Finite Group Databasebeta

← #6450140

Subgroups — #6450140

Order 512.

3214 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#3166C211yesmaximal, characteristic, complemented
3C2×#3078C211yesmaximal, characteristic, complemented
4#34946C211yesmaximal, characteristic
5#25716C214yesmaximal, complemented
6#25715C214yesmaximal, complemented
7#25708C218yesmaximal, complemented
8#23481C212yesmaximal, complemented
9#23481C212yesmaximal, complemented
10#21314C211yesmaximal, characteristic, complemented
11#21190C211yesmaximal, characteristic
12#20845C212yesmaximal, complemented
13#20844C212yesmaximal, complemented
14C4×#419C211yesmaximal, characteristic, complemented
15C2×#552—28no—
16C2×#552—28no—
17C2×#552—22no—
18C2×#552—22no—
19C2×#552C2214yes—
20C2×#552C2214yes—
21C2×#533C2211yescharacteristic
22C22×#126—21no—
23C22×#125—21no—
24C22×#125C2211yescharacteristic
25C22×#123—21no—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×#117—21no—
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#3166—28no—
34C2×#501C2212yescomplemented
35C2×#501C2212yescomplemented
36C2×#499C2214yescomplemented
37#3078—28no—
38C2×#479C2214yescomplemented
39C2×#477C2212yescomplemented
40C2×#475C2212yescomplemented
41C2×#428C2212yes—
42C2×#428C2212yes—
43C2×#428C2212yes—
44C2×#423C2211yescharacteristic
45C2×#419C2212yes—
46C2×#419C2212yes—
47C2×#407C2212yes—
48C2×#407C2212yes—
49C2×C4×M16C2211yescharacteristic
50#2563C2212yes—
51#2562C2214yes—
52#2561—21no—
53#2561—21no—
54#2547C2212yes—
55#2503—21no—
56#2435—21no—
57#2398—22no—
58C4×#124—22no—
59#2280C2212yes—
60#2280C2212yes—
61C4×#117C2211yescharacteristic
62#2036C2218yescomplemented
63#2036C2218yescomplemented
64#2035C2218yescomplemented
65#2035C2218yescomplemented
66#1989C2214yescomplemented
67#1989C2214yescomplemented
68#1955C2214yes—
69#1954C2214yes—
70#1854C2218yescomplemented
71#1854C2218yescomplemented
72#1853C2218yescomplemented
73#1852C2218yescomplemented
74#1763C2214yescomplemented
75C4×#103C2214yescomplemented
76#1736C2212yes—
77#1736C2212yes—
78#1736C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4D411yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×M16C2312yes—
85C22×M16C2312yes—
86#552—232no—
87#552—232no—
88#552—216no—
89#552—216no—
90#552—216no—
91#552—216no—
92#533—24no—
93#533—24no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#126—216no—
97C2×#126—24no—
98C2×#126—24no—
99C2×#125—216no—
100C2×#125—24no—
101C2×#125—24no—
102C2×#125—21no—
103C2×#125—21no—
104C2×#125—24no—
105C2×#125D418yes—
106C2×#125C2312yes—
107C2×#125C2312yes—
108C2×#125C2312yes—
109C2×#124—22no—
110C2×#123—216no—
111C2×#123—24no—
112C2×#123—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8C2314yes—
122C2×C22⋊Q8C2314yes—
123C2×C4×Q8—24no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8D418yes—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×D4—24no—
132C2×C4×D4—24no—
133C2×C4×D4D412yes—
134C2×C4×D4D412yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×#117—216no—
138C2×#117—22no—
139C2×#117—24no—
140C2×#117—24no—
141C2×#117—24no—
142C2×#117—24no—
143C2×#117C2314yes—
144C2×#117C2312yes—
145C2×#117C2312yes—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—22no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—24no—
154C22×C4⋊C4—24no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4C2314yes—
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C22⋊C4—21no—
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C22⋊C4D411yescharacteristic
164C22×C22⋊C4D411yescharacteristic
165C22×C42—21no—
166C22×C42—21no—
167C22×C42C2311yescharacteristic
168#501—28no—
169#501—28no—
170#499—216no—
171#479—216no—
172#477—28no—
173#475—28no—
174#428—28no—
175#428—28no—
176#428—28no—
177C2×#107C2314yes—
178C2×#107C2314yes—
179C2×#107C2314yes—
180C2×#106C2314yes—
181C2×#106C2314yes—
182C2×#106C2314yes—
183#423—24no—
184C2×C4⋊C8C2312yes—
185C2×C4⋊C8C2312yes—
186#419—28no—
187#419—28no—
188C2×#103C2314yes—
189C2×#103C2314yes—
190C2×#103C2318yes—
191#407—28no—
192#407—28no—
193C2×C22⋊C8C2318yes—
194C2×C22⋊C8C2318yes—
195C4×M16—24no—
196C2×#97C2312yes—
197C2×C4×C8C2312yes—
198#400—24no—
199#397—22no—
200#397—22no—
201#394—24no—
202#393—22no—
203#393—22no—
204#391—22no—
205#390—22no—
206#388—24no—
207#388—24no—
208#382—22no—
209#381—22no—
210#381—22no—
211#380C2314yes—
212#380C2314yes—
213#380C2314yes—
214#380C2314yes—
215#379—22no—
216#379—22no—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—24no—
221C4×C22⋊C4—24no—
222C4×C22⋊C4—21no—
223#376—21no—
224C43—21no—
225#340C2318yes—
226#340C2318yes—
227#340C2318yes—
228C25C2×D411yescharacteristic
229C25C2×D411yescharacteristic
230C22×Q8—22no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—24no—
235C22×Q8—24no—
236C22×Q8C2×D414yes—
237C22×Q8C2412yes—
238C22×D4—24no—
239C22×D4—44no—
240C22×D4C2×D412yes—
241C22×D4C2×D412yes—
242C23×C4—28no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—24no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—24no—
256C23×C4—48no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2412yes—
264C23×C4C2412yes—
265C23×C4C2411yescharacteristic
266C2×M16—24no—
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—24no—
271C2×M16#4118yes—
272C22×C8C2414yes—
273C22×C8C2414yes—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278#126—232no—
279#126—232no—
280#125—232no—
281#125—232no—
282#125—28no—
283#125—28no—
284#125—24no—
285#125—24no—
286#125—24no—
287#125—24no—
288#125C2×D418yes—
289#125C2×D418yes—
290#124—48no—
291#123—232no—
292#123—232no—
293C22⋊Q8—216no—
294C22⋊Q8—216no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C22⋊Q8—28no—
298C22⋊Q8—28no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301C4×Q8—28no—
302C4×Q8—28no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8C2×D418yes—
308C4×Q8C2×D418yes—
309C4×D4—28no—
310C4×D4—28no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—24no—
315#117—216no—
316#117—232no—
317#117—232no—
318#117—28no—
319#117—28no—
320#117—24no—
321#117—24no—
322#117—24no—
323#117—24no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—216no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—28no—
348C2×C4⋊C4—28no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2414yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C22⋊C4—28no—
365C2×C22⋊C4—28no—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4C2×D412yes—
379C2×C22⋊C4C2×D412yes—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C42—28no—
383C2×C42—28no—
384C2×C42—24no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D414yes—
394C2×C42C2×D414yes—
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107—28no—
398#107—28no—
399#107—28no—
400#106—28no—
401#106—28no—
402#106—28no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405#103—28no—
406#103—28no—
407#103—216no—
408C22⋊C8—216no—
409C22⋊C8—216no—
410#97—24no—
411C4×C8—24no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418C24—22no—
419C24—24no—
420C24—22no—
421C24—24no—
422C24—24no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24C22×D412yes—
427C24C22×D412yes—
428C24C22×D411yescharacteristic
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—28no—
432C2×Q8—28no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—44no—
439C2×Q8—48no—
440C2×Q8—82no—
441C2×Q8C22×D414yes—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—416no—
449C22×C4—216no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—432no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4#12014yes—
493C22×C4#12014yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C2×#4114yes—
504C22×C42−1+414yes—
505C22×C4C2511yescharacteristic
506M16—216no—
507M16—48no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8C2×#4118yes—
514C4⋊C4—216no—
515C4⋊C4—216no—
516C4⋊C4—216no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4C22×D414yes—
542C4⋊C4C22×D414yes—
543C4⋊C4C22×D414yes—
544C22⋊C4—216no—
545C22⋊C4—216no—
546C22⋊C4—216no—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—48no—
555C22⋊C4—84no—
556C22⋊C4—84no—
557C42—216no—
558C42—28no—
559C42—28no—
560C42—44no—
561C42—44no—
562C42—44no—
563C42—82no—
564C42#12014yes—
565C42#12014yes—
566C42C22×D412yes—
567C42C22×D412yes—
568C42C22×D412yes—
569C42C22×D412yes—
570C23—22no—
571C23—22no—
572C23—28no—
573C23—24no—
574C23—22no—
575C23—24no—
576C23—22no—
577C23—24no—
578C23—22no—
579C23—24no—
580C23—22no—
581C23—22no—
582C23—416no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—88no—
589C23C2×#12012yes—
590C23#54714yes—
591C23#57514yes—
592C23C23×D411yescharacteristic
593Q8—24no—
594Q8—24no—
595Q8—48no—
596Q8—44no—
597Q8—84no—
598Q8—84no—
599D4—48no—
600D4—48no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—48no—
618C2×C4—432no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4C2×#12012yes—
642C2×C4C2×#12012yes—
643C2×C4C2×#12012yes—
644C2×C4C2×#12012yes—
645C2×C4C2×#12012yes—
646C2×C4#53214yes—
647C2×C4#53614yes—
648C2×C4#54714yes—
649C2×C4#54818yes—
650C2×C4C23×D411yescharacteristic
651C2×C4C23×D411yescharacteristic
652C2×C4C22×#4112yes—
653C2×C4C2×2−1+412yes—
654C8—28no—
655C8—44no—
656C22—28no—
657C22—22no—
658C22—22no—
659C22—28no—
660C22—24no—
661C22—416no—
662C22—44no—
663C22—42no—
664C22—42no—
665C22—44no—
666C22—44no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—88no—
671C22#243214yes—
672C22C22×#12011yescharacteristic
673C22C2×#54712yes—
674C22C2×#57512yes—
675C22#358612yes—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—84no—
687C4—82no—
688C4—82no—
689C4#243214yes—
690C4C2×#53211yescharacteristic
691C4C2×#53611yescharacteristic
692C4C2×#54812yes—
693C2—24no—
694C2—44no—
695C2—82no—
696C2C2×#243211yescharacteristic
697C2#6035911yescharacteristic
698C2#6036811yescharacteristic
699C2#6125314yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×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×#3166C2semidirect
3. C2×#3078C2semidirect
5. #25716C2semidirect
6. #25715C2semidirect
7. #25708C2semidirect
8. #23481C2semidirect
9. #23481C2semidirect
10. #21314C2semidirect
12. #20845C2semidirect
13. #20844C2semidirect
14. C4×#419C2semidirect
34. C2×#501C22semidirect
35. C2×#501C22semidirect
36. C2×#499C22semidirect
38. C2×#479C22semidirect
39. C2×#477C22semidirect
40. C2×#475C22semidirect
62. #2036C22semidirect
63. #2036C22semidirect
64. #2035C22semidirect
65. #2035C22semidirect
66. #1989C22semidirect
67. #1989C22semidirect
70. #1854C22semidirect
71. #1854C22semidirect
72. #1853C22semidirect
73. #1852C22semidirect
74. #1763C22semidirect
75. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#31662
C2×#30782
#349462
#257162
#257152
#257082
#234812
#213142
#211902
#208452
#208442
C4×#4192