Finite Group Databasebeta

← #516091

Subgroups — #516091

Order 512.

3072 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#2293C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic, complemented
4#14193C214yesmaximal, complemented
5#14191C214yesmaximal, complemented
6#14129C211yesmaximal, characteristic, complemented
7#14104C214yesmaximal, complemented
8#14095C214yesmaximal
9#14042C212yesmaximal, complemented
10#13993C212yesmaximal, complemented
11#13924C211yesmaximal, characteristic, complemented
12#13871C211yesmaximal, characteristic
13#13777C212yesmaximal, complemented
14#13777C211yesmaximal, characteristic, complemented
15C4×#389C212yesmaximal, complemented
16C22×#123C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2486C4116yescomplemented
23C2×#398C2212yes—
24C2×#397C2212yes—
25#2293—28no—
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#388C2212yes—
31C2×#388C2212yes—
32C2×#387C2214yes—
33C2×#381C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38#2274—28no—
39C2×C4×C4⋊C4C2214yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2214yes—
45C2×C4×C22⋊C4C2214yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C43C2211yescharacteristic
49#1445C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1441C2214yes—
53#1440C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2212yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1437C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76C22⋊#95C2214yes—
77C4×#95C2214yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×#123—48no—
83C2×#123C2×C4116yes—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4C2×C418yes—
87C2×#117—24no—
88C2×#117—28no—
89C2×#117—24no—
90C2×#117—28no—
91C2×#117—28no—
92C2×#117D418yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4C2×C418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2314yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129#398C2×C4116yes—
130#397C2×C4116yes—
131#390—28no—
132#389—28no—
133#389—28no—
134#389C2×C4116yes—
135#388—28no—
136#388—28no—
137#387—48no—
138#381C2×C4116yes—
139#380—24no—
140#380—28no—
141#380C2×C4116yes—
142#380C2×C418yes—
143C4×C4⋊C4—216no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—48no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—216no—
151C4×C22⋊C4—24no—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C43—24no—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—42no—
181C22×D4C22×C414yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4C22×C414yes—
193C23×C4C22×C414yes—
194C23×C4C22×C418yes—
195C23×C4C2×D414yes—
196C23×C4#4114yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210#123—432no—
211C4×D4—232no—
212#117—232no—
213#117—232no—
214#117—416no—
215C2×C4⋊C4—232no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C22×C418yes—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C22×C4116yes—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4C2×D418yes—
248C2×C4⋊C4#4118yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4C22×C414yes—
275C2×C22⋊C4C22×C414yes—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C22×C4116yes—
278C2×C22⋊C4C2×D418yes—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—44no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C22×C414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42#4118yes—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95C22×C4116yes—
327#95C22×C418yes—
328#95C22×C4116yes—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24C23×C412yes—
344C24C23×C412yes—
345C24C22×D412yes—
346C24C2×#4112yes—
347C24C2511yescharacteristic
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—48no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—216no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—216no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—216no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4C4×D4116yes—
429C22×C4C23×C412yes—
430C22×C4C23×C412yes—
431C22×C4C23×C418yes—
432C22×C4C23×C418yes—
433C22×C4C23×C412yes—
434C22×C4C23×C412yes—
435C22×C4C23×C412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C2×#4118yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4114yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C42−1+414yes—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—232no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—48no—
457C4⋊C4—416no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C22⋊C4—216no—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—216no—
464C22⋊C4—48no—
465C22⋊C4—416no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—88no—
470C22⋊C4—84no—
471C42—216no—
472C42—28no—
473C42—28no—
474C42—48no—
475C42—48no—
476C42—416no—
477C42—44no—
478C42—44no—
479C42—88no—
480C42—88no—
481C42#123116yescomplemented
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C2×C4×D414yes—
524C23C24×C411yescharacteristic
525C23C23×D411yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4112yes—
528C23C22×#4112yes—
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+412yes—
531C23#58512yes—
532C23#58512yes—
533D4—416no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—232no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—416no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—816no—
610C2×C4C2×C4×D414yes—
611C2×C4C2×C4×D414yes—
612C2×C4C4×#4118yes—
613C2×C4#51814yes—
614C2×C4C2×#12318yes—
615C2×C4C2×#12314yes—
616C2×C4#52514yes—
617C2×C4#52918yes—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22C22×C4×D411yescharacteristic
637C22C2×C4×#4112yes—
638C22C2×#51811yescharacteristic
639C22#342811yescharacteristic
640C22C4×2−1+412yes—
641C22C22×#12311yescharacteristic
642C22C2×#52511yescharacteristic
643C22C2×#52912yes—
644C22#348812yes—
645C22#348912yes—
646C22#349412yes—
647C22#350612yes—
648C22#350712yes—
649C22#350911yescharacteristic
650C22#351011yescharacteristic
651C22#351112yes—
652C22#351212yes—
653C22#351612yes—
654C22#351712yes—
655C22#352012yes—
656C22#352112yes—
657C4—24no—
658C4—28no—
659C4—24no—
660C4—48no—
661C4—48no—
662C4—44no—
663C4—48no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—84no—
668C4—82no—
669C4C4×#12318yes—
670C2—44no—
671C2—82no—
672C2C2×C4×#12311yescharacteristic
673C2#3420411yescharacteristic
674C2#3426811yescharacteristic
675C2#3431612yes—
676C2#3431812yes—
677C2#3551712yes—
678C2#3552112yes—
679C2#3559912yes—
680C2#3561412yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#2274C2semidirect
4. #14193C2semidirect
5. #14191C2semidirect
6. #14129C2semidirect
7. #14104C2semidirect
9. #14042C2semidirect
10. #13993C2semidirect
11. #13924C2semidirect
13. #13777C2semidirect
14. #13777C2semidirect
15. C4×#389C2semidirect
22. #2486C4semidirect
481. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#22932
C2×#22742
#141932
#141912
#141292
#141042
#140952
#140422
#139932
#139242
#138712
#137772
C4×#3892