Finite Group Databasebeta

← #792611

Subgroups — #792611

Order 512.

2470 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2347C212yesmaximal, complemented
2#17015C212yesmaximal
3#16116C212yesmaximal, complemented
4#16029C214yesmaximal, complemented
5#15978C212yesmaximal
6#15151C211yesmaximal, characteristic, complemented
7#15131C212yesmaximal, complemented
8#15074C214yesmaximal, complemented
9#14238C212yesmaximal, complemented
10#14132C212yesmaximal, complemented
11#14129C212yesmaximal, complemented
12#13911C211yesmaximal, characteristic, complemented
13#13876C212yesmaximal, complemented
14#13871C212yesmaximal
15C4×#394C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C22×#117C2212yes—
18C2×#398C2212yes—
19C2×#398C2212yes—
20C2×#396C2214yes—
21C2×#394C2211yescharacteristic
22#2347—216no—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#383C2214yes—
32C2×#382C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2214yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42#1445C2212yes—
43#1445C2212yes—
44#1445C2214yes—
45#1442C2211yescharacteristic
46#1442C2214yes—
47#1442C2212yes—
48#1442C2212yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2211yescharacteristic
52#1441C2214yes—
53#1440C2212yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2211yescharacteristic
61#1438C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C4×#95C2211yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C2×C4×D4—28no—
79C2×C4×D4—42no—
80C2×#117—28no—
81C2×#117—28no—
82C2×#117—28no—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2314yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2314yes—
112C22×C42C2312yes—
113#398—28no—
114#398—44no—
115#396—48no—
116#394—24no—
117#389—28no—
118#389—28no—
119#389—28no—
120#388—216no—
121#388—216no—
122#388—48no—
123#384—42no—
124#384—42no—
125#383—216no—
126#382—44no—
127#380—28no—
128#380—28no—
129#380—44no—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—216no—
132C4×C4⋊C4—42no—
133C4×C4⋊C4D418yescomplemented
134C4×C22⋊C4—28no—
135C4×C22⋊C4—28no—
136C2×#95C2311yescharacteristic
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C25C2412yes—
164C22×D4—24no—
165C22×D4—22no—
166C22×D4—41no—
167C22×D4—42no—
168C22×D4—41no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4#4118yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C4×D4—416no—
191#117—264no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C2×D418yes—
230C2×C4⋊C4#4118yes—
231C2×C4⋊C4#4114yes—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
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—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
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—28no—
260C2×C42—28no—
261C2×C42—28no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—28no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42C2×D414yes—
277C2×C42C2×D418yes—
278#95—24no—
279#95—24no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—22no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—44no—
294#95—44no—
295#95—41no—
296#95—41no—
297#95—41no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95—44no—
304#95C2×D418yes—
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—
318C24—22no—
319C24C2×#4114yes—
320C24C2511yescharacteristic
321C2×D4—48no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—48no—
325C2×D4—44no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4C22×D412yes—
404C22×C4C22×D418yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4114yes—
409C22×C4C2×#4118yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4114yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C42−1+414yes—
418C22×C42−1+412yes—
419C4⋊C4—216no—
420C4⋊C4—216no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—416no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—416no—
427C4⋊C4—48no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C4⋊C4—88no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C22⋊C4—216no—
434C22⋊C4—232no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C42—232no—
443C42—28no—
444C42—48no—
445C42—48no—
446C42—44no—
447C42—44no—
448C42—88no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—42no—
457C23—42no—
458C23—42no—
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—
484C23C23×D411yescharacteristic
485C23C22×#4112yes—
486C23C22×#4112yes—
487C23C22×#4111yescharacteristic
488C23C2×H(5,𝔽2)11yescharacteristic
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×2−1+411yescharacteristic
492C23C2×2−1+412yes—
493C23#58512yes—
494C23#58511yescharacteristic
495D4—416no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—432no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
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—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4#52518yes—
578C2×C4#53714yes—
579C2×C4#55014yes—
580C2×C4#55114yes—
581C2×C4#55218yes—
582C2×C4#55514yes—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22C2×#52512yes—
599C22C2×#53711yescharacteristic
600C22C2×#55011yescharacteristic
601C22C2×#55111yescharacteristic
602C22C2×#55212yes—
603C22C2×#55511yescharacteristic
604C22#347512yes—
605C22#348212yes—
606C22#348912yes—
607C22#349212yes—
608C22#349312yes—
609C22#349611yescharacteristic
610C22#350511yescharacteristic
611C22#351112yes—
612C22#351111yescharacteristic
613C22#351412yes—
614C22#351612yes—
615C22#351712yes—
616C22#353711yescharacteristic
617C22#354011yescharacteristic
618C22#355011yescharacteristic
619C22#355512yes—
620C22#355911yescharacteristic
621C4—24no—
622C4—42no—
623C4—44no—
624C4—42no—
625C4—42no—
626C4—48no—
627C4—42no—
628C4—48no—
629C4—44no—
630C4—81no—
631C4—84no—
632C4—84no—
633C4—82no—
634C4#260918yes—
635C2—84no—
636C2C2×#260911yescharacteristic
637C2#3576312yes—
638C2#3650412yes—
639C2#3974211yescharacteristic
640C2#4292911yescharacteristic
641C2#4512812yes—
642C2#4533011yescharacteristic
643C2#4540811yescharacteristic
644C2#5920111yescharacteristic
645C2#5929312yes—
646C2#5995111yescharacteristic
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×#2347C2semidirect
3. #16116C2semidirect
4. #16029C2semidirect
6. #15151C2semidirect
7. #15131C2semidirect
8. #15074C2semidirect
9. #14238C2semidirect
10. #14132C2semidirect
11. #14129C2semidirect
12. #13911C2semidirect
13. #13876C2semidirect
133. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
#170152
#161162
#160292
#159782
#151512
#151312
#150742
#142382
#141322
#141292
#139112
#138762
#138712
C4×#3942