Finite Group Databasebeta

← #800066

Subgroups — #800066

Order 512.

2348 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×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2346C211yesmaximal, characteristic
3#16679C212yesmaximal
4#16001C214yesmaximal, complemented
5#15444C214yesmaximal
6#15242C211yesmaximal, characteristic, complemented
7#15202C211yesmaximal, characteristic, complemented
8#15150C211yesmaximal, characteristic
9#15122C212yesmaximal, complemented
10#14882C212yesmaximal
11#14030C212yesmaximal, complemented
12#13981C212yesmaximal, complemented
13#13973C212yesmaximal
14#13961C212yesmaximal, complemented
15#13921C211yesmaximal, characteristic, complemented
16#13879C212yesmaximal
17C4×#393C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×#117C2211yescharacteristic
21C2×#397C2212yes—
22C2×#397C2212yes—
23C2×#393C2211yescharacteristic
24Q8×C4⋊C4—28no—
25#2346—28no—
26C2×#389C2212yes—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2214yes—
30C2×#389C2212yes—
31C2×#388C2212yes—
32C2×#388C2212yes—
33C2×#387C2214yes—
34C2×#384C2211yescharacteristic
35C2×#384C2212yes—
36C2×#383C2212yes—
37C2×#382C2212yes—
38C2×#382C2212yes—
39C2×#380C2212yes—
40C2×#380C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×#376C2212yes—
47#1446C2212yes—
48#1446C2211yescharacteristic
49#1445C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1444C2212yes—
54#1444C2214yes—
55#1443C2214yes—
56#1443C2214yes—
57#1442C2211yescharacteristic
58#1442C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1439C2214yes—
65#1439C2211yescharacteristic
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82#1434C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C4×Q8—28no—
88C2×C4×Q8—24no—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—44no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—24no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125#397—44no—
126#397—44no—
127#393—24no—
128#389—24no—
129#389—24no—
130#389—28no—
131#389—28no—
132#389—216no—
133#388—28no—
134#388—28no—
135#387—48no—
136#384—28no—
137#384—24no—
138#383—44no—
139#382—28no—
140#382—28no—
141#380—28no—
142#380—24no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4D418yes—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—28no—
148#376—28no—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C25C2411yescharacteristic
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8#4114yes—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4#4114yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C4×Q8—232no—
210C4×Q8—432no—
211#117—232no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4#4114yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42C2×D418yes—
292C2×C42C2×D414yes—
293C2×C42#4118yes—
294#95—22no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—24no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—22no—
308#95—24no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—24no—
314#95—28no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—44no—
326#95C2×D418yes—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—21no—
336C24—22no—
337C24C2×#4112yes—
338C24C2511yescharacteristic
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—216no—
342C2×Q8—28no—
343C2×Q8—216no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4C22×D414yes—
426C22×C4C22×D412yes—
427C22×C4C22×D414yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4112yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C42−1+414yes—
442C22×C42−1+414yes—
443C4⋊C4—28no—
444C4⋊C4—216no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—416no—
449C4⋊C4—416no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—88no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C22⋊C4—216no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—44no—
462C22⋊C4—48no—
463C22⋊C4—84no—
464C22⋊C4—88no—
465C42—216no—
466C42—28no—
467C42—216no—
468C42—216no—
469C42—48no—
470C42—416no—
471C42—44no—
472C42—48no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23C23×D411yescharacteristic
497C23C22×#4112yes—
498C23C22×#4111yescharacteristic
499C23C22×#4111yescharacteristic
500C23C22×#4111yescharacteristic
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×2−1+412yes—
504C23C2×2−1+412yes—
505C23#58511yescharacteristic
506C23#58512yes—
507Q8—232no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—416no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4#52514yes—
597C2×C4#52514yes—
598C2×C4#53614yes—
599C2×C4#54614yes—
600C2×C4#55218yes—
601C2×C4#55218yes—
602C2×C4#55618yes—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—44no—
607C22—42no—
608C22—82no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—82no—
614C22C2×#52511yescharacteristic
615C22C2×#52511yescharacteristic
616C22C2×#53611yescharacteristic
617C22C2×#54611yescharacteristic
618C22C2×#55212yes—
619C22C2×#55212yes—
620C22C2×#55612yes—
621C22#348011yescharacteristic
622C22#348311yescharacteristic
623C22#348611yescharacteristic
624C22#348811yescharacteristic
625C22#349212yes—
626C22#349312yes—
627C22#350011yescharacteristic
628C22#350612yes—
629C22#350712yes—
630C22#350811yescharacteristic
631C22#351512yes—
632C22#351711yescharacteristic
633C22#351812yes—
634C22#354412yes—
635C22#354712yes—
636C22#355512yes—
637C4—24no—
638C4—28no—
639C4—44no—
640C4—42no—
641C4—44no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—44no—
646C4—82no—
647C4—81no—
648C4—84no—
649C4—84no—
650C4—82no—
651C4—82no—
652C4#259918yes—
653C2—82no—
654C2C2×#259911yescharacteristic
655C2#3571211yescharacteristic
656C2#3575611yescharacteristic
657C2#3616311yescharacteristic
658C2#3641011yescharacteristic
659C2#3895811yescharacteristic
660C2#4281311yescharacteristic
661C2#4490512yes—
662C2#4498312yes—
663C2#5925612yes—
664C2#5943512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×Q8×C4⋊C4C2semidirect
4. #16001C2semidirect
6. #15242C2semidirect
7. #15202C2semidirect
9. #15122C2semidirect
11. #14030C2semidirect
12. #13981C2semidirect
14. #13961C2semidirect
15. #13921C2semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#23462
#166792
#160012
#154442
#152422
#152022
#151502
#151222
#148822
#140302
#139812
#139732
#139612
#139212
#138792
C4×#3932