Finite Group Databasebeta

← #4306578

Subgroups — #4306578

Order 512.

3342 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#15197C211yesmaximal, characteristic, complemented
4#15187C212yesmaximal, complemented
5#15156C211yesmaximal, characteristic, complemented
6#15152C211yesmaximal, characteristic, complemented
7#15146C212yesmaximal, complemented
8#14843C212yesmaximal, complemented
9#14826C212yesmaximal, complemented
10#14822C212yesmaximal, complemented
11#14767C214yesmaximal, complemented
12#14738C214yesmaximal, complemented
13C4×#400C211yesmaximal, characteristic, complemented
14C2×#1441C212yesmaximal, complemented
15D4×#95C212yesmaximal
16D4×#95C212yesmaximal
17D4×#95C211yesmaximal, characteristic
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C23×C4⋊C4C2212yes—
25C2×#400C2211yescharacteristic
26C2×#400C2214yes—
27C2×#400C2212yes—
28C2×#400C2212yes—
29D4×C4⋊C4—28no—
30D4×C4⋊C4—28no—
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#385C2212yes—
35C2×#385C2214yes—
36C2×#385C2212yes—
37C2×#384C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C22×#95C2214yes—
44C22×#95C2214yes—
45C22×#95C2212yes—
46#1445C2211yescharacteristic
47#1445C2212yes—
48#1445C2214yes—
49#1443C2212yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2212yes—
53#1442C2211yescharacteristic
54#1442C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2211yescharacteristic
60#1441C2214yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C22⋊#95C2214yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C22⋊#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2314yes—
88C24×C4C2314yes—
89C24×C4C2312yes—
90C24×C4C2312yes—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—28no—
94C2×C4×D4—24no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C2×C4×D4—28no—
102C2×C4×D4D418yes—
103C2×C4×D4Q818yes—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—44no—
108C22×C4⋊C4—48no—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#400—24no—
136#400—44no—
137#400—48no—
138#400—44no—
139#389—24no—
140#389—28no—
141#389—24no—
142#385—28no—
143#385—28no—
144#385—216no—
145#384—24no—
146#384—24no—
147#380—28no—
148#380—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C2×#95—44no—
152C2×#95—48no—
153C2×#95—48no—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25C2412yes—
188C22×D4C2×D414yes—
189C22×D4C2×Q814yes—
190C22×D4#4114yes—
191C22×D4#4118yes—
192C22×D4#4118yes—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2×D418yes—
223C23×C4C2×Q818yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C4×D4—232no—
241C4×D4—232no—
242C4×D4—416no—
243C4×D4—432no—
244C4×D4—432no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—432no—
278C2×C4⋊C4—432no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×Q814yes—
289C2×C4⋊C4#4114yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4C2×D418yes—
302C2×C22⋊C4C2×Q818yes—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42C2×D414yes—
315C2×C42C2×Q814yes—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—42no—
342#95—44no—
343#95—42no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24C22×D414yes—
356C24C22×Q814yes—
357C24C2×#4114yes—
358C24C2×#4118yes—
359C24C2×#4118yes—
360C24C2511yescharacteristic
361C2×D4C22⋊Q8116yes—
362C2×D4#123116yes—
363C2×D4#124116yes—
364C2×D4#125132yes—
365C2×D4#126132yes—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—416no—
453C22×C4—48no—
454C22×C4—416no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—84no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4C22×D412yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×Q812yes—
480C22×C4C22×Q814yes—
481C22×C4C22×Q812yes—
482C22×C4C22×Q812yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4114yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C42−1+414yes—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C22⋊C4—216no—
510C22⋊C4—216no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C42—28no—
515C42—28no—
516C42—44no—
517C42—48no—
518C42—48no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23C2×C22⋊Q818yes—
538C23C2×#12318yes—
539C23C2×#12418yes—
540C23C2×#125116yes—
541C23C2×#126116yes—
542C23C23×D411yescharacteristic
543C23C23×Q811yescharacteristic
544C23C22×#4111yescharacteristic
545C23C22×#4112yes—
546C23C22×#4112yes—
547C23C22×#4111yescharacteristic
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×H(5,𝔽2)12yes—
551C23C2×2−1+412yes—
552C23#58511yescharacteristic
553D4#400164yescomplemented
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—432no—
584C2×C4—432no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—832no—
646C2×C4—832no—
647C2×C4—816no—
648C2×C4C2×C22⋊Q814yes—
649C2×C4C2×#12314yes—
650C2×C4C2×#12414yes—
651C2×C4C2×#12518yes—
652C2×C4C2×#12618yes—
653C2×C4#54614yes—
654C2×C4#55414yes—
655C22—216no—
656C22—28no—
657C22—28no—
658C22—216no—
659C22—28no—
660C22—216no—
661C22—28no—
662C22—28no—
663C22—216no—
664C22—28no—
665C22C2×#400116yes—
666C22C22×C22⋊Q811yescharacteristic
667C22C22×#12311yescharacteristic
668C22C22×#12411yescharacteristic
669C22C22×#12512yes—
670C22C22×#12612yes—
671C22C2×#54611yescharacteristic
672C22C2×#55411yescharacteristic
673C22#347511yescharacteristic
674C22#348112yes—
675C22#348212yes—
676C22#348411yescharacteristic
677C22#348811yescharacteristic
678C22#349911yescharacteristic
679C22#350111yescharacteristic
680C22#350312yes—
681C22#351312yes—
682C22#351512yes—
683C22#351912yes—
684C22#351911yescharacteristic
685C22#351912yes—
686C22#352012yes—
687C22#352012yes—
688C22#352012yes—
689C4—24no—
690C4—24no—
691C4—44no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—84no—
699C4—84no—
700C4—82no—
701C4—82no—
702C4—81no—
703C4—82no—
704C4C2×#40018yes—
705C2—216no—
706C2C22×#40011yescharacteristic
707C2#3573911yescharacteristic
708C2#3585812yes—
709C2#3590311yescharacteristic
710C2#3591412yes—
711C2#3599811yescharacteristic
712C2#3615411yescharacteristic
713C2#3623611yescharacteristic
714C2#3624112yes—
715C2#3629012yes—
716C2#3630311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #15197C2semidirect
4. #15187C2semidirect
5. #15156C2semidirect
6. #15152C2semidirect
7. #15146C2semidirect
8. #14843C2semidirect
9. #14826C2semidirect
10. #14822C2semidirect
11. #14767C2semidirect
12. #14738C2semidirect
13. C4×#400C2semidirect
14. C2×#1441C2semidirect
553. D4#400semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
#151972
#151872
#151562
#151522
#151462
#148432
#148262
#148222
#147672
#147382
C4×#4002
C2×#14412
D4×#952