Finite Group Databasebeta

← #521969

Subgroups — #521969

Order 512.

2834 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2347C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic
3C2×#2296C211yesmaximal, characteristic
4#14686C211yesmaximal, characteristic
5#14408C214yesmaximal
6#14402C214yesmaximal
7#14356C212yesmaximal, complemented
8#14046C212yesmaximal, complemented
9#14045C212yesmaximal, complemented
10#14005C212yesmaximal, complemented
11#14005C212yesmaximal, complemented
12#13964C212yesmaximal, complemented
13#13954C212yesmaximal, complemented
14#13951C211yesmaximal, characteristic
15#13951C211yesmaximal, characteristic
16#13783C212yesmaximal, complemented
17C4×#380C211yesmaximal, characteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2347—28no—
22#2346—28no—
23#2296C4116yescomplemented
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2214yes—
27C2×#389C2214yes—
28C2×#389C2214yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#384C2212yes—
40C2×#383C2212yes—
41C2×#382C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×#376C2212yes—
56#1444C2214yes—
57#1444C2214yes—
58#1444C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80#1434C2211yescharacteristic
81#1434C2212yes—
82#1434C2212yes—
83#1434C2211yescharacteristic
84C22⋊#95C2212yes—
85C4×#95C2212yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#117—44no—
90C2×#117C2×C418yes—
91C2×#117C2×C418yes—
92C2×#117D418yes—
93C2×#117Q818yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42—44no—
119C22×C42C2×C418yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130#389—28no—
131#389—28no—
132#389—216no—
133#389—48no—
134#389—48no—
135#389—44no—
136#389—44no—
137#389C2×C418yes—
138#389C2×C418yes—
139#389C2×C418yes—
140#389C2×C418yes—
141#388—216no—
142#388—216no—
143#385C2×C4116yes—
144#385C2×C4116yes—
145#384—44no—
146#383—28no—
147#382—28no—
148#380—24no—
149#380—24no—
150#380—28no—
151#380—44no—
152#380—44no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—44no—
157C4×C4⋊C4C2×C4116yes—
158C4×C4⋊C4C2×C4116yes—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—24no—
161#376—28no—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C25C2411yescharacteristic
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4C22×C414yes—
202C23×C4C22×C414yes—
203C23×C4C22×C414yes—
204C23×C4C2×D414yes—
205C23×C4C2×Q814yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#117—416no—
223#117C4⋊C4164yescomplemented
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
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—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C418yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C4⋊C4C2×Q814yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4C22×C418yes—
288C2×C22⋊C4C22×C418yes—
289C2×C22⋊C4C2×D418yes—
290C2×C22⋊C4C2×Q818yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—28no—
302C2×C42—416no—
303C2×C42—44no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42C22×C414yes—
310C2×C42C22×C414yes—
311C2×C42C22×C418yes—
312C2×C42C22×C414yes—
313C2×C42C22×C414yes—
314C2×C42C22×C418yes—
315C2×C42C2×D418yes—
316C2×C42C2×Q818yes—
317#95—22no—
318#95—24no—
319#95—28no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C22×C418yes—
341#95C22×C418yes—
342#95C22×C418yes—
343#95C22×C418yes—
344#95#4118yes—
345C24—22no—
346C24—21no—
347C24—21no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24C23×C412yes—
354C24C22×D412yes—
355C24C22×Q812yes—
356C24C2511yescharacteristic
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4C2×C4⋊C4116yes—
439C22×C4C23×C412yes—
440C22×C4C23×C414yes—
441C22×C4C23×C414yes—
442C22×C4C23×C414yes—
443C22×C4C23×C414yes—
444C22×C4C23×C412yes—
445C22×C4C23×C412yes—
446C22×C4C23×C412yes—
447C22×C4C23×C412yes—
448C22×C4C23×C412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×Q812yes—
454C22×C4C22×Q814yes—
455C22×C4C22×Q814yes—
456C22×C4C22×Q812yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C42−1+412yes—
464C22×C42−1+412yes—
465C22×C42−1+414yes—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—216no—
470C4⋊C4—216no—
471C4⋊C4—48no—
472C4⋊C4—416no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C4⋊C4C2×C4⋊C4116yes—
483C4⋊C4C2×C4⋊C4116yes—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4C2×C4⋊C4132yes—
490C42—28no—
491C42—28no—
492C42—44no—
493C42—44no—
494C42—48no—
495C42—416no—
496C42—48no—
497C42—48no—
498C42—88no—
499C42C2×C4⋊C4132yes—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23C22×C4⋊C414yes—
523C23C24×C411yescharacteristic
524C23C23×D411yescharacteristic
525C23C23×Q811yescharacteristic
526C23C22×#4112yes—
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×2−1+412yes—
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+411yescharacteristic
533C23#58512yes—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—232no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4C22×C4⋊C414yes—
621C2×C4C22×C4⋊C418yes—
622C2×C4C22×C4⋊C418yes—
623C2×C4C22×C4⋊C414yes—
624C2×C4C4×#4114yes—
625C2×C4#51814yes—
626C2×C4#51914yes—
627C2×C4#52014yes—
628C2×C4#52014yes—
629C2×C4#53718yes—
630C2×C4#54418yes—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22C23×C4⋊C411yescharacteristic
643C22C2×C4×#4111yescharacteristic
644C22C2×#51811yescharacteristic
645C22C2×#51911yescharacteristic
646C22C2×#52011yescharacteristic
647C22C2×#52011yescharacteristic
648C22#342811yescharacteristic
649C22C2×#53712yes—
650C22C2×#54412yes—
651C22D4×#4112yes—
652C22Q8×#4112yes—
653C22#349811yescharacteristic
654C22#349911yescharacteristic
655C22#350311yescharacteristic
656C22#350311yescharacteristic
657C22#350912yes—
658C22#351112yes—
659C22#351912yes—
660C22#352012yes—
661C22#354112yes—
662C22#354212yes—
663C22#354712yes—
664C22#355512yes—
665C4—24no—
666C4—28no—
667C4—24no—
668C4—28no—
669C4—24no—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—82no—
677C4—84no—
678C4—84no—
679C4—82no—
680C4#231618yes—
681C2—44no—
682C2C2×#231611yescharacteristic
683C2C4⋊C4×#4111yescharacteristic
684C2#3418311yescharacteristic
685C2#3422511yescharacteristic
686C2#3423311yescharacteristic
687C2#3439411yescharacteristic
688C2#3440411yescharacteristic
689C2#3995812yes—
690C2#3995912yes—
691C2#3998212yes—
692C2#3998312yes—
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
7. #14356C2semidirect
8. #14046C2semidirect
9. #14045C2semidirect
10. #14005C2semidirect
11. #14005C2semidirect
12. #13964C2semidirect
13. #13954C2semidirect
16. #13783C2semidirect
23. #2296C4semidirect
223. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
C2×#23462
C2×#22962
#146862
#144082
#144022
#143562
#140462
#140452
#140052
#139642
#139542
#139512
#137832
C4×#3802