic
12#13905C211yesmaximal, characteristic
13#13804C211yesmaximal, characteristic, complemented
14C4×#384C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic
16C22×#126C2214yes—
17C22×C4×D4C2211yescharacteristic
18#2496—216no—
19C2×#400C2214yes—
20C2×#397C2214yes—
21C2×#396C2214yes—
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#387C2214yes—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2214yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C43C2211yescharacteristic
43#1445C2214yes—
44#1445C2212yes—
45#1445C2212yes—
46#1444C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2212yes—
56#1438C2212yes—
57#1437C2212yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2212yes—
61#1437C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73#1434C2212yes—
74#1434C2214yes—
75C4×#95C2212yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C2×#126—216no—
79C2×#126—216no—
80C2×#126—48no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2314yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115#400—216no—
116#397—216no—
117#396—48no—
118#390—28no—
119#389—28no—
120#389—28no—
121#389—44no—
122#387—216no—
123#384—24no—
124#384D418yescomplemented
125#382—24no—
126#382—24no—
127#380—28no—
128#380—28no—
129#380—28no—
130#380—44no—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—48no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—44no—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C43—24no—
165C25C2412yes—
166C22×D4—24no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—42no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192#126—464no—
193C4×D4—416no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C2×D4116yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4#4114yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4#4118yes—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—28no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—44no—
278C2×C42—42no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42C2×D414yes—
282C2×C42#4114yes—
283C2×C42#4114yes—
284C2×C42#4114yes—
285#95—24no—
286#95—24no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—22no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24C2×#4114yes—
326C24C2511yescharacteristic
327C2×D4—28no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—44no—
331C2×D4—44no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
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—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
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—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4C22×D414yes—
405C22×C4C22×D414yes—
406C22×C4C22×D414yes—
407C22×C4C22×D412yes—
408C22×C4C2×#4114yes—
409C22×C4C2×#4114yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4H(5,𝔽2)14yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+414yes—
424C22×C42−1+412yes—
425C4⋊C4—232no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—48no—
430C4⋊C4—416no—
431C4⋊C4—416no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—416no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—88no—
438C22⋊C4—216no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C42—216no—
448C42—28no—
449C42—48no—
450C42—416no—
451C42—44no—
452C42—48no—
453C42—84no—
454C42—84no—
455C42#126116yescomplemented
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
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—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23C23×D411yescharacteristic
492C23C22×#4111yescharacteristic
493C23C22×#4111yescharacteristic
494C23C22×#4111yescharacteristic
495C23C22×#4111yescharacteristic
496C23C22×#4112yes—
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×H(5,𝔽2)12yes—
499C23C2×2−1+411yescharacteristic
500C23C2×2−1+412yes—
501C23#58512yes—
502D4—416no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—816no—
580C2×C4C2×#12614yes—
581C2×C4C2×#12614yes—
582C2×C4C2×#12614yes—
583C2×C4#53214yes—
584C2×C4#54214yes—
585C2×C4#54618yes—
586C2×C4#54714yes—
587C2×C4#55014yes—
588C2×C4#55314yes—
589C2×C4#55614yes—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22C22×#12611yescharacteristic
606C22C2×#53211yescharacteristic
607C22#344411yescharacteristic
608C22C2×#54211yescharacteristic
609C22#345511yescharacteristic
610C22C2×#54612yes—
611C22C2×#54711yescharacteristic
612C22C2×#55011yescharacteristic
613C22C2×#55311yescharacteristic
614C22C2×#55611yescharacteristic
615C22#347512yes—
616C22#347512yes—
617C22#348212yes—
618C22#349212yes—
619C22#349712yes—
620C22#350811yescharacteristic
621C22#350911yescharacteristic
622C22#351112yes—
623C22#351112yes—
624C22#351312yes—
625C22#352012yes—
626C22#354612yes—
627C22#356312yes—
628C4—28no—
629C4—24no—
630C4—24no—
631C4—44no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—82no—
637C4—84no—
638C4—82no—
639C4—84no—
640C4—82no—
641C4#251218yes—
642C2—84no—
643C2C2×#251211yescharacteristic
644C2#3481411yescharacteristic
645C2#3555711yescharacteristic
646C2#3614911yescharacteristic
647C2#3642412yes—
648C2#3650911yescharacteristic
649C2#3918511yescharacteristic
650C2#3937311yescharacteristic
651C2#3990212yes—
652C2#5941012yes—
653C2#5943112yes—
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×#2496C2semidirect
3. #15714C2semidirect
4. #15183C2semidirect
5. #15131C2semidirect
6. #14853C2semidirect
7. #14707C2semidirect
8. #14101C2semidirect
9. #13973C2semidirect
10. #13954C2semidirect
13. #13804C2semidirect
14. C4×#384C2semidirect
124. #384D4semidirect
455. C42#126semidirect

Maximal subgroups

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

subgroupindex
C2×#24962
#159132
#157142
#151832
#151312
#148532
#147072
#141012
#139732
#139542
#139052
#138042
C4×#3842
C4×#3902