Finite Group Databasebeta

← #6447638

Subgroups — #6447638

Order 512.

3264 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×#544C211yesmaximal, characteristic, complemented
2C2×#3142C211yesmaximal, characteristic, complemented
3C2×#3141C211yesmaximal, characteristic, complemented
4#34633C211yesmaximal, characteristic
5#25916C218yesmaximal, complemented
6#25915C214yesmaximal, complemented
7#25914C214yesmaximal, complemented
8#23453C212yesmaximal, complemented
9#23452C212yesmaximal, complemented
10#21325C211yesmaximal, characteristic, complemented
11#21324C211yesmaximal, characteristic, complemented
12#20910C212yesmaximal, complemented
13#20909C212yesmaximal, complemented
14#20377C211yesmaximal, characteristic, complemented
15C2×#544—28no—
16C2×#544—28no—
17C2×#544—22no—
18C2×#544—22no—
19C2×#544C2214yes—
20C2×#544C2214yes—
21C2×#532C2211yescharacteristic
22C22×C4⋊Q8—21no—
23C22×C4⋊Q8C2211yescharacteristic
24C22×C4⋊Q8C2211yescharacteristic
25C22×#125—21no—
26C22×#125C2211yescharacteristic
27C22×#125C2211yescharacteristic
28C2×#525C2212yes—
29C22×C22⋊Q8—21no—
30C22×C22⋊Q8—21no—
31C22×#117C2211yescharacteristic
32C22×#117C2211yescharacteristic
33C23×C4⋊C4C2211yescharacteristic
34#3142—28no—
35#3141—28no—
36C2×#491C2212yescomplemented
37C2×#490C2212yescomplemented
38C2×#489C2214yescomplemented
39C2×#488C2214yescomplemented
40C2×#487C2212yescomplemented
41C2×#486C2212yescomplemented
42C2×#417C2212yes—
43C2×#417C2212yes—
44C2×#417C2212yes—
45C2×#417C2212yes—
46C2×#407C2212yes—
47C2×C4×M16C2211yescharacteristic
48C2×C4×M16C2211yescharacteristic
49#2621C2212yes—
50#2620C2214yes—
51#2619C2212yes—
52#2618—21no—
53#2618—21no—
54#2617—21no—
55#2617—21no—
56#2392—22no—
57#2392—22no—
58#2297C2212yes—
59#2285C2211yescharacteristic
60#2026C2218yescomplemented
61#2025C2218yescomplemented
62#2024C2218yescomplemented
63#2023C2218yescomplemented
64#1993C2214yescomplemented
65#1992C2214yescomplemented
66#1991C2214yescomplemented
67#1990C2214yescomplemented
68#1875C2218yes—
69#1874C2218yes—
70#1873C2218yes—
71#1872C2218yes—
72#1750C2214yes—
73#1750C2214yes—
74#1736C2212yes—
75#1736C2212yes—
76#1736C2212yes—
77#1736C2212yes—
78#1728C2212yes—
79C22×#41C2312yes—
80C23×Q8—21no—
81C23×Q8—21no—
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84C22×M16C2312yes—
85C22×M16C2312yes—
86#544—232no—
87#544—232no—
88#544—216no—
89#544—216no—
90#544—216no—
91#544—216no—
92#532—24no—
93#532—24no—
94C2×C4⋊Q8—216no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8D418yescomplemented
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#127C2311yescharacteristic
109C2×#125—216no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125—24no—
113C2×#125—24no—
114C2×#125—22no—
115C2×#125—21no—
116C2×#125—21no—
117C2×#125—24no—
118C2×#125D418yescomplemented
119C2×#125C2314yes—
120C2×#125C2312yes—
121C2×#125C2312yes—
122C2×#124C2312yes—
123#525—216no—
124C2×#123—22no—
125C2×C22⋊Q8—216no—
126C2×C22⋊Q8—216no—
127C2×C22⋊Q8—24no—
128C2×C22⋊Q8—24no—
129C2×C22⋊Q8—24no—
130C2×C22⋊Q8—24no—
131C2×C22⋊Q8C2314yes—
132C2×#121—21no—
133C2×#121—21no—
134C2×#121C2314yes—
135C2×#120—22no—
136C2×C4×D4—24no—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×#117—22no—
140C2×#117—24no—
141C2×#117—24no—
142C2×#117—24no—
143C2×#117—24no—
144C2×#117D412yes—
145C2×#117D412yes—
146C2×#117C2314yes—
147C2×#117C2312yes—
148C2×#117C2312yes—
149C2×#117C2312yes—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—22no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4—21no—
156C22×C4⋊C4—21no—
157C22×C4⋊C4—21no—
158C22×C4⋊C4—24no—
159C22×C4⋊C4—24no—
160C22×C4⋊C4D411yescharacteristic
161C22×C4⋊C4D411yescharacteristic
162C22×C4⋊C4C2314yes—
163C22×C4⋊C4C2311yescharacteristic
164C22×C4⋊C4C2311yescharacteristic
165C22×C4⋊C4C2311yescharacteristic
166C22×C4⋊C4C2311yescharacteristic
167C22×C22⋊C4—21no—
168C22×C22⋊C4D411yescharacteristic
169C22×C22⋊C4D411yescharacteristic
170C22×C42—21no—
171C22×C42C2311yescharacteristic
172C22×C42C2311yescharacteristic
173#491—28no—
174#490—28no—
175#489—216no—
176#488—216no—
177#487—28no—
178#486—28no—
179#417—28no—
180#417—28no—
181#417—28no—
182#417—28no—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#102C2314yes—
188C2×#102C2314yes—
189C2×#102C2314yes—
190C2×#102C2314yes—
191#407—28no—
192C2×C22⋊C8C2318yes—
193C4×M16—24no—
194C4×M16—24no—
195C2×#97C2312yes—
196C2×#97C2312yes—
197C2×C4×C8C2312yes—
198C2×C4×C8C2312yes—
199#398—22no—
200#398—22no—
201#397—22no—
202#397—22no—
203#396—24no—
204#396—24no—
205#394—22no—
206#394—22no—
207#394—24no—
208#394—24no—
209#392—22no—
210#392—22no—
211#391C2314yes—
212#390C2314yes—
213#386—22no—
214#386—22no—
215#384C2312yes—
216#382C2312yes—
217#380—21no—
218#380—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C22⋊C4—24no—
222#340C2318yes—
223#340C2318yes—
224#340C2318yes—
225#340C2318yes—
226#336C2318yes—
227C25C2×D411yescharacteristic
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—28no—
231C2×#41—28no—
232C22×Q8—28no—
233C22×Q8—28no—
234C22×Q8—24no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—84no—
238C22×Q8—84no—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2412yes—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—28no—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4—84no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4#4118yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—24no—
271C22×C8C2414yes—
272C22×C8C2414yes—
273C4⋊Q8—232no—
274C4⋊Q8—232no—
275C4⋊Q8—216no—
276C4⋊Q8—28no—
277C4⋊Q8—28no—
278C4⋊Q8—28no—
279C4⋊Q8—28no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8C2×D418yes—
285C4⋊Q8C2×D418yes—
286#127—22no—
287#127—22no—
288#125—232no—
289#125—232no—
290#125—216no—
291#125—28no—
292#125—28no—
293#125—28no—
294#125—28no—
295#125—24no—
296#125—24no—
297#125C2×D418yes—
298#125C2×D418yes—
299#124—24no—
300#124—24no—
301#123—48no—
302C22⋊Q8—232no—
303C22⋊Q8—232no—
304C22⋊Q8—232no—
305C22⋊Q8—232no—
306C22⋊Q8—216no—
307#121—216no—
308#121—44no—
309#121—44no—
310#120—48no—
311C4×D4—416no—
312C4×D4—44no—
313C4×D4—44no—
314#117—216no—
315#117—28no—
316#117—28no—
317#117—28no—
318#117—28no—
319#117—24no—
320#117—24no—
321#117—24no—
322#117—24no—
323#117—28no—
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—28no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—28no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×D412yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2×D414yes—
366C2×C4⋊C4C2×D414yes—
367C2×C4⋊C4C2414yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4C2×D412yes—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D412yes—
387C2×C42—28no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—24no—
391C2×C42—22no—
392C2×C42C2×D412yes—
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D414yes—
396C2×C42C2×D414yes—
397C2×C42C2412yes—
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400#103—28no—
401#103—28no—
402#103—28no—
403#103—28no—
404#102—28no—
405#102—28no—
406#102—28no—
407#102—28no—
408C22⋊C8—216no—
409#97—24no—
410#97—24no—
411C4×C8—24no—
412C4×C8—24no—
413#95—24no—
414#95—24no—
415#95—22no—
416#95—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—42no—
421C24—42no—
422C24#12414yes—
423C24C22×D412yes—
424C24C22×D411yescharacteristic
425#41—416no—
426#41—416no—
427C2×Q8—216no—
428C2×Q8—28no—
429C2×Q8—28no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—84no—
435C2×Q8—84no—
436C2×Q8—88no—
437C2×Q8—816no—
438C2×Q8—816no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—44no—
444C2×D4—48no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4—84no—
452C2×D4—84no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—216no—
458C22×C4—216no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—216no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—88no—
487C22×C4—816no—
488C22×C4#12014yes—
489C22×C4#12014yes—
490C22×C4#12418yes—
491C22×C4#12414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C2×#4118yes—
503C22×C4C2×#4114yes—
504C22×C42−1+414yes—
505C22×C4C2511yescharacteristic
506M16—48no—
507M16—48no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C4⋊C4—216no—
515C4⋊C4—216no—
516C4⋊C4—216no—
517C4⋊C4—216no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—28no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—48no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4C22×D414yes—
543C4⋊C4C22×D414yes—
544C4⋊C4C22×D414yes—
545C4⋊C4C22×D414yes—
546C22⋊C4—216no—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—44no—
552C22⋊C4—48no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—48no—
556C22⋊C4—84no—
557C22⋊C4—84no—
558C42—216no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—44no—
563C42#12014yes—
564C42#12014yes—
565C42C22×D412yes—
566C42C22×D412yes—
567C42C22×D412yes—
568C42C22×D412yes—
569C23—22no—
570C23—22no—
571C23—24no—
572C23—24no—
573C23—22no—
574C23—22no—
575C23—28no—
576C23—48no—
577C23—44no—
578C23—48no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23C2×#12012yes—
586C23C2×#12414yes—
587C23C2×#12412yes—
588C23#53612yes—
589C23C23×D411yescharacteristic
590Q8—44no—
591Q8—44no—
592Q8—88no—
593Q8—88no—
594Q8—816no—
595D4—44no—
596D4—44no—
597D4—84no—
598D4—84no—
599D4—816no—
600D4—88no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—216no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—816no—
638C2×C4—88no—
639C2×C4C2×#12012yes—
640C2×C4C2×#12012yes—
641C2×C4C2×#12012yes—
642C2×C4C2×#12012yes—
643C2×C4C2×#12012yes—
644C2×C4C2×#12414yes—
645C2×C4C2×#12412yes—
646C2×C4#53612yes—
647C2×C4#54818yes—
648C2×C4#54818yes—
649C2×C4C23×D411yescharacteristic
650C2×C4C23×D411yescharacteristic
651C2×C4C22×#4112yes—
652C2×C4C2×2−1+412yes—
653C8—44no—
654C8—44no—
655C22—24no—
656C22—24no—
657C22—22no—
658C22—22no—
659C22—24no—
660C22—28no—
661C22—416no—
662C22—44no—
663C22—42no—
664C22—42no—
665C22—82no—
666C22—84no—
667C22—84no—
668C22—82no—
669C22—82no—
670C22#243614yes—
671C22C22×#12011yescharacteristic
672C22C22×#12411yescharacteristic
673C22C2×#53611yescharacteristic
674C22#358512yes—
675C22#358612yes—
676C4—24no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—42no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—44no—
687C4—82no—
688C4—84no—
689C4#243614yes—
690C4C2×#54812yes—
691C4C2×#54812yes—
692C2—24no—
693C2—44no—
694C2—162no—
695C2C2×#243611yescharacteristic
696C2#6036711yescharacteristic
697C2#6036811yescharacteristic
698C2#6062112yes—
699C2#6086712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34633C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#544C2semidirect
2. C2×#3142C2semidirect
3. C2×#3141C2semidirect
5. #25916C2semidirect
6. #25915C2semidirect
7. #25914C2semidirect
8. #23453C2semidirect
9. #23452C2semidirect
10. #21325C2semidirect
11. #21324C2semidirect
12. #20910C2semidirect
13. #20909C2semidirect
14. #20377C2semidirect
36. C2×#491C22semidirect
37. C2×#490C22semidirect
38. C2×#489C22semidirect
39. C2×#488C22semidirect
40. C2×#487C22semidirect
41. C2×#486C22semidirect
60. #2026C22semidirect
61. #2025C22semidirect
62. #2024C22semidirect
63. #2023C22semidirect
64. #1993C22semidirect
65. #1992C22semidirect
66. #1991C22semidirect
67. #1990C22semidirect
103. C2×C4⋊Q8D4semidirect
118. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#31422
C2×#31412
#346332
#259162
#259152
#259142
#234532
#234522
#213252
#213242
#209102
#209092
#203772