Finite Group Databasebeta

← #6343602

Subgroups — #6343602

Order 512.

4248 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3496C211yesmaximal, characteristic
2#60230C211yesmaximal, characteristic, complemented
3#34190C211yesmaximal, characteristic, complemented
4#17626C214yesmaximal, complemented
5#17620C218yesmaximal, complemented
6#17613C214yesmaximal, complemented
7#17589C212yesmaximal, complemented
8#17577C214yesmaximal, complemented
9#17561C212yesmaximal, complemented
10C2×#1510C214yesmaximal, complemented
11C22×H(5,𝔽2)—21no—
12#3496—24no—
13#3496—22no—
14#3496—21no—
15#3496—21no—
16#3496C418yescomplemented
17#3496C2212yes—
18#3496C2214yes—
19#3496C2212yes—
20C2×#547C414yescomplemented
21C2×#546—24no—
22C2×D42C414yescomplemented
23C2×#537—21no—
24C2×#534—22no—
25C2×#532C2212yes—
26C22×#121C2214yes—
27C22×C4×D4C2211yescharacteristic
28#2975C2212yescomplemented
29#2974—24no—
30#2973C2214yescomplemented
31#2972C2212yescomplemented
32D4×M16—24no—
33#2920C2212yescomplemented
34C2×#407C2214yescomplemented
35#2357C2212yescomplemented
36#2353C2214yescomplemented
37#2349C2212yescomplemented
38#2344—21no—
39#2342—22no—
40D4×C22⋊C4—21no—
41#2319—21no—
42#2318—22no—
43#2283C2211yescharacteristic, complemented
44#2283C2211yescharacteristic, complemented
45#2280C2212yescomplemented
46C2×#381C2212yescomplemented
47#2275—21no—
48#1519C22116yescomplemented
49#1517C2218yescomplemented
50#1514C2218yescomplemented
51#1510C2214yescomplemented
52#1510C2218yescomplemented
53#1510C2214yescomplemented
54#1510C22116yescomplemented
55#1507C22116yescomplemented
56#1505C2218yescomplemented
57#1502C2218yescomplemented
58C2×#327C22116yescomplemented
59C2×H(5,𝔽2)—28no—
60C2×H(5,𝔽2)—22no—
61C2×H(5,𝔽2)—24no—
62C2×H(5,𝔽2)—22no—
63C2×H(5,𝔽2)—416no—
64C22×#41—24no—
65C22×#41—22no—
66C23×D4—24no—
67C23×D4—24no—
68C23×D4C2×C414yescomplemented
69C23×D4D411yescharacteristic
70C24×C4C2312yes—
71C22×M16—24no—
72#547—24no—
73#547—28no—
74#547—24no—
75#547C2×C418yes—
76#547C2×C4116yescomplemented
77#547C2×C418yescomplemented
78#546—216no—
79#546—28no—
80#546—28no—
81#546—28no—
82#546—216no—
83#546—28no—
84D42—28no—
85D42—24no—
86D42—24no—
87D42C2×C418yes—
88D42C2×C418yescomplemented
89D42C2×C4116yescomplemented
90#537—28no—
91#537—22no—
92#537—24no—
93#537—22no—
94#534—216no—
95#534—24no—
96#534—24no—
97#534—24no—
98#534—24no—
99#532—22no—
100#532—22no—
101#532—22no—
102#532—22no—
103#532C2×C414yescomplemented
104#532C2×C414yescomplemented
105#532C2314yes—
106#532C2314yes—
107C2×C4⋊Q8C2×C412yescomplemented
108C2×#127C2×C412yescomplemented
109C2×#124—22no—
110C2×#123—28no—
111C2×C22⋊Q8—24no—
112C2×#121—24no—
113C2×#121—24no—
114C2×#121—24no—
115C2×#121—24no—
116C2×#121—24no—
117C2×#121—24no—
118C2×#121—22no—
119C2×#121—24no—
120C2×#121C2×C418yescomplemented
121C2×#121C2×C418yescomplemented
122C2×#121C2×C418yescomplemented
123C2×#121C2314yes—
124C2×#121C2318yes—
125C2×#121C2318yes—
126C2×#120—28no—
127C2×C4×D4—22no—
128C2×C4×D4—22no—
129C2×C4×D4C2×C414yes—
130C2×C4×D4C2×C414yes—
131C2×C4×D4C2×C414yescomplemented
132C2×C4×D4D414yes—
133C2×C4×D4D414yes—
134C2×C4×D4D412yes—
135C2×C4×D4D411yescharacteristic
136C2×C4×D4D411yescharacteristic
137C2×C4×D4C2311yescharacteristic
138C2×C4×D4C2312yes—
139C2×C4×D4C2311yescharacteristic
140C2×#117—22no—
141C22×C4⋊C4C2×C414yescomplemented
142C22×C4⋊C4D411yescharacteristic
143C22×C22⋊C4—24no—
144C22×C22⋊C4—22no—
145C22×C22⋊C4D412yes—
146C22×C42C2311yescharacteristic
147#436—28no—
148#435—216no—
149C8×D4—28no—
150#432—22no—
151#432C2314yescomplemented
152#431—22no—
153#431C2314yescomplemented
154#423—22no—
155#407—24no—
156#407—24no—
157#407C2314yes—
158#407C2318yescomplemented
159C2×C22⋊C8—24no—
160C2×C22⋊C8C2318yescomplemented
161C4×M16—22no—
162#390—22no—
163#389—22no—
164#388—24no—
165#388—24no—
166#387—24no—
167#386—24no—
168#385—24no—
169#383—21no—
170#383C2312yescomplemented
171#382—21no—
172#382C2312yescomplemented
173#381—22no—
174#381—22no—
175#381C2311yescharacteristic
176#381C2312yes—
177#381C2311yescharacteristic
178#381C2314yescomplemented
179#380—22no—
180#380—21no—
181#380—21no—
182#380C2314yescomplemented
183#379—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2312yescomplemented
187C4×C22⋊C4—22no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—24no—
191C4×C22⋊C4C2314yescomplemented
192#376—21no—
193#376C2312yescomplemented
194C2×#95—22no—
195C2×#95C2314yescomplemented
196#327C23116yescomplemented
197#327C23132yescomplemented
198#327C23116yescomplemented
199C25—24no—
200C25C2×D412yes—
201H(5,𝔽2)—432no—
202H(5,𝔽2)—464no—
203H(5,𝔽2)—416no—
204H(5,𝔽2)—416no—
205C2×#41—28no—
206C2×#41—28no—
207C2×#41—216no—
208C2×#41—24no—
209C2×#41—24no—
210C2×#41—24no—
211C2×#41—24no—
212C2×#41—416no—
213C2×#41—432no—
214C22×Q8—22no—
215C22×D4—28no—
216C22×D4—28no—
217C22×D4—28no—
218C22×D4—28no—
219C22×D4—24no—
220C22×D4—28no—
221C22×D4—28no—
222C22×D4—28no—
223C22×D4—216no—
224C22×D4—28no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—24no—
230C22×D4—432no—
231C22×D4—432no—
232C22×D4—88no—
233C22×D4C22⋊C414yes—
234C22×D4C22⋊C414yes—
235C22×D4C22×C414yes—
236C22×D4C22×C418yes—
237C22×D4C22×C418yes—
238C22×D4C2×D412yes—
239C22×D4C2×D411yescharacteristic
240C22×D4C2×D411yescharacteristic
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—84no—
245C23×C4C22×C414yes—
246C23×C4C22×C414yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D414yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2×D414yes—
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C23×C4C2412yes—
257C2×M16—24no—
258C2×M16—28no—
259C2×M16—88no—
260C22×C8—28no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8C22×C414yes—
264C4⋊Q8C22×C414yes—
265#127—22no—
266#127—22no—
267#127C22×C414yes—
268#127C22×C414yes—
269#124—24no—
270#124—24no—
271#124—24no—
272#124—24no—
273#123—216no—
274#123—216no—
275#123—216no—
276#123—216no—
277C22⋊Q8—28no—
278C22⋊Q8—216no—
279C22⋊Q8—28no—
280#121—28no—
281#121—28no—
282#121—28no—
283#121—28no—
284#121—28no—
285#121—28no—
286#121—24no—
287#121—24no—
288#121—24no—
289#121—24no—
290#121—28no—
291#121—28no—
292#121—28no—
293#121—28no—
294#121—24no—
295#121—24no—
296#121—28no—
297#121—216no—
298#121—28no—
299#121C22×C418yes—
300#121C22×C4116yes—
301#121C22×C4116yes—
302#121C2418yes—
303#120—216no—
304#120—216no—
305#120—216no—
306#120—216no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—48no—
314C4×D4C22⋊C418yes—
315C4×D4C22⋊C414yes—
316C4×D4C22⋊C414yes—
317C4×D4C22⋊C418yes—
318C4×D4C22⋊C414yes—
319C4×D4C22⋊C414yes—
320C4×D4C22×C418yes—
321C4×D4C22×C418yes—
322C4×D4C22×C414yes—
323C4×D4C22×C418yes—
324C4×D4C22×C414yes—
325C4×D4C2×D414yes—
326C4×D4C2×D418yes—
327C4×D4C2×D414yes—
328#117—84no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4C22⋊C414yes—
337C2×C4⋊C4C22⋊C414yes—
338C2×C4⋊C4C22×C414yes—
339C2×C4⋊C4C22×C418yes—
340C2×C4⋊C4C22×C418yes—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C22⋊C4—28no—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—28no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—28no—
350C2×C22⋊C4—28no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—84no—
358C2×C22⋊C4C22⋊C418yes—
359C2×C22⋊C4C22⋊C418yes—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42C22×C412yes—
368C2×C42C22×C412yes—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C4⋊C8—24no—
376C4⋊C8—24no—
377C22⋊C8—24no—
378C22⋊C8—28no—
379C22⋊C8—28no—
380C22⋊C8C2418yes—
381#97—24no—
382C4×C8—24no—
383#95—21no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95C2412yes—
393#95C2414yes—
394#95C2412yes—
395C24—28no—
396C24—24no—
397C24—28no—
398C24—28no—
399C24—44no—
400C24—416no—
401C24—88no—
402C24—82no—
403C24—88no—
404C24C2×C22⋊C414yes—
405C24C2×C22⋊C414yes—
406C24C22×D411yescharacteristic
407C24C22×D412yes—
408C24C22×D412yes—
409#41—416no—
410#41—416no—
411#41—416no—
412#41—464no—
413#41—432no—
414#41—432no—
415#41—416no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—48no—
419C2×D4—28no—
420C2×D4—216no—
421C2×D4—216no—
422C2×D4—216no—
423C2×D4—216no—
424C2×D4—28no—
425C2×D4—28no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—28no—
432C2×D4—28no—
433C2×D4—24no—
434C2×D4—48no—
435C2×D4—48no—
436C2×D4—464no—
437C2×D4—432no—
438C2×D4—432no—
439C2×D4—416no—
440C2×D4—432no—
441C2×D4—432no—
442C2×D4—432no—
443C2×D4—432no—
444C2×D4—432no—
445C2×D4—88no—
446C2×D4—88no—
447C2×D4—88no—
448C2×D4—88no—
449C2×D4—88no—
450C2×D4—84no—
451C2×D4C2×C22⋊C418yes—
452C2×D4C2×C22⋊C414yes—
453C2×D4C2×C22⋊C414yes—
454C2×D4AGL(1,ℤ/8)18yes—
455C2×D4C23×C418yes—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—42no—
468C22×C4—48no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—48no—
472C22×C4—84no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—84no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—81no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4C2×C22⋊C412yes—
482C22×C4C2×C22⋊C412yes—
483C22×C4C2×C22⋊C414yes—
484C22×C4C2×C22⋊C412yes—
485C22×C4C2×C22⋊C412yes—
486C22×C4C2×C22⋊C414yes—
487C22×C4C2×C22⋊C414yes—
488C22×C4C2×C22⋊C414yes—
489C22×C4C2×C22⋊C414yes—
490C22×C4C2×C22⋊C414yes—
491C22×C4C23×C411yescharacteristic
492C22×C4C23×C411yescharacteristic
493C22×C4C23×C414yes—
494C22×C4C23×C414yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D414yes—
504C22×C4C22×D414yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C2511yescharacteristic
507M16—48no—
508M16—88no—
509C2×C8—28no—
510C2×C8—88no—
511C4⋊C4—24no—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—44no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4C2×C22⋊C418yes—
520C4⋊C4C2×C22⋊C414yes—
521C4⋊C4C2×C22⋊C414yes—
522C4⋊C4C23×C418yes—
523C22⋊C4—28no—
524C22⋊C4—216no—
525C22⋊C4—216no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—28no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—48no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4C2×C22⋊C418yes—
536C22⋊C4C2×C22⋊C418yes—
537C22⋊C4C2×C22⋊C418yes—
538C22⋊C4C2×C22⋊C418yes—
539C42—44no—
540C42—82no—
541C42—84no—
542C42C2×C22⋊C412yes—
543C42C2×C22⋊C412yes—
544C42C2×C22⋊C412yes—
545C42C2×C22⋊C412yes—
546C42C23×C412yes—
547C42C23×C412yes—
548C42C22×D412yes—
549C42C22×D412yes—
550C23—22no—
551C23—28no—
552C23—28no—
553C23—22no—
554C23—28no—
555C23—24no—
556C23—28no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—48no—
566C23—48no—
567C23—416no—
568C23—48no—
569C23—416no—
570C23—88no—
571C23—88no—
572C23—88no—
573C23—88no—
574C23—816no—
575C23—1632no—
576C23C22×C22⋊C414yes—
577C23C22×C22⋊C411yescharacteristic
578C23C22×C22⋊C411yescharacteristic
579C23C22×C22⋊C414yes—
580C23C2×AGL(1,ℤ/8)18yes—
581C23C23×D411yescharacteristic
582Q8—48no—
583Q8—48no—
584D4—216no—
585D4—28no—
586D4—48no—
587D4—48no—
588D4—48no—
589D4—416no—
590D4—432no—
591D4—432no—
592D4—416no—
593D4—48no—
594D4—84no—
595D4—84no—
596D4—88no—
597D4—88no—
598D4#418116yes—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—416no—
626C2×C4—48no—
627C2×C4—416no—
628C2×C4—416no—
629C2×C4—416no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—168no—
640C2×C4C22×C22⋊C411yescharacteristic
641C2×C4C22×C22⋊C411yescharacteristic
642C2×C4C22×C22⋊C412yes—
643C2×C4C22×C22⋊C412yes—
644C2×C4C22×C22⋊C412yes—
645C2×C4C22×C22⋊C414yes—
646C2×C4C22×C22⋊C414yes—
647C2×C4C22×C22⋊C412yes—
648C2×C4C2×AGL(1,ℤ/8)14yes—
649C2×C4C24×C411yescharacteristic
650C2×C4C23×D411yescharacteristic
651C8—48no—
652C22—21no—
653C22—24no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—24no—
658C22—21no—
659C22—24no—
660C22—21no—
661C22—24no—
662C22—21no—
663C22—24no—
664C22—48no—
665C22—44no—
666C22—416no—
667C22—48no—
668C22—48no—
669C22—48no—
670C22—84no—
671C22—88no—
672C22—816no—
673C22—1616no—
674C22—1616no—
675C22#288312yes—
676C22C2×#41818yes—
677C22C23×C22⋊C411yescharacteristic
678C22C22×AGL(1,ℤ/8)12yes—
679C4—21no—
680C4—21no—
681C4—21no—
682C4—21no—
683C4—24no—
684C4—24no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—48no—
689C4—82no—
690C4—84no—
691C4—82no—
692C4C2×#41814yes—
693C2—22no—
694C2—22no—
695C2—28no—
696C2—41no—
697C2—44no—
698C2—88no—
699C2C2×#288311yescharacteristic
700C2C22×#41811yescharacteristic
701C2#6009311yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3496C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60230C2semidirect
3. #34190C2semidirect
4. #17626C2semidirect
5. #17620C2semidirect
6. #17613C2semidirect
7. #17589C2semidirect
8. #17577C2semidirect
9. #17561C2semidirect
10. C2×#1510C2semidirect
16. #3496C4semidirect
20. C2×#547C4semidirect
22. C2×D42C4semidirect
28. #2975C22semidirect
30. #2973C22semidirect
31. #2972C22semidirect
33. #2920C22semidirect
34. C2×#407C22semidirect
35. #2357C22semidirect
36. #2353C22semidirect
37. #2349C22semidirect
43. #2283C22semidirect
44. #2283C22semidirect
45. #2280C22semidirect
46. C2×#381C22semidirect
48. #1519C22semidirect
49. #1517C22semidirect
50. #1514C22semidirect
51. #1510C22semidirect
52. #1510C22semidirect
53. #1510C22semidirect
54. #1510C22semidirect
55. #1507C22semidirect
56. #1505C22semidirect
57. #1502C22semidirect
58. C2×#327C22semidirect
68. C23×D4C2×C4semidirect
76. #547C2×C4semidirect
77. #547C2×C4semidirect
88. D42C2×C4semidirect
89. D42C2×C4semidirect
103. #532C2×C4semidirect
104. #532C2×C4semidirect
107. C2×C4⋊Q8C2×C4semidirect
108. C2×#127C2×C4semidirect
120. C2×#121C2×C4semidirect
121. C2×#121C2×C4semidirect
122. C2×#121C2×C4semidirect
131. C2×C4×D4C2×C4semidirect
141. C22×C4⋊C4C2×C4semidirect
151. #432C23semidirect
153. #431C23semidirect
158. #407C23semidirect
160. C2×C22⋊C8C23semidirect
170. #383C23semidirect
172. #382C23semidirect
178. #381C23semidirect
182. #380C23semidirect
186. C4×C4⋊C4C23semidirect
191. C4×C22⋊C4C23semidirect
193. #376C23semidirect
195. C2×#95C23semidirect
196. #327C23semidirect
197. #327C23semidirect
198. #327C23semidirect

Maximal subgroups

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

subgroupindex
C2×#34962
#602302
#341902
#176262
#176202
#176132
#175892
#175772
#175612
C2×#15102