Finite Group Databasebeta

← #515307

Subgroups — #515307

Order 512.

4338 conjugacy classes of subgroups.

The subgroup lattice has 767 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic, complemented
4#14067C214yesmaximal, complemented
5#14063C214yesmaximal, complemented
6#13987C214yesmaximal, complemented
7#13983C214yesmaximal, complemented
8#13882C212yesmaximal, complemented
9#13878C212yesmaximal, complemented
10#13873C212yesmaximal
11#13868C212yesmaximal
12#13796C211yesmaximal, characteristic, complemented
13#13778C211yesmaximal, characteristic, complemented
14#13776C211yesmaximal, characteristic, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C22×#117C2212yes—
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C2×#388C2214yes—
24C2×#387C2214yescomplemented
25C2×#385C4116yescomplemented
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#385C2212yes—
29C2×#385C2212yes—
30#2280—28no—
31C2×#381C2212yes—
32C2×#381C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#379C2212yes—
37C2×#379C2212yes—
38#2274—28no—
39C2×C4×C4⋊C4C2214yescomplemented
40C2×C4×C4⋊C4C2214yescomplemented
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1445C2214yescomplemented
52#1444C2214yescomplemented
53#1442C2214yescomplemented
54#1441C2214yescomplemented
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69#1434C2214yescomplemented
70#1434C2214yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C26C2311yescharacteristic
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4C2×C418yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×#117—28no—
95C2×#117—28no—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117—44no—
99C2×#117—44no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4C2×C418yes—
104C22×C4⋊C4C2×C418yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—28no—
113C22×C22⋊C4—44no—
114C22×C22⋊C4C2×C418yes—
115C22×C22⋊C4C2×C418yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2312yes—
134#388—48no—
135#387—48no—
136#385C2×C4116yes—
137#385C2×C4116yes—
138#385C2×C4116yes—
139#385C2×C4116yes—
140#381—28no—
141#381—28no—
142#380—28no—
143#380—28no—
144#380—28no—
145#379—28no—
146#379—28no—
147C4×C4⋊C4—216no—
148C4×C4⋊C4—216no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—44no—
155C4×C22⋊C4—44no—
156C4×C22⋊C4D418yes—
157C2×#95C2×C418yes—
158C2×#95C2×C418yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C25—24no—
183C25—24no—
184C25—24no—
185C25—22no—
186C25—22no—
187C25—24no—
188C25—42no—
189C25—42no—
190C25C22×C414yes—
191C25C2411yescharacteristic
192C25C2411yescharacteristic
193C25C2411yescharacteristic
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—232no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—416no—
218C23×C4—42no—
219C23×C4—416no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—432no—
233C23×C4—432no—
234C23×C4C22×C414yes—
235C23×C4C22×C414yes—
236C23×C4C22×C414yes—
237C23×C4C22×C414yes—
238C23×C4C22×C414yes—
239C23×C4C22×C414yes—
240C23×C4C22×C414yes—
241C23×C4C22×C414yes—
242C23×C4C22×C414yes—
243C23×C4C22×C414yes—
244C23×C4C2×D414yes—
245C23×C4#4118yes—
246C23×C4#4114yes—
247C23×C4C2414yes—
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2414yes—
254C23×C4C2414yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259#117—264no—
260#117—432no—
261C2×C4⋊C4—232no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—416no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4C22×C418yes—
279C2×C4⋊C4C22×C418yes—
280C2×C4⋊C4C22×C418yes—
281C2×C4⋊C4C22×C418yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—232no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—416no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4C22×C418yes—
313C2×C22⋊C4C22×C418yes—
314C2×C22⋊C4C22×C418yes—
315C2×C22⋊C4C22×C418yes—
316C2×C22⋊C4C2×D418yes—
317C2×C22⋊C4#4118yes—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—28no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42C2×D418yes—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95C22×C418yes—
355#95C22×C418yes—
356#95C22×C418yes—
357#95C22×C418yes—
358#95C2×D418yes—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—232no—
364C24—28no—
365C24—28no—
366C24—28no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—21no—
374C24—21no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—28no—
382C24—28no—
383C24—44no—
384C24—44no—
385C24—48no—
386C24—48no—
387C24—48no—
388C24—48no—
389C24—416no—
390C24—416no—
391C24—416no—
392C24—48no—
393C24—48no—
394C24C23×C412yes—
395C24C23×C412yes—
396C24C23×C412yes—
397C24C22×D412yes—
398C24C2×#4114yes—
399C24C2×#4112yes—
400C24C2×#4114yes—
401C24H(5,𝔽2)14yes—
402C242−1+414yes—
403C24C2511yescharacteristic
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—28no—
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—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—4128no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—416no—
501C22×C4—84no—
502C22×C4—84no—
503C22×C4—88no—
504C22×C4#117116yescomplemented
505C22×C4C4×D4116yes—
506C22×C4C23×C414yes—
507C22×C4C23×C414yes—
508C22×C4C23×C412yes—
509C22×C4C23×C414yes—
510C22×C4C23×C414yes—
511C22×C4C23×C412yes—
512C22×C4C23×C412yes—
513C22×C4C23×C412yes—
514C22×C4C22×D418yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C2×#4118yes—
518C22×C4C2×#4114yes—
519C22×C4C2×#4114yes—
520C22×C4C2×#4112yes—
521C22×C4C2×#4112yes—
522C22×C4H(5,𝔽2)14yes—
523C4⋊C4—232no—
524C4⋊C4—216no—
525C4⋊C4—216no—
526C4⋊C4—216no—
527C4⋊C4—48no—
528C4⋊C4—416no—
529C4⋊C4—88no—
530C22⋊C4—216no—
531C22⋊C4—216no—
532C22⋊C4—216no—
533C22⋊C4—216no—
534C22⋊C4—216no—
535C22⋊C4—48no—
536C22⋊C4—416no—
537C22⋊C4—416no—
538C22⋊C4—416no—
539C22⋊C4—416no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C42—232no—
543C42—48no—
544C42—48no—
545C42—84no—
546C42—84no—
547C42—88no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—28no—
556C23—24no—
557C23—28no—
558C23—28no—
559C23—28no—
560C23—232no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—28no—
577C23—28no—
578C23—24no—
579C23—28no—
580C23—28no—
581C23—416no—
582C23—416no—
583C23—416no—
584C23—416no—
585C23—464no—
586C23—464no—
587C23—416no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—416no—
601C23C2×#11714yes—
602C23C2×C4×D414yes—
603C23C4×#4114yes—
604C23#51814yes—
605C23#51914yes—
606C23#52518yes—
607C23#52918yes—
608C23C24×C411yescharacteristic
609C23C23×D411yescharacteristic
610C23C22×#4112yes—
611C23C22×#4111yescharacteristic
612C23C22×#4112yes—
613C23C2×H(5,𝔽2)12yes—
614C23C2×H(5,𝔽2)12yes—
615C23C2×2−1+412yes—
616C23#58512yes—
617C2×C4—232no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—232no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—28no—
650C2×C4—28no—
651C2×C4—28no—
652C2×C4—28no—
653C2×C4—28no—
654C2×C4—28no—
655C2×C4—416no—
656C2×C4—416no—
657C2×C4—416no—
658C2×C4—416no—
659C2×C4—416no—
660C2×C4—416no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
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—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—88no—
692C2×C4—88no—
693C2×C4C2×#11718yes—
694C2×C4C2×C4×D414yes—
695C2×C4C2×C4×D414yes—
696C2×C4#51814yes—
697C22—216no—
698C22—216no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—24no—
703C22—24no—
704C22—24no—
705C22—24no—
706C22—24no—
707C22—24no—
708C22—24no—
709C22—24no—
710C22—24no—
711C22—464no—
712C22—48no—
713C22—48no—
714C22—48no—
715C22—48no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—42no—
723C22—42no—
724C22#229018yes—
725C22C22×#11711yescharacteristic
726C22C22×C4×D411yescharacteristic
727C22C2×C4×#4111yescharacteristic
728C22C2×#51811yescharacteristic
729C22C2×#51811yescharacteristic
730C22C2×#51911yescharacteristic
731C22#342811yescharacteristic
732C22C2×#52512yes—
733C22C2×#52912yes—
734C22#348412yes—
735C22#348912yes—
736C22#349012yes—
737C22#349214yes—
738C22#349512yes—
739C22#350614yes—
740C22#351314yes—
741C22#352014yes—
742C4—24no—
743C4—24no—
744C4—24no—
745C4—28no—
746C4—48no—
747C4—48no—
748C4—44no—
749C4—42no—
750C4—44no—
751C4—44no—
752C4—84no—
753C4—84no—
754C2—28no—
755C2—44no—
756C2—44no—
757C2C2×#229011yescharacteristic
758C2#3417811yescharacteristic
759C2#3418611yescharacteristic
760C2#3419811yescharacteristic
761C2#3420311yescharacteristic
762C2#3423811yescharacteristic
763C2#3424511yescharacteristic
764C2#3535314yes—
765C2#3537214yes—
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
3. C2×#2274C2semidirect
4. #14067C2semidirect
5. #14063C2semidirect
6. #13987C2semidirect
7. #13983C2semidirect
8. #13882C2semidirect
9. #13878C2semidirect
12. #13796C2semidirect
13. #13778C2semidirect
14. #13776C2semidirect
24. C2×#387C22semidirect
25. C2×#385C4semidirect
39. C2×C4×C4⋊C4C22semidirect
40. C2×C4×C4⋊C4C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
69. #1434C22semidirect
70. #1434C22semidirect
504. C22×C4#117semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
C2×#22802
C2×#22742
#140672
#140632
#139872
#139832
#138822
#138782
#138732
#138682
#137962
#137782
#137762
C4×#3792