Finite Group Databasebeta

← #6393382

Subgroups — #6393382

Order 512.

3466 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C2×#2973C211yesmaximal, characteristic
3C2×#2352C211yesmaximal, characteristic
4#22497C218yesmaximal, complemented
5#22425C218yesmaximal, complemented
6#21866C212yesmaximal, complemented
7#21765C212yesmaximal, complemented
8#21759C212yesmaximal, complemented
9#21394C212yesmaximal, complemented
10#20911C212yesmaximal, complemented
11C2×#1872C212yesmaximal, complemented
12#3501—28no—
13#3501C4116yescomplemented
14C2×#554—24no—
15C2×#554C2214yes—
16C2×D4×Q8C2214yes—
17C2×#544—21no—
18C2×#543—21no—
19C2×#543—21no—
20C2×#533C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2212yes—
23C2×#518—21no—
24C22×C4×D4C2211yescharacteristic
25C22×#117C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27#2973—28no—
28C2×#435C2218yes—
29C2×#432C2212yes—
30C2×#419C2214yescomplemented
31C22×#103C2214yescomplemented
32C2×#407C2212yes—
33C22×C22⋊C8C2212yes—
34#2352—24no—
35#2352C2218yes—
36C2×#388—21no—
37C2×#387—21no—
38C2×#386—21no—
39C2×#382C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4—21no—
44#1993C2214yescomplemented
45#1988C2214yescomplemented
46#1981C2214yescomplemented
47#1960C2214yescomplemented
48#1894C2218yescomplemented
49#1892C2218yescomplemented
50#1873C2218yescomplemented
51#1872C2214yes—
52#1872C2214yes—
53#1872C2214yes—
54#1872C2214yes—
55#1872C2218yescomplemented
56#1824C2214yescomplemented
57#1823C2214yescomplemented
58#1805C2218yescomplemented
59#1787C2214yescomplemented
60#1787C2214yescomplemented
61#1763C2218yescomplemented
62#1736C2214yescomplemented
63C2×#340C2214yescomplemented
64C23×Q8C2312yes—
65C23×Q8C2312yes—
66C23×D4D411yescharacteristic
67C24×C4—21no—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C22×M16C2314yes—
71C23×C8C2314yes—
72#554—264no—
73#554—216no—
74#554C2×C4132yescomplemented
75D4×Q8—216no—
76D4×Q8C2×C4132yescomplemented
77#544—216no—
78#543—216no—
79#543—216no—
80#533—28no—
81#533C2×C4116yescomplemented
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#125—24no—
86C2×#125—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8C2×C4116yes—
99C2×C22⋊Q8C2×C4116yescomplemented
100C2×C22⋊Q8C2314yes—
101C2×C22⋊Q8C2314yes—
102C2×C22⋊Q8C2314yes—
103#518—216no—
104C2×C4×Q8—22no—
105C2×C4×Q8C2314yes—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4C2×C418yes—
109C2×C4×D4D418yes—
110C2×C4×D4D414yes—
111C2×C4×D4C2314yes—
112C2×#117—22no—
113C2×#117—22no—
114C2×#117—44no—
115C2×#117C2314yes—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—44no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—22no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4D411yescharacteristic
135C22×C22⋊C4D411yescharacteristic
136C22×C42—21no—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#435—232no—
140#432—28no—
141C2×C4⋊C8C2314yes—
142#419—48no—
143C2×#103—48no—
144C2×#103C2318yes—
145C2×#103C2318yes—
146C2×#103C2318yes—
147C2×#103C2318yes—
148#407—28no—
149C2×C22⋊C8—28no—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×#97C2314yes—
154#388—28no—
155#387—28no—
156#386—28no—
157#382—22no—
158#382C2314yes—
159#380—22no—
160#380—22no—
161#380C2314yes—
162#380C2314yes—
163C4×C4⋊C4—22no—
164C4×C4⋊C4C2314yes—
165C4×C22⋊C4—28no—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171#340C2318yes—
172#340C2318yes—
173#340C2318yes—
174#340C2318yes—
175C25C2×D411yescharacteristic
176C25C2×D411yescharacteristic
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8C22×C418yes—
187C22×Q8C22×C418yescomplemented
188C22×Q8C2414yes—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4C22⋊C418yes—
192C22×D4C2×D414yes—
193C23×C4—21no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—24no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—82no—
209C23×C4C22×C414yes—
210C23×C4C22×C414yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C2×M16—28no—
223C2×M16—44no—
224C2×M16—44no—
225C22×C8—28no—
226C22×C8—44no—
227C22×C8—44no—
228C22×C8C2418yes—
229C4⋊Q8—232no—
230C4⋊Q8—24no—
231C4⋊Q8—24no—
232C4⋊Q8C22×C418yescomplemented
233C4⋊Q8C22×C418yescomplemented
234#125—232no—
235#125—216no—
236C22⋊Q8—216no—
237C22⋊Q8—216no—
238C22⋊Q8—216no—
239C22⋊Q8—216no—
240C22⋊Q8—216no—
241C22⋊Q8—28no—
242C22⋊Q8—28no—
243C22⋊Q8—28no—
244C22⋊Q8—232no—
245C22⋊Q8—232no—
246C22⋊Q8—232no—
247C22⋊Q8—232no—
248C22⋊Q8C22×C4116yes—
249C22⋊Q8C22×C4116yes—
250C22⋊Q8C22×C4116yescomplemented
251C4×Q8—216no—
252C4×Q8—28no—
253C4×Q8C22×C4116yescomplemented
254C4×D4—232no—
255C4×D4—232no—
256C4×D4C22⋊C4132yes—
257C4×D4C22×C4116yes—
258C4×D4C2×D4116yes—
259#117—216no—
260#117—28no—
261#117—44no—
262#117—48no—
263C2×C4⋊C4—216no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C22⋊C418yes—
290C2×C4⋊C4C22×C418yes—
291C2×C4⋊C4C22×C418yescomplemented
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—216no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4C22⋊C418yes—
310C2×C22⋊C4C22⋊C418yes—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D414yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42—82no—
318C2×C42C22×C414yes—
319C2×C42C2×D412yes—
320C2×C42C2×D414yes—
321C2×C42C2412yes—
322C4⋊C8—28no—
323#103—216no—
324#103—48no—
325#103—48no—
326#103—48no—
327C22⋊C8—28no—
328C22⋊C8—28no—
329C22⋊C8—28no—
330#97—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336C24—22no—
337C24—22no—
338C24—41no—
339C24—41no—
340C24—41no—
341C24—41no—
342C24—41no—
343C24—41no—
344C24—41no—
345C24—41no—
346C24C2×C22⋊C414yes—
347C24C2×C22⋊C414yes—
348C24#13714yes—
349C24#13714yes—
350C24C22×D412yes—
351C24C22×D412yes—
352C24C22×D411yescharacteristic
353C2×Q8—28no—
354C2×Q8—28no—
355C2×Q8—28no—
356C2×Q8—24no—
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—416no—
360C2×Q8—416no—
361C2×Q8—82no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×Q8—82no—
367C2×Q8—88no—
368C2×Q8—88no—
369C2×Q8C23×C418yes—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—232no—
373C2×D4C2×C22⋊C4116yes—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—48no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—48no—
419C22×C4—42no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—84no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—816no—
430C22×C4C2×C22⋊C414yes—
431C22×C4C2×C22⋊C414yes—
432C22×C4C2×C22⋊C414yes—
433C22×C4C2×C22⋊C414yes—
434C22×C4C2×C22⋊C418yes—
435C22×C4C2×C22⋊C418yes—
436C22×C4C23×C412yes—
437C22×C4C23×C414yes—
438C22×C4C23×C414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C2511yescharacteristic
448M16—88no—
449C2×C8—28no—
450C2×C8—28no—
451C2×C8—28no—
452C2×C8—88no—
453C4⋊C4—24no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—216no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—48no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4C2×C22⋊C4116yes—
473C4⋊C4C23×C418yes—
474C4⋊C4H(5,𝔽2)14yes—
475C22⋊C4—216no—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—84no—
481C22⋊C4C2×C22⋊C4116yes—
482C22⋊C4C2×C22⋊C4116yes—
483C42—216no—
484C42—48no—
485C42—84no—
486C42C2×C22⋊C418yes—
487C42C23×C414yes—
488C42C22×D414yes—
489C23—21no—
490C23—22no—
491C23—21no—
492C23—22no—
493C23—21no—
494C23—22no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—28no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23#419116yescomplemented
517C23C22×C22⋊C414yes—
518C23C22×C22⋊C412yes—
519C23C22×C22⋊C414yes—
520C23C2×#13714yes—
521C23C2×#13712yes—
522C23C2×#13712yes—
523C23C2×#13714yes—
524C23C23×D411yescharacteristic
525Q8—216no—
526Q8—48no—
527Q8—88no—
528Q8—88no—
529Q8—88no—
530D4—232no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—88no—
573C2×C4—88no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4—816no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4C22×C22⋊C414yes—
581C2×C4C22×C22⋊C412yes—
582C2×C4C22×C22⋊C414yes—
583C2×C4C22×C22⋊C414yes—
584C2×C4C22×C22⋊C414yes—
585C2×C4C24×C411yescharacteristic
586C2×C4C23×D411yescharacteristic
587C2×C4C2×H(5,𝔽2)12yes—
588C2×C4#58512yes—
589C8—48no—
590C22—22no—
591C22—24no—
592C22—22no—
593C22—24no—
594C22—22no—
595C22—24no—
596C22—22no—
597C22—24no—
598C22—22no—
599C22—24no—
600C22—22no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—416no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—88no—
611C22C2×#41914yes—
612C22C2×#41918yes—
613C22C23×C22⋊C411yescharacteristic
614C22#348411yescharacteristic
615C22#348511yescharacteristic
616C22C22×#13711yescharacteristic
617C22C22×#13711yescharacteristic
618C22#358511yescharacteristic
619C22#358611yescharacteristic
620C4—24no—
621C4—24no—
622C4—22no—
623C4—24no—
624C4—24no—
625C4—44no—
626C4—48no—
627C4—48no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4C4×H(5,𝔽2)12yes—
632C4#349612yes—
633C2—24no—
634C2—28no—
635C2—44no—
636C2#3425611yescharacteristic
637C2C22×#41911yescharacteristic
638C2#6008911yescharacteristic
639C2#6157011yescharacteristic
640C2#6157311yescharacteristic
641C2#6170511yescharacteristic
642C2#6170811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
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
4. #22497C2semidirect
5. #22425C2semidirect
6. #21866C2semidirect
7. #21765C2semidirect
8. #21759C2semidirect
9. #21394C2semidirect
10. #20911C2semidirect
11. C2×#1872C2semidirect
13. #3501C4semidirect
30. C2×#419C22semidirect
31. C22×#103C22semidirect
44. #1993C22semidirect
45. #1988C22semidirect
46. #1981C22semidirect
47. #1960C22semidirect
48. #1894C22semidirect
49. #1892C22semidirect
50. #1873C22semidirect
55. #1872C22semidirect
56. #1824C22semidirect
57. #1823C22semidirect
58. #1805C22semidirect
59. #1787C22semidirect
60. #1787C22semidirect
61. #1763C22semidirect
62. #1736C22semidirect
63. C2×#340C22semidirect
74. #554C2×C4semidirect
76. D4×Q8C2×C4semidirect
81. #533C2×C4semidirect
99. C2×C22⋊Q8C2×C4semidirect
187. C22×Q8C22×C4semidirect
232. C4⋊Q8C22×C4semidirect
233. C4⋊Q8C22×C4semidirect
250. C22⋊Q8C22×C4semidirect
253. C4×Q8C22×C4semidirect
291. C2×C4⋊C4C22×C4semidirect
516. C23#419semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#29732
C2×#23522
#224972
#224252
#218662
#217652
#217592
#213942
#209112
C2×#18722