Finite Group Databasebeta

← #6468286

Subgroups — #6468286

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic
3#29220C211yesmaximal, characteristic, complemented
4#29207C211yesmaximal, characteristic, complemented
5#29193C211yesmaximal, characteristic, complemented
6#29191C211yesmaximal, characteristic, complemented
7#29179C211yesmaximal, characteristic, complemented
8#29177C211yesmaximal, characteristic, complemented
9#29164C214yesmaximal
10#29144C211yesmaximal, characteristic, complemented
11#28515C211yesmaximal, characteristic, complemented
12#24842C214yesmaximal
13#23785C214yesmaximal, complemented
14#21618C214yesmaximal, complemented
15#21409C211yesmaximal, characteristic
16#21409C211yesmaximal, characteristic
17#20824C211yesmaximal, characteristic
18#20824C211yesmaximal, characteristic
19C4×#423C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#479C2214yes—
22C2×#423C2212yes—
23C2×C4×M16C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2864—21no—
26#2864—21no—
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2851—21no—
32#2851—21no—
33#2851C2212yes—
34#2851C2212yes—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×#125—21no—
40C4×#125—21no—
41C4×#125—21no—
42C4×#125—21no—
43C4×#125C2212yes—
44C4×#125C2212yes—
45C4×C22⋊Q8—21no—
46C4×C22⋊Q8—21no—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2212yes—
49#2296—21no—
50#2296—21no—
51#2296—21no—
52#2295—21no—
53#2295C2212yes—
54#2285C2211yescharacteristic
55#2285C2211yescharacteristic
56#2285C2211yescharacteristic
57C42×Q8—21no—
58C42×Q8C2212yes—
59C4×#117—21no—
60C2×C43C2211yescharacteristic
61#2101C2214yes—
62#2097C2214yes—
63#2088C2214yes—
64#2087C2214yes—
65#2068C2214yes—
66#2066C2214yes—
67#2059C2214yes—
68#2026C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1989C2214yes—
74#1956C2211yescharacteristic
75#1956C2211yescharacteristic
76#1956C2211yescharacteristic
77#1956C2211yescharacteristic
78#1949C2214yes—
79#1921C2214yes—
80#1919C2214yes—
81#1894C2214yes—
82#1894C2214yes—
83#1865C2214yes—
84#1863C2214yes—
85#1845C2211yescharacteristic
86#1845C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1834C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1809C2214yes—
94#1809C2214yes—
95#1805C2214yes—
96#1776C2214yes—
97#1776C2214yes—
98#1767C2212yes—
99#1767C2212yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1763C2214yes—
105#1763C2214yes—
106#1728C2212yes—
107C22×M16C2312yes—
108C22×M16C2312yes—
109#533—24no—
110#533Q818yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#125—21no—
114C2×#125—21no—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×#117—21no—
127C2×#117—21no—
128C2×#117—21no—
129C2×#117—21no—
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#479—216no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142#423—28no—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C4×M16—24no—
152C4×M16—24no—
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159#391—21no—
160#391—21no—
161#391C2312yes—
162#391C2312yes—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#385—22no—
204#385—22no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#382—21no—
212#382—21no—
213#382—21no—
214#382—21no—
215#382—21no—
216#382C2312yes—
217#382C2312yes—
218#380—21no—
219#380—21no—
220#380—21no—
221#380C2312yes—
222#380C2312yes—
223#380C2312yes—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C22⋊C4—22no—
249C4×C22⋊C4—22no—
250#376—22no—
251C43—22no—
252C43D411yescharacteristic
253C43D411yescharacteristic
254C43D411yescharacteristic
255C43D411yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8C2412yes—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C2×M16—24no—
282C2×M16—24no—
283C2×M16—24no—
284C2×M16—24no—
285C2×M16—42no—
286C2×M16—42no—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8C2×Q814yes—
294C4⋊Q8C2×Q814yes—
295#125—28no—
296#125—28no—
297#125—44no—
298#125—44no—
299#125—44no—
300C22⋊Q8—28no—
301C22⋊Q8—24no—
302C22⋊Q8C2×Q818yes—
303C4×Q8—28no—
304C4×Q8—24no—
305C4×Q8—28no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8C2×Q818yes—
310#117—28no—
311#117—44no—
312#117—44no—
313#117—44no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—42no—
365C2×C42—42no—
366C2×C42—42no—
367C2×C42—42no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×Q814yes—
391C2×C42#4114yes—
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398#107—28no—
399#107—44no—
400#106—28no—
401#106—44no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405C4⋊C8#4118yes—
406#103—28no—
407#103—28no—
408#103—44no—
409#103—44no—
410#97—24no—
411#97—24no—
412C4×C8—22no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438C24C22×D411yescharacteristic
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8C22×Q814yes—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4C22⋊Q814yes—
473C22×C4C22⋊Q814yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×Q812yes—
488C22×C4C2×#4112yes—
489C22×C4C2511yescharacteristic
490M16—48no—
491M16—48no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8C2×#4114yes—
503C2×C8C2×#4114yes—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4C22×Q814yes—
541C22⋊C4—28no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C42—28no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—44no—
569C42—44no—
570C42—44no—
571C42—44no—
572C42—44no—
573C42—44no—
574C42—44no—
575C42C22⋊Q814yes—
576C42C22⋊Q814yes—
577C42#13714yes—
578C42C22×Q812yes—
579C42C22×Q812yes—
580C42C2×#4112yes—
581C42C2×#4112yes—
582C42C2×#4114yes—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23C2×C22⋊Q812yes—
590C23C23×D411yescharacteristic
591Q8—28no—
592Q8—48no—
593Q8—44no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4C2×C22⋊Q812yes—
653C2×C4C2×C22⋊Q812yes—
654C2×C4C2×C22⋊Q812yes—
655C2×C4C2×C22⋊Q812yes—
656C2×C4C2×C22⋊Q812yes—
657C2×C4#53212yes—
658C2×C4#53212yes—
659C2×C4C2×#13712yes—
660C2×C4C2×#13712yes—
661C2×C4C2×#13712yes—
662C2×C4#57512yes—
663C2×C4#57512yes—
664C2×C4#57512yes—
665C2×C4C23×Q811yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4112yes—
668C2×C4#58512yes—
669C8—24no—
670C8—24no—
671C8—42no—
672C8—42no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22C22×C22⋊Q811yescharacteristic
681C22C2×#53211yescharacteristic
682C22#344911yescharacteristic
683C22C22×#13711yescharacteristic
684C22C2×#57511yescharacteristic
685C22C2×#57511yescharacteristic
686C22C2×#57511yescharacteristic
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—44no—
699C4—44no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4#307814yes—
707C4#308014yes—
708C4#347512yes—
709C4Q8×#4112yes—
710C2—42no—
711C2#3494711yescharacteristic
712C2C2×#307811yescharacteristic
713C2C2×#308011yescharacteristic
714C2#6069311yescharacteristic
715C2#6070111yescharacteristic
716C2#6096611yescharacteristic
717C2#6097211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29220C2semidirect
4. #29207C2semidirect
5. #29193C2semidirect
6. #29191C2semidirect
7. #29179C2semidirect
8. #29177C2semidirect
10. #29144C2semidirect
11. #28515C2semidirect
13. #23785C2semidirect
14. #21618C2semidirect

Maximal subgroups

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

subgroupindex
#366402
C4×#5332
#292202
#292072
#291932
#291912
#291792
#291772
#291642
#291442
#285152
#248422
#237852
#216182
#214092
#208242
C4×#4232