Finite Group Databasebeta

← #183232

Subgroups — #183232

Order 512.

1610 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34204C211yesmaximal, characteristic
2#20379C211yesmaximal, characteristic, complemented
3C2×#1510C211yesmaximal, characteristic
4C22×#327C211yesmaximal, characteristic
5#13799C211yesmaximal, characteristic, complemented
6#8220C212yesmaximal, complemented
7#8218C212yesmaximal, complemented
8#8216C212yesmaximal, complemented
9C4×#327C212yesmaximal, complemented
10#8024C211yesmaximal, characteristic, complemented
11#8020C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C2×#518C411yescharacteristic, complemented
14C22×#117C2211yescharacteristic
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#405—21no—
18#2363—22no—
19#2361—22no—
20#2342—21no—
21#2342—21no—
22D4×C22⋊C4—21no—
23D4×C22⋊C4—21no—
24#2324C412yescomplemented
25#2324C412yescomplemented
26#2316C411yescharacteristic, complemented
27#2309C411yescharacteristic, complemented
28C4×#121C2212yes—
29C4×#121C2212yes—
30#2280C2211yescharacteristic
31C2×#381C2211yescharacteristic
32#2274C411yescharacteristic, complemented
33C2×C4×C22⋊C4—21no—
34C22×#95C2211yescharacteristic
35#1750—22no—
36#1749C2212yescomplemented
37C4×C22⋊C8C2212yescomplemented
38#1729—21no—
39#1510C418yescomplemented
40C2×#327C418yescomplemented
41C2×#327C2211yescharacteristic
42C2×#327C2212yes—
43C2×#327C2211yescharacteristic
44C2×#327C2211yescharacteristic
45C2×#327C2212yes—
46C2×#327C2211yescharacteristic
47#1436—22no—
48#1434C2212yescomplemented
49C22⋊#95—21no—
50C4×#95C2212yescomplemented
51#1294C2212yescomplemented
52#1294C2212yescomplemented
53#1294C2212yescomplemented
54#1294C2212yescomplemented
55C23×D4—21no—
56C23×D4C2×C411yescharacteristic
57C24×C4—21no—
58C24×C4C2311yescharacteristic
59C22×M16—21no—
60C23×C8—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121C2×C414yes—
66C2×#121C2×C414yes—
67C2×#121C2×C411yescharacteristic
68C2×#121C2×C411yescharacteristic
69C2×#121C2311yescharacteristic
70C2×#121C2311yescharacteristic
71#518—44no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4C2×C412yescomplemented
76C2×C4×D4C2×C412yescomplemented
77C2×#117—44no—
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C412yes—
83C2×#117D411yescharacteristic
84C2×#117D411yescharacteristic
85C2×#117D412yes—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4D412yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—22no—
95C22×C42C2312yes—
96#407—22no—
97#407C2×C414yescomplemented
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8C2×C414yescomplemented
102C2×C22⋊C8C2311yescharacteristic
103C2×C22⋊C8C2311yescharacteristic
104C2×C22⋊C8C2312yes—
105#405—44no—
106C2×#97—22no—
107C2×C4×C8—22no—
108#390—22no—
109#390—22no—
110#390—22no—
111#390C2×C412yescomplemented
112#390C2×C412yescomplemented
113#389—22no—
114#389C2×C412yescomplemented
115#389C2×C412yescomplemented
116#388—22no—
117#388—22no—
118#388—22no—
119#388—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#385—22no—
125#385—22no—
126#381—21no—
127#381—22no—
128#381C2×C414yescomplemented
129#380—21no—
130#380—21no—
131#380C2×C412yes—
132#380C2312yes—
133#379—21no—
134#379—21no—
135#379—21no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4C2×C412yescomplemented
138C4×C4⋊C4C2×C412yescomplemented
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—44no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95C2×C414yescomplemented
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156#336—22no—
157#336—22no—
158#327C2×C414yescomplemented
159#327C2×C414yescomplemented
160#327C2×C414yescomplemented
161#327C2×C414yescomplemented
162#327C2×C418yescomplemented
163#327C2×C418yescomplemented
164C25—21no—
165C25C2×D411yescharacteristic
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—21no—
173C22×D4—22no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—82no—
177C22×D4—82no—
178C22×D4C4214yescomplemented
179C22×D4C22×C411yescharacteristic
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—82no—
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C412yes—
197C23×C4C22×C412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D412yes—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4#4112yes—
204C23×C4#4112yes—
205C23×C4C2411yescharacteristic
206C2×M16—24no—
207C2×M16—42no—
208C2×M16—42no—
209C22×C8—24no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—22no—
213C22×C8—42no—
214C22×C8—42no—
215#121—22no—
216#121—22no—
217#121—24no—
218#121—24no—
219#121—24no—
220#121—24no—
221#121—21no—
222#121—21no—
223#121—21no—
224#121—21no—
225#121C4214yescomplemented
226#121C4214yescomplemented
227#121C22×C412yes—
228#121C22×C412yes—
229#121C22×C412yes—
230#121C22×C412yes—
231C4×D4—44no—
232C4×D4—44no—
233C4×D4—48no—
234C4×D4—82no—
235C4×D4—82no—
236C4×D4—82no—
237C4×D4—82no—
238#117—24no—
239#117—24no—
240#117—44no—
241#117—44no—
242#117—42no—
243#117—42no—
244#117—42no—
245#117—42no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C4214yescomplemented
252C2×C4⋊C4C22⋊C412yes—
253C2×C4⋊C4C22⋊C412yes—
254C2×C4⋊C4C22⋊C412yes—
255C2×C4⋊C4C22⋊C412yes—
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C412yes—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22⋊C411yescharacteristic
285C2×C22⋊C4C22⋊C411yescharacteristic
286C2×C22⋊C4C22⋊C412yes—
287C2×C22⋊C4C22⋊C412yes—
288C2×C22⋊C4C22⋊C411yescharacteristic
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—82no—
297C2×C42C22×C412yes—
298C2×C42C22×C412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—22no—
306C22⋊C8C22×C412yes—
307C22⋊C8C22×C412yes—
308C22⋊C8C22×C414yes—
309#97—44no—
310C4×C8—44no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—21no—
320#95—21no—
321#95—22no—
322#95—44no—
323#95—44no—
324#95—82no—
325#95C22×C412yes—
326#95C22×C412yes—
327#95C22×C414yes—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—21no—
333C24—21no—
334C24—24no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24—82no—
339C24—82no—
340C24C2×C22⋊C411yescharacteristic
341C24C2×C22⋊C411yescharacteristic
342C24C4×D412yes—
343C24C4×D412yes—
344C24C22×D411yescharacteristic
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—24no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—48no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—48no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—88no—
369C2×D4—84no—
370C2×D4—84no—
371C2×D4C2×C4212yes—
372C2×D4C2×C4212yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—48no—
407C22×C4—44no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—48no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—84no—
428C22×C4—82no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4C2×C4212yes—
432C22×C4C2×C4212yes—
433C22×C4C2×C22⋊C412yes—
434C22×C4C2×C22⋊C412yes—
435C22×C4C2×C22⋊C412yes—
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C412yes—
439C22×C4#11712yes—
440C22×C4#11712yes—
441C22×C4C4×D412yes—
442C22×C4C4×D412yes—
443C22×C4C4×D412yes—
444C22×C4C4×D412yes—
445C22×C4C4×D412yes—
446C22×C4C4×D412yes—
447C22×C4C23×C411yescharacteristic
448C22×C4C23×C411yescharacteristic
449C22×C4C23×C411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453M16—48no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—41no—
458C2×C8—42no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—48no—
463C2×C8—42no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4C2×C4212yes—
476C4⋊C4C2×C4212yes—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C22⋊C4—48no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4#9914yes—
501C42—24no—
502C42—24no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—82no—
508C42—82no—
509C42—88no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—48no—
541C23—84no—
542C23—84no—
543C23—164no—
544C23—164no—
545C23C4×C22⋊C412yes—
546C23C4×C22⋊C412yes—
547C23C2×#9912yes—
548C23#41012yes—
549C23C22×C22⋊C411yescharacteristic
550C23C2×C4×D411yescharacteristic
551C23C2×C4×D411yescharacteristic
552C23#57612yes—
553C23#57712yes—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—48no—
559D4—44no—
560D4—44no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—84no—
566D4—84no—
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—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
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—44no—
622C2×C4—42no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—84no—
626C2×C4—816no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—164no—
634C2×C4—164no—
635C2×C4C4×C22⋊C412yes—
636C2×C4C4×C22⋊C412yes—
637C2×C4C2×#9914yes—
638C2×C4C22×C4211yescharacteristic
639C2×C4C2×#11711yescharacteristic
640C2×C4C2×C4×D411yescharacteristic
641C2×C4C2×C4×D411yescharacteristic
642C8—42no—
643C8—42no—
644C8—44no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—24no—
654C22—22no—
655C22—22no—
656C22—24no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—44no—
661C22—48no—
662C22—44no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—168no—
676C22C4×#9914yes—
677C22C2×C4×C22⋊C411yescharacteristic
678C22C22×#9911yescharacteristic
679C22C2×#41011yescharacteristic
680C22#289712yes—
681C22#290112yes—
682C22#294612yes—
683C22#294811yescharacteristic
684C22#294911yescharacteristic
685C4—21no—
686C4—21no—
687C4—21no—
688C4—22no—
689C4—22no—
690C4—21no—
691C4—24no—
692C4—44no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—84no—
699C4—82no—
700C4—82no—
701C4—82no—
702C2—21no—
703C2—22no—
704C2—24no—
705C2—21no—
706C2—42no—
707C2—82no—
708C2—82no—
709C2#1760312yes—
710C2#1760412yes—
711C2C2×C4×#9911yescharacteristic
712C2#2040511yescharacteristic
713C2#2041611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20379C2semidirect
5. #13799C2semidirect
6. #8220C2semidirect
7. #8218C2semidirect
8. #8216C2semidirect
9. C4×#327C2semidirect
10. #8024C2semidirect
11. #8020C2semidirect
13. C2×#518C4semidirect
24. #2324C4semidirect
25. #2324C4semidirect
26. #2316C4semidirect
27. #2309C4semidirect
32. #2274C4semidirect
36. #1749C22semidirect
37. C4×C22⋊C8C22semidirect
39. #1510C4semidirect
40. C2×#327C4semidirect
48. #1434C22semidirect
50. C4×#95C22semidirect
51. #1294C22semidirect
52. #1294C22semidirect
53. #1294C22semidirect
54. #1294C22semidirect
75. C2×C4×D4C2×C4semidirect
76. C2×C4×D4C2×C4semidirect
97. #407C2×C4semidirect
101. C2×C22⋊C8C2×C4semidirect
111. #390C2×C4semidirect
112. #390C2×C4semidirect
114. #389C2×C4semidirect
115. #389C2×C4semidirect
128. #381C2×C4semidirect
137. C4×C4⋊C4C2×C4semidirect
138. C4×C4⋊C4C2×C4semidirect
152. C2×#95C2×C4semidirect
158. #327C2×C4semidirect
159. #327C2×C4semidirect
160. #327C2×C4semidirect
161. #327C2×C4semidirect
162. #327C2×C4semidirect
163. #327C2×C4semidirect
178. C22×D4C42semidirect
225. #121C42semidirect
226. #121C42semidirect
251. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
#342042
#203792
C2×#15102
C22×#3272
#137992
#82202
#82182
#82162
C4×#3272
#80242
#80202