Finite Group Databasebeta

← #189533

Subgroups — #189533

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2929C211yesmaximal, characteristic
2#21874C212yesmaximal
3#20658C211yesmaximal, characteristic, complemented
4C2×#1808C211yesmaximal, characteristic, complemented
5#8610C214yesmaximal, complemented
6#8609C214yesmaximal, complemented
7#7983C212yesmaximal, complemented
8C22×#131C411yescharacteristic, complemented
9C24×C8C2211yescharacteristic
10C2×#525C2211yescharacteristic
11#2929—21no—
12#2929—21no—
13#2929—21no—
14#2929—21no—
15#2929—21no—
16#2929—21no—
17#2929—21no—
18#2929—21no—
19C2×#435C412yescomplemented
20C2×#435C2212yes—
21C2×C8×D4C412yescomplemented
22C2×C8×D4C2212yes—
23C2×#427—21no—
24C22×#107—21no—
25C2×#424C411yescharacteristic, complemented
26C2×#408C411yescharacteristic, complemented
27C2×#407C2211yescharacteristic
28C2×#405C411yescharacteristic, complemented
29#2280C2212yes—
30C2×#380C2212yes—
31#1996—22no—
32#1964—22no—
33#1898—24no—
34#1808—21no—
35#1808—21no—
36#1808—21no—
37#1808C2212yes—
38#1808C2212yes—
39#1808C2212yes—
40#1807—21no—
41#1807C2212yes—
42#1799C2218yes—
43#1735—21no—
44C2×#340—21no—
45#1289C2218yes—
46C22×#41C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C2×#131—21no—
49C2×#131—21no—
50C2×#131—21no—
51C2×#131—21no—
52C2×#131—44no—
53C22×M16C2×C411yescharacteristic
54C22×M16C2×C411yescharacteristic
55C22×M16C2×C412yescomplemented
56C23×C8—44no—
57C23×C8C2×C411yescharacteristic
58C23×C8C2×C411yescharacteristic
59C23×C8C2×C411yescharacteristic
60C23×C8C2×C411yescharacteristic
61C23×C8C2×C412yescomplemented
62C23×C8D411yescharacteristic
63C23×C8D411yescharacteristic
64C23×C8D411yescharacteristic
65C23×C8D411yescharacteristic
66C23×C8D411yescharacteristic
67C23×C8D411yescharacteristic
68C23×C8D411yescharacteristic
69C23×C8D411yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72#525—21no—
73#525—21no—
74#525—21no—
75#525—21no—
76#525—21no—
77#525—21no—
78#525—21no—
79#525—21no—
80C2×#123Q812yes—
81C2×C22⋊Q8D411yescharacteristic
82C2×C22⋊Q8Q811yescharacteristic
83C2×#121D411yescharacteristic
84C2×#121Q811yescharacteristic
85C2×#120D412yes—
86C2×C4×D4C2×C412yes—
87C2×C4×D4C2312yes—
88C2×#117—22no—
89C2×#117C2×C411yescharacteristic
90C22×C4⋊C4—22no—
91C22×C22⋊C4—22no—
92#435—22no—
93#435—22no—
94#435—22no—
95#435—22no—
96#435—22no—
97#435—22no—
98#435—22no—
99#435—22no—
100C8×D4—22no—
101C8×D4—22no—
102C8×D4—22no—
103C8×D4—22no—
104C8×D4—22no—
105C8×D4—22no—
106C8×D4—22no—
107C8×D4—22no—
108#427—44no—
109C2×#107—22no—
110C2×#107—22no—
111C2×#107—22no—
112C2×#107—44no—
113C2×#106—22no—
114#424—21no—
115#424—21no—
116#424—21no—
117#424—21no—
118C2×C4⋊C8C2×C412yescomplemented
119C2×C4⋊C8C2×C412yescomplemented
120C2×#99—28no—
121#408—21no—
122#408—21no—
123#408—21no—
124#408—21no—
125#407—21no—
126#407—21no—
127#407—21no—
128#407—21no—
129C2×C22⋊C8C2×C412yes—
130C2×C22⋊C8C2×C412yescomplemented
131C2×C22⋊C8C2×C412yescomplemented
132C2×C22⋊C8C2312yes—
133#405—21no—
134#405—21no—
135#405—21no—
136#405—21no—
137C2×#97C2×C412yescomplemented
138C2×C4×C8C2×C412yescomplemented
139#381—22no—
140#380—22no—
141#380—22no—
142#380—22no—
143#380—22no—
144#380C2314yes—
145#380C2314yes—
146#379—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C2×#95—22no—
150#342—28no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155C25C2×D411yescharacteristic
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—42no—
161C2×#41—42no—
162C22×Q8C4⋊C411yescharacteristic
163C22×D4C4⋊C412yes—
164C22×D4C4⋊C412yes—
165C22×D4C4⋊C411yescharacteristic
166C22×D4C2×D412yes—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×Q811yescharacteristic
179C23×C4C2×Q811yescharacteristic
180C23×C4#4111yescharacteristic
181C23×C4#4111yescharacteristic
182C23×C4#4111yescharacteristic
183C23×C4#4111yescharacteristic
184C23×C4C2411yescharacteristic
185#131—44no—
186#131—44no—
187#131—44no—
188#131—44no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—44no—
196C2×M16—42no—
197C2×M16—42no—
198C22×C8—24no—
199C22×C8—24no—
200C22×C8—24no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8—44no—
215C22×C8C22⋊C411yescharacteristic
216C22×C8C22⋊C411yescharacteristic
217C22×C8C22⋊C411yescharacteristic
218C22×C8C22⋊C411yescharacteristic
219C22×C8C22⋊C411yescharacteristic
220C22×C8C22⋊C411yescharacteristic
221C22×C8C22⋊C411yescharacteristic
222C22×C8C22⋊C411yescharacteristic
223C22×C8C22⋊C411yescharacteristic
224C22×C8C22⋊C411yescharacteristic
225C22×C8C22⋊C411yescharacteristic
226C22×C8C22⋊C411yescharacteristic
227C22×C8C22⋊C411yescharacteristic
228C22×C8C22⋊C411yescharacteristic
229C22×C8C22⋊C411yescharacteristic
230C22×C8C22⋊C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C22×C8C2×D411yescharacteristic
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D411yescharacteristic
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245#123—22no—
246#123—22no—
247#123—22no—
248#123—22no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257#121—21no—
258#121—21no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264#121—21no—
265#120—22no—
266#120—22no—
267#120—22no—
268#120—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—44no—
282#117—44no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4C4⋊C411yescharacteristic
289C2×C4⋊C4C4⋊C411yescharacteristic
290C2×C4⋊C4C4⋊C412yes—
291C2×C4⋊C4C2×Q812yes—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—84no—
298C2×C22⋊C4C4⋊C412yes—
299C2×C22⋊C4C4⋊C412yes—
300C2×C22⋊C4C4⋊C412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C22⋊C4C2×Q812yes—
303C2×C42—42no—
304C2×C42C22×C412yes—
305#107—44no—
306#107—44no—
307#107—44no—
308#106—44no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313#99—88no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321C22⋊C8—22no—
322#97—22no—
323#97—22no—
324C4×C8—22no—
325C4×C8—22no—
326#95—24no—
327#95—24no—
328#95—42no—
329#95—42no—
330C24—22no—
331C24—22no—
332C24—42no—
333C24—42no—
334C24C2×C4⋊C411yescharacteristic
335C24C2×C4⋊C411yescharacteristic
336C24C2×C4⋊C412yes—
337C24#12111yescharacteristic
338C24#12111yescharacteristic
339C24#12111yescharacteristic
340C24#12111yescharacteristic
341C24C22×D411yescharacteristic
342#41—42no—
343#41—44no—
344#41—44no—
345#41—42no—
346#41—42no—
347#41—42no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—42no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—42no—
362C2×D4—44no—
363C2×D4—44no—
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—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C4⋊C412yes—
399C22×C4C2×C4⋊C411yescharacteristic
400C22×C4C2×C4⋊C412yes—
401C22×C4C2×C4⋊C412yes—
402C22×C4C2×C4⋊C411yescharacteristic
403C22×C4#11711yescharacteristic
404C22×C4#11711yescharacteristic
405C22×C4#11711yescharacteristic
406C22×C4#11711yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4C22⋊Q811yescharacteristic
412C22×C4C22⋊Q811yescharacteristic
413C22×C4C22⋊Q811yescharacteristic
414C22×C4C22⋊Q811yescharacteristic
415C22×C4C22⋊Q811yescharacteristic
416C22×C4C22⋊Q811yescharacteristic
417C22×C4C22⋊Q811yescharacteristic
418C22×C4C22⋊Q811yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×Q811yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425M16—44no—
426M16—42no—
427M16—42no—
428M16—42no—
429M16—42no—
430M16—44no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—44no—
461C2×C8—44no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8#9914yescomplemented
465C2×C8C2×C22⋊C411yescharacteristic
466C2×C8C2×C22⋊C411yescharacteristic
467C2×C8C2×C22⋊C411yescharacteristic
468C2×C8C2×C22⋊C411yescharacteristic
469C2×C8C2×C22⋊C411yescharacteristic
470C2×C8C2×C22⋊C411yescharacteristic
471C2×C8C2×C22⋊C411yescharacteristic
472C2×C8C2×C22⋊C411yescharacteristic
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—84no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C42—22no—
503C42—22no—
504C42—84no—
505C23—22no—
506C23—22no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—48no—
530C23—48no—
531C23—42no—
532C23—42no—
533C23#38011yescharacteristic
534C23#38011yescharacteristic
535C23#38011yescharacteristic
536C23#38011yescharacteristic
537C23C22×C4⋊C411yescharacteristic
538C23C2×#12111yescharacteristic
539C23C2×#12111yescharacteristic
540C23#57612yes—
541C23#57812yes—
542Q8—42no—
543Q8—42no—
544D4—42no—
545D4—44no—
546D4—44no—
547D4—44no—
548D4—44no—
549D4—42no—
550D4—42no—
551D4—44no—
552D4—42no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
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—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
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—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—168no—
602C2×C4#38011yescharacteristic
603C2×C4#38011yescharacteristic
604C2×C4#38011yescharacteristic
605C2×C4#38011yescharacteristic
606C2×C4C2×#9912yes—
607C2×C4#41012yes—
608C2×C4C22×C22⋊C411yescharacteristic
609C2×C4C2×#11711yescharacteristic
610C2×C4C2×C22⋊Q811yescharacteristic
611C2×C4C2×C22⋊Q811yescharacteristic
612C8—21no—
613C8—21no—
614C8—21no—
615C8—21no—
616C8—21no—
617C8—21no—
618C8—42no—
619C8—42no—
620C8C2×#9912yes—
621C8C2×#9912yes—
622C22—24no—
623C22—24no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—84no—
652C22#179912yes—
653C22#181212yes—
654C22C2×#38011yescharacteristic
655C22#291312yes—
656C22#305811yescharacteristic
657C22#305911yescharacteristic
658C22#306112yes—
659C4—21no—
660C4—21no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—164no—
669C4—164no—
670C4#179912yes—
671C4C22×#9911yescharacteristic
672C4C2×#41011yescharacteristic
673C2—21no—
674C2—22no—
675C2—22no—
676C2—22no—
677C2—21no—
678C2—22no—
679C2—41no—
680C2—41no—
681C2—82no—
682C2C2×#179911yescharacteristic
683C2#2066311yescharacteristic
684C2C2×#181211yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20658C2semidirect
4. C2×#1808C2semidirect
5. #8610C2semidirect
6. #8609C2semidirect
7. #7983C2semidirect
8. C22×#131C4semidirect
19. C2×#435C4semidirect
21. C2×C8×D4C4semidirect
25. C2×#424C4semidirect
26. C2×#408C4semidirect
28. C2×#405C4semidirect
55. C22×M16C2×C4semidirect
61. C23×C8C2×C4semidirect
118. C2×C4⋊C8C2×C4semidirect
119. C2×C4⋊C8C2×C4semidirect
130. C2×C22⋊C8C2×C4semidirect
131. C2×C22⋊C8C2×C4semidirect
137. C2×#97C2×C4semidirect
138. C2×C4×C8C2×C4semidirect
464. C2×C8#99semidirect

Maximal subgroups

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

subgroupindex
C2×#29292
#218742
#206582
C2×#18082
#86102
#86092
#79832