Finite Group Databasebeta

← #214740

Subgroups — #214740

Order 512.

1167 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36009C211yesmaximal, characteristic
2#20633C211yesmaximal, characteristic, complemented
3#17367C211yesmaximal, characteristic, complemented
4#9474C212yesmaximal, complemented
5#9386C212yesmaximal, complemented
6#9203C212yesmaximal, complemented
7#9025C212yesmaximal, complemented
8#8761C212yesmaximal, complemented
9#8376C212yesmaximal, complemented
10C2×#547C2211yescharacteristic
11C2×#546—21no—
12C2×#518C2211yescharacteristic
13#2973C2212yes—
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#1466—21no—
44#1465—21no—
45#1464—21no—
46#1462—21no—
47#1459C2212yes—
48C2×#323C2212yes—
49#1294C2214yescomplemented
50#1289C2214yescomplemented
51C22×#41—21no—
52C23×D4D411yescharacteristic
53#547—44no—
54#546—48no—
55C2×C4⋊Q8C2×C411yescharacteristic, complemented
56C2×#124—21no—
57C2×#123—21no—
58C2×#123—21no—
59C2×#123—21no—
60C2×#123—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121C2×C411yescharacteristic, complemented
67C2×#121C2311yescharacteristic
68C2×#120—21no—
69C2×#120—21no—
70#518—44no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic, complemented
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×#117C2×C411yescharacteristic
83C2×#117D411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic, complemented
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4C2×C411yescharacteristic
90C22×C22⋊C4D411yescharacteristic
91C22×C22⋊C4D411yescharacteristic
92C22×C22⋊C4C2311yescharacteristic
93#435—44no—
94#432—22no—
95C2×#99—22no—
96C2×#99—22no—
97C2×#99—44no—
98#407—22no—
99C2×C22⋊C8C2314yes—
100#398—21no—
101#398C2×C411yescharacteristic, complemented
102#398C2×C411yescharacteristic, complemented
103#398C2×C411yescharacteristic, complemented
104#397—21no—
105#397—21no—
106#397—21no—
107#397—21no—
108#397—21no—
109#397C2×C411yescharacteristic
110#397C2×C411yescharacteristic
111#397C2×C411yescharacteristic
112#397C2×C411yescharacteristic
113#397C2×C411yescharacteristic
114#397C2×C411yescharacteristic
115#397C2×C411yescharacteristic, complemented
116#397C2×C411yescharacteristic, complemented
117#397C2×C411yescharacteristic, complemented
118#396—21no—
119#394—21no—
120#394—21no—
121#394—21no—
122#394—21no—
123#394—21no—
124#394C2×C411yescharacteristic, complemented
125#392C2×C411yescharacteristic
126#392C2×C411yescharacteristic
127#392C2×C411yescharacteristic, complemented
128#391—21no—
129#391—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#388—21no—
134#388—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#385—21no—
144#385—21no—
145#385—44no—
146#382C2×C411yescharacteristic, complemented
147#381—21no—
148#381—21no—
149#380—21no—
150#380—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—22no—
155#380C2×C411yescharacteristic, complemented
156#380C2314yes—
157#380C2311yescharacteristic
158C4×C4⋊C4C2×C411yescharacteristic, complemented
159C4×C22⋊C4—21no—
160C2×#95—21no—
161C2×#95—21no—
162C2×#95C2×C411yescharacteristic, complemented
163C2×#95C2311yescharacteristic
164#342—22no—
165#342—22no—
166#340—44no—
167#327—44no—
168#324—22no—
169#323—22no—
170#323—22no—
171#323—22no—
172C25C22⋊C411yescharacteristic
173C25C2×D411yescharacteristic
174C2×#41—44no—
175C2×#41—44no—
176C2×#41—42no—
177C2×#41—42no—
178C22×Q8—21no—
179C22×D4—24no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—42no—
188C22×D4C22⋊C411yescharacteristic
189C22×D4C22⋊C411yescharacteristic
190C22×D4C22×C411yescharacteristic
191C22×D4C2×D411yescharacteristic
192C22×D4C2×D411yescharacteristic
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—42no—
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2411yescharacteristic
208C2×M16—42no—
209C22×C8—24no—
210C4⋊Q8—42no—
211#124—44no—
212#123—44no—
213#123—44no—
214#123—44no—
215#123—44no—
216C22⋊Q8—44no—
217C22⋊Q8—44no—
218#121—24no—
219#121—42no—
220#121—44no—
221#121—44no—
222#121—44no—
223#120—44no—
224#120—44no—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—42no—
228C4×D4—42no—
229C4×D4—42no—
230C4×D4—42no—
231C4×D4—42no—
232C4×D4—42no—
233C4×D4—42no—
234C4×D4—44no—
235C4×D4—44no—
236#117—42no—
237#117—42no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C22⋊C4—24no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
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—24no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4C22⋊C411yescharacteristic
289C2×C22⋊C4C22⋊C411yescharacteristic
290C2×C22⋊C4C22⋊C411yescharacteristic
291C2×C22⋊C4C22⋊C411yescharacteristic
292C2×C22⋊C4C22×C411yescharacteristic
293C2×C22⋊C4C22×C411yescharacteristic
294C2×C22⋊C4C22×C411yescharacteristic
295C2×C22⋊C4C2×D411yescharacteristic
296C2×C22⋊C4C2×D411yescharacteristic
297C2×C22⋊C4C2×D411yescharacteristic
298C2×C22⋊C4C2×D411yescharacteristic
299C2×C22⋊C4C2×D411yescharacteristic
300C2×C42—21no—
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22⋊C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C4⋊C8—42no—
306#99—44no—
307#99—44no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—42no—
311#97—42no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
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—24no—
328#95—42no—
329#95C22×C411yescharacteristic
330#95C22×C411yescharacteristic
331#95C22×C411yescharacteristic
332#95C22×C411yescharacteristic
333C24—22no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—41no—
340C24—81no—
341C24—81no—
342C24—81no—
343C24—81no—
344C24C2×C22⋊C411yescharacteristic
345C24C2×C22⋊C411yescharacteristic
346C24C2×C22⋊C411yescharacteristic
347C24C2×C22⋊C411yescharacteristic
348C24AGL(1,ℤ/8)14yes—
349C24C22×D411yescharacteristic
350#41—88no—
351#41—88no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4—81no—
387C2×D4AGL(1,ℤ/8)12yes—
388C2×D4#13712yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—84no—
432C22×C4—162no—
433C22×C4—162no—
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4H(5,𝔽2)14yes—
445M16—162no—
446C2×C8—44no—
447C2×C8—84no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—84no—
466C4⋊C4—162no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—81no—
500C22⋊C4—84no—
501C42—22no—
502C42—22no—
503C42—41no—
504C42—41no—
505C42—42no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—24no—
513C23—24no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—41no—
518C23—41no—
519C23—44no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—82no—
536C23—82no—
537C23—84no—
538C23—84no—
539C23—161no—
540C23—161no—
541C23—161no—
542C23—161no—
543C23—162no—
544C23#41712yes—
545C23#41812yes—
546C23#41812yes—
547C23C22×C22⋊C411yescharacteristic
548C23#53412yes—
549C23C2×AGL(1,ℤ/8)12yes—
550C23C2×AGL(1,ℤ/8)11yescharacteristic
551C23C2×#13711yescharacteristic
552Q8—84no—
553D4—24no—
554D4—24no—
555D4—44no—
556D4—44no—
557D4—42no—
558D4—42no—
559D4—82no—
560D4—84no—
561D4—84no—
562D4—82no—
563D4—82no—
564D4—162no—
565D4—162no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
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—82no—
609C2×C4—82no—
610C2×C4—84no—
611C2×C4—82no—
612C2×C4—84no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—161no—
619C2×C4—161no—
620C2×C4—161no—
621C2×C4—161no—
622C2×C4—162no—
623C2×C4—162no—
624C2×C4—168no—
625C2×C4#41712yes—
626C2×C4#51812yes—
627C2×C4#53412yes—
628C8—84no—
629C22—21no—
630C22—22no—
631C22—21no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—42no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—84no—
654C22—81no—
655C22—81no—
656C22—84no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—164no—
666C22#230811yescharacteristic
667C22C2×#41811yescharacteristic
668C22C2×#41711yescharacteristic
669C22#319212yes—
670C22#319312yes—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—164no—
689C2—21no—
690C2—22no—
691C2—21no—
692C2—22no—
693C2—42no—
694C2—42no—
695C2—42no—
696C2—81no—
697C2—161no—
698C2#2176411yescharacteristic
699C2#2177711yescharacteristic
700C2#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. #17367C2semidirect
4. #9474C2semidirect
5. #9386C2semidirect
6. #9203C2semidirect
7. #9025C2semidirect
8. #8761C2semidirect
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
55. C2×C4⋊Q8C2×C4semidirect
66. C2×#121C2×C4semidirect
75. C2×C4×D4C2×C4semidirect
84. C22×C4⋊C4C2×C4semidirect
101. #398C2×C4semidirect
102. #398C2×C4semidirect
103. #398C2×C4semidirect
115. #397C2×C4semidirect
116. #397C2×C4semidirect
117. #397C2×C4semidirect
124. #394C2×C4semidirect
127. #392C2×C4semidirect
146. #382C2×C4semidirect
155. #380C2×C4semidirect
158. C4×C4⋊C4C2×C4semidirect
162. 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
#173672
#94742
#93862
#92032
#90252
#87612
#83762