Finite Group Databasebeta

← #521960

Subgroups — #521960

Order 512.

3058 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×#2357C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3C2×#2293C211yesmaximal, characteristic
4#14689C211yesmaximal, characteristic, complemented
5#14488C214yesmaximal, complemented
6#14405C214yesmaximal, complemented
7#14355C212yesmaximal, complemented
8#14046C212yesmaximal, complemented
9#14045C212yesmaximal, complemented
10#14018C212yesmaximal
11#14018C212yesmaximal
12#13962C212yesmaximal, complemented
13#13955C211yesmaximal, characteristic, complemented
14#13954C211yesmaximal, characteristic, complemented
15#13951C212yesmaximal, complemented
16#13782C212yesmaximal, complemented
17C4×#384C211yesmaximal, characteristic, complemented
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21#2357—28no—
22#2349—28no—
23#2293C4116yescomplemented
24C2×#390C2212yes—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2214yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#388C2214yes—
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#384C2212yes—
40C2×#382C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C22×#95C2212yes—
53C2×C43C2211yescharacteristic
54#1444C2214yes—
55#1444C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78#1434C2212yes—
79#1434C2212yes—
80#1434C2212yes—
81#1434C2212yes—
82#1434C2212yes—
83#1434C2212yes—
84C22⋊#95C2214yes—
85C22⋊#95C2214yes—
86C23×D4C2311yescharacteristic
87C24×C4C2314yes—
88C24×C4C2312yes—
89C2×C4×D4—28no—
90C2×C4×D4—44no—
91C2×C4×D4—44no—
92C2×C4×D4C2×C418yes—
93C2×C4×D4D418yes—
94C2×C4×D4Q818yes—
95C2×#117—48no—
96C2×#117C2×C4116yescomplemented
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#390—28no—
133#390C2×C418yes—
134#390C2×C418yes—
135#389—28no—
136#389—48no—
137#389—44no—
138#389—44no—
139#389C2×C418yescomplemented
140#389C2×C418yescomplemented
141#388—216no—
142#388C2×C4116yescomplemented
143#388C2×C4116yescomplemented
144#385—216no—
145#385—216no—
146#385—48no—
147#384—28no—
148#382—24no—
149#382—24no—
150#381—28no—
151#380—28no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—44no—
156C4×C4⋊C4—44no—
157C4×C4⋊C4C2×C418yescomplemented
158C4×C22⋊C4—28no—
159C4×C22⋊C4C2×C4116yescomplemented
160C2×#95—28no—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C43C2×C418yescomplemented
189C25C2412yes—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4C22×C414yes—
195C22×D4C2×D414yes—
196C22×D4C2×Q814yes—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C22×C418yes—
207C23×C4C2×D418yes—
208C23×C4C2×Q818yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C4×D4—432no—
226C4×D4C4⋊C4164yescomplemented
227#117—432no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C414yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C418yes—
270C2×C4⋊C4C22×C414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C4⋊C4#4118yes—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C22×C418yes—
288C2×C22⋊C4C22×C418yes—
289C2×C22⋊C4C22×C418yes—
290C2×C22⋊C4C22×C418yes—
291C2×C22⋊C4C22×C418yes—
292C2×C22⋊C4C2×D418yes—
293C2×C22⋊C4C2×Q818yes—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—44no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42C22×C414yes—
313C2×C42C22×C414yes—
314C2×C42C22×C414yes—
315C2×C42C22×C414yes—
316C2×C42C22×C418yes—
317C2×C42C22×C414yes—
318C2×C42C22×C414yes—
319C2×C42C22×C414yes—
320C2×C42C2×D414yes—
321C2×C42C2×Q814yes—
322#95—22no—
323#95—24no—
324#95—28no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—42no—
339#95—42no—
340#95—42no—
341#95—42no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95C22×C418yes—
347#95C22×C418yes—
348#95C22×C418yes—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24C23×C414yes—
358C24C22×D414yes—
359C24C22×Q814yes—
360C24C2511yescharacteristic
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4C2×C4⋊C4116yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
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—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4C2×C4⋊C4132yes—
448C22×C4C23×C412yes—
449C22×C4C23×C414yes—
450C22×C4C23×C414yes—
451C22×C4C23×C412yes—
452C22×C4C23×C412yes—
453C22×C4C23×C414yes—
454C22×C4C23×C412yes—
455C22×C4C23×C412yes—
456C22×C4C23×C412yes—
457C22×C4C23×C412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×Q812yes—
463C22×C4C22×Q812yes—
464C22×C4C22×Q814yes—
465C22×C4C22×Q812yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+412yes—
473C22×C42−1+412yes—
474C4⋊C4—216no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C4⋊C4C2×C4⋊C4116yes—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—88no—
497C22⋊C4C2×C4⋊C4132yes—
498C42—28no—
499C42—28no—
500C42—28no—
501C42—28no—
502C42—44no—
503C42—44no—
504C42—48no—
505C42—48no—
506C42—48no—
507C42—88no—
508C42C2×C4⋊C4116yes—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23C22×C4⋊C418yes—
532C23C24×C411yescharacteristic
533C23C23×D411yescharacteristic
534C23C23×Q811yescharacteristic
535C23C22×#4112yes—
536C23C2×H(5,𝔽2)12yes—
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×2−1+411yescharacteristic
540C23C2×2−1+411yescharacteristic
541C23#58512yes—
542C23#58512yes—
543D4—232no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—432no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—832no—
628C2×C4C22×C4⋊C414yes—
629C2×C4C22×C4⋊C414yes—
630C2×C4C22×C4⋊C418yes—
631C2×C4C22×C4⋊C414yes—
632C2×C4C4×#4114yes—
633C2×C4#51814yes—
634C2×C4#52014yes—
635C2×C4#52014yes—
636C2×C4#53514yes—
637C2×C4#53614yes—
638C2×C4#54414yes—
639C2×C4#54414yes—
640C22—28no—
641C22—28no—
642C22—28no—
643C22—48no—
644C22—48no—
645C22—48no—
646C22—48no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22C23×C4⋊C411yescharacteristic
652C22C2×C4×#4111yescharacteristic
653C22C2×#51811yescharacteristic
654C22C2×#52011yescharacteristic
655C22C2×#52011yescharacteristic
656C22#342811yescharacteristic
657C22#342811yescharacteristic
658C22C2×#53511yescharacteristic
659C22C2×#53611yescharacteristic
660C22C2×#54411yescharacteristic
661C22C2×#54411yescharacteristic
662C22D4×#4112yes—
663C22Q8×#4112yes—
664C22#349612yes—
665C22#350112yes—
666C22#350512yes—
667C22#350712yes—
668C22#351912yes—
669C22#352212yes—
670C22#354612yes—
671C22#355112yes—
672C22#355412yes—
673C22#355712yes—
674C4—24no—
675C4—24no—
676C4—28no—
677C4—24no—
678C4—24no—
679C4—44no—
680C4—44no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—44no—
685C4—82no—
686C4—82no—
687C4—84no—
688C4—84no—
689C4#231718yes—
690C2—48no—
691C2C2×#231711yescharacteristic
692C2C4⋊C4×#4111yescharacteristic
693C2#3418111yescharacteristic
694C2#3422411yescharacteristic
695C2#3423611yescharacteristic
696C2#3439311yescharacteristic
697C2#3439711yescharacteristic
698C2#3994812yes—
699C2#3994912yes—
700C2#3997212yes—
701C2#3997312yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14689C2semidirect
5. #14488C2semidirect
6. #14405C2semidirect
7. #14355C2semidirect
8. #14046C2semidirect
9. #14045C2semidirect
12. #13962C2semidirect
13. #13955C2semidirect
14. #13954C2semidirect
15. #13951C2semidirect
16. #13782C2semidirect
17. C4×#384C2semidirect
23. #2293C4semidirect
96. C2×#117C2×C4semidirect
139. #389C2×C4semidirect
140. #389C2×C4semidirect
142. #388C2×C4semidirect
143. #388C2×C4semidirect
157. C4×C4⋊C4C2×C4semidirect
159. C4×C22⋊C4C2×C4semidirect
188. C43C2×C4semidirect
226. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23572
C2×#23492
C2×#22932
#146892
#144882
#144052
#143552
#140462
#140452
#140182
#139622
#139552
#139542
#139512
#137822
C4×#3842