Finite Group Databasebeta

← #515294

Subgroups — #515294

Order 512.

4562 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic, complemented
2C2×#2280C211yesmaximal, characteristic
3C2×#2280C211yesmaximal, characteristic
4C22×#380C211yesmaximal, characteristic
5C2×#2274C211yesmaximal, characteristic, complemented
6#13994C214yesmaximal, complemented
7#13993C214yesmaximal, complemented
8#13972C214yesmaximal, complemented
9#13971C214yesmaximal
10#13876C214yesmaximal, complemented
11#13875C214yesmaximal, complemented
12#13795C211yesmaximal, characteristic, complemented
13#13795C211yesmaximal, characteristic, complemented
14C25×C4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22#2317—28no—
23C2×#389C2212yescomplemented
24C2×#389C2212yescomplemented
25C2×#388C2218yes—
26C2×#385C2214yes—
27C2×#384C2212yescomplemented
28C2×#382C2212yescomplemented
29#2280—28no—
30#2280C4116yescomplemented
31C2×#381C2212yes—
32C2×#381C2212yes—
33C2×#380—28no—
34C2×#380C2212yes—
35C2×#380C2214yes—
36C2×#380C2212yes—
37C2×#380C2214yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#379C2212yes—
41C2×#379C2212yes—
42#2274—28no—
43C2×C4×C4⋊C4C2214yescomplemented
44C2×C4×C4⋊C4C2214yescomplemented
45C2×C4×C22⋊C4C2214yes—
46C2×C4×C22⋊C4C2214yes—
47C2×C4×C22⋊C4C2214yes—
48C22×#95C2212yes—
49#1443C2218yescomplemented
50#1442C2218yescomplemented
51#1440C2218yes—
52#1439C2218yes—
53#1437C2218yes—
54#1436C2218yescomplemented
55C4⋊#95C2212yescomplemented
56C4⋊#95C2214yescomplemented
57C4⋊#95C2212yescomplemented
58C4⋊#95C2212yescomplemented
59C4⋊#95C2214yescomplemented
60C4⋊#95C2212yescomplemented
61#1434C2218yescomplemented
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C26C2311yescharacteristic
65C24×C4—24no—
66C24×C4—24no—
67C24×C4—24no—
68C24×C4—24no—
69C24×C4—24no—
70C24×C4—24no—
71C24×C4C2×C418yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C2×#117—24no—
79C2×#117—24no—
80C2×#117—24no—
81C2×#117—24no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117C2×C418yes—
88C2×#117C2×C418yes—
89C2×#117D418yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—48no—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2318yes—
124#389—28no—
125#389—28no—
126#388—416no—
127#385—216no—
128#384—28no—
129#382—28no—
130#381—28no—
131#381C2×C4116yes—
132#380—216no—
133#380—216no—
134#380—28no—
135#380—28no—
136#380C2×C4116yescomplemented
137#380C2×C4116yescomplemented
138#379—28no—
139#379C2×C4116yes—
140C4×C4⋊C4—216no—
141C4×C4⋊C4—216no—
142C4×C22⋊C4—216no—
143C4×C22⋊C4—216no—
144C4×C22⋊C4C2×C4132yescomplemented
145C2×#95—28no—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2318yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2318yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C25—22no—
159C25—22no—
160C25—24no—
161C25—24no—
162C25—22no—
163C25—22no—
164C25—22no—
165C25—22no—
166C25—42no—
167C25—42no—
168C25C22×C414yes—
169C25C2412yes—
170C25C2411yescharacteristic
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—232no—
180C23×C4—232no—
181C23×C4—232no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—416no—
202C23×C4—416no—
203C23×C4—416no—
204C23×C4—416no—
205C23×C4C22×C418yes—
206C23×C4C22×C418yes—
207C23×C4C22×C414yes—
208C23×C4C22×C414yes—
209C23×C4C2×D418yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4#4114yes—
213C23×C4#4114yes—
214C23×C4#4114yes—
215C23×C4#4114yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#117—232no—
228#117—232no—
229#117—232no—
230#117—416no—
231C2×C4⋊C4—232no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—416no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4—48no—
261C2×C4⋊C4C22×C418yes—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×D418yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—432no—
290C2×C22⋊C4C22×C418yes—
291C2×C22⋊C4C22×C418yes—
292C2×C22⋊C4C22×C4116yes—
293C2×C22⋊C4C22×C4116yes—
294C2×C22⋊C4C2×D418yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—216no—
305C2×C42—48no—
306C2×C42—48no—
307C2×C42C22×C418yes—
308C2×C42C22×C418yes—
309C2×C42C2×D418yes—
310#95—216no—
311#95—216no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—48no—
317#95—48no—
318#95—48no—
319#95—48no—
320#95C22×C4116yes—
321#95C22×C4116yes—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—28no—
327C24—28no—
328C24—216no—
329C24—28no—
330C24—216no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—28no—
344C24—28no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—48no—
349C24—48no—
350C24—48no—
351C24—48no—
352C24—48no—
353C24—48no—
354C24—44no—
355C24—44no—
356C24—48no—
357C24—44no—
358C24—44no—
359C24—44no—
360C24—44no—
361C24C23×C414yes—
362C24C23×C412yes—
363C24C22×D412yes—
364C24C22×D412yes—
365C24C2×#4112yes—
366C24C2×#4112yes—
367C24C2×#4112yes—
368C24C2×#4112yes—
369C24H(5,𝔽2)14yes—
370C242−1+414yes—
371C24C2511yescharacteristic
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—4128no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4#117116yes—
450C22×C4C4×D4116yes—
451C22×C4C4×D4116yes—
452C22×C4#121116yes—
453C22×C4#123116yes—
454C22×C4#124116yes—
455C22×C4#126116yes—
456C22×C4C23×C414yes—
457C22×C4C23×C414yes—
458C22×C4C23×C418yes—
459C22×C4C23×C414yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C22×D418yes—
463C22×C4C22×D418yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—232no—
491C22⋊C4—216no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—432no—
496C22⋊C4—48no—
497C42—216no—
498C42—216no—
499C42—216no—
500C42—48no—
501C42—816no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—232no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—432no—
538C23—432no—
539C23—416no—
540C23—416no—
541C23—416no—
542C23—416no—
543C23—416no—
544C23—432no—
545C23—432no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—416no—
563C23C2×#11714yes—
564C23C2×C4×D414yes—
565C23C2×C4×D414yes—
566C23#52018yes—
567C23C2×#12114yes—
568C23C2×#12314yes—
569C23C2×#12414yes—
570C23C2×#12614yes—
571C23C24×C411yescharacteristic
572C23C23×D411yescharacteristic
573C23C23×D411yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4111yescharacteristic
576C23C22×#4111yescharacteristic
577C23C22×#4111yescharacteristic
578C23C2×H(5,𝔽2)12yes—
579C23C2×2−1+412yes—
580C23#58512yes—
581C23#58512yes—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—416no—
619C2×C4—432no—
620C2×C4—432no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—48no—
634C2×C4—48no—
635C2×C4—44no—
636C2×C4—816no—
637C2×C4—816no—
638C2×C4#388164yescomplemented
639C2×C4C2×#11714yes—
640C2×C4C2×#11714yes—
641C2×C4C2×C4×D414yes—
642C2×C4C2×C4×D414yes—
643C2×C4C2×C4×D414yes—
644C2×C4C2×C4×D414yes—
645C2×C4C2×#12114yes—
646C2×C4C2×#12114yes—
647C2×C4C2×#12314yes—
648C2×C4C2×#12314yes—
649C2×C4C2×#12414yes—
650C2×C4C2×#12414yes—
651C2×C4C2×#12614yes—
652C2×C4C2×#12614yes—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—28no—
657C22—24no—
658C22—24no—
659C22—28no—
660C22—24no—
661C22—28no—
662C22—28no—
663C22—28no—
664C22—28no—
665C22—28no—
666C22—464no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22C2×#38818yes—
680C22C22×#11711yescharacteristic
681C22C22×C4×D411yescharacteristic
682C22C22×C4×D411yescharacteristic
683C22C2×#52012yes—
684C22#342812yes—
685C22C22×#12111yescharacteristic
686C22C22×#12311yescharacteristic
687C22C22×#12411yescharacteristic
688C22C22×#12611yescharacteristic
689C22#348411yescharacteristic
690C22#348511yescharacteristic
691C22#348611yescharacteristic
692C22#348711yescharacteristic
693C22#348812yes—
694C22#348912yes—
695C22#349212yes—
696C22#349312yes—
697C22#349412yes—
698C22#349412yes—
699C22#349412yes—
700C22#349512yes—
701C22#349512yes—
702C22#349512yes—
703C4—44no—
704C4—44no—
705C4—48no—
706C4—48no—
707C4—48no—
708C4—44no—
709C4—84no—
710C4—82no—
711C4—82no—
712C4C2×#38818yes—
713C4C2×#38818yes—
714C2—28no—
715C2—48no—
716C2C22×#38811yescharacteristic
717C2#3418012yes—
718C2#3420011yescharacteristic
719C2#3420111yescharacteristic
720C2#3420212yes—
721C2#3551411yescharacteristic
722C2#3551612yes—
723C2#3554412yes—
724C2#3558411yescharacteristic
725C2#3564411yescharacteristic
726C2#3566111yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2317C2semidirect
5. C2×#2274C2semidirect
6. #13994C2semidirect
7. #13993C2semidirect
8. #13972C2semidirect
10. #13876C2semidirect
11. #13875C2semidirect
12. #13795C2semidirect
13. #13795C2semidirect
23. C2×#389C22semidirect
24. C2×#389C22semidirect
27. C2×#384C22semidirect
28. C2×#382C22semidirect
30. #2280C4semidirect
43. C2×C4×C4⋊C4C22semidirect
44. C2×C4×C4⋊C4C22semidirect
49. #1443C22semidirect
50. #1442C22semidirect
54. #1436C22semidirect
55. C4⋊#95C22semidirect
56. C4⋊#95C22semidirect
57. C4⋊#95C22semidirect
58. C4⋊#95C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect
61. #1434C22semidirect
136. #380C2×C4semidirect
137. #380C2×C4semidirect
144. C4×C22⋊C4C2×C4semidirect
638. C2×C4#388semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#22802
C22×#3802
C2×#22742
#139942
#139932
#139722
#139712
#138762
#138752
#137952