Finite Group Databasebeta

← #636392

Subgroups — #636392

Order 512.

3102 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3C2×#2621C212yesmaximal
4C2×C4×C22⋊Q8C211yesmaximal, characteristic
5C2×#2285C211yesmaximal, characteristic
6#16517C214yesmaximal, complemented
7#16399C214yesmaximal
8#15237C212yesmaximal, complemented
9#15180C212yesmaximal, complemented
10#15133C214yesmaximal, complemented
11#15132C214yesmaximal
12#14258C212yesmaximal
13#14155C211yesmaximal, characteristic, complemented
14#13926C211yesmaximal, characteristic, complemented
15#13911C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#117C2212yes—
22C23×C42C2211yescharacteristic
23#2857—28no—
24#2621—216no—
25#2546—28no—
26C2×#398C2214yes—
27C2×#398C2218yes—
28C2×#397C2218yes—
29C4×C22⋊Q8—28no—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2214yes—
39C2×#389C2211yescharacteristic
40C2×#387C2212yes—
41C2×#385C2212yes—
42#2285—28no—
43C2×#384C2211yescharacteristic
44C2×#384C2212yes—
45C2×#382C2212yes—
46C2×#382C2212yes—
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2212yes—
58#1445C2214yes—
59#1443C2218yes—
60#1442C2214yes—
61#1442C2218yes—
62#1441C2218yes—
63#1441C2218yes—
64#1441C2214yes—
65#1441C2218yes—
66#1440C2218yes—
67C4⋊#95C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C23×Q8C2312yes—
73C23×Q8C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—28no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—44no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8D418yes—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×#117—28no—
93C2×#117—28no—
94C2×#117—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2318yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C42—24no—
117C22×C42—24no—
118C22×C42D418yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#398—232no—
128#398—216no—
129#397—232no—
130#391—28no—
131#391—28no—
132#391—28no—
133#391—28no—
134#391—24no—
135#391—44no—
136#389—216no—
137#389—28no—
138#389—28no—
139#389—24no—
140#387—28no—
141#385—28no—
142#384—28no—
143#384—24no—
144#382—28no—
145#382—28no—
146#380—28no—
147#380—28no—
148#380—24no—
149#380—28no—
150#380—28no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4D418yes—
156C4×C22⋊C4—28no—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2312yes—
174C25C2411yescharacteristic
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8C2×D414yes—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—42no—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4#4114yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C4⋊Q8—432no—
221C4⋊Q8—416no—
222C22⋊Q8—232no—
223C22⋊Q8—432no—
224C4×Q8—416no—
225#117—432no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4C2×D418yes—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—232no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—28no—
302C2×C42C2×D418yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42#4114yes—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—216no—
319#95—216no—
320#95—216no—
321#95—216no—
322#95—48no—
323#95—48no—
324#95—48no—
325#95C2×D418yes—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—21no—
333C24—22no—
334C24C22×D412yes—
335C24C22×D412yes—
336C24C2×#4112yes—
337C24C2511yescharacteristic
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—44no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—44no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—44no—
349C2×Q8—48no—
350C2×Q8—44no—
351C2×Q8—48no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4#121116yes—
427C22×C4C22×D412yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D418yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+414yes—
448C22×C42−1+414yes—
449C22×C42−1+414yes—
450C22×C42−1+412yes—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—416no—
455C4⋊C4—416no—
456C4⋊C4—432no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C22⋊C4—216no—
469C22⋊C4—416no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C42—28no—
473C42—28no—
474C42—28no—
475C42—216no—
476C42—48no—
477C42—48no—
478C42—48no—
479C42#121116yes—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23C2×#12114yes—
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C23×D411yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4111yescharacteristic
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×2−1+412yes—
510C23C2×2−1+412yes—
511C23C2×2−1+412yes—
512C23C2×2−1+411yescharacteristic
513C23#58512yes—
514Q8—416no—
515Q8—816no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—232no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—416no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—416no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4C2×#12114yes—
587C2×C4C2×#12114yes—
588C2×C4C2×#12114yes—
589C2×C4C2×#12114yes—
590C2×C4C2×#12114yes—
591C2×C4#53614yes—
592C2×C4#53614yes—
593C2×C4#53618yes—
594C2×C4#53614yes—
595C2×C4#53714yes—
596C2×C4#54118yes—
597C2×C4D4214yes—
598C2×C4#54714yes—
599C2×C4#54814yes—
600C2×C4#55314yes—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22C22×#12111yescharacteristic
613C22C22×#12111yescharacteristic
614C22C2×#53611yescharacteristic
615C22C2×#53611yescharacteristic
616C22C2×#53611yescharacteristic
617C22C2×#53612yes—
618C22C2×#53711yescharacteristic
619C22#344911yescharacteristic
620C22C2×#54112yes—
621C22C2×D4211yescharacteristic
622C22C2×#54711yescharacteristic
623C22C2×#54811yescharacteristic
624C22C2×#55311yescharacteristic
625C22#348912yes—
626C22#349512yes—
627C22#349712yes—
628C22#349912yes—
629C22#350712yes—
630C22#350912yes—
631C22#354712yes—
632C22#355012yes—
633C22#355512yes—
634C22#355612yes—
635C4—24no—
636C4—24no—
637C4—24no—
638C4—42no—
639C4—44no—
640C4—48no—
641C4—44no—
642C4—82no—
643C4—84no—
644C4—84no—
645C4—84no—
646C4—82no—
647C4#241618yes—
648C4#262118yescomplemented
649C2—44no—
650C2C2×#262111yescharacteristic
651C2C2×#241611yescharacteristic
652C2#3517811yescharacteristic
653C2#3530611yescharacteristic
654C2#3531611yescharacteristic
655C2#3763311yescharacteristic
656C2#3764111yescharacteristic
657C2#3934312yes—
658C2#3935312yes—
659C2#3972412yes—
660C2#3973412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2546C2semidirect
6. #16517C2semidirect
8. #15237C2semidirect
9. #15180C2semidirect
10. #15133C2semidirect
13. #14155C2semidirect
14. #13926C2semidirect
15. #13911C2semidirect
648. C4#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#25462
C2×#26212
C2×C4×C22⋊Q82
C2×#22852
#165172
#163992
#152372
#151802
#151332
#151322
#142582
#141552
#139262
#139112