Finite Group Databasebeta

← #214744

Subgroups — #214744

Order 512.

1083 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35702C211yesmaximal, characteristic
2#20633C211yesmaximal, characteristic
3#17367C211yesmaximal, characteristic
4#9471C212yesmaximal, complemented
5#9385C212yesmaximal, complemented
6#9199C212yesmaximal, complemented
7#9013C212yesmaximal, complemented
8#8760C212yesmaximal, complemented
9#8379C212yesmaximal, complemented
10C2×D4×Q8C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2973C2212yes—
13C22×#99—21no—
14#2628C411yescharacteristic, complemented
15#2626C411yescharacteristic, complemented
16#2626C2211yescharacteristic
17#2580C411yescharacteristic, complemented
18#2564C411yescharacteristic, complemented
19#2564C2211yescharacteristic
20#2563C411yescharacteristic, complemented
21#2452—21no—
22#2450—21no—
23#2450—21no—
24#2439—21no—
25#2437—21no—
26#2437—21no—
27#2409C411yescharacteristic, complemented
28#2409C2211yescharacteristic
29#2396C411yescharacteristic, complemented
30C2×#393C411yescharacteristic, complemented
31C2×#393C2211yescharacteristic
32#2349—21no—
33#2349C2211yescharacteristic
34D4×C22⋊C4—21no—
35#2315C2212yes—
36#1799—21no—
37#1799—21no—
38#1799C2212yes—
39#1799C2212yes—
40#1787C2214yescomplemented
41C2×#342—21no—
42#1520C2214yescomplemented
43#1466—21no—
44#1465—21no—
45#1464—21no—
46#1462—21no—
47#1459C2212yes—
48C2×#323C2212yes—
49#1294C2214yescomplemented
50#1289C2214yescomplemented
51C23×Q8C2×C411yescharacteristic, complemented
52C23×Q8C2311yescharacteristic
53C23×D4D411yescharacteristic
54D4×Q8—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic, complemented
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2×C411yescharacteristic, complemented
62C2×C22⋊Q8C2311yescharacteristic
63#518—44no—
64C2×C4×Q8C2×C411yescharacteristic, complemented
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117D411yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4C2311yescharacteristic
84#435—44no—
85#432—22no—
86C2×#103—44no—
87C2×#99—22no—
88C2×#99—22no—
89C2×#99—44no—
90#407—22no—
91C2×C22⋊C8C2314yes—
92#400—21no—
93#400C2×C411yescharacteristic, complemented
94#399—21no—
95#399C2×C411yescharacteristic
96#399C2×C411yescharacteristic
97#399C2×C411yescharacteristic, complemented
98#398—21no—
99#396—21no—
100#396—21no—
101#396—21no—
102#396—21no—
103#396—21no—
104#395C2×C411yescharacteristic, complemented
105#395C2×C411yescharacteristic, complemented
106#395C2×C411yescharacteristic, complemented
107#393—21no—
108#393—21no—
109#393—21no—
110#393—21no—
111#393C2×C411yescharacteristic
112#393C2×C411yescharacteristic
113#393C2×C411yescharacteristic
114#393C2×C411yescharacteristic
115#393C2×C411yescharacteristic
116#393C2×C411yescharacteristic
117#393C2×C411yescharacteristic, complemented
118#393C2×C411yescharacteristic, complemented
119#393C2×C411yescharacteristic, complemented
120#391—21no—
121#391—21no—
122#390—21no—
123#390—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#387—21no—
128#387—21no—
129#386—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—44no—
144#382—21no—
145#382C2×C411yescharacteristic, complemented
146#380—21no—
147#380—21no—
148#380—21no—
149#380—21no—
150#380—21no—
151#380—22no—
152#380C2×C411yescharacteristic, complemented
153#380C2314yes—
154#380C2311yescharacteristic
155#379—21no—
156#379—21no—
157C4×C4⋊C4—21no—
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#328—44no—
167#324—22no—
168#323—22no—
169#323—22no—
170#323—22no—
171C25C22⋊C411yescharacteristic
172C25C2×D411yescharacteristic
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—24no—
176C22×Q8—42no—
177C22×Q8—82no—
178C22×Q8—82no—
179C22×Q8C22×C411yescharacteristic
180C22×Q8C22×C411yescharacteristic
181C22×Q8C22×C411yescharacteristic
182C22×D4—24no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—42no—
186C22×D4C22⋊C411yescharacteristic
187C22×D4C22⋊C411yescharacteristic
188C22×D4C2×D411yescharacteristic
189C22×D4C2×D411yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—42no—
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22×C411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2411yescharacteristic
204C2×M16—42no—
205C22×C8—24no—
206C4⋊Q8—42no—
207C4⋊Q8—44no—
208C22⋊Q8—24no—
209C22⋊Q8—44no—
210C22⋊Q8—44no—
211C22⋊Q8—42no—
212C22⋊Q8—82no—
213C22⋊Q8—82no—
214C4×Q8—42no—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—44no—
224C4×D4—42no—
225C4×D4—42no—
226#117—42no—
227#117—42no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C22⋊C4—24no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—21no—
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—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—82no—
275C2×C22⋊C4—82no—
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22×C411yescharacteristic
281C2×C22⋊C4C22×C411yescharacteristic
282C2×C22⋊C4C22×C411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C22⋊C4C2×D411yescharacteristic
285C2×C22⋊C4C2×D411yescharacteristic
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C42—21no—
289C2×C42—21no—
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C4⋊C8—42no—
295#103—88no—
296#99—44no—
297#99—44no—
298C22⋊C8—24no—
299C22⋊C8—24no—
300C22⋊C8—42no—
301#97—42no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—24no—
320#95—42no—
321#95C22×C411yescharacteristic
322#95C22×C411yescharacteristic
323#95C22×C411yescharacteristic
324#95C22×C411yescharacteristic
325C24—22no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—22no—
330C24—41no—
331C24—81no—
332C24—81no—
333C24—81no—
334C24—81no—
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C24#13714yes—
340C24C22×D411yescharacteristic
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—48no—
350C2×Q8—81no—
351C2×Q8—81no—
352C2×Q8—81no—
353C2×Q8—81no—
354C2×Q8—84no—
355C2×Q8—84no—
356C2×Q8—84no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4AGL(1,ℤ/8)12yes—
372C2×D4#13712yes—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—84no—
414C22×C4—162no—
415C22×C4—162no—
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C23×C411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4H(5,𝔽2)14yes—
427M16—162no—
428C2×C8—44no—
429C2×C8—84no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—84no—
451C4⋊C4—162no—
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—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—42no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—84no—
484C42—22no—
485C42—22no—
486C42—41no—
487C42—41no—
488C42—42no—
489C42—42no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—24no—
497C23—24no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—41no—
502C23—41no—
503C23—44no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—82no—
513C23—82no—
514C23—84no—
515C23—84no—
516C23—161no—
517C23—161no—
518C23—161no—
519C23—161no—
520C23—162no—
521C23#41712yes—
522C23#41912yes—
523C23#41912yes—
524C23C22×C22⋊C411yescharacteristic
525C23#53412yes—
526C23C2×AGL(1,ℤ/8)11yescharacteristic
527C23C2×#13712yes—
528C23C2×#13711yescharacteristic
529Q8—24no—
530Q8—42no—
531Q8—44no—
532Q8—84no—
533Q8—82no—
534Q8—84no—
535D4—24no—
536D4—24no—
537D4—44no—
538D4—44no—
539D4—42no—
540D4—42no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—161no—
595C2×C4—161no—
596C2×C4—161no—
597C2×C4—161no—
598C2×C4—161no—
599C2×C4—161no—
600C2×C4—162no—
601C2×C4—168no—
602C2×C4#41712yes—
603C2×C4#51812yes—
604C2×C4#53412yes—
605C8—84no—
606C22—21no—
607C22—22no—
608C22—21no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—84no—
631C22—81no—
632C22—81no—
633C22—84no—
634C22—81no—
635C22—164no—
636C22#230811yescharacteristic
637C22C2×#41911yescharacteristic
638C22C2×#41711yescharacteristic
639C22#319312yes—
640C22#319512yes—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—41no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—164no—
660C2—21no—
661C2—22no—
662C2—21no—
663C2—22no—
664C2—42no—
665C2—42no—
666C2—42no—
667C2—81no—
668C2#2176511yescharacteristic
669C2#2177611yescharacteristic
670C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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
4. #9471C2semidirect
5. #9385C2semidirect
6. #9199C2semidirect
7. #9013C2semidirect
8. #8760C2semidirect
9. #8379C2semidirect
14. #2628C4semidirect
15. #2626C4semidirect
17. #2580C4semidirect
18. #2564C4semidirect
20. #2563C4semidirect
27. #2409C4semidirect
29. #2396C4semidirect
30. C2×#393C4semidirect
40. #1787C22semidirect
42. #1520C22semidirect
49. #1294C22semidirect
50. #1289C22semidirect
51. C23×Q8C2×C4semidirect
56. C2×C4⋊Q8C2×C4semidirect
61. C2×C22⋊Q8C2×C4semidirect
64. C2×C4×Q8C2×C4semidirect
93. #400C2×C4semidirect
97. #399C2×C4semidirect
104. #395C2×C4semidirect
105. #395C2×C4semidirect
106. #395C2×C4semidirect
117. #393C2×C4semidirect
118. #393C2×C4semidirect
119. #393C2×C4semidirect
145. #382C2×C4semidirect
152. #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
#357022
#206332
#173672
#94712
#93852
#91992
#90132
#87602
#83792