Finite Group Databasebeta

← #792605

Subgroups — #792605

Order 512.

2994 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2609C212yesmaximal
2C2×#2344C212yesmaximal, complemented
3#16688C212yesmaximal, complemented
4#15980C212yesmaximal, complemented
5#15953C214yesmaximal, complemented
6#15952C212yesmaximal
7#15151C211yesmaximal, characteristic, complemented
8#15131C212yesmaximal, complemented
9#14891C214yesmaximal, complemented
10#14133C212yesmaximal, complemented
11#14130C212yesmaximal, complemented
12#13911C211yesmaximal, characteristic, complemented
13#13876C212yesmaximal, complemented
14#13872C212yesmaximal
15C4×#394C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C22×#117C2214yes—
18C23×C4⋊C4C2212yes—
19#2609—216no—
20C2×#398C2212yes—
21C2×#398C2212yes—
22C2×#396C2214yes—
23C2×#394C2211yescharacteristic
24#2344—216no—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#388C2214yes—
29C2×#387C2214yes—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#386C2214yes—
33C2×#385C2214yes—
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#382C2212yes—
37C2×#381C2214yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46#1445C2212yes—
47#1445C2212yes—
48#1444C2214yes—
49#1443C2214yes—
50#1442C2211yescharacteristic
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2211yescharacteristic
59#1441C2214yes—
60#1440C2212yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2211yescharacteristic
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2314yes—
80C2×C4×D4—28no—
81C2×C4×D4—42no—
82C2×#117—216no—
83C2×#117—216no—
84C2×#117—48no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#398—28no—
118#398—44no—
119#396—216no—
120#394D418yes—
121#389—28no—
122#389—28no—
123#389—28no—
124#388—216no—
125#387—216no—
126#387—216no—
127#387—216no—
128#386—216no—
129#385—216no—
130#384—42no—
131#384—42no—
132#382—28no—
133#381—216no—
134#380—28no—
135#380—28no—
136#380—44no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—42no—
139C4×C4⋊C4D418yescomplemented
140C4×C22⋊C4—28no—
141C4×C22⋊C4—28no—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C25C2412yes—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4C2×D414yes—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—24no—
178C23×C4—28no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4#4118yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C4×D4—416no—
204#117—464no—
205C2×C4⋊C4—264no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4C2×D418yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4#4118yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×D418yes—
272C2×C22⋊C4C2×D418yes—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—28no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—44no—
287C2×C42C2×D418yes—
288C2×C42C2×D414yes—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—41no—
308#95—44no—
309#95—41no—
310#95—41no—
311#95—44no—
312#95—42no—
313#95—44no—
314#95C2×D418yes—
315#95C2×D414yes—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24C22×D414yes—
329C24C2×#4114yes—
330C242−1+414yes—
331C24C2511yescharacteristic
332C2×D4—216no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—44no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—232no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D418yes—
425C22×C4C22×D414yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4118yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+412yes—
436C22×C42−1+412yes—
437C4⋊C4—232no—
438C4⋊C4—216no—
439C4⋊C4—216no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—416no—
443C4⋊C4—416no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—48no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C22⋊C4—216no—
453C22⋊C4—216no—
454C22⋊C4—416no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—88no—
460C42—28no—
461C42—216no—
462C42—416no—
463C42—48no—
464C42—44no—
465C42—44no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23#54718yes—
501C23C23×D411yescharacteristic
502C23C23×D411yescharacteristic
503C23C22×#4112yes—
504C23C22×#4112yes—
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×2−1+411yescharacteristic
508C23C2×2−1+412yes—
509C23C2×2−1+411yescharacteristic
510C23#58512yes—
511C23#58511yescharacteristic
512D4—416no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—464no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—432no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4#52518yes—
596C2×C4#53714yes—
597C2×C4D4214yes—
598C2×C4#55114yes—
599C2×C4#55218yes—
600C2×C4#55514yes—
601C22—28no—
602C22—28no—
603C22—28no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22C2×#52512yes—
617C22C2×#53711yescharacteristic
618C22C2×D4211yescharacteristic
619C22C2×#54712yes—
620C22D4×#4112yes—
621C22C2×#55111yescharacteristic
622C22C2×#55212yes—
623C22C2×#55511yescharacteristic
624C22#348912yes—
625C22#349512yes—
626C22#349512yes—
627C22#349611yescharacteristic
628C22#349711yescharacteristic
629C22#349911yescharacteristic
630C22#350612yes—
631C22#352012yes—
632C22#352312yes—
633C22#352412yes—
634C22#353611yescharacteristic
635C22#353811yescharacteristic
636C22#355111yescharacteristic
637C22#355911yescharacteristic
638C22#356312yes—
639C4—24no—
640C4—44no—
641C4—48no—
642C4—48no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—84no—
649C4—84no—
650C4—81no—
651C4—82no—
652C4#260918yescomplemented
653C2—48no—
654C2C2×#260911yescharacteristic
655C2#3583212yes—
656C2#3610012yes—
657C2#3935911yescharacteristic
658C2#4298011yescharacteristic
659C2#4525111yescharacteristic
660C2#4525312yes—
661C2#4538411yescharacteristic
662C2#5919811yescharacteristic
663C2#5929012yes—
664C2#5984111yescharacteristic
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×#2344C2semidirect
3. #16688C2semidirect
4. #15980C2semidirect
5. #15953C2semidirect
7. #15151C2semidirect
8. #15131C2semidirect
9. #14891C2semidirect
10. #14133C2semidirect
11. #14130C2semidirect
12. #13911C2semidirect
13. #13876C2semidirect
139. C4×C4⋊C4D4semidirect
652. C4#2609semidirect

Maximal subgroups

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

subgroupindex
C2×#26092
C2×#23442
#166882
#159802
#159532
#159522
#151512
#151312
#148912
#141332
#141302
#139112
#138762
#138722
C4×#3942