Finite Group Databasebeta

← #876617

Subgroups — #876617

Order 512.

4150 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic, complemented
2C2×#2697C212yesmaximal, complemented
3C2×#2679C211yesmaximal, characteristic
4C2×#2534C212yesmaximal, complemented
5#17071C212yesmaximal, complemented
6#17015C212yesmaximal, complemented
7#15549C214yesmaximal, complemented
8#15283C214yesmaximal, complemented
9#15163C212yesmaximal, complemented
10#14839C214yesmaximal, complemented
11#14700C212yesmaximal, complemented
12#13995C214yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic, complemented
14C22×#123C2212yes—
15C22×#120C2212yes—
16C22×#120C2214yes—
17C22×C4×D4C2211yescharacteristic
18C23×C22⋊C4C2212yescomplemented
19#2724—28no—
20#2697—216no—
21#2679—28no—
22#2534—216no—
23C2×#399C2214yescomplemented
24C2×#398C2212yescomplemented
25C2×#397C2214yescomplemented
26C2×#397C2214yescomplemented
27C2×#396C2214yescomplemented
28C2×#396C2214yescomplemented
29C2×#394C2214yes—
30C2×#394C2212yes—
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#389C2212yescomplemented
40C2×#388C2214yescomplemented
41C2×#388C2214yescomplemented
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#385C2214yescomplemented
45C2×#381C2212yescomplemented
46C2×#380C2212yescomplemented
47C2×#379C2212yescomplemented
48C2×#379C2214yescomplemented
49C2×C4×C4⋊C4C2211yescharacteristic, complemented
50C2×C4×C22⋊C4C2214yescomplemented
51C2×C43C2211yescharacteristic, complemented
52#1445C2212yescomplemented
53#1445C2214yescomplemented
54#1444C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1438C2212yes—
60#1438C2214yes—
61#1438C2214yes—
62#1437C2214yescomplemented
63#1437C2214yescomplemented
64#1437C2214yescomplemented
65#1437C2214yescomplemented
66C4⋊#95C2212yescomplemented
67C4⋊#95C2212yescomplemented
68C4⋊#95C2212yescomplemented
69C4⋊#95C2211yescharacteristic, complemented
70C4⋊#95C2211yescharacteristic, complemented
71C26C2312yes—
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C2×#123—28no—
79C2×#123—28no—
80C2×#123—28no—
81C2×#120—28no—
82C2×#120—28no—
83C2×#120—28no—
84C2×#120—216no—
85C2×#120—216no—
86C2×#120—48no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C2×C4×D4D418yescomplemented
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—28no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#399—48no—
127#398—28no—
128#397—216no—
129#397—216no—
130#396—216no—
131#396—216no—
132#394—28no—
133#394—216no—
134#394—216no—
135#394—216no—
136#390—28no—
137#390—28no—
138#390—44no—
139#390—44no—
140#390D418yes—
141#390D418yescomplemented
142#389—28no—
143#388—216no—
144#388—216no—
145#386—216no—
146#386—216no—
147#385—216no—
148#381—28no—
149#380—28no—
150#379—216no—
151#379—28no—
152C4×C4⋊C4—42no—
153C4×C22⋊C4—48no—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C43—42no—
168C25—24no—
169C25—28no—
170C25—24no—
171C25—24no—
172C25—42no—
173C25—42no—
174C25—44no—
175C25—44no—
176C25—44no—
177C25—42no—
178C25—42no—
179C25C2414yes—
180C25C2412yes—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4C2×D414yes—
204C22×D4C2×D414yes—
205C22×D4C2×D414yes—
206C22×D4#4118yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4C2×D418yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230#123—432no—
231#120—464no—
232#120—432no—
233C4×D4—416no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—432no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D4116yes—
314C2×C22⋊C4C2×D418yes—
315C2×C22⋊C4C2×D4116yes—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—41no—
323C2×C42—41no—
324C2×C42—42no—
325C2×C42—41no—
326C2×C42—41no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—44no—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—44no—
345#95—42no—
346C24—24no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—416no—
369C24—48no—
370C24—416no—
371C24—416no—
372C24—48no—
373C24—416no—
374C24—48no—
375C24—48no—
376C24—48no—
377C24—88no—
378C24—88no—
379C24—84no—
380C24—84no—
381C24—84no—
382C24—84no—
383C24—88no—
384C24—88no—
385C24—88no—
386C24—88no—
387C24—84no—
388C24—84no—
389C24—88no—
390C24—88no—
391C24C22×D414yes—
392C24C22×D414yes—
393C24C22×D414yes—
394C24C2×#4118yes—
395C24C2511yescharacteristic
396C2×D4—216no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—416no—
406C2×D4—416no—
407C2×D4—48no—
408C2×D4—416no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—84no—
412C2×D4—84no—
413C2×D4—84no—
414C2×D4—84no—
415C2×D4—84no—
416C2×D4—88no—
417C2×D4—88no—
418C2×D4—88no—
419C2×D4—88no—
420C2×D4—84no—
421C2×D4#127116yes—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—216no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—416no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—88no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4C22×D412yes—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D414yes—
497C22×C4C2×#4114yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4H(5,𝔽2)14yes—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—416no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—88no—
509C4⋊C4—88no—
510C4⋊C4—84no—
511C22⋊C4—232no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—416no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—416no—
521C22⋊C4—416no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C42—28no—
529C42—44no—
530C42—44no—
531C42—48no—
532C42—48no—
533C42—84no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—28no—
552C23—28no—
553C23—44no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—432no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—832no—
593C23—816no—
594C23—816no—
595C23—816no—
596C23—832no—
597C23—832no—
598C23—832no—
599C23—816no—
600C23—816no—
601C23—816no—
602C23C2×#12718yes—
603C23C23×D411yescharacteristic
604C23C23×D411yescharacteristic
605C23C23×D411yescharacteristic
606C23C22×#4112yes—
607C23C2×H(5,𝔽2)11yescharacteristic
608C23C2×H(5,𝔽2)11yescharacteristic
609C23C2×H(5,𝔽2)12yes—
610C23#58511yescharacteristic
611C23#58511yescharacteristic
612C23#58512yes—
613C23#58512yes—
614D4—416no—
615D4—816no—
616D4—816no—
617D4—832no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—28no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—28no—
642C2×C4—24no—
643C2×C4—28no—
644C2×C4—24no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—816no—
683C2×C4—816no—
684C2×C4C2×#12714yes—
685C2×C4#53414yes—
686C2×C4#54618yes—
687C22—216no—
688C22—28no—
689C22—44no—
690C22—48no—
691C22—44no—
692C22—48no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—48no—
698C22—48no—
699C22—48no—
700C22—48no—
701C22—48no—
702C22—48no—
703C22—48no—
704C22—48no—
705C22—864no—
706C22—84no—
707C22—84no—
708C22—84no—
709C22—84no—
710C22—84no—
711C22—84no—
712C22—84no—
713C22—84no—
714C22C22×#12711yescharacteristic
715C22C2×#53411yescharacteristic
716C22#344911yescharacteristic
717C22#345912yes—
718C22C2×#54612yes—
719C22D4×#4112yes—
720C22D4×#4112yes—
721C22#348111yescharacteristic
722C22#349112yes—
723C22#349612yes—
724C22#349611yescharacteristic
725C22#349612yes—
726C22#349611yescharacteristic
727C22#349611yescharacteristic
728C22#349611yescharacteristic
729C22#350012yes—
730C22#351412yes—
731C22#352911yescharacteristic
732C22#353412yes—
733C22#354311yescharacteristic
734C22#354511yescharacteristic
735C22#356012yes—
736C22#356812yes—
737C4—24no—
738C4—24no—
739C4—28no—
740C4—44no—
741C4—44no—
742C4—44no—
743C4—42no—
744C4—44no—
745C4—84no—
746C4—84no—
747C4—84no—
748C2—48no—
749C2—88no—
750C2#3500111yescharacteristic
751C2#3626012yes—
752C2#3712911yescharacteristic
753C2#3843212yes—
754C2#3941412yes—
755C2#3945212yes—
756C2#4623711yescharacteristic
757C2#4628011yescharacteristic
758C2#5216311yescharacteristic
759C2#5942011yescharacteristic
760C2#5983611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2679C2
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×#2724C2semidirect
2. C2×#2697C2semidirect
4. C2×#2534C2semidirect
5. #17071C2semidirect
6. #17015C2semidirect
7. #15549C2semidirect
8. #15283C2semidirect
9. #15163C2semidirect
10. #14839C2semidirect
11. #14700C2semidirect
12. #13995C2semidirect
13. C4×#390C2semidirect
18. C23×C22⋊C4C22semidirect
23. C2×#399C22semidirect
24. C2×#398C22semidirect
25. C2×#397C22semidirect
26. C2×#397C22semidirect
27. C2×#396C22semidirect
28. C2×#396C22semidirect
39. C2×#389C22semidirect
40. C2×#388C22semidirect
41. C2×#388C22semidirect
44. C2×#385C22semidirect
45. C2×#381C22semidirect
46. C2×#380C22semidirect
47. C2×#379C22semidirect
48. C2×#379C22semidirect
49. C2×C4×C4⋊C4C22semidirect
50. C2×C4×C22⋊C4C22semidirect
51. C2×C43C22semidirect
52. #1445C22semidirect
53. #1445C22semidirect
54. #1444C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. #1437C22semidirect
65. #1437C22semidirect
66. C4⋊#95C22semidirect
67. C4⋊#95C22semidirect
68. C4⋊#95C22semidirect
69. C4⋊#95C22semidirect
70. C4⋊#95C22semidirect
89. C2×C4×D4D4semidirect
141. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#26972
C2×#26792
C2×#25342
#170712
#170152
#155492
#152832
#151632
#148392
#147002
#139952
C4×#3902