Finite Group Databasebeta

← #182685

Subgroups — #182685

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34205C211yesmaximal, characteristic
2#20524C211yesmaximal, characteristic, complemented
3C2×#1510C211yesmaximal, characteristic
4C2×#1507C211yesmaximal, characteristic
5#13807C211yesmaximal, characteristic, complemented
6#8220C212yesmaximal, complemented
7#8218C212yesmaximal, complemented
8#8216C212yesmaximal, complemented
9C4×#327C212yesmaximal, complemented
10#8019C211yesmaximal, characteristic, complemented
11#8015C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C2×#518—21no—
14C2×#407C2211yescharacteristic
15C2×#407C2211yescharacteristic
16#2364—22no—
17#2361C412yescomplemented
18#2361C412yescomplemented
19#2344C412yescomplemented
20#2342—22no—
21D4×C22⋊C4C412yescomplemented
22#2325—21no—
23#2325—21no—
24#2308—21no—
25C4×#121C2212yes—
26C4×#121C2212yes—
27C2×#381C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1753—22no—
32#1753—22no—
33#1749C2212yescomplemented
34C4×C22⋊C8C2212yescomplemented
35#1510C418yescomplemented
36#1507C418yescomplemented
37C2×#327C2212yes—
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2212yes—
41#1437—21no—
42#1437—21no—
43#1434—21no—
44#1434C2211yescharacteristic, complemented
45#1434C2211yescharacteristic, complemented
46C4×#95—21no—
47C4×#95C2211yescharacteristic, complemented
48C4×#95C2211yescharacteristic, complemented
49#1294C2212yescomplemented
50#1294C2212yescomplemented
51#1294C2212yescomplemented
52#1294C2212yescomplemented
53C23×D4—21no—
54C23×D4C2×C411yescharacteristic
55C24×C4C2311yescharacteristic
56C22×M16—21no—
57C22×M16—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121C2×C414yes—
63C2×#121C2×C414yes—
64C2×#121C2×C412yes—
65C2×#121C2312yes—
66#518—44no—
67#518—44no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2×C412yes—
74C2×#117—22no—
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#407—22no—
91#407—22no—
92#407C2×C414yescomplemented
93#407C2×C414yescomplemented
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8C2312yes—
97C2×C22⋊C8C2312yes—
98C2×#97—22no—
99C2×C4×C8—22no—
100#390C2×C412yes—
101#390C2×C412yes—
102#389C2×C412yes—
103#389C2×C412yes—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#387—22no—
109#387—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#385—22no—
117#385—22no—
118#381—22no—
119#381—22no—
120#381C2×C414yescomplemented
121#380—21no—
122#380—21no—
123#380C2×C412yes—
124#380C2312yes—
125#379—22no—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C412yes—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4C2×C412yes—
132C4×C22⋊C4C2×C412yes—
133C4×C22⋊C4C2×C414yescomplemented
134C4×C22⋊C4D412yes—
135C4×C22⋊C4D412yes—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95D411yescharacteristic
145C2×#95D411yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150#336—22no—
151#336—22no—
152#327C2×C418yescomplemented
153#327C2×C418yescomplemented
154#327C2×C418yescomplemented
155#327C2×C418yescomplemented
156C25—21no—
157C25C2×D411yescharacteristic
158C22×D4—22no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—22no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—81no—
169C22×D4—81no—
170C22×D4C4214yescomplemented
171C22×D4C22×C412yes—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—81no—
182C23×C4—81no—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C412yes—
186C23×C4C22×C412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4#4112yes—
195C23×C4#4112yes—
196C23×C4C2411yescharacteristic
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—81no—
200C2×M16—81no—
201C2×M16—81no—
202C2×M16—81no—
203C22×C8—22no—
204C22×C8—22no—
205#121—24no—
206#121—24no—
207#121—24no—
208#121—24no—
209#121—24no—
210#121—22no—
211#121—22no—
212#121C4218yescomplemented
213#121C22×C414yes—
214#121C22×C414yes—
215C4×D4—48no—
216C4×D4—48no—
217C4×D4—44no—
218C4×D4—44no—
219C4×D4—44no—
220C4×D4—44no—
221C4×D4—82no—
222C4×D4—82no—
223#117—48no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—81no—
231C2×C4⋊C4—81no—
232C2×C4⋊C4C4214yescomplemented
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—81no—
260C2×C22⋊C4—81no—
261C2×C22⋊C4C22⋊C412yes—
262C2×C22⋊C4C22⋊C412yes—
263C2×C22⋊C4C22⋊C412yes—
264C2×C22⋊C4C22⋊C412yes—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—81no—
273C2×C42—81no—
274C2×C42—81no—
275C2×C42—81no—
276C2×C42—81no—
277C2×C42—81no—
278C2×C42C22×C412yes—
279C2×C42C22×C412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8C22×C414yes—
287C22⋊C8C22×C414yes—
288#97—82no—
289C4×C8—82no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—22no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—42no—
302#95—42no—
303#95—42no—
304#95—42no—
305#95—81no—
306#95—81no—
307#95C22⋊C412yes—
308#95C22⋊C412yes—
309#95C22⋊C412yes—
310#95C22⋊C412yes—
311#95C22×C412yes—
312#95C22×C412yes—
313#95C22×C412yes—
314#95C22×C412yes—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—24no—
321C24—41no—
322C24—41no—
323C24—41no—
324C24—41no—
325C24—42no—
326C24—42no—
327C24—81no—
328C24—81no—
329C24—81no—
330C24—81no—
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C24C4×D412yes—
334C24C4×D412yes—
335C24C22×D411yescharacteristic
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—28no—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—44no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—81no—
354C2×D4—81no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4C2×C4214yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—44no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4C2×C4212yes—
435C22×C4C2×C4212yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4C2×C22⋊C412yes—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C412yes—
442C22×C4#11712yes—
443C22×C4#11712yes—
444C22×C4C4×D412yes—
445C22×C4C4×D412yes—
446C22×C4C4×D412yes—
447C22×C4C4×D412yes—
448C22×C4C4×D412yes—
449C22×C4C4×D412yes—
450C22×C4AGL(1,ℤ/8)12yes—
451C22×C4AGL(1,ℤ/8)12yes—
452C22×C4C23×C411yescharacteristic
453C22×C4C23×C411yescharacteristic
454C22×C4C23×C411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4111yescharacteristic
457C22×C4C2×#4111yescharacteristic
458M16—84no—
459M16—84no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4C2×C4214yes—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—84no—
493C42—42no—
494C42—42no—
495C42—42no—
496C42—42no—
497C42—44no—
498C42—44no—
499C42—84no—
500C42—84no—
501C42—84no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—48no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—81no—
535C23—81no—
536C23—81no—
537C23—81no—
538C23—81no—
539C23—81no—
540C23—81no—
541C23—81no—
542C23—84no—
543C23—84no—
544C23—162no—
545C23—162no—
546C23—162no—
547C23—162no—
548C23C4×C22⋊C412yes—
549C23C4×C22⋊C412yes—
550C23C22×C22⋊C411yescharacteristic
551C23C2×C4×D411yescharacteristic
552C23C2×C4×D411yescharacteristic
553C23C2×AGL(1,ℤ/8)11yescharacteristic
554C23C2×AGL(1,ℤ/8)11yescharacteristic
555C23#57511yescharacteristic
556C23#57511yescharacteristic
557D4—44no—
558D4—44no—
559D4—416no—
560D4—44no—
561D4—44no—
562D4—44no—
563D4—44no—
564D4—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—164no—
645C2×C4—164no—
646C2×C4—162no—
647C2×C4—162no—
648C2×C4C4×C22⋊C412yes—
649C2×C4C4×C22⋊C412yes—
650C2×C4#41814yes—
651C2×C4C22×C4211yescharacteristic
652C2×C4C2×#11711yescharacteristic
653C2×C4C2×C4×D411yescharacteristic
654C2×C4C2×C4×D411yescharacteristic
655C8—44no—
656C8—44no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—168no—
695C22C2×C4×C22⋊C411yescharacteristic
696C22#288311yescharacteristic
697C22#288311yescharacteristic
698C22C2×#41811yescharacteristic
699C22#289611yescharacteristic
700C22#290111yescharacteristic
701C22#290111yescharacteristic
702C22C4×AGL(1,ℤ/8)11yescharacteristic
703C22C4×AGL(1,ℤ/8)11yescharacteristic
704C22C4×AGL(1,ℤ/8)11yescharacteristic
705C22C4×AGL(1,ℤ/8)11yescharacteristic
706C4—21no—
707C4—21no—
708C4—21no—
709C4—22no—
710C4—22no—
711C4—21no—
712C4—24no—
713C4—44no—
714C4—44no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C2—21no—
724C2—21no—
725C2—21no—
726C2—21no—
727C2—42no—
728C2—42no—
729C2—84no—
730C2#1762011yescharacteristic
731C2#1762311yescharacteristic
732C2#1762511yescharacteristic
733C2#1762611yescharacteristic
734C2#2039211yescharacteristic
735C2#2040511yescharacteristic
736C2C4×#41811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20524C2semidirect
5. #13807C2semidirect
6. #8220C2semidirect
7. #8218C2semidirect
8. #8216C2semidirect
9. C4×#327C2semidirect
10. #8019C2semidirect
11. #8015C2semidirect
17. #2361C4semidirect
18. #2361C4semidirect
19. #2344C4semidirect
21. D4×C22⋊C4C4semidirect
33. #1749C22semidirect
34. C4×C22⋊C8C22semidirect
35. #1510C4semidirect
36. #1507C4semidirect
44. #1434C22semidirect
45. #1434C22semidirect
47. C4×#95C22semidirect
48. C4×#95C22semidirect
49. #1294C22semidirect
50. #1294C22semidirect
51. #1294C22semidirect
52. #1294C22semidirect
92. #407C2×C4semidirect
93. #407C2×C4semidirect
120. #381C2×C4semidirect
133. C4×C22⋊C4C2×C4semidirect
152. #327C2×C4semidirect
153. #327C2×C4semidirect
154. #327C2×C4semidirect
155. #327C2×C4semidirect
170. C22×D4C42semidirect
212. #121C42semidirect
232. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
#342052
#205242
C2×#15102
C2×#15072
#138072
#82202
#82182
#82162
C4×#3272
#80192
#80152