Finite Group Databasebeta

← #518010

Subgroups — #518010

Order 512.

2678 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#2370C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic
4#14327C212yesmaximal
5#14241C212yesmaximal
6#14237C214yesmaximal
7#14211C214yesmaximal
8#14186C212yesmaximal
9#14161C211yesmaximal, characteristic
10#14160C211yesmaximal, characteristic
11#14151C212yesmaximal
12#14150C212yesmaximal
13C4×#395C212yesmaximal
14#13856C211yesmaximal, characteristic
15#13804C212yesmaximal
16C4⋊C42C212yesmaximal
17#13750C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#2660C4116yescomplemented
22C2×#398C2214yes—
23C2×#395C2212yes—
24C2×#395C2212yes—
25#2370—28no—
26#2365—28no—
27C2×#391C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2214yes—
33C2×#387C2214yes—
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#384C2212yes—
37C2×#382C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2214yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×#376C2211yescharacteristic
52#1446C2212yes—
53#1446C2212yes—
54#1444C2211yescharacteristic
55#1444C2211yescharacteristic
56#1444C2212yes—
57#1444C2212yes—
58#1444C2211yescharacteristic
59#1444C2211yescharacteristic
60#1444C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2212yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2214yes—
81#1434C2214yes—
82C4×#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2214yes—
86C23×Q8C2312yes—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8C2×C418yes—
90C2×#125—24no—
91C2×#125—24no—
92C2×#125C2×C418yes—
93C2×C4×Q8—28no—
94C2×C4×Q8—48no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129#398C2×C4132yes—
130#395C2×C4116yes—
131#395C2×C4116yes—
132#391—28no—
133#389—28no—
134#389—24no—
135#389—24no—
136#389—28no—
137#389—48no—
138#387—28no—
139#387—216no—
140#387—28no—
141#384C2×C4116yes—
142#382C2×C4116yes—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—48no—
154C4×C4⋊C4D418yes—
155C4×C4⋊C4Q8116yescomplemented
156#376C2×C418yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—44no—
190C22×Q8—44no—
191C22×Q8C22×C418yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C4⋊Q8—232no—
210#125—232no—
211C4×Q8—432no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
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—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C22×C414yes—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C22×C418yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C418yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C4116yes—
268C2×C4⋊C4C22×C4116yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×Q818yes—
272C2×C4⋊C4C2×Q818yes—
273C2×C42—28no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42C22×C414yes—
298C2×C42C22×C418yes—
299C2×C42C2×D418yes—
300C2×C42C2×D414yes—
301C2×C42C2×Q8116yes—
302C2×C42C2×Q818yes—
303C2×C42#4114yes—
304C2×C42#4118yes—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95C22×C414yes—
329#95C22×C414yes—
330#95C22×C418yes—
331#95C22×C4116yes—
332#95C22×C4116yes—
333#95C2×D418yes—
334#95C2×Q8116yes—
335C24C2511yescharacteristic
336C2×Q8—216no—
337C2×Q8—416no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—416no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4C23×C414yes—
416C22×C4C23×C418yes—
417C22×C4C23×C414yes—
418C22×C4C23×C412yes—
419C22×C4C23×C412yes—
420C22×C4C23×C414yes—
421C22×C4C23×C414yes—
422C22×C4C23×C412yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D414yes—
427C22×C4C22×D412yes—
428C22×C4C22×Q818yes—
429C22×C4C22×Q814yes—
430C22×C4C22×Q814yes—
431C22×C4C22×Q818yes—
432C22×C4C22×Q814yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4114yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C42−1+412yes—
441C22×C42−1+412yes—
442C4⋊C4—216no—
443C4⋊C4—28no—
444C4⋊C4—216no—
445C4⋊C4—216no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—28no—
451C4⋊C4—216no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—416no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—84no—
463C4⋊C4C4⋊Q8116yes—
464C42—48no—
465C42—416no—
466C42—48no—
467C42—416no—
468C42—48no—
469C42—44no—
470C42—44no—
471C42—84no—
472C42—88no—
473C42—84no—
474C42C4×D4116yes—
475C42C4×Q8132yescomplemented
476C23C24×C411yescharacteristic
477C23C23×D411yescharacteristic
478C23C23×Q812yes—
479C23C22×#4111yescharacteristic
480C23C22×#4112yes—
481C23C2×H(5,𝔽2)12yes—
482C23C2×2−1+411yescharacteristic
483C23C2×2−1+411yescharacteristic
484C23#58512yes—
485C23#58511yescharacteristic
486C23#58511yescharacteristic
487Q8—432no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4C2×C4×D414yes—
580C2×C4C2×C4×D414yes—
581C2×C4C2×C4×D414yes—
582C2×C4C2×C4×Q818yes—
583C2×C4C2×C4×Q818yes—
584C2×C4C2×C4×Q818yes—
585C2×C4C2×C4⋊Q818yes—
586C2×C4C2×C4⋊Q814yes—
587C2×C4#53214yes—
588C2×C4#53318yes—
589C22C22×C4×D411yescharacteristic
590C22C22×C4×Q812yes—
591C22C4×H(5,𝔽2)12yes—
592C22C4×2−1+411yescharacteristic
593C22C4×2−1+411yescharacteristic
594C22C22×C4⋊Q811yescharacteristic
595C22C2×#53211yescharacteristic
596C22C2×#53312yes—
597C22#348411yescharacteristic
598C22#348711yescharacteristic
599C22#348911yescharacteristic
600C22#348911yescharacteristic
601C22#349212yes—
602C22#349312yes—
603C22#349412yes—
604C22#349512yes—
605C22#350112yes—
606C22#350212yes—
607C22#350312yes—
608C22#350412yes—
609C22#350612yes—
610C22#350911yescharacteristic
611C22#351011yescharacteristic
612C4—28no—
613C4—24no—
614C4—24no—
615C4—28no—
616C4—28no—
617C4—44no—
618C4—44no—
619C4—42no—
620C4—48no—
621C4—44no—
622C4—48no—
623C4—44no—
624C4—42no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4C4×C4⋊Q818yes—
629C2C2×C4×C4⋊Q811yescharacteristic
630C2#3426511yescharacteristic
631C2#3426811yescharacteristic
632C2#3427112yes—
633C2#3427212yes—
634C2#3592311yescharacteristic
635C2#3592811yescharacteristic
636C2#3593612yes—
637C2#3595212yes—
638C2#3595711yescharacteristic
639C2#3606511yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #2660C4semidirect
155. C4×C4⋊C4Q8semidirect
475. C42C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
C2×#23702
C2×#23652
#143272
#142412
#142372
#142112
#141862
#141612
#141602
#141512
#141502
C4×#3952
#138562
#138042
C4⋊C422
#137502