Finite Group Databasebeta

← #532280

Subgroups — #532280

Order 512.

3054 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2374C212yesmaximal
2C2×C4×#123C211yesmaximal, characteristic
3#14514C212yesmaximal, complemented
4#14513C212yesmaximal
5#14165C212yesmaximal, complemented
6#14114C214yesmaximal, complemented
7#14113C214yesmaximal, complemented
8#13906C211yesmaximal, characteristic
9#13884C212yesmaximal, complemented
10#13876C212yesmaximal, complemented
11#13876C212yesmaximal, complemented
12#13872C211yesmaximal, characteristic
13C4⋊C42C212yesmaximal, complemented
14C22⋊C4×C4⋊C4C212yesmaximal, complemented
15C4×#380C212yesmaximal, complemented
16C22×#123C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2374—216no—
21C4×#123C4116yescomplemented
22C2×#390C2212yes—
23C2×#388C2214yes—
24C2×#388C2214yes—
25C2×#388C2212yes—
26C2×#387C2214yes—
27C2×#387C2212yes—
28C2×#386C2212yes—
29C2×#386C2211yescharacteristic
30C2×#385C2211yescharacteristic
31C2×#384C2212yes—
32C2×#384C2211yescharacteristic
33C2×#381C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×C4×C4⋊C4C2214yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2214yes—
45C2×C4×C22⋊C4C2214yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2214yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#123C2×C418yes—
84C2×#123D418yes—
85C2×#123Q818yes—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4C2×C418yes—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C42—42no—
114C22×C42—42no—
115C22×C42C2×C418yescomplemented
116C22×C42C2314yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2314yes—
121C22×C42C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124#390—28no—
125#388—216no—
126#388—216no—
127#388C2×C4116yes—
128#387—216no—
129#387C2×C4116yescomplemented
130#386—28no—
131#386C2×C418yes—
132#385C2×C418yescomplemented
133#384—28no—
134#384—42no—
135#381C2×C418yescomplemented
136#380—28no—
137#380—28no—
138#380—28no—
139#380—42no—
140C4×C4⋊C4—216no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—42no—
143C4×C4⋊C4—44no—
144C4×C4⋊C4—44no—
145C4×C4⋊C4C2×C4116yescomplemented
146C4×C22⋊C4—216no—
147C4×C22⋊C4—216no—
148C4×C22⋊C4C2×C4116yes—
149C4×C22⋊C4C2×C418yescomplemented
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4C22×C414yes—
180C22×D4C2×D414yes—
181C22×D4C2×Q814yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—41no—
193C23×C4—41no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—42no—
203C23×C4C22×C414yes—
204C23×C4C22×C414yes—
205C23×C4C22×C414yes—
206C23×C4C22×C414yes—
207C23×C4C2×D414yes—
208C23×C4C2×Q814yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#123C4⋊C4164yescomplemented
223C4×D4—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
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—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4C22×C414yes—
253C2×C4⋊C4C22×C418yes—
254C2×C4⋊C4C22×C418yes—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4C2×Q818yes—
257C2×C4⋊C4#4118yes—
258C2×C4⋊C4#4118yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4C22×C414yes—
280C2×C22⋊C4C22×C414yes—
281C2×C22⋊C4C22×C418yes—
282C2×C22⋊C4C22×C414yes—
283C2×C22⋊C4C22×C418yes—
284C2×C22⋊C4C22×C414yes—
285C2×C22⋊C4C2×D418yes—
286C2×C22⋊C4C2×D414yes—
287C2×C22⋊C4C2×Q818yes—
288C2×C22⋊C4C2×Q814yes—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—24no—
300C2×C42—416no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—42no—
310C2×C42C22×C414yes—
311C2×C42C22×C418yes—
312C2×C42C22×C418yes—
313C2×C42C22×C418yes—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—42no—
331#95—42no—
332#95C22×C418yes—
333#95C22×C418yes—
334#95C22×C418yes—
335#95C22×C418yes—
336#95C22×C418yes—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—21no—
349C24—22no—
350C24—21no—
351C24—21no—
352C24—21no—
353C24C23×C412yes—
354C24C23×C412yes—
355C24C22×D412yes—
356C24C22×D412yes—
357C24C22×Q812yes—
358C24C22×Q812yes—
359C24C2511yescharacteristic
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4C2×C4⋊C4116yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—24no—
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—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—216no—
416C22×C4—216no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—88no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4C2×C4⋊C4116yescomplemented
456C22×C4C23×C414yes—
457C22×C4C23×C412yes—
458C22×C4C23×C412yes—
459C22×C4C23×C414yes—
460C22×C4C23×C412yes—
461C22×C4C23×C414yes—
462C22×C4C23×C414yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×Q814yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q812yes—
471C22×C4C2×#4118yes—
472C22×C4C2×#4118yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—416no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—84no—
486C4⋊C4—88no—
487C4⋊C4—84no—
488C4⋊C4C2×C4⋊C4132yescomplemented
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—84no—
497C22⋊C4C2×C4⋊C4132yes—
498C22⋊C4C2×C4⋊C4116yescomplemented
499C42—216no—
500C42—216no—
501C42—28no—
502C42—48no—
503C42—416no—
504C42—88no—
505C42—88no—
506C42—84no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C22×C4⋊C414yes—
552C23C22×C4⋊C414yes—
553C23C24×C411yescharacteristic
554C23C23×D411yescharacteristic
555C23C23×Q811yescharacteristic
556C23C22×#4112yes—
557C23C22×#4112yes—
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23#58512yes—
561C23#58512yes—
562C23#58511yescharacteristic
563C23#58511yescharacteristic
564D4—416no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—416no—
629C2×C4—416no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—816no—
645C2×C4C22×C4⋊C418yes—
646C2×C4C22×C4⋊C418yes—
647C2×C4C22×C4⋊C414yes—
648C2×C4C4×#4118yes—
649C2×C4#51814yes—
650C2×C4#55114yes—
651C2×C4#55318yes—
652C2×C4#55914yes—
653C22—28no—
654C22—28no—
655C22—24no—
656C22—24no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—44no—
666C22—44no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22C23×C4⋊C411yescharacteristic
676C22C2×C4×#4112yes—
677C22C2×#51811yescharacteristic
678C22#342812yes—
679C22#342811yescharacteristic
680C22D4×#4112yes—
681C22D4×#4112yes—
682C22C2×#55111yescharacteristic
683C22C2×#55312yes—
684C22Q8×#4112yes—
685C22Q8×#4112yes—
686C22C2×#55911yescharacteristic
687C22#349611yescharacteristic
688C22#349611yescharacteristic
689C22#350111yescharacteristic
690C22#350111yescharacteristic
691C22#352112yes—
692C22#352112yes—
693C22#352412yes—
694C22#352412yes—
695C22#353912yes—
696C22#354011yescharacteristic
697C22#354011yescharacteristic
698C4—28no—
699C4—28no—
700C4—24no—
701C4—48no—
702C4—42no—
703C4—44no—
704C4—42no—
705C4—42no—
706C4—44no—
707C4—48no—
708C4—84no—
709C4—84no—
710C4#237418yescomplemented
711C2—44no—
712C2—44no—
713C2C4⋊C4×#4112yes—
714C2C2×#237411yescharacteristic
715C2#3418111yescharacteristic
716C2#3424412yes—
717C2#3436111yescharacteristic
718C2#5937511yescharacteristic
719C2#5937611yescharacteristic
720C2#5939212yes—
721C2#5939312yes—
722C2#5962811yescharacteristic
723C2#5962911yescharacteristic
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
3. #14514C2semidirect
5. #14165C2semidirect
6. #14114C2semidirect
7. #14113C2semidirect
9. #13884C2semidirect
10. #13876C2semidirect
11. #13876C2semidirect
13. C4⋊C42C2semidirect
14. C22⋊C4×C4⋊C4C2semidirect
15. C4×#380C2semidirect
21. C4×#123C4semidirect
115. C22×C42C2×C4semidirect
129. #387C2×C4semidirect
132. #385C2×C4semidirect
135. #381C2×C4semidirect
145. C4×C4⋊C4C2×C4semidirect
149. C4×C22⋊C4C2×C4semidirect
222. #123C4⋊C4semidirect
455. C22×C4C2×C4⋊C4semidirect
488. C4⋊C4C2×C4⋊C4semidirect
498. C22⋊C4C2×C4⋊C4semidirect
710. C4#2374semidirect

Maximal subgroups

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

subgroupindex
C2×#23742
C2×C4×#1232
#145142
#145132
#141652
#141142
#141132
#139062
#138842
#138762
#138722
C4⋊C422
C22⋊C4×C4⋊C42
C4×#3802