Finite Group Databasebeta

← #6393384

Subgroups — #6393384

Order 512.

4626 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3496C211yesmaximal, characteristic
2C2×#2973C211yesmaximal, characteristic, complemented
3C2×#2352C211yesmaximal, characteristic, complemented
4#22495C218yesmaximal, complemented
5#22423C218yesmaximal, complemented
6#21865C212yesmaximal, complemented
7#21764C212yesmaximal, complemented
8#21758C212yesmaximal, complemented
9#21395C212yesmaximal, complemented
10#20912C212yesmaximal, complemented
11C2×#1874C212yesmaximal, complemented
12C22×H(5,𝔽2)—21no—
13#3496—28no—
14#3496C4116yescomplemented
15C2×#547C2214yes—
16C2×#546—24no—
17C2×D42C2214yes—
18C2×#537—21no—
19C2×#534—21no—
20C2×#534—21no—
21C2×#532C2212yes—
22C22×#121C2212yes—
23C22×#121C2212yes—
24C22×C4×D4C2211yescharacteristic
25C22×#117C2211yescharacteristic, complemented
26C23×C4⋊C4C2211yescharacteristic, complemented
27#2973—28no—
28C2×#435C2218yescomplemented
29C2×#432C2212yescomplemented
30C2×#418C2214yescomplemented
31C22×#102C2214yescomplemented
32C2×#407C2212yescomplemented
33C22×C22⋊C8C2212yescomplemented
34#2352—24no—
35#2352C2218yescomplemented
36C2×#388—21no—
37C2×#387—21no—
38C2×#386—21no—
39C2×#382C2211yescharacteristic, complemented
40C2×#380C2211yescharacteristic, complemented
41C2×#380C2211yescharacteristic, complemented
42C2×C4×C4⋊C4C2211yescharacteristic, complemented
43C2×C4×C22⋊C4—21no—
44#1993C2214yescomplemented
45#1987C2214yescomplemented
46#1982C2214yescomplemented
47#1961C2214yescomplemented
48#1895C2218yescomplemented
49#1893C2218yescomplemented
50#1875C2218yescomplemented
51#1874C2214yescomplemented
52#1874C2214yescomplemented
53#1874C2214yescomplemented
54#1874C2214yescomplemented
55#1874C2218yescomplemented
56#1824C2214yescomplemented
57#1823C2214yescomplemented
58#1804C2218yescomplemented
59#1786C2214yescomplemented
60#1786C2214yescomplemented
61#1762C2218yescomplemented
62#1736C2214yescomplemented
63C2×#340C2214yescomplemented
64C2×H(5,𝔽2)—216no—
65C2×H(5,𝔽2)—48no—
66C2×H(5,𝔽2)—48no—
67C22×#41—24no—
68C22×#41—22no—
69C23×D4—22no—
70C23×D4—22no—
71C23×D4—24no—
72C23×D4D411yescharacteristic
73C23×D4C2312yes—
74C23×D4C2312yes—
75C24×C4—21no—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C22×M16C2314yes—
79C23×C8C2314yes—
80#547—216no—
81#547C2×C4132yescomplemented
82#546—264no—
83D42—216no—
84D42C2×C4132yescomplemented
85#537—216no—
86#534—216no—
87#534—216no—
88#532—28no—
89#532C2×C4116yescomplemented
90C2×C4⋊Q8C2312yes—
91C2×#127C2312yes—
92C2×#124—22no—
93C2×#123—24no—
94C2×#123—24no—
95C2×C22⋊Q8—24no—
96C2×#121—22no—
97C2×#121—22no—
98C2×#121—22no—
99C2×#121—22no—
100C2×#121—28no—
101C2×#121—28no—
102C2×#121—22no—
103C2×#121—24no—
104C2×#121C2×C4116yes—
105C2×#121C2×C4116yescomplemented
106C2×#121C2314yes—
107C2×#121C2314yes—
108C2×#121C2314yes—
109C2×#120—24no—
110C2×#120—24no—
111C2×C4×D4—22no—
112C2×C4×D4C2×C418yes—
113C2×C4×D4D418yes—
114C2×C4×D4D414yes—
115C2×C4×D4C2314yes—
116C2×C4×D4C2314yes—
117C2×#117—22no—
118C2×#117—44no—
119C2×#117C2314yes—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—44no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4D411yescharacteristic
137C22×C22⋊C4D411yescharacteristic
138C22×C42—21no—
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141#435—232no—
142#432—28no—
143C2×C4⋊C8C2314yes—
144#418—48no—
145C2×#102—48no—
146C2×#102C2318yes—
147C2×#102C2318yes—
148C2×#102C2318yes—
149C2×#102C2318yes—
150#407—28no—
151C2×C22⋊C8—28no—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×#97C2314yes—
156#388—28no—
157#387—28no—
158#386—28no—
159#382—22no—
160#382C2314yes—
161#380—22no—
162#380—22no—
163#380C2314yes—
164#380C2314yes—
165C4×C4⋊C4—22no—
166C4×C4⋊C4C2314yes—
167C4×C22⋊C4—28no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173#340C2318yes—
174#340C2318yes—
175#340C2318yes—
176#340C2318yes—
177C25—22no—
178C25—22no—
179C25C2×D411yescharacteristic
180C25C2×D411yescharacteristic
181H(5,𝔽2)—4128no—
182C2×#41—232no—
183C2×#41—216no—
184C2×#41—48no—
185C2×#41—48no—
186C2×#41—416no—
187C2×#41—416no—
188C22×Q8—22no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—216no—
192C22×D4—216no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—232no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—416no—
208C22×D4—416no—
209C22×D4—48no—
210C22×D4—48no—
211C22×D4—48no—
212C22×D4—48no—
213C22×D4C22⋊C418yes—
214C22×D4C22×C418yes—
215C22×D4C22×C418yescomplemented
216C22×D4C2×D414yes—
217C22×D4C2414yes—
218C23×C4—21no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—82no—
233C23×C4C22×C414yes—
234C23×C4C22×C414yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D412yes—
237C23×C4C2×D412yes—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2412yes—
246C2×M16—28no—
247C2×M16—44no—
248C2×M16—44no—
249C22×C8—28no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8C2418yes—
253C4⋊Q8—24no—
254C4⋊Q8C22×C418yescomplemented
255#127—24no—
256#127C22×C418yescomplemented
257#124—216no—
258#123—232no—
259#123—232no—
260C22⋊Q8—232no—
261#121—216no—
262#121—216no—
263#121—216no—
264#121—216no—
265#121—216no—
266#121—28no—
267#121—28no—
268#121—28no—
269#121—232no—
270#121C22×C4116yes—
271#121C22×C4116yes—
272#121C22×C4116yescomplemented
273#120—232no—
274#120—232no—
275C4×D4—216no—
276C4×D4—28no—
277C4×D4C22⋊C4132yes—
278C4×D4C22×C4116yes—
279C4×D4C22×C4116yescomplemented
280C4×D4C2×D4116yes—
281#117—28no—
282#117—44no—
283#117—48no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—48no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4C22⋊C418yes—
303C2×C4⋊C4C22×C418yes—
304C2×C4⋊C4C22×C418yescomplemented
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—216no—
312C2×C22⋊C4—216no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4C22⋊C418yes—
324C2×C22⋊C4C22⋊C418yes—
325C2×C22⋊C4C2×D414yes—
326C2×C22⋊C4C2×D414yes—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—44no—
330C2×C42—82no—
331C2×C42C22×C414yes—
332C2×C42C2×D412yes—
333C2×C42C2×D414yes—
334C2×C42C2412yes—
335C4⋊C8—28no—
336#102—216no—
337#102—48no—
338#102—48no—
339#102—48no—
340C22⋊C8—28no—
341C22⋊C8—28no—
342C22⋊C8—28no—
343#97—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—28no—
353C24—28no—
354C24—41no—
355C24—41no—
356C24—41no—
357C24—41no—
358C24—41no—
359C24—41no—
360C24—41no—
361C24—41no—
362C24—44no—
363C24—44no—
364C24—44no—
365C24—44no—
366C24—82no—
367C24—82no—
368C24—82no—
369C24—82no—
370C24—82no—
371C24—82no—
372C24—82no—
373C24—82no—
374C24C2×C22⋊C414yes—
375C24C2×C22⋊C414yes—
376C24AGL(1,ℤ/8)14yes—
377C24AGL(1,ℤ/8)14yes—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C22×D411yescharacteristic
381#41—464no—
382#41—4128no—
383C2×Q8—28no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—232no—
389C2×D4—216no—
390C2×D4—216no—
391C2×D4—216no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—416no—
400C2×D4—416no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—4128no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—464no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—464no—
412C2×D4—88no—
413C2×D4—88no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4C2×C22⋊C4116yes—
421C2×D4C23×C418yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—216no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—48no—
471C22×C4—42no—
472C22×C4—84no—
473C22×C4—816no—
474C22×C4C2×C22⋊C414yes—
475C22×C4C2×C22⋊C414yes—
476C22×C4C2×C22⋊C418yes—
477C22×C4C2×C22⋊C418yes—
478C22×C4C2×C22⋊C414yes—
479C22×C4C2×C22⋊C414yes—
480C22×C4C23×C412yes—
481C22×C4C23×C414yes—
482C22×C4C23×C414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C2511yescharacteristic
492M16—88no—
493C2×C8—28no—
494C2×C8—28no—
495C2×C8—28no—
496C2×C8—88no—
497C4⋊C4—24no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4C2×C22⋊C4116yes—
511C4⋊C4C23×C418yes—
512C4⋊C4H(5,𝔽2)14yescomplemented
513C22⋊C4—216no—
514C22⋊C4—216no—
515C22⋊C4—216no—
516C22⋊C4—216no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—84no—
520C22⋊C4C2×C22⋊C4116yes—
521C22⋊C4C2×C22⋊C4116yes—
522C42—48no—
523C42—84no—
524C42C2×C22⋊C418yes—
525C42C23×C414yes—
526C42C22×D414yes—
527C23—21no—
528C23—22no—
529C23—21no—
530C23—22no—
531C23—21no—
532C23—22no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—28no—
542C23—28no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—416no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—416no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23—82no—
563C23—88no—
564C23—88no—
565C23—88no—
566C23—88no—
567C23—164no—
568C23—164no—
569C23—164no—
570C23—164no—
571C23—164no—
572C23—164no—
573C23—164no—
574C23—164no—
575C23#418116yescomplemented
576C23C22×C22⋊C414yes—
577C23C22×C22⋊C414yes—
578C23C22×C22⋊C412yes—
579C23C2×AGL(1,ℤ/8)14yes—
580C23C2×AGL(1,ℤ/8)12yes—
581C23C2×AGL(1,ℤ/8)12yes—
582C23C2×AGL(1,ℤ/8)14yes—
583C23C23×D411yescharacteristic
584Q8—416no—
585D4—232no—
586D4—216no—
587D4—416no—
588D4—432no—
589D4—432no—
590D4—432no—
591D4—48no—
592D4—88no—
593D4—88no—
594D4—88no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—432no—
620C2×C4—48no—
621C2×C4—416no—
622C2×C4—42no—
623C2×C4—416no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—816no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4C22×C22⋊C412yes—
641C2×C4C22×C22⋊C414yes—
642C2×C4C22×C22⋊C414yes—
643C2×C4C22×C22⋊C414yes—
644C2×C4C22×C22⋊C414yes—
645C2×C4C24×C411yescharacteristic
646C2×C4C23×D411yescharacteristic
647C2×C4C2×H(5,𝔽2)12yes—
648C2×C4#58512yes—
649C8—48no—
650C22—22no—
651C22—24no—
652C22—22no—
653C22—24no—
654C22—22no—
655C22—24no—
656C22—22no—
657C22—24no—
658C22—22no—
659C22—24no—
660C22—22no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—416no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—48no—
671C22—48no—
672C22—48no—
673C22—88no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—1616no—
679C22—1616no—
680C22C2×#41814yes—
681C22C2×#41818yes—
682C22C23×C22⋊C411yescharacteristic
683C22#348411yescharacteristic
684C22#348511yescharacteristic
685C22C22×AGL(1,ℤ/8)11yescharacteristic
686C22C22×AGL(1,ℤ/8)11yescharacteristic
687C22#358511yescharacteristic
688C22#358611yescharacteristic
689C4—24no—
690C4—24no—
691C4—22no—
692C4—24no—
693C4—24no—
694C4—44no—
695C4—48no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4C4×H(5,𝔽2)12yes—
700C4#349612yes—
701C2—24no—
702C2—28no—
703C2—44no—
704C2—88no—
705C2#3425611yescharacteristic
706C2C22×#41811yescharacteristic
707C2#6008911yescharacteristic
708C2#6156911yescharacteristic
709C2#6157211yescharacteristic
710C2#6170411yescharacteristic
711C2#6170711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3496C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2973C2semidirect
3. C2×#2352C2semidirect
4. #22495C2semidirect
5. #22423C2semidirect
6. #21865C2semidirect
7. #21764C2semidirect
8. #21758C2semidirect
9. #21395C2semidirect
10. #20912C2semidirect
11. C2×#1874C2semidirect
14. #3496C4semidirect
25. C22×#117C22semidirect
26. C23×C4⋊C4C22semidirect
28. C2×#435C22semidirect
29. C2×#432C22semidirect
30. C2×#418C22semidirect
31. C22×#102C22semidirect
32. C2×#407C22semidirect
33. C22×C22⋊C8C22semidirect
35. #2352C22semidirect
39. C2×#382C22semidirect
40. C2×#380C22semidirect
41. C2×#380C22semidirect
42. C2×C4×C4⋊C4C22semidirect
44. #1993C22semidirect
45. #1987C22semidirect
46. #1982C22semidirect
47. #1961C22semidirect
48. #1895C22semidirect
49. #1893C22semidirect
50. #1875C22semidirect
51. #1874C22semidirect
52. #1874C22semidirect
53. #1874C22semidirect
54. #1874C22semidirect
55. #1874C22semidirect
56. #1824C22semidirect
57. #1823C22semidirect
58. #1804C22semidirect
59. #1786C22semidirect
60. #1786C22semidirect
61. #1762C22semidirect
62. #1736C22semidirect
63. C2×#340C22semidirect
81. #547C2×C4semidirect
84. D42C2×C4semidirect
89. #532C2×C4semidirect
105. C2×#121C2×C4semidirect
215. C22×D4C22×C4semidirect
254. C4⋊Q8C22×C4semidirect
256. #127C22×C4semidirect
272. #121C22×C4semidirect
279. C4×D4C22×C4semidirect
304. C2×C4⋊C4C22×C4semidirect
512. C4⋊C4H(5,𝔽2)semidirect
575. C23#418semidirect

Maximal subgroups

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

subgroupindex
C2×#34962
C2×#29732
C2×#23522
#224952
#224232
#218652
#217642
#217582
#213952
#209122
C2×#18742