Finite Group Databasebeta

← #214708

Subgroups — #214708

Order 512.

1184 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36009C211yesmaximal, characteristic
2#20633C211yesmaximal, characteristic, complemented
3#17350C211yesmaximal, characteristic, complemented
4#9470C212yesmaximal, complemented
5#9384C212yesmaximal, complemented
6#9203C212yesmaximal, complemented
7#9025C212yesmaximal, complemented
8#8760C212yesmaximal, complemented
9#8376C212yesmaximal, complemented
10C2×#547C2211yescharacteristic
11C2×#546—21no—
12C2×#518C2211yescharacteristic
13#2972C2212yes—
14C22×#99—21no—
15#2621C411yescharacteristic, complemented
16#2617C411yescharacteristic, complemented
17#2617C2211yescharacteristic
18#2547C411yescharacteristic, complemented
19#2543C411yescharacteristic, complemented
20#2543C2211yescharacteristic
21#2535C411yescharacteristic, complemented
22#2486C411yescharacteristic, complemented
23#2486C2211yescharacteristic
24#2427—21no—
25#2425—21no—
26#2425—21no—
27#2415—21no—
28#2412—21no—
29#2412—21no—
30#2392C411yescharacteristic, complemented
31C2×#397C411yescharacteristic, complemented
32C2×#397C2211yescharacteristic
33#2349C2211yescharacteristic
34#2339—21no—
35#2315C2212yes—
36#1823C2214yescomplemented
37#1799—21no—
38#1799—21no—
39#1799C2212yes—
40#1799C2212yes—
41C2×#342—21no—
42#1519C2214yescomplemented
43#1460—21no—
44#1459—21no—
45#1459C2212yes—
46#1458—21no—
47C2×#323—21no—
48C2×#323C2212yes—
49#1294C2214yescomplemented
50#1289C2214yescomplemented
51C22×#41—21no—
52C23×D4D411yescharacteristic
53#547—44no—
54#546—44no—
55#546—44no—
56C2×C4⋊Q8C2×C411yescharacteristic, complemented
57C2×#124—21no—
58C2×#123—21no—
59C2×#123—21no—
60C2×#123—21no—
61C2×#123—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121C2×C411yescharacteristic, complemented
68C2×#121C2311yescharacteristic
69C2×#120—21no—
70C2×#120—21no—
71#518—44no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic, complemented
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×C4×D4D411yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×#117C2×C411yescharacteristic
84C2×#117D411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic, complemented
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4C2×C411yescharacteristic
91C22×C22⋊C4D411yescharacteristic
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4C2311yescharacteristic
94C8×D4—44no—
95#431—22no—
96C2×#99—22no—
97C2×#99—22no—
98C2×#99—44no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8C2314yes—
101#398—21no—
102#398C2×C411yescharacteristic, complemented
103#398C2×C411yescharacteristic, complemented
104#398C2×C411yescharacteristic, complemented
105#397—21no—
106#397—21no—
107#397—21no—
108#397—21no—
109#397—21no—
110#397C2×C411yescharacteristic
111#397C2×C411yescharacteristic
112#397C2×C411yescharacteristic
113#397C2×C411yescharacteristic
114#397C2×C411yescharacteristic
115#397C2×C411yescharacteristic
116#397C2×C411yescharacteristic, complemented
117#397C2×C411yescharacteristic, complemented
118#397C2×C411yescharacteristic, complemented
119#396—21no—
120#394—21no—
121#394—21no—
122#394—21no—
123#394—21no—
124#394—21no—
125#394C2×C411yescharacteristic, complemented
126#392C2×C411yescharacteristic
127#392C2×C411yescharacteristic
128#392C2×C411yescharacteristic, complemented
129#391—21no—
130#391—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#388—21no—
135#388—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#386—21no—
144#385—21no—
145#385—21no—
146#385—44no—
147#382C2×C411yescharacteristic, complemented
148#381—21no—
149#381—21no—
150#380—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—22no—
156#380C2×C411yescharacteristic, complemented
157#380C2314yes—
158#380C2311yescharacteristic
159C4×C4⋊C4C2×C411yescharacteristic, complemented
160C4×C22⋊C4—21no—
161C2×#95—21no—
162C2×#95—21no—
163C2×#95C2×C411yescharacteristic, complemented
164C2×#95C2311yescharacteristic
165#342—22no—
166#342—22no—
167#340—44no—
168#327—44no—
169#324—22no—
170#323—22no—
171#323—22no—
172#323—22no—
173C25C22⋊C411yescharacteristic
174C25C2×D411yescharacteristic
175C2×#41—44no—
176C2×#41—44no—
177C2×#41—42no—
178C2×#41—42no—
179C22×Q8—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4C22⋊C411yescharacteristic
188C22×D4C22⋊C411yescharacteristic
189C22×D4D814yes—
190C22×D4SD1614yes—
191C22×D4C22×C411yescharacteristic
192C22×D4C2×D411yescharacteristic
193C22×D4C2×D411yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—42no—
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C22⋊C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2411yescharacteristic
209C22×C8—24no—
210C22×C8—42no—
211C4⋊Q8—42no—
212#124—44no—
213#123—44no—
214#123—44no—
215#123—44no—
216#123—42no—
217#123—42no—
218C22⋊Q8—44no—
219C22⋊Q8—44no—
220#121—24no—
221#121—42no—
222#121—44no—
223#121—44no—
224#121—42no—
225#121—42no—
226#120—44no—
227#120—42no—
228#120—42no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—42no—
232C4×D4—42no—
233C4×D4—42no—
234C4×D4—42no—
235C4×D4—42no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—44no—
241#117—42no—
242#117—42no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4C22⋊C411yescharacteristic
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C22⋊C4—24no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22×C411yescharacteristic
299C2×C22⋊C4C22×C411yescharacteristic
300C2×C22⋊C4C22×C411yescharacteristic
301C2×C22⋊C4C2×D411yescharacteristic
302C2×C22⋊C4C2×D411yescharacteristic
303C2×C22⋊C4C2×D411yescharacteristic
304C2×C22⋊C4C2×D411yescharacteristic
305C2×C22⋊C4C2×D411yescharacteristic
306C2×C42—21no—
307C2×C42C22⋊C411yescharacteristic
308C2×C42C22⋊C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C4⋊C8—42no—
312#99—44no—
313#99—44no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316C22⋊C8—42no—
317C4×C8—42no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—24no—
334#95—42no—
335#95C22×C411yescharacteristic
336#95C22×C411yescharacteristic
337#95C22×C411yescharacteristic
338#95C22×C411yescharacteristic
339C24—22no—
340C24—22no—
341C24—21no—
342C24—41no—
343C24—41no—
344C24—81no—
345C24—81no—
346C24—81no—
347C24—81no—
348C24#10214yes—
349C24C2×C22⋊C411yescharacteristic
350C24C2×C22⋊C411yescharacteristic
351C24C2×C22⋊C411yescharacteristic
352C24C2×C22⋊C411yescharacteristic
353C24C2×D812yes—
354C24C2×SD1612yes—
355C24C22×D411yescharacteristic
356#41—88no—
357#41—88no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—81no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4—81no—
387C2×D4—81no—
388C2×D4—81no—
389C2×D4#10214yes—
390C2×D4#10214yes—
391C2×D4C2×D812yes—
392C2×D4C2×D812yes—
393C2×D4C2×SD1612yes—
394C2×D4C2×SD1612yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—84no—
438C22×C4—162no—
439C22×C4—162no—
440C22×C4#10214yes—
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C23×C411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4H(5,𝔽2)14yes—
452C2×C8—44no—
453C2×C8—84no—
454C2×C8—162no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—84no—
473C4⋊C4—162no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—84no—
509C42—22no—
510C42—22no—
511C42—41no—
512C42—41no—
513C42—42no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—24no—
523C23—24no—
524C23—41no—
525C23—41no—
526C23—44no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—82no—
543C23—82no—
544C23—82no—
545C23—82no—
546C23—84no—
547C23—161no—
548C23—161no—
549C23—162no—
550C23—162no—
551C23C2×#10212yes—
552C23C2×#10212yes—
553C23C2×#10212yes—
554C23C2×#10212yes—
555C23C22×C22⋊C411yescharacteristic
556C23#53412yes—
557C23C22×D811yescharacteristic
558C23C22×SD1611yescharacteristic
559C23#57512yes—
560Q8—84no—
561D4—24no—
562D4—24no—
563D4—24no—
564D4—24no—
565D4—44no—
566D4—44no—
567D4—82no—
568D4—84no—
569D4—84no—
570D4—82no—
571D4—82no—
572D4—162no—
573D4—162no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—84no—
618C2×C4—82no—
619C2×C4—84no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—161no—
626C2×C4—161no—
627C2×C4—161no—
628C2×C4—161no—
629C2×C4—161no—
630C2×C4—161no—
631C2×C4—162no—
632C2×C4—168no—
633C2×C4C2×#10212yes—
634C2×C4C2×#10212yes—
635C2×C4#51812yes—
636C2×C4#53412yes—
637C8—84no—
638C22—21no—
639C22—22no—
640C22—21no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—84no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—84no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—164no—
675C22#230811yescharacteristic
676C22C22×#10211yescharacteristic
677C22#289611yescharacteristic
678C22#318612yes—
679C22#318712yes—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—164no—
698C2—21no—
699C2—22no—
700C2—21no—
701C2—22no—
702C2—42no—
703C2—42no—
704C2—42no—
705C2—42no—
706C2—161no—
707C2#2175811yescharacteristic
708C2#2177711yescharacteristic
709C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36009C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20633C2semidirect
3. #17350C2semidirect
4. #9470C2semidirect
5. #9384C2semidirect
6. #9203C2semidirect
7. #9025C2semidirect
8. #8760C2semidirect
9. #8376C2semidirect
15. #2621C4semidirect
16. #2617C4semidirect
18. #2547C4semidirect
19. #2543C4semidirect
21. #2535C4semidirect
22. #2486C4semidirect
30. #2392C4semidirect
31. C2×#397C4semidirect
36. #1823C22semidirect
42. #1519C22semidirect
49. #1294C22semidirect
50. #1289C22semidirect
56. C2×C4⋊Q8C2×C4semidirect
67. C2×#121C2×C4semidirect
76. C2×C4×D4C2×C4semidirect
85. C22×C4⋊C4C2×C4semidirect
102. #398C2×C4semidirect
103. #398C2×C4semidirect
104. #398C2×C4semidirect
116. #397C2×C4semidirect
117. #397C2×C4semidirect
118. #397C2×C4semidirect
125. #394C2×C4semidirect
128. #392C2×C4semidirect
147. #382C2×C4semidirect
156. #380C2×C4semidirect
159. C4×C4⋊C4C2×C4semidirect
163. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#360092
#206332
#173502
#94702
#93842
#92032
#90252
#87602
#83762