Finite Group Databasebeta

← #520640

Subgroups — #520640

Order 512.

2990 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C4⋊C4C211yesmaximal, characteristic
2#14685C211yesmaximal, characteristic
3#14684C211yesmaximal, characteristic
4#14683C211yesmaximal, characteristic
5#13964C212yesmaximal, complemented
6#13950C212yesmaximal, complemented
7#13949C212yesmaximal, complemented
8#13932C212yesmaximal, complemented
9#13929C212yesmaximal, complemented
10#13928C212yesmaximal, complemented
11#13882C212yesmaximal
12#13874C212yesmaximal
13#13870C212yesmaximal
14#13819C212yesmaximal, complemented
15#13806C212yesmaximal, complemented
16#13799C212yesmaximal
17C4×#381C211yesmaximal, characteristic
18#13756C211yesmaximal, characteristic
19#13753C211yesmaximal, characteristic
20C22×#117C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24C23×C42C2211yescharacteristic
25C23×C42C2211yescharacteristic
26C2×#387C2214yes—
27C2×#383C2212yes—
28C2×#381C2212yes—
29C2×#381C2212yes—
30C2×#381C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4C4116yescomplemented
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C2×#376C2212yes—
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1445C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62#1434C2214yes—
63#1434C2212yes—
64#1434C2212yes—
65#1434C2212yes—
66#1434C2212yes—
67#1434C2212yes—
68#1434C2212yes—
69#1434C2212yes—
70#1434C2212yes—
71#1434C2212yes—
72#1434C2212yes—
73#1434C2212yes—
74#1434C2214yes—
75#1434C2212yes—
76#1434C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2211yescharacteristic
84C22⋊#95C2211yescharacteristic
85C22⋊#95C2211yescharacteristic
86C4×#95C2212yes—
87C4×#95C2212yes—
88C4×#95C2212yes—
89C4×#95C2212yes—
90C4×#95C2212yes—
91C4×#95C2214yes—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C24×C4C2311yescharacteristic
99C2×#117—28no—
100C2×#117—28no—
101C2×#117—44no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4C2×C418yes—
106C22×C4⋊C4C2×C418yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42—28no—
122C22×C42—44no—
123C22×C42—44no—
124C22×C42C2×C418yes—
125C22×C42C2×C418yes—
126C22×C42C2×C418yes—
127C22×C42C2314yes—
128C22×C42C2314yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#387—216no—
143#383—28no—
144#381—28no—
145#381—28no—
146#381—44no—
147#380—44no—
148C4×C4⋊C4C2×C418yes—
149C4×C4⋊C4C2×C418yes—
150C4×C4⋊C4C2×C418yes—
151C4×C4⋊C4C2×C418yes—
152C4×C4⋊C4C2×C418yes—
153C4×C4⋊C4C2×C418yes—
154C4×C4⋊C4C2×C418yes—
155C4×C4⋊C4C2×C418yes—
156C4×C22⋊C4—28no—
157C4×C22⋊C4—44no—
158#376—28no—
159C2×#95C2×C418yes—
160C2×#95C2×C418yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C25C2411yescharacteristic
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C22×C414yes—
220C23×C4C22×C414yes—
221C23×C4C22×C414yes—
222C23×C4C22×C414yes—
223C23×C4C22×C414yes—
224C23×C4C22×C414yes—
225C23×C4C22×C414yes—
226C23×C4#4114yes—
227C23×C4#4114yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246#117—432no—
247C2×C4⋊C4—232no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—232no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C414yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C414yes—
270C2×C4⋊C4C22×C414yes—
271C2×C4⋊C4C22×C414yes—
272C2×C4⋊C4C22×C414yes—
273C2×C4⋊C4C22×C414yes—
274C2×C4⋊C4#4118yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4#4118yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—232no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—44no—
297C2×C42—416no—
298C2×C42—416no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42C22×C414yes—
311C2×C42C22×C414yes—
312C2×C42C22×C414yes—
313C2×C42C22×C414yes—
314C2×C42C22×C414yes—
315C2×C42C22×C414yes—
316C2×C42C22×C414yes—
317C2×C42C22×C414yes—
318C2×C42C22×C414yes—
319C2×C42C22×C414yes—
320C2×C42C22×C414yes—
321C2×C42C22×C414yes—
322C2×C42#4118yes—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95C22×C414yes—
351#95C22×C414yes—
352#95C22×C414yes—
353#95C22×C414yes—
354#95C22×C414yes—
355#95C22×C414yes—
356#95C22×C414yes—
357#95C22×C414yes—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24C23×C412yes—
363C24C2×#4112yes—
364C24C2×#4112yes—
365C24C2×#4114yes—
366C24H(5,𝔽2)14yes—
367C24C2511yescharacteristic
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—232no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4#117116yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C23×C412yes—
467C22×C4C23×C412yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C23×C412yes—
471C22×C4C23×C412yes—
472C22×C4C23×C412yes—
473C22×C4C23×C412yes—
474C22×C4C23×C412yes—
475C22×C4C23×C412yes—
476C22×C4C23×C412yes—
477C22×C4C2×#4118yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C42−1+414yes—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—416no—
497C4⋊C4—88no—
498C4⋊C4#117116yescomplemented
499C22⋊C4—216no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C42—28no—
504C42—28no—
505C42—28no—
506C42—44no—
507C42—44no—
508C42—44no—
509C42—44no—
510C42—44no—
511C42—44no—
512C42—44no—
513C42—44no—
514C42—416no—
515C42—416no—
516C42—88no—
517C42#117116yescomplemented
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23C2×#11714yes—
531C23C4×#4114yes—
532C23#51814yes—
533C23#55118yes—
534C23#55318yes—
535C23C24×C411yescharacteristic
536C23C22×#4111yescharacteristic
537C23C22×#4111yescharacteristic
538C23C22×#4112yes—
539C23C22×#4112yes—
540C23C22×#4112yes—
541C23C2×H(5,𝔽2)12yes—
542C23C2×2−1+412yes—
543C23#58512yes—
544C2×C4—232no—
545C2×C4—232no—
546C2×C4—232no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4C2×#11714yes—
633C2×C4C2×#11714yes—
634C2×C4C2×#11714yes—
635C2×C4C2×#11714yes—
636C2×C4C2×#11714yes—
637C2×C4C2×#11714yes—
638C2×C4C2×#11714yes—
639C2×C4C2×#11714yes—
640C2×C4#51914yes—
641C22—216no—
642C22—216no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22#235318yes—
650C22C22×#11711yescharacteristic
651C22C22×#11711yescharacteristic
652C22C22×#11711yescharacteristic
653C22C2×C4×#4111yescharacteristic
654C22C2×#51811yescharacteristic
655C22C2×#51911yescharacteristic
656C22#342811yescharacteristic
657C22#344412yes—
658C22#344412yes—
659C22C2×#55112yes—
660C22C2×#55312yes—
661C22#347514yes—
662C22#348214yes—
663C22#349214yes—
664C22#350712yes—
665C22#350712yes—
666C22#351314yes—
667C4—24no—
668C4—24no—
669C4—24no—
670C4—24no—
671C4—24no—
672C4—24no—
673C4—24no—
674C4—24no—
675C4—24no—
676C4—48no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—84no—
683C4—84no—
684C4—84no—
685C2—28no—
686C2#3401211yescharacteristic
687C2#3410911yescharacteristic
688C2C2×#235311yescharacteristic
689C2#3412811yescharacteristic
690C2#3417811yescharacteristic
691C2#3418711yescharacteristic
692C2#3419011yescharacteristic
693C2#3635214yes—
694C2#3635314yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13964C2semidirect
6. #13950C2semidirect
7. #13949C2semidirect
8. #13932C2semidirect
9. #13929C2semidirect
10. #13928C2semidirect
14. #13819C2semidirect
15. #13806C2semidirect
32. C2×C4×C4⋊C4C4semidirect
498. C4⋊C4#117semidirect
517. C42#117semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C4⋊C42
#146852
#146842
#146832
#139642
#139502
#139492
#139322
#139292
#139282
#138822
#138742
#138702
#138192
#138062
#137992
C4×#3812
#137562
#137532