Finite Group Databasebeta

← #183233

Subgroups — #183233

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34202C211yesmaximal, characteristic
2#20379C211yesmaximal, characteristic, complemented
3C2×#1513C211yesmaximal, characteristic
4C2×#1501C211yesmaximal, characteristic
5#13801C211yesmaximal, characteristic, complemented
6#8221C212yesmaximal, complemented
7#8219C212yesmaximal, complemented
8#8216C212yesmaximal, complemented
9C4×#327C212yesmaximal, complemented
10#8023C211yesmaximal, characteristic, complemented
11#8019C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13C2×#520C411yescharacteristic, complemented
14C22×#117C2211yescharacteristic
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#405—21no—
18#2359—22no—
19#2358—22no—
20#2354—21no—
21#2352—21no—
22#2352—21no—
23D4×C4⋊C4—21no—
24#2324C412yescomplemented
25#2323C412yescomplemented
26#2317C411yescharacteristic, complemented
27#2311C411yescharacteristic, complemented
28C4×C22⋊Q8C2212yes—
29C4×#121C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32#2274C411yescharacteristic, complemented
33C2×C4×C4⋊C4—21no—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1750—22no—
36#1749C2212yes—
37C4×C22⋊C8C2212yes—
38#1729—21no—
39#1513C418yescomplemented
40#1501C418yescomplemented
41C2×#328C2211yescharacteristic
42C2×#328C2212yes—
43C2×#328C2211yescharacteristic
44C2×#327C2211yescharacteristic
45C2×#327C2212yes—
46C2×#327C2211yescharacteristic
47#1437—22no—
48C4⋊#95—21no—
49#1434C2212yes—
50C4×#95C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55C22×#41C2×C411yescharacteristic
56C24×C4C2311yescharacteristic
57C22×M16—21no—
58C23×C8—21no—
59#525—22no—
60#525—22no—
61#525C2×C414yes—
62#525C2×C414yes—
63C2×#123—21no—
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×C22⋊Q8C2311yescharacteristic
66C2×#121C2×C411yescharacteristic
67C2×#121C2311yescharacteristic
68C2×#120—21no—
69#520—44no—
70C2×C4×Q8C2×C412yes—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4C2×C412yes—
76C2×#117—21no—
77C2×#117—44no—
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C412yes—
82C2×#117D411yescharacteristic
83C2×#117D411yescharacteristic
84C2×#117D412yes—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4D411yescharacteristic
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4D412yes—
95C22×C42—21no—
96C22×C42—22no—
97C22×C42C2312yes—
98#407—22no—
99#407C2×C414yescomplemented
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8C2×C414yescomplemented
104C2×C22⋊C8C2311yescharacteristic
105C2×C22⋊C8C2311yescharacteristic
106C2×C22⋊C8C2312yes—
107#405—44no—
108C2×#97—22no—
109C2×C4×C8—22no—
110#391C2×C412yes—
111#390—22no—
112#390C2×C412yes—
113#389—22no—
114#389C2×C412yes—
115#389C2×C412yes—
116#388—22no—
117#388—22no—
118#388—22no—
119#388—22no—
120#387—22no—
121#387—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#385—22no—
126#385—22no—
127#385—22no—
128#384—21no—
129#382—21no—
130#380—21no—
131#380—21no—
132#380—22no—
133#380—21no—
134#380—21no—
135#380C2×C412yes—
136#380C2×C414yescomplemented
137#380C2312yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—44no—
142C4×C4⋊C4C2×C412yes—
143C4×C4⋊C4C2×C412yes—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4C2×C414yescomplemented
149C2×#95—21no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156#336—22no—
157#336—22no—
158#328C2×C414yescomplemented
159#328C2×C414yescomplemented
160#328C2×C418yescomplemented
161#327C2×C414yescomplemented
162#327C2×C414yescomplemented
163#327C2×C418yescomplemented
164C25C2×D411yescharacteristic
165C2×#41—22no—
166C2×#41—42no—
167C2×#41—42no—
168C2×#41C4214yescomplemented
169C22×Q8C22×C411yescharacteristic
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C412yes—
186C23×C4C22×C412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4C2411yescharacteristic
195C2×M16—24no—
196C2×M16—42no—
197C2×M16—42no—
198C22×C8—24no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—42no—
203C22×C8—42no—
204#123—24no—
205#123—24no—
206C22⋊Q8—22no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8C4214yescomplemented
210C22⋊Q8C22×C412yes—
211C22⋊Q8C22×C412yes—
212#121—22no—
213#121—21no—
214#121—21no—
215#121C4214yescomplemented
216#121C22×C412yes—
217#121C22×C412yes—
218#120—24no—
219#120—24no—
220C4×Q8—44no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—44no—
226C4×D4—48no—
227C4×D4—82no—
228C4×D4—82no—
229#117—22no—
230#117—24no—
231#117—24no—
232#117—44no—
233#117—44no—
234#117—42no—
235#117—42no—
236#117—44no—
237#117C4214yescomplemented
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22⋊C412yes—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C412yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—81no—
279C2×C22⋊C4—81no—
280C2×C22⋊C4—81no—
281C2×C22⋊C4—81no—
282C2×C22⋊C4C22⋊C412yes—
283C2×C22⋊C4C22⋊C412yes—
284C2×C22⋊C4C22⋊C412yes—
285C2×C22⋊C4C22⋊C412yes—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—44no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—82no—
297C2×C42C22×C412yes—
298C2×C42C22×C412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—22no—
306C22⋊C8C22×C412yes—
307C22⋊C8C22×C412yes—
308C22⋊C8C22×C414yes—
309#97—44no—
310C4×C8—44no—
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—22no—
322#95—44no—
323#95—44no—
324#95—82no—
325#95C22×C412yes—
326#95C22×C412yes—
327#95C22×C414yes—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—41no—
334C24—41no—
335C24—41no—
336C24—41no—
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C24C4×D412yes—
340C24C4×D412yes—
341C24C22×D411yescharacteristic
342#41—48no—
343#41—48no—
344C2×Q8—21no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8C2×C4212yes—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—81no—
362C2×D4—81no—
363C2×D4C2×C4212yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4C2×C4212yes—
427C22×C4C2×C4212yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4C2×C22⋊C412yes—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C412yes—
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4#11712yes—
435C22×C4#11712yes—
436C22×C4C4×D412yes—
437C22×C4C4×D412yes—
438C22×C4C4×D412yes—
439C22×C4C4×D412yes—
440C22×C4C4×D412yes—
441C22×C4C4×D412yes—
442C22×C4C23×C411yescharacteristic
443C22×C4C23×C411yescharacteristic
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448M16—48no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—41no—
453C2×C8—42no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—48no—
458C2×C8—42no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—48no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4#9914yes—
478C4⋊C4C2×C4212yes—
479C4⋊C4C2×C4212yes—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—84no—
499C42—24no—
500C42—24no—
501C42—24no—
502C42—42no—
503C42—42no—
504C42—48no—
505C42—44no—
506C42—82no—
507C42—82no—
508C42—88no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—81no—
541C23—81no—
542C23—81no—
543C23—81no—
544C23—81no—
545C23—81no—
546C23—81no—
547C23—81no—
548C23C4×C22⋊C412yes—
549C23C4×C22⋊C412yes—
550C23C22×C22⋊C411yescharacteristic
551C23C2×C4×D411yescharacteristic
552C23C2×C4×D411yescharacteristic
553C23#57611yescharacteristic
554C23#57711yescharacteristic
555C23#57711yescharacteristic
556C23#57811yescharacteristic
557Q8—42no—
558Q8—42no—
559D4—42no—
560D4—42no—
561D4—48no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—84no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—82no—
635C2×C4—81no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—81no—
639C2×C4—164no—
640C2×C4—162no—
641C2×C4—162no—
642C2×C4C4×C22⋊C412yes—
643C2×C4C4×C22⋊C412yes—
644C2×C4C2×#9914yes—
645C2×C4C2×#9912yes—
646C2×C4#41012yes—
647C2×C4C22×C4211yescharacteristic
648C2×C4C2×#11711yescharacteristic
649C2×C4C2×C4×D411yescharacteristic
650C2×C4C2×C4×D411yescharacteristic
651C8—42no—
652C8—42no—
653C8—44no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22C2×C4×C22⋊C411yescharacteristic
692C22C22×#9911yescharacteristic
693C22C2×#41011yescharacteristic
694C22#289711yescharacteristic
695C22#289811yescharacteristic
696C22#290111yescharacteristic
697C22#290111yescharacteristic
698C22#294611yescharacteristic
699C22#294711yescharacteristic
700C22#295111yescharacteristic
701C22#295211yescharacteristic
702C4—21no—
703C4—21no—
704C4—21no—
705C4—22no—
706C4—21no—
707C4—24no—
708C4—44no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—84no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4C4×#9914yes—
720C2—21no—
721C2—21no—
722C2—21no—
723C2—21no—
724C2—42no—
725C2—42no—
726C2—82no—
727C2#1760311yescharacteristic
728C2#1760511yescharacteristic
729C2#1760611yescharacteristic
730C2#1760711yescharacteristic
731C2C2×C4×#9911yescharacteristic
732C2#2040511yescharacteristic
733C2#2041511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20379C2semidirect
5. #13801C2semidirect
6. #8221C2semidirect
7. #8219C2semidirect
8. #8216C2semidirect
9. C4×#327C2semidirect
10. #8023C2semidirect
11. #8019C2semidirect
13. C2×#520C4semidirect
24. #2324C4semidirect
25. #2323C4semidirect
26. #2317C4semidirect
27. #2311C4semidirect
32. #2274C4semidirect
39. #1513C4semidirect
40. #1501C4semidirect
99. #407C2×C4semidirect
103. C2×C22⋊C8C2×C4semidirect
136. #380C2×C4semidirect
148. C4×C22⋊C4C2×C4semidirect
158. #328C2×C4semidirect
159. #328C2×C4semidirect
160. #328C2×C4semidirect
161. #327C2×C4semidirect
162. #327C2×C4semidirect
163. #327C2×C4semidirect
168. C2×#41C42semidirect
209. C22⋊Q8C42semidirect
215. #121C42semidirect
237. #117C42semidirect

Maximal subgroups

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

subgroupindex
#342022
#203792
C2×#15132
C2×#15012
#138012
#82212
#82192
#82162
C4×#3272
#80232
#80192