Finite Group Databasebeta

← #189531

Subgroups — #189531

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 674 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×#1807C211yesmaximal, characteristic, complemented
5#8610C214yesmaximal, complemented
6#8608C214yesmaximal, 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×#106—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#1808C2212yes—
36#1807—21no—
37#1807—21no—
38#1807—21no—
39#1807C2212yes—
40#1807C2212yes—
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×#106—22no—
111C2×#106—22no—
112C2×#106—22no—
113C2×#106—44no—
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#106—44no—
307#106—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—24no—
331C24—42no—
332C24—42no—
333C24C2×C4⋊C411yescharacteristic
334C24C2×C4⋊C411yescharacteristic
335C24C2×C4⋊C412yes—
336C24#12111yescharacteristic
337C24#12111yescharacteristic
338C24#12111yescharacteristic
339C24#12111yescharacteristic
340C24C22×D411yescharacteristic
341#41—42no—
342#41—44no—
343#41—44no—
344#41—42no—
345#41—42no—
346#41—42no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—42no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—28no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—44no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C4⋊C412yes—
397C22×C4C2×C4⋊C411yescharacteristic
398C22×C4C2×C4⋊C412yes—
399C22×C4C2×C4⋊C412yes—
400C22×C4C2×C4⋊C411yescharacteristic
401C22×C4#11711yescharacteristic
402C22×C4#11711yescharacteristic
403C22×C4#11711yescharacteristic
404C22×C4#11711yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12111yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4C22⋊Q811yescharacteristic
410C22×C4C22⋊Q811yescharacteristic
411C22×C4C22⋊Q811yescharacteristic
412C22×C4C22⋊Q811yescharacteristic
413C22×C4C22⋊Q811yescharacteristic
414C22×C4C22⋊Q811yescharacteristic
415C22×C4C22⋊Q811yescharacteristic
416C22×C4C22⋊Q811yescharacteristic
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×Q811yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423M16—44no—
424M16—42no—
425M16—42no—
426M16—42no—
427M16—42no—
428M16—44no—
429C2×C8—22no—
430C2×C8—22no—
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—41no—
447C2×C8—41no—
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—44no—
459C2×C8—44no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8#9914yescomplemented
463C2×C8C2×C22⋊C411yescharacteristic
464C2×C8C2×C22⋊C411yescharacteristic
465C2×C8C2×C22⋊C411yescharacteristic
466C2×C8C2×C22⋊C411yescharacteristic
467C2×C8C2×C22⋊C411yescharacteristic
468C2×C8C2×C22⋊C411yescharacteristic
469C2×C8C2×C22⋊C411yescharacteristic
470C2×C8C2×C22⋊C411yescharacteristic
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—84no—
484C22⋊C4—22no—
485C22⋊C4—22no—
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—44no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C42—22no—
501C42—22no—
502C42—84no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—416no—
526C23—42no—
527C23—42no—
528C23#38011yescharacteristic
529C23#38011yescharacteristic
530C23#38011yescharacteristic
531C23#38011yescharacteristic
532C23C22×C4⋊C411yescharacteristic
533C23C2×#12111yescharacteristic
534C23C2×#12111yescharacteristic
535C23#57714yes—
536Q8—42no—
537Q8—42no—
538D4—42no—
539D4—44no—
540D4—44no—
541D4—44no—
542D4—44no—
543D4—44no—
544D4—44no—
545D4—42no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
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—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—168no—
595C2×C4#38011yescharacteristic
596C2×C4#38011yescharacteristic
597C2×C4#38011yescharacteristic
598C2×C4#38011yescharacteristic
599C2×C4C2×#9912yes—
600C2×C4#41012yes—
601C2×C4C22×C22⋊C411yescharacteristic
602C2×C4C2×#11711yescharacteristic
603C2×C4C2×C22⋊Q811yescharacteristic
604C2×C4C2×C22⋊Q811yescharacteristic
605C8—21no—
606C8—21no—
607C8—21no—
608C8—21no—
609C8—21no—
610C8—21no—
611C8—42no—
612C8—42no—
613C8C2×#9912yes—
614C8C2×#9912yes—
615C22—28no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—84no—
642C22#179912yes—
643C22#181212yes—
644C22C2×#38011yescharacteristic
645C22#291212yes—
646C22#305712yes—
647C22#306012yes—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—164no—
658C4—164no—
659C4#179912yes—
660C4C22×#9911yescharacteristic
661C4C2×#41011yescharacteristic
662C2—22no—
663C2—22no—
664C2—22no—
665C2—22no—
666C2—22no—
667C2—41no—
668C2—41no—
669C2—82no—
670C2C2×#179911yescharacteristic
671C2#2066211yescharacteristic
672C2C2×#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×#1807C2semidirect
5. #8610C2semidirect
6. #8608C2semidirect
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
462. 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×#18072
#86102
#86082
#79832