Finite Group Databasebeta

← #355542

Subgroups — #355542

Order 512.

2146 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#1964C212yesmaximal
3C4×#380C211yesmaximal, characteristic, complemented
4#13753C211yesmaximal, characteristic, complemented
5#10960C214yesmaximal, complemented
6#10854C212yesmaximal, complemented
7#10761C214yesmaximal, complemented
8C22×#121C2211yescharacteristic
9C22×C4×D4C411yescharacteristic, complemented
10C22×C4×D4C2212yes—
11C23×C42C2211yescharacteristic
12C22×#99C2218yes—
13C4×#121—21no—
14C4×#121—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121—21no—
20C4×#121—21no—
21C2×#390C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#388C412yescomplemented
24C2×#386C2212yes—
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C411yescharacteristic, complemented
29C2×C4×C22⋊C4C412yescomplemented
30C2×C4×C22⋊C4C2212yes—
31C2×C4×C22⋊C4C2212yes—
32C2×#376—21no—
33#1964C4116yescomplemented
34C2×#342C2218yes—
35C4⋊#95—21no—
36#1434—21no—
37#1434—21no—
38#1434—21no—
39C22⋊#95C2212yes—
40C22⋊#95C2212yes—
41C22⋊#95C2212yes—
42C4×#95—21no—
43C4×#95—21no—
44C23×D4C2×C411yescharacteristic
45C23×D4C2312yes—
46C24×C4C2×C412yes—
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—24no—
55C2×#121C2×C418yes—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—44no—
65C2×C4×D4—48no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—44no—
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C412yes—
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C2×#99—48no—
95C2×#99—48no—
96C2×#99C2×C4132yescomplemented
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—24no—
118#380C2×C418yescomplemented
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—44no—
124C4×C22⋊C4—24no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—44no—
130C4×C22⋊C4C2×C418yescomplemented
131#376—44no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95C2×C412yes—
142C2×#95C2×C412yes—
143C2×#95C2×C412yes—
144#342—48no—
145C25C22×C412yes—
146C25C2411yescharacteristic
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—24no—
150C22×D4—28no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4C4214yescomplemented
159C22×D4C22×C418yes—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C22×C412yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C412yes—
182C23×C4C22×C412yes—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C414yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4#4114yes—
199#121—28no—
200#121—28no—
201#121—28no—
202#121—28no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—48no—
207C4×D4—48no—
208C4×D4—48no—
209C4×D4—48no—
210C4×D4—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C4214yescomplemented
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4C4218yescomplemented
251C2×C22⋊C4C22×C418yes—
252C2×C22⋊C4C22×C418yes—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—28no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290#99—432no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—28no—
298#95—28no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24—41no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24C2×C4214yes—
319C24C23×C411yescharacteristic
320C24C23×C412yes—
321C24C22×D411yescharacteristic
322C24C22×D411yescharacteristic
323C24C2×#4112yes—
324C24H(5,𝔽2)14yes—
325C24H(5,𝔽2)12yes—
326C242−1+412yes—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—216no—
330C2×D4—216no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—42no—
345C2×D4—42no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
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—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—416no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—816no—
434C22×C4C2×C4214yes—
435C22×C4C2×C4212yes—
436C22×C4C2×C4212yes—
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4#11712yes—
450C22×C4#11712yes—
451C22×C4C4×D414yes—
452C22×C4C4×D414yes—
453C22×C4C4×D414yes—
454C22×C4C4×D414yes—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—42no—
458C4⋊C4—416no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—28no—
467C22⋊C4—216no—
468C22⋊C4—28no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—416no—
496C42—44no—
497C42—44no—
498C42—88no—
499C42#9914yescomplemented
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—24no—
513C23—28no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23C22×C4211yescharacteristic
540C23C22×C22⋊C411yescharacteristic
541C23C2×#11711yescharacteristic
542C23C2×C4×D412yes—
543C23C2×C4×D412yes—
544C23#51812yes—
545C23#51814yes—
546C23#52012yes—
547C23#52014yes—
548C23#53512yes—
549C23#53512yes—
550C23#53511yescharacteristic
551C23#53611yescharacteristic
552C23#53712yes—
553C23#54014yes—
554C23#54214yes—
555D4—48no—
556D4—416no—
557D4—48no—
558D4—48no—
559D4—48no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—88no—
626C2×C4—88no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4C4×C22⋊C412yes—
638C2×C4C4×C22⋊C412yes—
639C2×C4C4×C22⋊C412yes—
640C2×C4C4×C22⋊C412yes—
641C2×C4C2×#9912yes—
642C2×C4C2×#9912yes—
643C2×C4C2×#9912yes—
644C2×C4#41012yes—
645C2×C4#41012yes—
646C2×C4#41012yes—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—21no—
654C22—21no—
655C22—22no—
656C22—22no—
657C22—21no—
658C22—22no—
659C22—22no—
660C22—21no—
661C22—22no—
662C22—22no—
663C22—21no—
664C22—21no—
665C22—22no—
666C22—22no—
667C22—21no—
668C22—21no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—88no—
677C22—88no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22C2×C4×C22⋊C411yescharacteristic
683C22#227512yes—
684C22#230911yescharacteristic
685C22#231111yescharacteristic
686C22#231912yes—
687C22#232414yes—
688C22#232612yes—
689C22#232812yes—
690C22C22×#9911yescharacteristic
691C22C2×#41011yescharacteristic
692C22C2×#41011yescharacteristic
693C22C2×#41011yescharacteristic
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—44no—
706C4—88no—
707C4—88no—
708C4—82no—
709C4—82no—
710C4C4×#9914yes—
711C2—22no—
712C2—24no—
713C2—22no—
714C2—22no—
715C2—22no—
716C2—84no—
717C2#1378511yescharacteristic
718C2C2×C4×#9911yescharacteristic
719C2C4×#41011yescharacteristic
720C2#2179311yescharacteristic
721C2#2179411yescharacteristic
722C2#2187411yescharacteristic
723C2#2187511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#380C2semidirect
4. #13753C2semidirect
5. #10960C2semidirect
6. #10854C2semidirect
7. #10761C2semidirect
9. C22×C4×D4C4semidirect
21. C2×#390C4semidirect
22. C2×#389C4semidirect
23. C2×#388C4semidirect
28. C2×C4×C4⋊C4C4semidirect
29. C2×C4×C22⋊C4C4semidirect
33. #1964C4semidirect
96. C2×#99C2×C4semidirect
118. #380C2×C4semidirect
130. C4×C22⋊C4C2×C4semidirect
158. C22×D4C42semidirect
226. C2×C4⋊C4C42semidirect
250. C2×C22⋊C4C42semidirect
499. C42#99semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#19642
C4×#3802
#137532
#109602
#108542
#107612