Finite Group Databasebeta

← #615772

Subgroups — #615772

Order 512.

2334 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2507C212yesmaximal
2#15137C212yesmaximal
3#15137C212yesmaximal
4#15068C214yesmaximal
5#15008C214yesmaximal
6#14712C211yesmaximal, characteristic
7#14712C212yesmaximal
8#14699C212yesmaximal
9#14151C211yesmaximal, characteristic
10#14045C212yesmaximal
11#14005C212yesmaximal
12#13933C212yesmaximal
13#13910C211yesmaximal, characteristic
14#13910C211yesmaximal, characteristic
15#13910C211yesmaximal, characteristic
16C4×#389C211yesmaximal, characteristic
17C42×C4⋊C4C211yesmaximal, characteristic
18#2507—216no—
19C2×#398C2214yes—
20C2×#398C2214yes—
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2214yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#389C2211yescharacteristic
36C2×#387C2214yes—
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×#384C2211yescharacteristic
40C2×#384C2212yes—
41C2×#384C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C43C2211yescharacteristic
52C2×C43C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66#1436C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2214yes—
79#1434C2214yes—
80C4×#95C2212yes—
81C4×#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123#398—216no—
124#398—216no—
125#389—28no—
126#389—28no—
127#389—28no—
128#389—24no—
129#389—28no—
130#389—28no—
131#389—216no—
132#389—28no—
133#389—44no—
134#389—48no—
135#389—44no—
136#389—44no—
137#389—44no—
138#389—44no—
139#389Q818yes—
140#387—216no—
141#384—28no—
142#384—28no—
143#384—24no—
144#384—24no—
145#384—44no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4D418yes—
153C4×C4⋊C4Q818yes—
154C4×C4⋊C4Q818yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C43—24no—
182C43—44no—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
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⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C4⋊C4C2×Q814yes—
264C2×C4⋊C4C2×Q818yes—
265C2×C4⋊C4C2×Q814yes—
266C2×C4⋊C4C2×Q814yes—
267C2×C4⋊C4C2×Q814yes—
268C2×C4⋊C4C2×Q814yes—
269C2×C4⋊C4#4114yes—
270C2×C4⋊C4#4114yes—
271C2×C4⋊C4#4114yes—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—41no—
307C2×C42—41no—
308C2×C42C2×D414yes—
309C2×C42C2×D418yes—
310C2×C42C2×Q814yes—
311C2×C42C2×Q818yes—
312C2×C42C2×Q814yes—
313C2×C42C2×Q818yes—
314C2×C42C2×Q814yes—
315C2×C42#4114yes—
316C2×C42#4114yes—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—28no—
333#95—44no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95C2×D418yes—
340#95C2×Q818yes—
341#95C2×Q818yes—
342#95C2×Q818yes—
343C24C2511yescharacteristic
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4C22×D412yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×Q812yes—
433C22×C4C22×Q814yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q814yes—
436C22×C4C22×Q812yes—
437C22×C4C22×Q814yes—
438C22×C4C22×Q814yes—
439C22×C4C22×Q814yes—
440C22×C4C22×Q812yes—
441C22×C4C22×Q812yes—
442C22×C4C22×Q812yes—
443C22×C4C22×Q812yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C22×Q812yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—84no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C4⋊C4—88no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C4⋊C4#125116yes—
486C4⋊C4C4⋊Q8116yes—
487C42—28no—
488C42—28no—
489C42—216no—
490C42—28no—
491C42—48no—
492C42—48no—
493C42—44no—
494C42—44no—
495C42—48no—
496C42—48no—
497C42—48no—
498C42—48no—
499C42—48no—
500C42—44no—
501C42#125116yes—
502C23C23×D411yescharacteristic
503C23C23×Q811yescharacteristic
504C23C23×Q811yescharacteristic
505C23C23×Q811yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4111yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4111yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×2−1+411yescharacteristic
515C23#58511yescharacteristic
516C23#58511yescharacteristic
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4C2×#12514yes—
618C2×C4C2×#12518yes—
619C2×C4C2×#12514yes—
620C2×C4C2×#12514yes—
621C2×C4C2×#12514yes—
622C2×C4C2×C4⋊Q818yes—
623C2×C4C2×C4⋊Q814yes—
624C2×C4#53214yes—
625C2×C4#53214yes—
626C2×C4#53314yes—
627C2×C4#53314yes—
628C2×C4#53314yes—
629C2×C4#54014yes—
630C2×C4D4×Q814yes—
631C2×C4#55314yes—
632C2×C4#55414yes—
633C2×C4#55414yes—
634C2×C4Q8214yes—
635C2×C4Q8214yes—
636C22C22×#12511yescharacteristic
637C22C22×#12511yescharacteristic
638C22C22×C4⋊Q811yescharacteristic
639C22C2×#53211yescharacteristic
640C22C2×#53211yescharacteristic
641C22C2×#53311yescharacteristic
642C22C2×#53311yescharacteristic
643C22C2×#53311yescharacteristic
644C22C2×#54011yescharacteristic
645C22#346011yescharacteristic
646C22#346111yescharacteristic
647C22C2×D4×Q811yescharacteristic
648C22D4×#4111yescharacteristic
649C22C2×#55311yescharacteristic
650C22C2×#55411yescharacteristic
651C22C2×#55411yescharacteristic
652C22C2×Q8211yescharacteristic
653C22C2×Q8211yescharacteristic
654C22Q8×#4111yescharacteristic
655C22Q8×#4111yescharacteristic
656C22Q8×#4111yescharacteristic
657C22#348111yescharacteristic
658C22#348211yescharacteristic
659C22#349411yescharacteristic
660C22#349511yescharacteristic
661C22#349611yescharacteristic
662C22#349811yescharacteristic
663C22#350111yescharacteristic
664C22#350311yescharacteristic
665C22#350311yescharacteristic
666C22#350411yescharacteristic
667C22#351711yescharacteristic
668C22#351711yescharacteristic
669C22#351911yescharacteristic
670C22#352011yescharacteristic
671C4—28no—
672C4—24no—
673C4—28no—
674C4—24no—
675C4—24no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—42no—
680C4—44no—
681C4—42no—
682C4—44no—
683C4—82no—
684C4—84no—
685C4—84no—
686C4#250718yescomplemented
687C4#285918yes—
688C2C2×#250711yescharacteristic
689C2#3492911yescharacteristic
690C2#3495011yescharacteristic
691C2#3517111yescharacteristic
692C2#3517611yescharacteristic
693C2#3526511yescharacteristic
694C2#3626111yescharacteristic
695C2#3629011yescharacteristic
696C2#3647211yescharacteristic
697C2#3648111yescharacteristic
698C2Q8×#12511yescharacteristic
699C2Q8×C4⋊Q811yescharacteristic
700C2#3656111yescharacteristic
701C2#3656411yescharacteristic
702C2C2×#285911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
686. C4#2507semidirect

Maximal subgroups

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

subgroupindex
C2×#25072
#151372
#150682
#150082
#147122
#146992
#141512
#140452
#140052
#139332
#139102
C4×#3892
C42×C4⋊C42