Finite Group Databasebeta

← #6393416

Subgroups — #6393416

Order 512.

3332 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3538C211yesmaximal, characteristic
2C2×#2977C211yesmaximal, characteristic
3#34200C211yesmaximal, characteristic
4#22491C214yesmaximal, complemented
5#22486C214yesmaximal, complemented
6#22425C214yesmaximal, complemented
7D4×#103C214yesmaximal, complemented
8#21873C212yesmaximal, complemented
9#21771C212yesmaximal, complemented
10#21762C212yesmaximal, complemented
11#21398C212yesmaximal, complemented
12#20913C212yesmaximal, complemented
13C2×#1872C212yesmaximal, complemented
14#3538—28no—
15#3538C4116yescomplemented
16C2×#563—22no—
17C2×#552C2214yes—
18C2×#551—22no—
19C2×D4×Q8C2214yes—
20C2×#544—22no—
21C2×#541C2212yes—
22C2×#539—22no—
23C2×#536C2212yes—
24C2×#525C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C2×#520C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28#2977—28no—
29C2×#435C2214yes—
30C2×C8×D4C2214yes—
31C2×#433C2212yes—
32C2×#419C2212yescomplemented
33C22×#103C2214yes—
34C22×#103C2212yescomplemented
35C2×#408C2212yes—
36C22×C22⋊C8C2212yes—
37#2361—22no—
38#2360—22no—
39#2352C2214yes—
40D4×C4⋊C4C2214yes—
41#2322—22no—
42#2317C2211yescharacteristic
43#2311C2211yescharacteristic
44C4×#120—22no—
45#2280C2211yescharacteristic
46#2274C2211yescharacteristic
47#1992C2212yescomplemented
48#1991C2212yescomplemented
49#1989C2214yescomplemented
50#1981C2214yescomplemented
51#1960C2212yescomplemented
52#1956C2212yescomplemented
53#1894C2214yescomplemented
54#1892C2214yes—
55#1892C2214yes—
56#1892C2214yescomplemented
57#1873C2214yes—
58#1873C2214yes—
59#1873C2214yescomplemented
60#1872C2214yes—
61#1872C2214yes—
62#1872C2214yes—
63#1872C2214yes—
64#1872C2214yescomplemented
65#1825C2214yescomplemented
66#1823C2212yes—
67#1823C2212yes—
68#1805C2214yescomplemented
69#1803C2214yescomplemented
70#1787C2212yes—
71#1787C2212yes—
72#1787C2214yescomplemented
73#1763C2214yescomplemented
74C4×#103C2214yescomplemented
75#1736C2212yescomplemented
76#1735C2212yescomplemented
77C2×#340C2214yes—
78C22×#41C2311yescharacteristic
79C23×Q8C2312yes—
80C23×Q8C2312yes—
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C22×M16C2312yes—
84C23×C8C2314yes—
85C23×C8C2312yes—
86#563—232no—
87#552—216no—
88#552C2×C4132yescomplemented
89#551—232no—
90D4×Q8—216no—
91D4×Q8C2×C4132yescomplemented
92#544—232no—
93#541—28no—
94#541C2×C4116yescomplemented
95#539—232no—
96#536—28no—
97#536C2×C4116yescomplemented
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8C2312yes—
101C2×#126—24no—
102C2×#126—24no—
103C2×#125—24no—
104C2×#125—24no—
105C2×#125C2312yes—
106C2×#124—22no—
107#525C2×C418yes—
108#525D418yes—
109C2×#123—22no—
110C2×#123—22no—
111C2×#123—24no—
112C2×#123D412yes—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—28no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8C2×C4116yescomplemented
119C2×C22⋊Q8D412yes—
120C2×C22⋊Q8C2314yes—
121C2×C22⋊Q8C2314yes—
122C2×C22⋊Q8C2314yes—
123C2×#121D412yes—
124C2×#120D412yes—
125#520—44no—
126C2×C4×Q8—21no—
127C2×C4×Q8C2314yes—
128C2×C4×D4—24no—
129C2×C4×D4—22no—
130C2×C4×D4—21no—
131C2×C4×D4—24no—
132C2×C4×D4C2314yes—
133C2×#117—24no—
134C2×#117C2312yes—
135C2×#117C2312yes—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—22no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—22no—
142C22×C4⋊C4—44no—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2314yes—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2311yescharacteristic
148C22×C22⋊C4—22no—
149#435—216no—
150C8×D4—216no—
151#433—28no—
152C2×C4⋊C8C2314yes—
153#419—44no—
154C2×#103—48no—
155C2×#103—44no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164#408—28no—
165C2×C22⋊C8—28no—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2314yes—
169C2×C22⋊C8C2314yes—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#391—21no—
173#390—21no—
174#389—22no—
175#389—24no—
176#388—24no—
177#387—24no—
178#386—22no—
179#386—24no—
180#385—22no—
181#385—24no—
182#385—24no—
183#384C2312yes—
184#382C2312yes—
185#381—21no—
186#380C2314yes—
187#380C2312yes—
188#380C2312yes—
189#379—21no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4C2314yes—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—24no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203C25C2×D411yescharacteristic
204C2×#41—42no—
205C2×#41—42no—
206C2×#41C22×C414yes—
207C2×#41C2×D414yes—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—44no—
213C22×Q8—42no—
214C22×Q8—42no—
215C22×Q8—44no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8C22×C418yes—
219C22×Q8C22×C418yescomplemented
220C22×Q8C2×D411yescharacteristic
221C22×Q8C2414yes—
222C22×D4C2×D414yes—
223C22×D4C2×D412yes—
224C22×D4C2×D411yescharacteristic
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4—84no—
236C23×C4C22×C414yes—
237C23×C4C2×D412yes—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D414yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C2×M16—24no—
244C2×M16—42no—
245C2×M16—42no—
246C22×C8—28no—
247C22×C8—24no—
248C22×C8—42no—
249C22×C8—44no—
250C22×C8—42no—
251C22×C8—44no—
252C22×C8C2414yes—
253C22×C8C2414yes—
254C4⋊Q8—216no—
255C4⋊Q8—232no—
256C4⋊Q8—24no—
257C4⋊Q8C22×C418yescomplemented
258#126—232no—
259#126—232no—
260#125—232no—
261#125—232no—
262#125—24no—
263#125C22×C418yescomplemented
264#124—216no—
265#123—216no—
266#123—216no—
267#123—232no—
268#123C22⋊C4116yes—
269C22⋊Q8—216no—
270C22⋊Q8—216no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274C22⋊Q8—232no—
275C22⋊Q8—232no—
276C22⋊Q8C22⋊C4116yes—
277C22⋊Q8C22×C4116yes—
278C22⋊Q8C22×C4116yes—
279C22⋊Q8C22×C4116yescomplemented
280#121C22⋊C4116yes—
281#120C22⋊C4116yes—
282C4×Q8—28no—
283C4×Q8—44no—
284C4×Q8C22×C4116yescomplemented
285C4×D4—232no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—88no—
289C4×D4C22×C4116yes—
290C4×D4C2×D4116yes—
291#117—24no—
292#117—232no—
293#117—24no—
294#117—42no—
295#117—42no—
296#117—42no—
297#117C22×C414yes—
298#117C22×C418yescomplemented
299#117C2×D414yes—
300C2×C4⋊C4—216no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C22×C418yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—216no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4C2×D414yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D414yes—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—24no—
351C2×C42C2412yes—
352C4⋊C8—28no—
353#103—28no—
354#103—28no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360#103—44no—
361C22⋊C8—28no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364C22⋊C8—28no—
365#97—24no—
366C4×C8—24no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372C24—24no—
373C24—24no—
374C24—41no—
375C24—41no—
376C24C2×C22⋊C414yes—
377C24C22×D411yescharacteristic
378C24C22×D412yes—
379C24C22×D412yes—
380#41—48no—
381#41—48no—
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×Q8—416no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8—82no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×Q8—88no—
399C2×Q8—84no—
400C2×Q8—84no—
401C2×Q8C2×C22⋊C414yes—
402C2×Q8C23×C418yes—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—41no—
406C2×D4—42no—
407C2×D4—41no—
408C2×D4—42no—
409C2×D4C2×C22⋊C4116yes—
410C2×D4C2×C22⋊C418yes—
411C2×D4C2×C22⋊C414yes—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—48no—
443C22×C4—82no—
444C22×C4—84no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—84no—
451C22×C4—816no—
452C22×C4C2×C22⋊C418yes—
453C22×C4C2×C22⋊C414yes—
454C22×C4C23×C414yes—
455C22×C4C23×C412yes—
456C22×C4C23×C414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C2511yescharacteristic
465M16—84no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—84no—
473C2×C8—88no—
474C4⋊C4—24no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—84no—
492C4⋊C4C2×C22⋊C418yes—
493C4⋊C4C2×C22⋊C4116yes—
494C4⋊C4C23×C418yes—
495C4⋊C4H(5,𝔽2)14yes—
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—28no—
499C22⋊C4—216no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—84no—
506C22⋊C4C2×C22⋊C4116yes—
507C22⋊C4C2×C22⋊C418yes—
508C22⋊C4C2×C22⋊C4116yes—
509C42—28no—
510C42—216no—
511C42—44no—
512C42—42no—
513C42—42no—
514C42—84no—
515C42C23×C414yes—
516C42C22×D414yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—216no—
522C23—21no—
523C23—22no—
524C23—21no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—44no—
531C23—44no—
532C23—48no—
533C23—44no—
534C23C22×C22⋊C412yes—
535C23C22×C22⋊C414yes—
536C23C22×C22⋊C414yes—
537C23#57714yes—
538C23#57814yes—
539C23C23×D411yescharacteristic
540Q8—28no—
541Q8—28no—
542Q8—44no—
543Q8—44no—
544Q8—44no—
545Q8—88no—
546Q8—88no—
547D4—232no—
548D4—44no—
549D4—48no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—416no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—88no—
587C2×C4—88no—
588C2×C4—84no—
589C2×C4—816no—
590C2×C4—84no—
591C2×C4—164no—
592C2×C4C22×C22⋊C418yes—
593C2×C4C22×C22⋊C414yes—
594C2×C4C22×C22⋊C414yes—
595C2×C4C22×C22⋊C412yes—
596C2×C4C24×C411yescharacteristic
597C2×C4C23×D411yescharacteristic
598C2×C4C2×H(5,𝔽2)12yes—
599C2×C4#58512yes—
600C8—44no—
601C8—44no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—28no—
609C22—24no—
610C22—22no—
611C22—24no—
612C22—416no—
613C22—44no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—41no—
619C22#289818yes—
620C22C23×C22⋊C411yescharacteristic
621C22#348411yescharacteristic
622C22#348511yescharacteristic
623C22C2×#57712yes—
624C22C2×#57812yes—
625C4—28no—
626C4—22no—
627C4—24no—
628C4—24no—
629C4—48no—
630C4—44no—
631C4—48no—
632C4—82no—
633C4—82no—
634C4—81no—
635C4—81no—
636C4C4×H(5,𝔽2)12yes—
637C4#349612yes—
638C2—28no—
639C2—44no—
640C2—42no—
641C2#3425611yescharacteristic
642C2C2×#289812yes—
643C2#6163111yescharacteristic
644C2#6163411yescharacteristic
645C2#6176712yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22491C2semidirect
5. #22486C2semidirect
6. #22425C2semidirect
7. D4×#103C2semidirect
8. #21873C2semidirect
9. #21771C2semidirect
10. #21762C2semidirect
11. #21398C2semidirect
12. #20913C2semidirect
13. C2×#1872C2semidirect
15. #3538C4semidirect
32. C2×#419C22semidirect
34. C22×#103C22semidirect
47. #1992C22semidirect
48. #1991C22semidirect
49. #1989C22semidirect
50. #1981C22semidirect
51. #1960C22semidirect
52. #1956C22semidirect
53. #1894C22semidirect
56. #1892C22semidirect
59. #1873C22semidirect
64. #1872C22semidirect
65. #1825C22semidirect
68. #1805C22semidirect
69. #1803C22semidirect
72. #1787C22semidirect
73. #1763C22semidirect
74. C4×#103C22semidirect
75. #1736C22semidirect
76. #1735C22semidirect
88. #552C2×C4semidirect
91. D4×Q8C2×C4semidirect
94. #541C2×C4semidirect
97. #536C2×C4semidirect
118. C2×C22⋊Q8C2×C4semidirect
219. C22×Q8C22×C4semidirect
257. C4⋊Q8C22×C4semidirect
263. #125C22×C4semidirect
279. C22⋊Q8C22×C4semidirect
284. C4×Q8C22×C4semidirect
298. #117C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35382
C2×#29772
#342002
#224912
#224862
#224252
D4×#1032
#218732
#217712
#217622
#213982
#209132
C2×#18722