Finite Group Databasebeta

← #4306441

Subgroups — #4306441

Order 512.

2902 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×#2396C212yesmaximal
2C2×#2365C211yesmaximal, characteristic, complemented
3C2×#2344C211yesmaximal, characteristic
4#15156C211yesmaximal, characteristic, complemented
5#15149C212yesmaximal, complemented
6#15143C211yesmaximal, characteristic, complemented
7#15131C212yesmaximal
8#15125C211yesmaximal, characteristic
9#14887C212yesmaximal
10#14876C212yesmaximal, complemented
11#14872C212yesmaximal, complemented
12#14794C214yesmaximal, complemented
13#14773C214yesmaximal
14C4×#399C211yesmaximal, characteristic
15#13821C211yesmaximal, characteristic, complemented
16C4⋊C42C212yesmaximal, complemented
17C22⋊C4×C4⋊C4C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×#117C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22#2396—216no—
23C2×#400C2212yes—
24C2×#399C2211yescharacteristic
25C2×#398C2214yes—
26C2×#396C2214yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#393C2212yes—
30#2365—28no—
31#2344—28no—
32C2×#391C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#387C2214yes—
40C2×#384C2212yes—
41C2×#381C2212yes—
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2214yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2214yes—
55#1446C2212yes—
56#1445C2211yescharacteristic
57#1445C2212yes—
58#1444C2211yescharacteristic
59#1444C2214yes—
60#1443C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2211yescharacteristic
71#1439C2212yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C22⋊#95C2214yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—24no—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—44no—
94C2×#117—28no—
95C2×#117—28no—
96C2×#117—44no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4Q818yescomplemented
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#400—28no—
133#399D418yes—
134#398—216no—
135#396—216no—
136#395—28no—
137#395—44no—
138#393—28no—
139#391—28no—
140#389—24no—
141#389—28no—
142#389—24no—
143#389—28no—
144#387—28no—
145#387—28no—
146#387—216no—
147#384—28no—
148#381—28no—
149#380—24no—
150#380—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—216no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4D418yes—
159C4×C22⋊C4—216no—
160C4×C22⋊C4—24no—
161C4×C22⋊C4—28no—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C25C2411yescharacteristic
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—28no—
196C22×Q8—24no—
197C22×Q8—44no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4C2×Q814yes—
209C23×C4C2×Q818yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C4⋊Q8—232no—
227C4×Q8—432no—
228#117—432no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×Q818yes—
276C2×C4⋊C4C2×Q814yes—
277C2×C4⋊C4C2×Q814yes—
278C2×C4⋊C4#4118yes—
279C2×C4⋊C4#4114yes—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D418yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—28no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—42no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42C2×D418yes—
318C2×C42C2×D414yes—
319C2×C42#4118yes—
320#95—22no—
321#95—22no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—22no—
342#95—24no—
343#95—42no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95C2×D414yes—
349#95C2×D418yes—
350C24—21no—
351C24—22no—
352C24—22no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24C22×D412yes—
359C24C22×Q812yes—
360C242−1+412yes—
361C24C2511yescharacteristic
362C2×Q8—216no—
363C2×Q8—416no—
364C2×Q8—416no—
365C2×Q8—48no—
366C2×Q8—48no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—216no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
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—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—216no—
419C22×C4—28no—
420C22×C4—216no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×Q812yes—
463C22×C4C22×Q814yes—
464C22×C4C22×Q814yes—
465C22×C4C22×Q812yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4118yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4114yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+414yes—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—216no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—416no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—416no—
491C4⋊C4—88no—
492C4⋊C4C22⋊Q8116yes—
493C22⋊C4—28no—
494C22⋊C4—216no—
495C22⋊C4—416no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C42—28no—
500C42—216no—
501C42—216no—
502C42—416no—
503C42—48no—
504C42—416no—
505C42—44no—
506C42—48no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—24no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23D4×Q814yes—
530C23C23×D411yescharacteristic
531C23C23×D411yescharacteristic
532C23C23×Q811yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4112yes—
535C23C22×#4112yes—
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×2−1+412yes—
538C23C2×2−1+411yescharacteristic
539C23#58512yes—
540C23#58511yescharacteristic
541Q8—432no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—432no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—416no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
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×C4C2×C22⋊Q818yes—
633C2×C4C2×C22⋊Q814yes—
634C2×C4#52514yes—
635C2×C4#52918yes—
636C2×C4#53214yes—
637C2×C4#53318yes—
638C2×C4D4214yes—
639C2×C4#54718yes—
640C22—28no—
641C22—24no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—42no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22C22×C22⋊Q811yescharacteristic
652C22C2×#52511yescharacteristic
653C22C2×#52912yes—
654C22C2×#53211yescharacteristic
655C22C2×#53312yes—
656C22C2×D4211yescharacteristic
657C22C2×#54712yes—
658C22C2×D4×Q811yescharacteristic
659C22D4×#4112yes—
660C22D4×#4111yescharacteristic
661C22#348711yescharacteristic
662C22#348911yescharacteristic
663C22#349312yes—
664C22#349512yes—
665C22#349911yescharacteristic
666C22#350111yescharacteristic
667C22#350212yes—
668C22#351112yes—
669C22#351312yes—
670C22#351612yes—
671C22#351612yes—
672C22#352011yescharacteristic
673C22#352312yes—
674C4—24no—
675C4—28no—
676C4—28no—
677C4—48no—
678C4—44no—
679C4—42no—
680C4—44no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—82no—
686C4—84no—
687C4—82no—
688C4—81no—
689C4#239618yescomplemented
690C2—44no—
691C2C2×#239611yescharacteristic
692C2#3584312yes—
693C2#3584511yescharacteristic
694C2#3599312yes—
695C2#3599711yescharacteristic
696C2#3606511yescharacteristic
697C2D4×C22⋊Q811yescharacteristic
698C2#3610211yescharacteristic
699C2#3611812yes—
700C2#3612711yescharacteristic
701C2#3613312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2365C2semidirect
4. #15156C2semidirect
5. #15149C2semidirect
6. #15143C2semidirect
10. #14876C2semidirect
11. #14872C2semidirect
12. #14794C2semidirect
15. #13821C2semidirect
16. C4⋊C42C2semidirect
99. C22×C4⋊C4Q8semidirect
689. C4#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#23962
C2×#23652
C2×#23442
#151562
#151492
#151432
#151312
#151252
#148872
#148762
#148722
#147942
#147732
C4×#3992
#138212
C4⋊C422
C22⋊C4×C4⋊C42