Finite Group Databasebeta

← #192365

Subgroups — #192365

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34205C211yesmaximal, characteristic
2C2×#1874C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic, complemented
4C2×#1519C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic, complemented
6#8869C214yesmaximal, complemented
7#8866C214yesmaximal, complemented
8#8023C211yesmaximal, characteristic, complemented
9#7907C211yesmaximal, characteristic, complemented
10C22×#121C2211yescharacteristic
11C2×#518—21no—
12C23×C4⋊C4C2211yescharacteristic
13C22×#107—21no—
14C22×#102—21no—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2364—22no—
18#2361—21no—
19#2361—21no—
20#2344—21no—
21#2342—22no—
22D4×C22⋊C4—21no—
23#2325C2212yes—
24#2325C2212yes—
25#2308—21no—
26C4×#121—21no—
27C4×#121—21no—
28C2×#389—21no—
29C2×#385—21no—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1874—22no—
34#1874C2214yes—
35#1852—22no—
36#1852C2214yescomplemented
37C2×#340—21no—
38C2×#340—21no—
39#1519C418yescomplemented
40C2×#327C2214yes—
41#1444C2212yescomplemented
42#1442C2212yescomplemented
43C4⋊#95—21no—
44C22⋊#95—21no—
45#1294C2214yescomplemented
46#1280C2212yescomplemented
47#1280C2212yescomplemented
48C23×D4—21no—
49C23×D4D411yescharacteristic
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8—21no—
53C23×C8—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121C2×C414yes—
59C2×#121D412yes—
60C2×#121Q814yes—
61C2×#121C2312yes—
62#518—44no—
63#518—44no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×#117—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4D411yescharacteristic
88C22×C22⋊C4D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C2×#107—24no—
92C2×#107—44no—
93C2×#102—24no—
94C2×#102—44no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8C2×C414yescomplemented
101C2×C22⋊C8C2312yes—
102C2×C22⋊C8C2312yes—
103#390—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—44no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#387—22no—
113#387—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#385—22no—
121#385—22no—
122#385—44no—
123#381—22no—
124#380—21no—
125#380—21no—
126#380—22no—
127#380C2×C414yescomplemented
128#380D412yes—
129#380C2312yes—
130#379—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C2×#95—21no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146#340—24no—
147#340—24no—
148#327C2×C4116yescomplemented
149C25—21no—
150C25C2×D411yescharacteristic
151C22×D4—24no—
152C22×D4—22no—
153C22×D4—24no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4C4⋊C414yescomplemented
164C22×D4C2×D412yes—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—82no—
180C23×C4C22×C412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×Q812yes—
191C23×C4#4112yes—
192C23×C4#4112yes—
193C23×C4C2411yescharacteristic
194C22×C8—24no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8—42no—
202#121—24no—
203#121—24no—
204#121—24no—
205#121—24no—
206#121—24no—
207#121—22no—
208#121—22no—
209#121C4⋊C418yescomplemented
210#121C22×C414yes—
211#121C2×Q814yes—
212C4×D4—48no—
213C4×D4—48no—
214C4×D4—44no—
215C4×D4—44no—
216C4×D4—44no—
217C4×D4—44no—
218C4×D4—82no—
219C4×D4—82no—
220#117—48no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4C4⋊C414yescomplemented
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4D814yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—42no—
285C2×C42—42no—
286#107—48no—
287#102—48no—
288C22⋊C8—24no—
289C22⋊C8—22no—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—42no—
293C22⋊C8C22×C414yes—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—22no—
305#95—44no—
306#95—44no—
307#95—82no—
308#95C22×C414yes—
309C24—22no—
310C24—24no—
311C24—41no—
312C24—41no—
313C24—41no—
314C24—41no—
315C24—42no—
316C24—42no—
317C24—81no—
318C24—81no—
319C24—81no—
320C24—81no—
321C24C4×D412yes—
322C24#12011yescharacteristic
323C24#12011yescharacteristic
324C24C22⋊Q812yes—
325C24C2×D812yes—
326C24#13512yes—
327C24AGL(1,ℤ/8)12yes—
328C24#13712yes—
329C24C22×D411yescharacteristic
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—28no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—48no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—84no—
351C2×D4C2×C4⋊C414yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—88no—
421C22×C4—88no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4C2×C4⋊C412yes—
427C22×C4C2×C4⋊C412yes—
428C22×C4C4×D412yes—
429C22×C4C4×D412yes—
430C22×C4C4×D412yes—
431C22×C4#12011yescharacteristic
432C22×C4#12011yescharacteristic
433C22×C4C22⋊Q812yes—
434C22×C4C22⋊Q812yes—
435C22×C4C22⋊Q812yes—
436C22×C4#12312yes—
437C22×C4#12312yes—
438C22×C4C2×D812yes—
439C22×C4C2×D812yes—
440C22×C4C23×C411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×Q811yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C2×C8—24no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—48no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—84no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—88no—
470C4⋊C4C2×C4⋊C414yes—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—28no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—48no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C42—44no—
489C42—44no—
490C42—42no—
491C42—42no—
492C42—81no—
493C42—81no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—24no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—48no—
526C23—84no—
527C23—84no—
528C23—162no—
529C23—162no—
530C23—162no—
531C23—162no—
532C23#38512yes—
533C23#38512yes—
534C23C4×D814yes—
535C23#44014yes—
536C23#44712yes—
537C23#44712yes—
538C23#45212yes—
539C23#45212yes—
540C23#47414yes—
541C23#47814yes—
542C23C2×C4×D411yescharacteristic
543C23C2×#12011yescharacteristic
544C23C2×C22⋊Q811yescharacteristic
545C23C22×D811yescharacteristic
546C23C2×#13511yescharacteristic
547C23C2×AGL(1,ℤ/8)11yescharacteristic
548C23C2×#13711yescharacteristic
549D4—44no—
550D4—44no—
551D4—416no—
552D4—44no—
553D4—44no—
554D4—44no—
555D4—48no—
556D4—82no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—88no—
618C2×C4—88no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—164no—
624C2×C4—164no—
625C2×C4—162no—
626C2×C4—162no—
627C2×C4#38512yes—
628C2×C4#38512yes—
629C2×C4B(4,𝔽2)12yes—
630C2×C4#45812yes—
631C2×C4C22×C4⋊C411yescharacteristic
632C2×C4C2×C4×D411yescharacteristic
633C2×C4C2×C22⋊Q811yescharacteristic
634C2×C4C2×#12311yescharacteristic
635C8—44no—
636C8—82no—
637C22—21no—
638C22—21no—
639C22—22no—
640C22—22no—
641C22—21no—
642C22—21no—
643C22—22no—
644C22—22no—
645C22—21no—
646C22—21no—
647C22—22no—
648C22—22no—
649C22—21no—
650C22—21no—
651C22—22no—
652C22—22no—
653C22—21no—
654C22—22no—
655C22—21no—
656C22—22no—
657C22—21no—
658C22—22no—
659C22—21no—
660C22—22no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—168no—
673C22#186214yes—
674C22#186214yes—
675C22C2×#38511yescharacteristic
676C22C2×C4×D811yescharacteristic
677C22C2×#44011yescharacteristic
678C22C2×#44711yescharacteristic
679C22C2×#45211yescharacteristic
680C22#302111yescharacteristic
681C22#302211yescharacteristic
682C22C2×B(4,𝔽2)11yescharacteristic
683C22C2×#45811yescharacteristic
684C22C2×#47411yescharacteristic
685C22C2×#47811yescharacteristic
686C4—21no—
687C4—21no—
688C4—21no—
689C4—21no—
690C4—22no—
691C4—42no—
692C4—42no—
693C4—44no—
694C4—84no—
695C4—84no—
696C4—82no—
697C4—81no—
698C4—82no—
699C4—81no—
700C4#186714yes—
701C2—21no—
702C2—24no—
703C2—21no—
704C2—24no—
705C2—21no—
706C2—21no—
707C2—84no—
708C2#1819411yescharacteristic
709C2#1819911yescharacteristic
710C2#1820211yescharacteristic
711C2#1820711yescharacteristic
712C2#2086511yescharacteristic
713C2C2×#186211yescharacteristic
714C2C2×#186711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1852C2semidirect
5. #14130C2semidirect
6. #8869C2semidirect
7. #8866C2semidirect
8. #8023C2semidirect
9. #7907C2semidirect
36. #1852C22semidirect
39. #1519C4semidirect
41. #1444C22semidirect
42. #1442C22semidirect
45. #1294C22semidirect
46. #1280C22semidirect
47. #1280C22semidirect
100. C2×C22⋊C8C2×C4semidirect
127. #380C2×C4semidirect
148. #327C2×C4semidirect
163. C22×D4C4⋊C4semidirect
209. #121C4⋊C4semidirect
248. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342052
C2×#18742
C2×#18522
C2×#15192
#141302
#88692
#88662
#80232
#79072