Finite Group Databasebeta

← #533183

Subgroups — #533183

Order 512.

3582 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2360C212yesmaximal, complemented
3C2×C4×C22⋊Q8C212yesmaximal, complemented
4#14551C212yesmaximal, complemented
5#14440C212yesmaximal
6#14215C212yesmaximal, complemented
7#14178C214yesmaximal, complemented
8#14029C214yesmaximal, complemented
9#14024C212yesmaximal, complemented
10#13997C212yesmaximal, complemented
11#13997C212yesmaximal, complemented
12#13872C211yesmaximal, characteristic
13#13824C212yesmaximal
14#13818C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2542C4116yescomplemented
24C2×#399C2212yes—
25C2×#399C2212yes—
26#2360—216no—
27C4×C22⋊Q8—216no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2212yes—
32C2×#389C2214yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#387C2214yes—
36C2×#387C2214yes—
37C2×#386C2212yes—
38C2×#386C2211yescharacteristic
39C2×#386C2211yescharacteristic
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#384C2212yes—
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2214yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2214yes—
50C2×C4×C22⋊C4C2214yes—
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53#1445C2214yes—
54#1445C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2212yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71C4⋊#95C2214yes—
72C4⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2211yescharacteristic
76C4×#95C2214yes—
77C4×#95C2212yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C2×C4⋊Q8—42no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8C2×C418yescomplemented
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8C2×C4116yescomplemented
93C2×C22⋊Q8C2×C4116yescomplemented
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—44no—
97C2×C4×D4—42no—
98C2×C4×D4—42no—
99C2×C4×D4C2×C418yes—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—42no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42—28no—
125C22×C42—42no—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2312yes—
134#399C2×C4116yescomplemented
135#399C2×C4116yescomplemented
136#391—28no—
137#391—28no—
138#391—44no—
139#390C2×C4116yes—
140#389—216no—
141#389—28no—
142#389D4116yescomplemented
143#387—216no—
144#387—216no—
145#386—24no—
146#386—28no—
147#386Q818yescomplemented
148#385—216no—
149#385—216no—
150#384C2×C4116yescomplemented
151#380—28no—
152#380—42no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—216no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4C2×C418yescomplemented
157C4×C22⋊C4—28no—
158C4×C22⋊C4—216no—
159C4×C22⋊C4—216no—
160C4×C22⋊C4—42no—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8C22×C418yes—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4C22×C414yes—
197C22×D4C2×Q814yes—
198C22×D4#4114yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—41no—
214C23×C4—42no—
215C23×C4—41no—
216C23×C4—41no—
217C23×C4—41no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—42no—
221C23×C4C22×C414yes—
222C23×C4C22×C414yes—
223C23×C4C2×Q814yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C4⋊Q8—416no—
238C22⋊Q8—264no—
239C22⋊Q8—264no—
240C4×Q8—432no—
241C4×D4—416no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C22×C4116yes—
269C2×C4⋊C4C22×C4116yes—
270C2×C4⋊C4C22×C414yes—
271C2×C4⋊C4C22×C418yes—
272C2×C4⋊C4C22×C418yes—
273C2×C4⋊C4C22×C418yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D4116yes—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4#4118yes—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4C22×C4116yes—
303C2×C22⋊C4C22×C4116yes—
304C2×C22⋊C4C22×C414yes—
305C2×C22⋊C4C22×C414yes—
306C2×C22⋊C4C2×Q818yes—
307C2×C22⋊C4C2×Q814yes—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—28no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—28no—
322C2×C42—416no—
323C2×C42—44no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42C22×C414yes—
331C2×C42C22×C418yes—
332C2×C42C2×D418yes—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—42no—
347#95—44no—
348#95—42no—
349#95—42no—
350#95—44no—
351#95—44no—
352#95C22×C418yes—
353#95C2×D4116yes—
354#95C2×Q818yes—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—21no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24C23×C412yes—
372C24C23×C412yes—
373C24C22×Q812yes—
374C24C22×Q812yes—
375C24C2×#4112yes—
376C24C2×#4112yes—
377C24C2511yescharacteristic
378C2×Q8—216no—
379C2×Q8—216no—
380C2×Q8—216no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4C4×Q8116yescomplemented
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—216no—
441C22×C4—216no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—216no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—416no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—88no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4C23×C412yes—
480C22×C4C23×C418yes—
481C22×C4C23×C412yes—
482C22×C4C23×C412yes—
483C22×C4C23×C418yes—
484C22×C4C23×C412yes—
485C22×C4C23×C412yes—
486C22×C4C22×D418yes—
487C22×C4C22×D414yes—
488C22×C4C22×D418yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×Q814yes—
492C22×C4C22×Q812yes—
493C22×C4C22×Q814yes—
494C22×C4C2×#4118yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4114yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C42−1+414yes—
500C4⋊C4—232no—
501C4⋊C4—232no—
502C4⋊C4—216no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—416no—
507C4⋊C4—48no—
508C4⋊C4—416no—
509C4⋊C4—48no—
510C4⋊C4—88no—
511C4⋊C4C4×D4132yes—
512C22⋊C4—232no—
513C22⋊C4—232no—
514C22⋊C4—216no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—416no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—84no—
522C42—28no—
523C42—216no—
524C42—48no—
525C42—416no—
526C42—48no—
527C42—416no—
528C42—48no—
529C42—88no—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23C2×C4×Q814yes—
575C23C2×C4×Q814yes—
576C23C24×C411yescharacteristic
577C23C23×D412yes—
578C23C23×Q811yescharacteristic
579C23C22×#4112yes—
580C23C22×#4111yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×2−1+412yes—
584C23#58511yescharacteristic
585C23#58511yescharacteristic
586C23#58512yes—
587Q8—432no—
588D4—416no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—28no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—416no—
626C2×C4—416no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—416no—
653C2×C4—416no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—816no—
669C2×C4C2×C4×D4116yes—
670C2×C4C2×C4×D418yes—
671C2×C4C2×C4×Q814yes—
672C2×C4#51814yes—
673C2×C4D4214yes—
674C2×C4D4×Q818yes—
675C2×C4#55018yes—
676C2×C4#55914yes—
677C22—28no—
678C22—28no—
679C22—24no—
680C22—24no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—44no—
698C22—44no—
699C22C22×C4×D412yes—
700C22C22×C4×Q811yescharacteristic
701C22C2×#51811yescharacteristic
702C22#342811yescharacteristic
703C22C4×2−1+412yes—
704C22C2×D4211yescharacteristic
705C22C2×D4×Q812yes—
706C22C2×#55012yes—
707C22D4×#4112yes—
708C22Q8×#4112yes—
709C22C2×#55911yescharacteristic
710C22#348212yes—
711C22#348512yes—
712C22#348912yes—
713C22#350111yescharacteristic
714C22#350111yescharacteristic
715C22#350511yescharacteristic
716C22#350511yescharacteristic
717C22#351512yes—
718C22#352112yes—
719C22#353811yescharacteristic
720C22#354011yescharacteristic
721C22#354712yes—
722C4—28no—
723C4—24no—
724C4—216no—
725C4—42no—
726C4—42no—
727C4—48no—
728C4—42no—
729C4—44no—
730C4—48no—
731C4—44no—
732C4—82no—
733C4—84no—
734C4#236618yes—
735C2—44no—
736C2—44no—
737C2C2×#236611yescharacteristic
738C2C4×D4×Q812yes—
739C2#3421211yescharacteristic
740C2#3428912yes—
741C2#3441211yescharacteristic
742C2#5899811yescharacteristic
743C2#5900011yescharacteristic
744C2#5905212yes—
745C2#5905412yes—
746C2#5962911yescharacteristic
747C2#5963111yescharacteristic
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
2. C2×#2360C2semidirect
3. C2×C4×C22⋊Q8C2semidirect
4. #14551C2semidirect
6. #14215C2semidirect
7. #14178C2semidirect
8. #14029C2semidirect
9. #14024C2semidirect
10. #13997C2semidirect
11. #13997C2semidirect
14. #13818C2semidirect
23. #2542C4semidirect
87. C2×C4⋊Q8C2×C4semidirect
92. C2×C22⋊Q8C2×C4semidirect
93. C2×C22⋊Q8C2×C4semidirect
134. #399C2×C4semidirect
135. #399C2×C4semidirect
142. #389D4semidirect
147. #386Q8semidirect
150. #384C2×C4semidirect
156. C4×C4⋊C4C2×C4semidirect
391. C2×D4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#23602
C2×C4×C22⋊Q82
#145512
#144402
#142152
#141782
#140292
#140242
#139972
#138722
#138242
#138182
C4×#3862