Finite Group Databasebeta

← #188530

Subgroups — #188530

Order 512.

1412 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#21934C211yesmaximal, characteristic, complemented
3C2×#1510C211yesmaximal, characteristic
4C2×#1507C211yesmaximal, characteristic
5#13953C211yesmaximal, characteristic, complemented
6#8577C212yesmaximal, complemented
7#8575C212yesmaximal, complemented
8#8573C212yesmaximal, complemented
9#8571C212yesmaximal, 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#1851—22no—
32#1851—22no—
33#1797C2212yescomplemented
34#1797C2212yescomplemented
35#1510—24no—
36#1507—24no—
37C2×#327C2212yes—
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2212yes—
41#1439—21no—
42#1439—21no—
43#1436—21no—
44#1436—21no—
45C4⋊#95C2211yescharacteristic, complemented
46C4⋊#95C2211yescharacteristic, complemented
47#1434C2211yescharacteristic, complemented
48#1434C2211yescharacteristic, 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×C412yes—
63C2×#121D414yes—
64C2×#121Q814yes—
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⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92#407—22no—
93#407—22no—
94#407—22no—
95#407—22no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8C2312yes—
99C2×C22⋊C8C2312yes—
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#381—22no—
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⋊C4—22no—
132C4×C22⋊C4C2×C412yes—
133C4×C22⋊C4C2×C412yes—
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#327—24no—
153#327—24no—
154#327—24no—
155#327—24no—
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×D4C4⋊C414yescomplemented
171C22×D4C22×C412yes—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—81no—
182C23×C4—81no—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×Q812yes—
194C23×C4#4112yes—
195C23×C4#4112yes—
196C23×C4C2411yescharacteristic
197C2×M16—42no—
198C2×M16—42no—
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#121C4⋊C418yescomplemented
213#121C2×D414yes—
214#121C2×Q814yes—
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—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—81no—
235C2×C4⋊C4—81no—
236C2×C4⋊C4—81no—
237C2×C4⋊C4—81no—
238C2×C4⋊C4C4⋊C414yescomplemented
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C412yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—81no—
266C2×C22⋊C4—81no—
267C2×C22⋊C4C22⋊C412yes—
268C2×C22⋊C4C22⋊C412yes—
269C2×C22⋊C4C22⋊C412yes—
270C2×C22⋊C4C22⋊C412yes—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—42no—
274C2×C42—81no—
275C2×C42—81no—
276C2×C42—81no—
277C2×C42—81no—
278C2×C42C22×C412yes—
279C2×C42C22×C412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C4⋊C8—82no—
283C4⋊C8—82no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—42no—
308#95—42no—
309#95—81no—
310#95—81no—
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
333C24#12112yes—
334C24C22⋊Q812yes—
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×C4⋊C414yes—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
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—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—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
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—41no—
420C22×C4—41no—
421C22×C4—41no—
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×C22⋊C412yes—
435C22×C4C2×C22⋊C412yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C412yes—
440C22×C4C2×C4⋊C412yes—
441C22×C4C2×C4⋊C412yes—
442C22×C4#11712yes—
443C22×C4#11712yes—
444C22×C4#12112yes—
445C22×C4#12112yes—
446C22×C4#12112yes—
447C22×C4C22⋊Q812yes—
448C22×C4C22⋊Q812yes—
449C22×C4C22⋊Q812yes—
450C22×C4AGL(1,ℤ/8)12yes—
451C22×C4AGL(1,ℤ/8)12yes—
452C22×C4C23×C411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×Q811yescharacteristic
456C22×C4C2×#4111yescharacteristic
457C22×C4C2×#4111yescharacteristic
458M16—84no—
459M16—84no—
460C2×C8—42no—
461C2×C8—42no—
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⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4C2×C4⋊C414yes—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—28no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—84no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—44no—
500C42—44no—
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—
548C23#38012yes—
549C23#38012yes—
550C23C22×C22⋊C411yescharacteristic
551C23C2×#12111yescharacteristic
552C23C2×C22⋊Q811yescharacteristic
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—22no—
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—24no—
585C2×C4—42no—
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—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
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—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
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×C4#38012yes—
649C2×C4#38012yes—
650C2×C4#41814yes—
651C2×C4C22×C4⋊C411yescharacteristic
652C2×C4C2×#11711yescharacteristic
653C2×C4C2×#12111yescharacteristic
654C2×C4C2×C22⋊Q811yescharacteristic
655C8—82no—
656C8—82no—
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×#38011yescharacteristic
696C22#288311yescharacteristic
697C22#288311yescharacteristic
698C22C2×#41811yescharacteristic
699C22#289611yescharacteristic
700C22#290111yescharacteristic
701C22#290111yescharacteristic
702C22#303611yescharacteristic
703C22#303611yescharacteristic
704C22#307711yescharacteristic
705C22#307711yescharacteristic
706C4—21no—
707C4—21no—
708C4—21no—
709C4—21no—
710C4—22no—
711C4—22no—
712C4—24no—
713C4—44no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—161no—
722C4—161no—
723C2—21no—
724C2—21no—
725C2—21no—
726C2—21no—
727C2—42no—
728C2—42no—
729C2—84no—
730C2#1762011yescharacteristic
731C2#1762311yescharacteristic
732C2#1762511yescharacteristic
733C2#1762611yescharacteristic
734C2#2063311yescharacteristic
735C2#2064611yescharacteristic
736C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×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. #21934C2semidirect
5. #13953C2semidirect
6. #8577C2semidirect
7. #8575C2semidirect
8. #8573C2semidirect
9. #8571C2semidirect
10. #8019C2semidirect
11. #8015C2semidirect
17. #2361C4semidirect
18. #2361C4semidirect
19. #2344C4semidirect
21. D4×C22⋊C4C4semidirect
33. #1797C22semidirect
34. #1797C22semidirect
45. C4⋊#95C22semidirect
46. C4⋊#95C22semidirect
47. #1434C22semidirect
48. #1434C22semidirect
49. #1294C22semidirect
50. #1294C22semidirect
51. #1294C22semidirect
52. #1294C22semidirect
170. C22×D4C4⋊C4semidirect
212. #121C4⋊C4semidirect
238. 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
#219342
C2×#15102
C2×#15072
#139532
#85772
#85752
#85732
#85712
#80192
#80152