Finite Group Databasebeta

← #181366

Subgroups — #181366

Order 512.

1497 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2#60230C211yesmaximal, characteristic
3#34185C211yesmaximal, characteristic
4#8110C212yesmaximal, complemented
5#8108C212yesmaximal, complemented
6#8107C212yesmaximal, complemented
7#8106C212yesmaximal, complemented
8#8104C212yesmaximal, complemented
9C2×#1305C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11#2975—21no—
12#2975—21no—
13#2974—24no—
14#2974C418yescomplemented
15#2973—21no—
16#2973—21no—
17#2973—21no—
18#2973—21no—
19#2972—21no—
20#2972—21no—
21D4×M16—24no—
22D4×M16C418yescomplemented
23#2920C2212yes—
24#2920C2212yes—
25C2×#407C2212yes—
26C2×#407C2212yes—
27C2×#407C2212yes—
28C2×#407C2212yes—
29#2353—21no—
30#2353—21no—
31#2353—21no—
32#2353—21no—
33#2352—21no—
34#2352—21no—
35#2352—21no—
36#2352—21no—
37#2313C412yes—
38#2312C412yes—
39#2308C412yes—
40#2283C2211yescharacteristic
41#2283C2211yescharacteristic
42#2280C2211yescharacteristic
43#2280C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46#2273C412yes—
47#1305C2214yes—
48#1305C2214yes—
49#1305C2214yes—
50#1305C2214yes—
51C23×D4C2×C411yescharacteristic
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C22×M16—22no—
56C22×M16C2×C414yes—
57C22×M16C2×C414yescomplemented
58C2×C4×D4C2×C414yes—
59C2×C4×D4C2×C414yes—
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D414yes—
69C2×C4×D4Q814yes—
70C2×#117—21no—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C42C2311yescharacteristic
80#436—28no—
81#436—28no—
82#435—28no—
83#435—28no—
84#435—28no—
85#435—28no—
86C8×D4—28no—
87C8×D4—28no—
88#432—22no—
89#432—22no—
90#432—22no—
91#432C2×C414yescomplemented
92#431—22no—
93#431—22no—
94#431—22no—
95#431C2×C414yescomplemented
96#423—22no—
97#423C2×C414yescomplemented
98#407—22no—
99#407—22no—
100#407—22no—
101#407—22no—
102#407—22no—
103#407—22no—
104#407—22no—
105#407—22no—
106#407C2×C414yes—
107#407C2×C414yes—
108#407C2×C414yescomplemented
109#407C2×C414yescomplemented
110#407C2314yes—
111#407C2314yes—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8—22no—
118C2×C22⋊C8C2×C414yes—
119C2×C22⋊C8C2×C414yescomplemented
120C4×M16—22no—
121C4×M16C2×C414yescomplemented
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#387—22no—
127#387—22no—
128#387—22no—
129#387—22no—
130#386—22no—
131#386—22no—
132#386—22no—
133#386—22no—
134#383—21no—
135#383—21no—
136#382—21no—
137#382—21no—
138#381—21no—
139#381—21no—
140#381C2×C412yes—
141#381C2×C412yes—
142#380—21no—
143#380—21no—
144#380—21no—
145#380—21no—
146#380—21no—
147#380—21no—
148#380C2×C412yes—
149#380C2×C412yes—
150#380C2311yescharacteristic
151#380C2311yescharacteristic
152#380C2311yescharacteristic
153#380C2311yescharacteristic
154#379C2×C412yes—
155#379C2×C412yes—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—22no—
161C4×C22⋊C4—22no—
162C4×C22⋊C4—22no—
163C4×C22⋊C4—22no—
164C4×C22⋊C4C2×C412yes—
165C4×C22⋊C4C2×C412yes—
166C4×C22⋊C4C2×C412yes—
167C4×C22⋊C4C2×C412yes—
168#376—21no—
169#376—21no—
170C2×#95—21no—
171C2×#95—21no—
172C25C22×C411yescharacteristic
173C25C22×C411yescharacteristic
174C22×D4—42no—
175C22×D4—42no—
176C22×D4C4214yes—
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C411yescharacteristic
179C22×D4C22⋊C411yescharacteristic
180C22×D4C22⋊C411yescharacteristic
181C22×D4C4⋊C414yes—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—82no—
185C23×C4—82no—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C412yes—
188C23×C4C22×C412yes—
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C412yes—
193C23×C4C22×C412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×Q812yes—
205C23×C4C2×Q812yes—
206C23×C4C2411yescharacteristic
207C2×M16—24no—
208C2×M16—24no—
209C2×M16—24no—
210C2×M16—24no—
211C2×M16—24no—
212C2×M16—84no—
213C2×M16—84no—
214C2×M16—84no—
215C2×M16—84no—
216C2×M16C22×C418yes—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8—24no—
220C22×C8—24no—
221C4×D4C22⋊C414yes—
222C4×D4C22⋊C414yes—
223C4×D4C22⋊C414yes—
224C4×D4C22⋊C414yes—
225C4×D4C22⋊C414yes—
226C4×D4C22⋊C414yes—
227C4×D4C22⋊C414yes—
228C4×D4C22⋊C414yes—
229C4×D4C4⋊C414yes—
230C4×D4C4⋊C414yes—
231C4×D4C4⋊C414yes—
232C4×D4C4⋊C414yes—
233C4×D4C4⋊C414yes—
234C4×D4C4⋊C414yes—
235C4×D4C4⋊C414yes—
236C4×D4C4⋊C414yes—
237#117—82no—
238#117—82no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4—82no—
247C2×C4⋊C4C4214yes—
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C4⋊C414yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4C4214yes—
262C2×C22⋊C4C4214yes—
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C22⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C4⋊C414yes—
272C2×C22⋊C4C4⋊C414yes—
273C2×C22⋊C4C22×C412yes—
274C2×C22⋊C4C22×C412yes—
275C2×C22⋊C4C22×C412yes—
276C2×C22⋊C4C22×C412yes—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42C22×C412yes—
282C2×C42C22×C412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×Q812yes—
289C4⋊C8—24no—
290C4⋊C8—24no—
291C4⋊C8—24no—
292C4⋊C8—24no—
293C22⋊C8—24no—
294C22⋊C8—24no—
295C22⋊C8—24no—
296C22⋊C8—24no—
297C22⋊C8—24no—
298C22⋊C8—24no—
299C22⋊C8—24no—
300C22⋊C8—24no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C22⋊C8—24no—
304C22⋊C8C22×C418yes—
305#97—24no—
306#97—24no—
307C4×C8—24no—
308C4×C8—24no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323C24—22no—
324C24—22no—
325C24—41no—
326C24—41no—
327C24—81no—
328C24—81no—
329C24—81no—
330C24—81no—
331C24C2×C4212yes—
332C24C2×C4212yes—
333C24C2×C22⋊C411yescharacteristic
334C24C2×C22⋊C411yescharacteristic
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C4⋊C412yes—
338C24C2×C4⋊C412yes—
339C24C23×C411yescharacteristic
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4#9514yes—
351C2×D4#9514yes—
352C2×D4#9514yes—
353C2×D4#9514yes—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—81no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4C2×C4212yes—
387C22×C4C2×C4212yes—
388C22×C4C2×C4212yes—
389C22×C4C2×C4212yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C4⋊C412yes—
407C22×C4C2×C4⋊C412yes—
408C22×C4C2×C4⋊C412yes—
409C22×C4C2×C4⋊C412yes—
410C22×C4C2×C4⋊C412yes—
411C22×C4C2×C4⋊C412yes—
412C22×C4C2×C4⋊C412yes—
413C22×C4C2×C4⋊C412yes—
414C22×C4C2×C4⋊C412yes—
415C22×C4C2×C4⋊C412yes—
416C22×C4C2×C4⋊C412yes—
417C22×C4C2×C4⋊C412yes—
418C22×C4C23×C411yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×Q811yescharacteristic
424M16—48no—
425M16—48no—
426M16—84no—
427M16—84no—
428M16—84no—
429M16—84no—
430C2×C8—28no—
431C2×C8—28no—
432C2×C8—84no—
433C2×C8—84no—
434C2×C8—84no—
435C2×C8—84no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4#9514yes—
443C4⋊C4#9514yes—
444C4⋊C4#9514yes—
445C4⋊C4#9514yes—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4#9514yes—
457C22⋊C4#9514yes—
458C22⋊C4#9514yes—
459C22⋊C4#9514yes—
460C22⋊C4#9514yes—
461C22⋊C4#9514yes—
462C22⋊C4#9514yes—
463C22⋊C4#9514yes—
464C42—82no—
465C42—82no—
466C42—82no—
467C42—82no—
468C42C2×C22⋊C412yes—
469C42C2×C22⋊C412yes—
470C42C2×C22⋊C412yes—
471C42C2×C22⋊C412yes—
472C42C2×C4⋊C412yes—
473C42C2×C4⋊C412yes—
474C42C2×C4⋊C412yes—
475C42C2×C4⋊C412yes—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—24no—
485C23—24no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23C2×#9512yes—
511C23C2×#9512yes—
512C23C2×#9512yes—
513C23C2×#9512yes—
514C23#41312yes—
515C23#41312yes—
516C23C22×C4211yescharacteristic
517C23C22×C22⋊C411yescharacteristic
518C23C22×C4⋊C411yescharacteristic
519D4—28no—
520D4—28no—
521D4—44no—
522D4—44no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—164no—
576C2×C4—164no—
577C2×C4C2×#9512yes—
578C2×C4C2×#9512yes—
579C2×C4C2×#9512yes—
580C2×C4C2×#9512yes—
581C2×C4C2×#9512yes—
582C2×C4C2×#9512yes—
583C2×C4C2×#9512yes—
584C2×C4C2×#9512yes—
585C2×C4C22×C22⋊C411yescharacteristic
586C2×C4C22×C22⋊C411yescharacteristic
587C2×C4C22×C4⋊C411yescharacteristic
588C2×C4C22×C4⋊C411yescharacteristic
589C8—48no—
590C8—48no—
591C22—21no—
592C22—22no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—21no—
601C22—21no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22#174214yes—
625C22C22×#9511yescharacteristic
626C22C2×#41311yescharacteristic
627C22C2×#41311yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4#288312yes—
644C2—21no—
645C2—21no—
646C2—24no—
647C2—42no—
648C2—41no—
649C2—41no—
650C2—42no—
651C2—42no—
652C2#2035411yescharacteristic
653C2C2×#174211yescharacteristic
654C2#2036011yescharacteristic
subgroupG/H
center, socleC22C22×#95
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8110C2semidirect
5. #8108C2semidirect
6. #8107C2semidirect
7. #8106C2semidirect
8. #8104C2semidirect
9. C2×#1305C2semidirect
14. #2974C4semidirect
22. D4×M16C4semidirect
57. C22×M16C2×C4semidirect
91. #432C2×C4semidirect
95. #431C2×C4semidirect
97. #423C2×C4semidirect
108. #407C2×C4semidirect
109. #407C2×C4semidirect
119. C2×C22⋊C8C2×C4semidirect
121. C4×M16C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602302
#341852
#81102
#81082
#81072
#81062
#81042
C2×#13052