Finite Group Databasebeta

← #189427

Subgroups — #189427

Order 512.

934 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60214C211yesmaximal, characteristic
2C4×#428C211yesmaximal, characteristic, complemented
3#17797C211yesmaximal, characteristic, complemented
4#17537C211yesmaximal, characteristic
5C2×#1509C211yesmaximal, characteristic
6#8697C212yesmaximal, complemented
7#8696C212yesmaximal, complemented
8#8695C212yesmaximal, complemented
9#8694C212yesmaximal, complemented
10#7950C211yesmaximal, characteristic, complemented
11#7948C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13#2969C412yescomplemented
14#2969C412yescomplemented
15#2968—22no—
16#2967—22no—
17#2966C2212yes—
18#2966C2212yes—
19Q8×M16C411yescharacteristic, complemented
20Q8×M16C411yescharacteristic, complemented
21D4×M16—21no—
22D4×M16—21no—
23D4×M16C411yescharacteristic, complemented
24D4×M16C411yescharacteristic, complemented
25#2921—21no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#428—21no—
30#2873—21no—
31C2×C4×M16C2211yescharacteristic
32C4×#117C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1776—22no—
35C4×#107C2212yescomplemented
36C4×#106C2212yescomplemented
37#1736—21no—
38#1569—22no—
39#1568C2212yescomplemented
40#1568C2212yescomplemented
41#1565—21no—
42#1565—21no—
43#1509—21no—
44#1509—21no—
45#1509C2212yes—
46#1509C2212yes—
47#1509C2211yescharacteristic
48#1509C2211yescharacteristic
49#1509C2211yescharacteristic
50#1509C2211yescharacteristic
51#1508—21no—
52#1507—21no—
53#1500—21no—
54#1500—21no—
55C2×#328—21no—
56C2×#327—21no—
57#1285C2212yescomplemented
58#1285C2212yescomplemented
59#1285C2212yescomplemented
60#1285C2212yescomplemented
61C22×M16—21no—
62C22×M16—21no—
63C22×M16D411yescharacteristic
64C22×M16D411yescharacteristic
65#529—21no—
66#529—21no—
67#529—21no—
68#529—21no—
69#529C2×C411yescharacteristic
70#529C2×C411yescharacteristic
71#529D411yescharacteristic
72#529D411yescharacteristic
73#529Q811yescharacteristic
74#529Q811yescharacteristic
75#529C2311yescharacteristic
76#529C2311yescharacteristic
77C2×#126—21no—
78C2×#125Q811yescharacteristic
79C2×#124Q811yescharacteristic
80C2×#123—21no—
81C2×C22⋊Q8D411yescharacteristic
82C2×#121D411yescharacteristic
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×#117—21no—
87C2×#117—21no—
88C22×C4⋊C4—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#446—22no—
92#446—22no—
93#446C2×C412yescomplemented
94#446C2×C412yescomplemented
95C8×Q8C2×C412yescomplemented
96C8×Q8C2×C412yescomplemented
97#436—22no—
98#436—22no—
99#436C2×C412yescomplemented
100#436C2×C412yescomplemented
101#435—22no—
102#435—22no—
103#435—22no—
104#435—22no—
105#435—22no—
106#435—22no—
107C8×D4—22no—
108C8×D4—22no—
109C8×D4C2×C412yescomplemented
110C8×D4C2×C412yescomplemented
111#433—22no—
112#433—22no—
113#433—22no—
114#433—22no—
115#432—21no—
116#432—21no—
117#432—21no—
118#432—21no—
119#432—21no—
120#432C2312yes—
121#432C2311yescharacteristic
122#432C2311yescharacteristic
123#432C2312yes—
124#431—21no—
125#431—21no—
126#431—21no—
127#431C2×C412yes—
128#428—28no—
129C2×#107—22no—
130C2×#106—22no—
131#423—21no—
132#423—21no—
133#423—21no—
134#423C2×C411yescharacteristic
135#423C2×C411yescharacteristic
136C2×C4⋊C8—21no—
137#407—21no—
138#407—21no—
139#407—21no—
140#407—21no—
141C2×C22⋊C8—21no—
142#405—22no—
143#405—22no—
144#405—22no—
145#405—22no—
146C4×M16—21no—
147C4×M16—21no—
148C4×M16—21no—
149C4×M16—21no—
150C4×M16C2×C411yescharacteristic
151C4×M16C2×C411yescharacteristic
152C4×M16C2×C412yes—
153C4×M16D411yescharacteristic
154C4×M16D411yescharacteristic
155C4×M16D412yes—
156C2×#97—21no—
157C2×#97D412yes—
158C2×C4×C8C2312yes—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4C2312yes—
164C4×C4⋊C4C2311yescharacteristic
165C4×C4⋊C4C2311yescharacteristic
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168#376—21no—
169C2×#95—21no—
170C43—21no—
171#340—22no—
172#340—22no—
173#335—22no—
174#335—22no—
175#334—22no—
176#334—22no—
177#328—22no—
178#328—22no—
179#328—21no—
180#328—21no—
181#328—21no—
182#328—21no—
183#327—22no—
184#327—22no—
185#327—21no—
186#327—21no—
187#327—21no—
188#327—21no—
189C22×Q8C4⋊C411yescharacteristic
190C22×D4—21no—
191C22×D4C4⋊C411yescharacteristic
192C23×C4—21no—
193C23×C4—41no—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4#4111yescharacteristic
197C2×M16—24no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—42no—
203C2×M16—44no—
204C2×M16—44no—
205C2×M16—41no—
206C2×M16C22⋊C411yescharacteristic
207C2×M16C22⋊C411yescharacteristic
208C2×M16C22⋊C411yescharacteristic
209C2×M16C22⋊C412yes—
210C2×M16C22⋊C412yes—
211C2×M16C22⋊C411yescharacteristic
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—41no—
215C22×C8C2×D412yes—
216C22×C8C2×D412yes—
217#126—21no—
218#126—21no—
219#126—21no—
220#126—21no—
221#126—21no—
222#126—21no—
223#126—21no—
224#126—21no—
225#125—21no—
226#125—21no—
227#125C4⋊C411yescharacteristic
228#125C4⋊C411yescharacteristic
229#125C2×Q811yescharacteristic
230#125C2×Q811yescharacteristic
231#124—21no—
232#124—21no—
233#124C4⋊C411yescharacteristic
234#124C4⋊C411yescharacteristic
235#124C2×Q811yescharacteristic
236#124C2×Q811yescharacteristic
237#123—21no—
238#123—21no—
239#123—21no—
240#123—21no—
241#123—21no—
242#123—21no—
243#123—21no—
244#123—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8C4⋊C411yescharacteristic
248C22⋊Q8C4⋊C411yescharacteristic
249C22⋊Q8C2×D411yescharacteristic
250C22⋊Q8C2×D411yescharacteristic
251#121—21no—
252#121—21no—
253#121C4⋊C411yescharacteristic
254#121C4⋊C411yescharacteristic
255#121C2×D411yescharacteristic
256#121C2×D411yescharacteristic
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8C4⋊C411yescharacteristic
260C4×Q8C4⋊C411yescharacteristic
261C4×Q8C22×C411yescharacteristic
262C4×Q8C22×C411yescharacteristic
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4C4⋊C411yescharacteristic
274C4×D4C4⋊C411yescharacteristic
275C4×D4C22×C411yescharacteristic
276C4×D4C22×C411yescharacteristic
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—24no—
286#117—42no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—41no—
297C2×C4⋊C4C4⋊C411yescharacteristic
298C2×C4⋊C4C4⋊C411yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—41no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—22no—
309C2×C42—41no—
310C2×C42—41no—
311C2×C42—41no—
312C2×C42—82no—
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×Q811yescharacteristic
320C2×C42C2×Q811yescharacteristic
321C2×C42#4111yescharacteristic
322C2×C42#4111yescharacteristic
323C2×C42#4111yescharacteristic
324C2×C42C2411yescharacteristic
325#107—28no—
326#106—28no—
327C4⋊C8—22no—
328C4⋊C8—22no—
329C4⋊C8—22no—
330C4⋊C8—22no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—22no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—41no—
337C4⋊C8—41no—
338C4⋊C8C22×C412yes—
339C4⋊C8C22×C412yes—
340C22⋊C8—22no—
341C22⋊C8—22no—
342C22⋊C8—22no—
343C22⋊C8—22no—
344C22⋊C8—22no—
345C22⋊C8—22no—
346C22⋊C8—22no—
347C22⋊C8—21no—
348C22⋊C8—21no—
349C22⋊C8—41no—
350C22⋊C8—41no—
351#97—22no—
352#97—22no—
353#97—22no—
354#97—22no—
355#97—21no—
356#97—21no—
357#97—41no—
358#97C22⋊C412yes—
359#97C22⋊C412yes—
360#97C22⋊C412yes—
361#97C22⋊C412yes—
362C4×C8—22no—
363C4×C8—22no—
364C4×C8—22no—
365C4×C8—41no—
366C4×C8C22×C412yes—
367C4×C8C22×C412yes—
368C4×C8C2×D412yes—
369C4×C8C2×D412yes—
370#95—22no—
371#95—21no—
372#95—21no—
373#95—41no—
374#95—41no—
375C24—21no—
376C24#12111yescharacteristic
377C2×Q8—21no—
378C2×Q8—81no—
379C2×Q8C2×C4⋊C411yescharacteristic
380C2×Q8C2×C4⋊C411yescharacteristic
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—81no—
387C2×D4—81no—
388C2×D4—81no—
389C2×D4C2×C4⋊C411yescharacteristic
390C2×D4C2×C4⋊C411yescharacteristic
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—81no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C4⋊C411yescharacteristic
414C22×C4C2×C4⋊C411yescharacteristic
415C22×C4#11711yescharacteristic
416C22×C4#11711yescharacteristic
417C22×C4#12111yescharacteristic
418C22×C4#12111yescharacteristic
419C22×C4#12111yescharacteristic
420C22×C4#12111yescharacteristic
421C22×C4#12111yescharacteristic
422C22×C4C22⋊Q811yescharacteristic
423C22×C4C22⋊Q811yescharacteristic
424C22×C4C22⋊Q811yescharacteristic
425C22×C4C22⋊Q811yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4111yescharacteristic
429M16—22no—
430M16—22no—
431M16—22no—
432M16—24no—
433M16—42no—
434M16—44no—
435M16—42no—
436M16—42no—
437M16—44no—
438M16—82no—
439M16#10414yes—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—42no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—42no—
449C2×C8—82no—
450C2×C8C2×C22⋊C412yes—
451C2×C8C2×C22⋊C412yes—
452C2×C8C2×C22⋊C412yes—
453C2×C8C2×C22⋊C412yes—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4C2×C4⋊C411yescharacteristic
480C4⋊C4C2×C4⋊C411yescharacteristic
481C4⋊C4C2×C4⋊C411yescharacteristic
482C4⋊C4C2×C4⋊C411yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—44no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—44no—
509C42—81no—
510C42—81no—
511C42—82no—
512C42—82no—
513C42—82no—
514C42C2×C22⋊C411yescharacteristic
515C42C2×C22⋊C411yescharacteristic
516C42C2×C4⋊C411yescharacteristic
517C42C2×C4⋊C411yescharacteristic
518C42#11711yescharacteristic
519C42#11711yescharacteristic
520C42#12111yescharacteristic
521C42#12111yescharacteristic
522C42C22⋊Q811yescharacteristic
523C42C22⋊Q811yescharacteristic
524C42C22⋊Q811yescharacteristic
525C42C22⋊Q811yescharacteristic
526C42C23×C411yescharacteristic
527C42C22×D411yescharacteristic
528C42C22×Q811yescharacteristic
529C42C2×#4111yescharacteristic
530C23—22no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—41no—
536C23—42no—
537C23—81no—
538C23—81no—
539C23#38011yescharacteristic
540C23#38011yescharacteristic
541C23C2×#12111yescharacteristic
542Q8—41no—
543Q8—41no—
544Q8—81no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—81no—
552D4—81no—
553D4—81no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
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—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—82no—
596C2×C4—88no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4#38011yescharacteristic
601C2×C4#38011yescharacteristic
602C2×C4#38011yescharacteristic
603C2×C4#38011yescharacteristic
604C2×C4#38011yescharacteristic
605C2×C4#38011yescharacteristic
606C2×C4C2×#10412yes—
607C2×C4#42112yes—
608C2×C4C22×C22⋊C411yescharacteristic
609C2×C4C22×C4⋊C411yescharacteristic
610C2×C4C2×#11711yescharacteristic
611C2×C4C2×#12111yescharacteristic
612C2×C4C2×C22⋊Q811yescharacteristic
613C2×C4C2×C22⋊Q811yescharacteristic
614C8—22no—
615C8—22no—
616C8—22no—
617C8—42no—
618C8—42no—
619C8—82no—
620C8—81no—
621C8—81no—
622C8C2×#10414yes—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—42no—
631C22—42no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—162no—
639C22#180612yes—
640C22C2×#38011yescharacteristic
641C22#306811yescharacteristic
642C22#306911yescharacteristic
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4#180612yes—
663C4C22×#10411yescharacteristic
664C4C2×#42111yescharacteristic
665C2—21no—
666C2—21no—
667C2—22no—
668C2—41no—
669C2—81no—
670C2—81no—
671C2C2×#180611yescharacteristic
672C2#2067111yescharacteristic
673C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#428C2semidirect
3. #17797C2semidirect
6. #8697C2semidirect
7. #8696C2semidirect
8. #8695C2semidirect
9. #8694C2semidirect
10. #7950C2semidirect
11. #7948C2semidirect
13. #2969C4semidirect
14. #2969C4semidirect
19. Q8×M16C4semidirect
20. Q8×M16C4semidirect
23. D4×M16C4semidirect
24. D4×M16C4semidirect
35. C4×#107C22semidirect
36. C4×#106C22semidirect
39. #1568C22semidirect
40. #1568C22semidirect
57. #1285C22semidirect
58. #1285C22semidirect
59. #1285C22semidirect
60. #1285C22semidirect
93. #446C2×C4semidirect
94. #446C2×C4semidirect
95. C8×Q8C2×C4semidirect
96. C8×Q8C2×C4semidirect
99. #436C2×C4semidirect
100. #436C2×C4semidirect
109. C8×D4C2×C4semidirect
110. C8×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602142
C4×#4282
#177972
#175372
C2×#15092
#86972
#86962
#86952
#86942
#79502
#79482