Finite Group Databasebeta

← #674290

Subgroups — #674290

Order 512.

4574 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2541C211yesmaximal, characteristic
2C2×#2501C212yesmaximal, complemented
3C2×#2445C214yesmaximal, complemented
4C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
5#15370C218yesmaximal, complemented
6#15148C214yesmaximal, complemented
7#15130C212yesmaximal, complemented
8#15130C212yesmaximal, complemented
9#14713C211yesmaximal, characteristic, complemented
10#14154C211yesmaximal, characteristic, complemented
11#14137C212yesmaximal, complemented
12#13910C211yesmaximal, characteristic, complemented
13C4×#384C211yesmaximal, characteristic, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C22×#127C2211yescharacteristic
16C22×#127C2212yes—
17C22×#123C2218yes—
18C22×#121C2212yes—
19C22×#121C2212yes—
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23#2541—28no—
24#2501—216no—
25#2445—232no—
26C2×#399C2218yes—
27C2×#394C2218yes—
28C2×#394C2218yes—
29C2×#392C2214yes—
30D4×C4⋊C4—28no—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2214yes—
34C2×#390C2214yes—
35C2×#390C2212yes—
36C2×#390C2214yes—
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#389C2212yes—
41C2×#389C2214yes—
42C2×#386C2218yes—
43C2×#385C2214yes—
44C2×#384C2211yescharacteristic
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2214yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×C43C2211yescharacteristic
56#1444C2214yes—
57#1439C2218yes—
58#1439C2218yes—
59#1439C2218yes—
60#1438C2214yes—
61#1437C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C23×D4C2312yes—
68C23×D4C2312yes—
69C23×D4C2314yes—
70C23×D4C2314yes—
71C23×D4C2311yescharacteristic
72C23×D4C2312yes—
73C24×C4C2312yes—
74C24×C4C2314yes—
75C2×#127—216no—
76C2×#127—28no—
77C2×#127—24no—
78C2×#127—24no—
79C2×#127—24no—
80C2×#123—232no—
81C2×#123—232no—
82C2×#123—416no—
83C2×#121—28no—
84C2×#121—28no—
85C2×#121—28no—
86C2×#121—28no—
87C2×#121—44no—
88C2×#121—44no—
89C2×C4×D4—28no—
90C2×C4×D4—216no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4D418yes—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2318yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2318yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2318yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#399—232no—
129#394—232no—
130#394—232no—
131#392—216no—
132#390—216no—
133#390—28no—
134#390—28no—
135#390—216no—
136#390—28no—
137#390—48no—
138#390—44no—
139#390—44no—
140#390D4116yescomplemented
141#389—28no—
142#389—216no—
143#386—232no—
144#385—216no—
145#384—28no—
146#384—24no—
147#384—24no—
148#384D418yescomplemented
149#380—28no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—216no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—44no—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2318yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C43—24no—
166C25C2414yes—
167C25C2412yes—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—216no—
171C22×D4—216no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—28no—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—48no—
195C22×D4—44no—
196C22×D4—48no—
197C22×D4—44no—
198C22×D4C2×D414yes—
199C22×D4C2×D418yes—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—216no—
208C23×C4—28no—
209C23×C4—44no—
210C23×C4—48no—
211C23×C4C2×D418yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224#127—432no—
225#127—416no—
226#123—4128no—
227#121—432no—
228#121—432no—
229C4×D4—232no—
230C4×D4—432no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—432no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D4116yes—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4#4114yes—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4C2×D418yes—
296C2×C22⋊C4C2×D4116yes—
297C2×C22⋊C4C2×D4116yes—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—42no—
315C2×C42C2×D414yes—
316C2×C42C2×D418yes—
317C2×C42C2×D414yes—
318C2×C42#4114yes—
319C2×C42#4114yes—
320#95—28no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—216no—
327#95—24no—
328#95—44no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—28no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—28no—
345C24—24no—
346C24—24no—
347C24—28no—
348C24—28no—
349C24C22×D418yes—
350C24C22×D414yes—
351C24C2511yescharacteristic
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—216no—
355C2×D4—216no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—432no—
359C2×D4—432no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—88no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—816no—
380C2×D4—816no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4#127116yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—216no—
421C22×C4—216no—
422C22×C4—216no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—416no—
434C22×C4—432no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—416no—
440C22×C4—816no—
441C22×C4—88no—
442C22×C4C22×D414yes—
443C22×C4C22×D418yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+414yes—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—232no—
470C4⋊C4—28no—
471C4⋊C4—216no—
472C4⋊C4—416no—
473C4⋊C4—416no—
474C4⋊C4—432no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—84no—
479C22⋊C4—216no—
480C22⋊C4—232no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—432no—
485C22⋊C4—416no—
486C22⋊C4—432no—
487C22⋊C4—816no—
488C22⋊C4—816no—
489C42—28no—
490C42—216no—
491C42—48no—
492C42—48no—
493C42—48no—
494C42—44no—
495C42—48no—
496C42#123116yescomplemented
497C23—24no—
498C23—24no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—24no—
507C23—28no—
508C23—24no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—48no—
525C23—44no—
526C23—44no—
527C23—48no—
528C23—48no—
529C23—44no—
530C23—48no—
531C23—44no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—44no—
537C23—44no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23C2×#12718yes—
543C23C23×D411yescharacteristic
544C23C23×D412yes—
545C23C23×D411yescharacteristic
546C23C22×#4111yescharacteristic
547C23C22×#4111yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×H(5,𝔽2)12yes—
552C23C2×2−1+412yes—
553C23#58512yes—
554D4—232no—
555D4—816no—
556D4—832no—
557D4—832no—
558D4—816no—
559D4—816no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—432no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—832no—
618C2×C4C2×#12314yes—
619C2×C4C2×#12314yes—
620C2×C4C2×#12314yes—
621C2×C4C2×#12714yes—
622C2×C4#53214yes—
623C2×C4#53614yes—
624C2×C4#54014yes—
625C2×C4D4218yes—
626C2×C4D4214yes—
627C2×C4#54614yes—
628C2×C4#54718yes—
629C2×C4#54718yes—
630C2×C4#55014yes—
631C2×C4#55914yes—
632C22—28no—
633C22—28no—
634C22—28no—
635C22—48no—
636C22—44no—
637C22—44no—
638C22—48no—
639C22—48no—
640C22—44no—
641C22—44no—
642C22—48no—
643C22—84no—
644C22—88no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—88no—
649C22—88no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—88no—
654C22C22×#12311yescharacteristic
655C22C22×#12711yescharacteristic
656C22C2×#53211yescharacteristic
657C22C2×#53611yescharacteristic
658C22C2×#54011yescharacteristic
659C22C2×D4212yes—
660C22C2×D4211yescharacteristic
661C22C2×#54611yescharacteristic
662C22C2×#54712yes—
663C22C2×#54712yes—
664C22C2×#55011yescharacteristic
665C22D4×#4112yes—
666C22C2×#55911yescharacteristic
667C22#349412yes—
668C22#349612yes—
669C22#349712yes—
670C22#349811yescharacteristic
671C22#349812yes—
672C22#350011yescharacteristic
673C22#350612yes—
674C22#352212yes—
675C22#354812yes—
676C22#355412yes—
677C4—28no—
678C4—24no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—48no—
684C4—44no—
685C4—84no—
686C4—82no—
687C4#244518yescomplemented
688C4#250118yescomplemented
689C2—48no—
690C2—164no—
691C2C2×#244511yescharacteristic
692C2C2×#250111yescharacteristic
693C2#3553111yescharacteristic
694C2D4×#12711yescharacteristic
695C2#3612612yes—
696C2#3626011yescharacteristic
697C2#3895211yescharacteristic
698C2#3935212yes—
699C2#3936311yescharacteristic
700C2#5964112yes—
701C2#5981512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2541C2
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×#2501C2semidirect
3. C2×#2445C2semidirect
4. C2×D4×C4⋊C4C2semidirect
5. #15370C2semidirect
6. #15148C2semidirect
7. #15130C2semidirect
8. #15130C2semidirect
9. #14713C2semidirect
10. #14154C2semidirect
11. #14137C2semidirect
12. #13910C2semidirect
13. C4×#384C2semidirect
14. C4×#390C2semidirect
140. #390D4semidirect
148. #384D4semidirect
496. C42#123semidirect
687. C4#2445semidirect
688. C4#2501semidirect

Maximal subgroups

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

subgroupindex
C2×#25412
C2×#25012
C2×#24452
C2×D4×C4⋊C42
#153702
#151482
#151302
#147132
#141542
#141372
#139102
C4×#3842
C4×#3902