Finite Group Databasebeta

← #188957

Subgroups — #188957

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2929C211yesmaximal, characteristic
2C2×#1810C211yesmaximal, characteristic, complemented
3C2×#1808C211yesmaximal, characteristic, complemented
4#17592C212yesmaximal
5#8628C214yesmaximal, complemented
6#8627C214yesmaximal, complemented
7#8107C212yesmaximal, complemented
8C22×#131C411yescharacteristic, complemented
9C24×C8C2211yescharacteristic
10C2×#525C2211yescharacteristic
11#2929—21no—
12#2929—21no—
13#2929—21no—
14#2929—21no—
15#2929C412yescomplemented
16#2929C412yescomplemented
17#2929C2212yes—
18#2929C2212yes—
19C2×#435—21no—
20C2×#435—21no—
21C2×C8×D4—21no—
22C2×C8×D4—21no—
23C22×#108—21no—
24C22×#107—21no—
25C2×#424C411yescharacteristic, complemented
26C2×#408C411yescharacteristic, complemented
27C2×#407C2212yes—
28C2×#407C2211yescharacteristic
29C2×#405C411yescharacteristic, complemented
30C2×#380C2212yes—
31#1810—21no—
32#1810—21no—
33#1810C2212yescomplemented
34#1810C2212yescomplemented
35#1808—21no—
36#1808—21no—
37#1808C2212yescomplemented
38#1808C2212yescomplemented
39C2×#341—21no—
40C2×#340—21no—
41#1520—22no—
42#1519—22no—
43#1513C2218yes—
44#1504—24no—
45#1305C2218yescomplemented
46C22×#41C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C2×#131—21no—
49C2×#131—21no—
50C2×#131—44no—
51C2×#131C2×C412yescomplemented
52C2×#131C2×C412yescomplemented
53C22×M16—21no—
54C22×M16—22no—
55C22×M16C2×C411yescharacteristic
56C22×M16C2×C411yescharacteristic
57C23×C8—21no—
58C23×C8—44no—
59C23×C8C2×C411yescharacteristic
60C23×C8C2×C411yescharacteristic
61C23×C8C2×C411yescharacteristic
62C23×C8C2×C411yescharacteristic
63C23×C8D411yescharacteristic
64C23×C8D411yescharacteristic
65C23×C8D411yescharacteristic
66C23×C8D411yescharacteristic
67C23×C8D411yescharacteristic
68C23×C8D411yescharacteristic
69C23×C8D411yescharacteristic
70C23×C8D411yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73#525—21no—
74#525—21no—
75#525—21no—
76#525—21no—
77#525C2×C412yes—
78#525D412yes—
79#525Q812yes—
80#525C2312yes—
81C2×#123—21no—
82C2×C22⋊Q8D411yescharacteristic
83C2×C22⋊Q8Q811yescharacteristic
84C2×#121D411yescharacteristic
85C2×#121Q811yescharacteristic
86C2×#120—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
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—
108C2×#108—22no—
109C2×#108—22no—
110C2×#108—44no—
111C2×#107—22no—
112C2×#107—22no—
113C2×#107—44no—
114#424—21no—
115#424—21no—
116#424C2×C412yescomplemented
117#424C2×C412yescomplemented
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120#408—21no—
121#408—21no—
122#408C2×C412yescomplemented
123#408C2×C412yescomplemented
124#407—21no—
125#407—21no—
126#407—22no—
127#407—22no—
128#407—22no—
129#407C2×C412yes—
130#407C2314yes—
131#407C2312yes—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—22no—
137C2×C22⋊C8—22no—
138#405—21no—
139#405—21no—
140#405C2×C412yescomplemented
141#405C2×C412yescomplemented
142C2×#97—21no—
143C2×C4×C8—21no—
144#380—22no—
145#380—22no—
146#380—22no—
147#380C2314yes—
148C2×#95—22no—
149#341—22no—
150#341—22no—
151#340—22no—
152#340—22no—
153#328—28no—
154#327—28no—
155C25#4111yescharacteristic
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—42no—
159C2×#41—42no—
160C2×#41C4⋊C412yes—
161C2×#41C22×C412yes—
162C22×Q8C4⋊C411yescharacteristic
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C4⋊C411yescharacteristic
167C23×C4—21no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×Q811yescharacteristic
180C23×C4C2×Q811yescharacteristic
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—22no—
192C2×M16—22no—
193C2×M16—24no—
194C2×M16—44no—
195C2×M16—42no—
196C2×M16—42no—
197C2×M16—42no—
198C2×M16—44no—
199C2×M16—44no—
200C2×M16C22×C412yes—
201C2×M16C22×C412yes—
202C22×C8—24no—
203C22×C8—24no—
204C22×C8—24no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—42no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8—42no—
219C22×C8—42no—
220C22×C8—44no—
221C22×C8—42no—
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⋊C411yescharacteristic
234C22×C8C22⋊C411yescharacteristic
235C22×C8C22⋊C411yescharacteristic
236C22×C8C22⋊C411yescharacteristic
237C22×C8C22⋊C411yescharacteristic
238C22×C8C22×C411yescharacteristic
239C22×C8C22×C411yescharacteristic
240C22×C8C22×C411yescharacteristic
241C22×C8C22×C411yescharacteristic
242C22×C8C2×D411yescharacteristic
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C22×C8C2×D411yescharacteristic
248C22×C8C2×D411yescharacteristic
249C22×C8C2×D411yescharacteristic
250#123—22no—
251#123—22no—
252#123—22no—
253#123—22no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8C4⋊C412yes—
259C22⋊Q8C4⋊C412yes—
260C22⋊Q8C2×D412yes—
261C22⋊Q8C2×Q812yes—
262#121—21no—
263#121—21no—
264#121—21no—
265#121—21no—
266#121C4⋊C412yes—
267#121C4⋊C412yes—
268#121C2×D412yes—
269#121C2×Q812yes—
270#120—22no—
271#120—22no—
272#120—22no—
273#120—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282#117—21no—
283#117—21no—
284#117C4⋊C412yes—
285#117C22×C412yes—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4C4⋊C411yescharacteristic
293C2×C4⋊C4C4⋊C411yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—84no—
302C2×C42—21no—
303#108—44no—
304#108—44no—
305#107—44no—
306#107—44no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—22no—
319C22⋊C8—24no—
320C22⋊C8—24no—
321C22⋊C8—42no—
322C22⋊C8—42no—
323#97—22no—
324#97—22no—
325C4×C8—22no—
326C4×C8—22no—
327#95—24no—
328#95—42no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—42no—
333C24—42no—
334C24#11711yescharacteristic
335C24#11711yescharacteristic
336C24#12111yescharacteristic
337C24#12111yescharacteristic
338C24C22⋊Q811yescharacteristic
339C24C22⋊Q811yescharacteristic
340C24C2×#4111yescharacteristic
341#41—44no—
342#41—44no—
343#41—42no—
344#41—42no—
345#41—42no—
346#41—42no—
347C2×Q8—21no—
348C2×Q8—42no—
349C2×Q8C2×C4⋊C412yes—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4C2×C4⋊C412yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C4⋊C411yescharacteristic
400C22×C4C2×C4⋊C411yescharacteristic
401C22×C4C2×C4⋊C411yescharacteristic
402C22×C4C2×C4⋊C411yescharacteristic
403C22×C4#11711yescharacteristic
404C22×C4#11711yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12111yescharacteristic
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×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×Q811yescharacteristic
422C22×C4C2×#4111yescharacteristic
423M16—44no—
424M16—42no—
425M16—42no—
426M16—44no—
427M16—42no—
428M16—42no—
429M16—44no—
430M16—44no—
431M16—84no—
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—22no—
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—41no—
461C2×C8—44no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—44no—
465C2×C8—44no—
466C2×C8—84no—
467C2×C8#9914yes—
468C2×C8C2×C22⋊C411yescharacteristic
469C2×C8C2×C22⋊C411yescharacteristic
470C2×C8C2×C22⋊C411yescharacteristic
471C2×C8C2×C22⋊C411yescharacteristic
472C2×C8C2×C22⋊C411yescharacteristic
473C2×C8C2×C22⋊C411yescharacteristic
474C2×C8C2×C22⋊C411yescharacteristic
475C2×C8C2×C22⋊C411yescharacteristic
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—84no—
485C4⋊C4C2×C4⋊C412yes—
486C4⋊C4C2×C4⋊C412yes—
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—22no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C42—22no—
500C42—22no—
501C23—22no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—48no—
525C23—48no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23#38011yescharacteristic
530C23#38011yescharacteristic
531C23#38011yescharacteristic
532C23#38011yescharacteristic
533C23#47212yes—
534C23#47312yes—
535C23C2×#11711yescharacteristic
536C23C2×#12111yescharacteristic
537C23C2×C22⋊Q811yescharacteristic
538Q8—42no—
539Q8—42no—
540D4—42no—
541D4—44no—
542D4—44no—
543D4—44no—
544D4—44no—
545D4—44no—
546D4—42no—
547D4—42no—
548D4—42no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
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—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
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—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
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×C4C22×C4⋊C411yescharacteristic
603C2×C4C2×#12111yescharacteristic
604C2×C4C2×C22⋊Q811yescharacteristic
605C8—21no—
606C8—21no—
607C8—21no—
608C8—21no—
609C8—21no—
610C8—21no—
611C8—42no—
612C8—42no—
613C8—84no—
614C8C2×#9912yes—
615C8C2×#9912yes—
616C22—24no—
617C22—24no—
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—21no—
633C22—21no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—84no—
646C22#179912yes—
647C22#181412yes—
648C22#181612yes—
649C22#181612yes—
650C22C2×#38011yescharacteristic
651C22C2×#47211yescharacteristic
652C22C2×#47311yescharacteristic
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—164no—
663C4#179912yes—
664C4C22×#9911yescharacteristic
665C4C2×#41011yescharacteristic
666C2—21no—
667C2—22no—
668C2—22no—
669C2—21no—
670C2—22no—
671C2—22no—
672C2—41no—
673C2—41no—
674C2—82no—
675C2C2×#179911yescharacteristic
676C2C2×#181411yescharacteristic
677C2C2×#181611yescharacteristic
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
2. C2×#1810C2semidirect
3. C2×#1808C2semidirect
5. #8628C2semidirect
6. #8627C2semidirect
7. #8107C2semidirect
8. C22×#131C4semidirect
15. #2929C4semidirect
16. #2929C4semidirect
25. C2×#424C4semidirect
26. C2×#408C4semidirect
29. C2×#405C4semidirect
33. #1810C22semidirect
34. #1810C22semidirect
37. #1808C22semidirect
38. #1808C22semidirect
45. #1305C22semidirect
51. C2×#131C2×C4semidirect
52. C2×#131C2×C4semidirect
116. #424C2×C4semidirect
117. #424C2×C4semidirect
122. #408C2×C4semidirect
123. #408C2×C4semidirect
140. #405C2×C4semidirect
141. #405C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29292
C2×#18102
C2×#18082
#175922
#86282
#86272
#81072