Finite Group Databasebeta

← #6392658

Subgroups — #6392658

Order 512.

2246 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3C2×#2352C211yesmaximal, characteristic
4#22712C214yesmaximal, complemented
5#22710C214yesmaximal, complemented
6#22662C214yesmaximal, complemented
7#22660C214yesmaximal, complemented
8#21986C211yesmaximal, characteristic, complemented
9#21986C211yesmaximal, characteristic, complemented
10#21944C212yesmaximal, complemented
11#21941C212yesmaximal, complemented
12#21365C211yesmaximal, characteristic, complemented
13#21365C211yesmaximal, characteristic, complemented
14#20844C212yesmaximal, complemented
15#20843C212yesmaximal, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2973C4116yescomplemented
20C2×#435C2214yes—
21C2×#435C2214yes—
22C2×#432C2211yescharacteristic
23C2×#432C2211yescharacteristic
24C2×#428C2214yescomplemented
25C2×#427C2214yescomplemented
26C2×#407C2212yes—
27C22×C22⋊C8C2212yes—
28#2357—22no—
29#2357—22no—
30#2354C2212yes—
31#2354C2212yes—
32#2354C2212yes—
33#2354C2212yes—
34#2352—22no—
35#2352—22no—
36#2352C2214yes—
37#2352C2214yes—
38#2312—21no—
39#2312—21no—
40#2309—21no—
41C2×#388—21no—
42C2×#387—21no—
43C2×#386—21no—
44#2285C2211yescharacteristic
45#2285C2211yescharacteristic
46C2×#382C2211yescharacteristic
47#2280C2212yes—
48#2280C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51#2274—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4—21no—
54#1949C2214yescomplemented
55#1949C2214yescomplemented
56#1948C2212yescomplemented
57#1948C2212yescomplemented
58#1947C2212yescomplemented
59#1947C2212yescomplemented
60#1854C2214yescomplemented
61#1854C2214yescomplemented
62#1854C2214yescomplemented
63#1854C2214yescomplemented
64#1854C2214yescomplemented
65#1854C2214yescomplemented
66#1853C2212yes—
67#1853C2212yes—
68#1853C2212yes—
69#1853C2212yes—
70#1853C2214yescomplemented
71#1852C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1852C2214yescomplemented
76#1847C2212yescomplemented
77#1847C2212yescomplemented
78#1836C2212yescomplemented
79#1836C2212yescomplemented
80#1809C2214yescomplemented
81#1809C2214yescomplemented
82#1808C2214yescomplemented
83#1807C2214yescomplemented
84#1776C2214yescomplemented
85#1776C2214yescomplemented
86#1736C2214yescomplemented
87#1735C2214yescomplemented
88C23×D4D411yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×M16C2314yes—
93C23×C8C2314yes—
94C2×C4×D4C2×C418yes—
95C2×C4×D4D412yes—
96C2×C4×D4D412yes—
97C2×C4×D4Q818yes—
98C2×C4×D4C2312yes—
99C2×C4×D4C2312yes—
100C2×#117—21no—
101C2×#117—21no—
102C2×#117—22no—
103C2×#117—44no—
104C2×#117C2314yes—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—44no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4D411yescharacteristic
122C22×C22⋊C4D411yescharacteristic
123C22×C42—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126#435C2×C4132yescomplemented
127#435C2×C4132yescomplemented
128#432C2×C418yescomplemented
129#432C2×C418yescomplemented
130#428—48no—
131#427—48no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142#407C2×C4116yes—
143C2×C22⋊C8C2×C4116yescomplemented
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C2×#97C2312yes—
149C2×#97C2312yes—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—24no—
163#388—24no—
164#387—24no—
165#387—24no—
166#386—24no—
167#386—24no—
168#384C2311yescharacteristic
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#382—21no—
173#382—21no—
174#382—21no—
175#382—21no—
176#382C2312yes—
177#382C2312yes—
178#381—22no—
179#381—22no—
180#380—21no—
181#380—21no—
182#380—22no—
183#380—22no—
184#380C2312yes—
185#380C2314yes—
186#380C2312yes—
187#380C2314yes—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191#380C2312yes—
192#379—22no—
193#379—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—24no—
211C4×C22⋊C4—24no—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95—21no—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225C25C2×D411yescharacteristic
226C25C2×D411yescharacteristic
227C22×D4—42no—
228C22×D4—42no—
229C22×D4C4⋊C418yes—
230C22×D4C2×D412yes—
231C22×D4C2×D412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4—82no—
244C23×C4C22×C414yes—
245C23×C4C22×C414yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2×Q814yes—
253C23×C4C2×Q814yes—
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C23×C4C2412yes—
257C2×M16—44no—
258C2×M16—44no—
259C2×M16C22×C4116yes—
260C22×C8—44no—
261C22×C8—44no—
262C22×C8C22×C4116yescomplemented
263C22×C8C2414yes—
264C22×C8C2414yes—
265C4×D4C4⋊C4116yes—
266C4×D4C4⋊C4116yes—
267C4×D4C22×C418yes—
268C4×D4C22×C418yes—
269C4×D4C2×Q818yes—
270C4×D4C2×Q818yes—
271#117—28no—
272#117—44no—
273#117—48no—
274#117—44no—
275#117—44no—
276#117—44no—
277#117—44no—
278#117—44no—
279#117—44no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C4⋊C418yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4C4⋊C418yes—
329C2×C22⋊C4C4⋊C418yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—44no—
344C2×C42—82no—
345C2×C42C22×C414yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×Q814yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—28no—
352#107—44no—
353#107—44no—
354#107—44no—
355#106—28no—
356#106—44no—
357#106—44no—
358#106—44no—
359C4⋊C8C22×C418yescomplemented
360C4⋊C8C22×C418yescomplemented
361C22⋊C8C22×C418yes—
362C22⋊C8C22×C418yes—
363C22⋊C8C22×C4116yes—
364C22⋊C8C22×C4116yescomplemented
365#97C22×C418yescomplemented
366#97C22×C418yescomplemented
367#95—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—41no—
393C24—41no—
394C24—41no—
395C24C2×C4⋊C414yes—
396C24C2×C4⋊C414yes—
397C24C22×D412yes—
398C24C22×D412yes—
399C24C22×D411yescharacteristic
400C2×D4—416no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4C2×C4⋊C418yes—
406C2×D4C2×C4⋊C418yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—48no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—84no—
452C22×C4—816no—
453C22×C4C2×C4⋊C414yes—
454C22×C4C2×C4⋊C414yes—
455C22×C4C2×C4⋊C414yes—
456C22×C4C2×C4⋊C414yes—
457C22×C4C2×C4⋊C418yes—
458C22×C4C2×C4⋊C418yes—
459C22×C4C23×C412yes—
460C22×C4C23×C414yes—
461C22×C4C23×C414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×Q812yes—
470C22×C4C22×Q814yes—
471C22×C4C22×Q814yes—
472C22×C4C2511yescharacteristic
473M16—416no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—416no—
479C2×C8C23×C418yes—
480C2×C8C23×C418yes—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4C2×C4⋊C418yes—
504C4⋊C4C2×C4⋊C418yes—
505C4⋊C4H(5,𝔽2)14yes—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—84no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4C2×C4⋊C418yes—
520C22⋊C4C2×C4⋊C418yes—
521C22⋊C4C2×C4⋊C418yes—
522C22⋊C4C2×C4⋊C418yes—
523C42—48no—
524C42—44no—
525C42—44no—
526C42—82no—
527C42—82no—
528C42—82no—
529C42—82no—
530C42—82no—
531C42—82no—
532C42C2×C4⋊C414yes—
533C42C2×C4⋊C414yes—
534C42C23×C412yes—
535C42C23×C412yes—
536C42C22×Q812yes—
537C42C22×Q812yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—28no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—28no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23—82no—
568C23C22×C4⋊C414yes—
569C23C22×C4⋊C414yes—
570C23C22×C4⋊C412yes—
571C23#57512yes—
572C23#57512yes—
573C23C23×D411yescharacteristic
574D4—48no—
575D4—48no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—816no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4C22×C4⋊C412yes—
632C2×C4C22×C4⋊C412yes—
633C2×C4C22×C4⋊C412yes—
634C2×C4C22×C4⋊C412yes—
635C2×C4C22×C4⋊C412yes—
636C2×C4C22×C4⋊C414yes—
637C2×C4C22×C4⋊C414yes—
638C2×C4C24×C411yescharacteristic
639C2×C4C23×Q811yescharacteristic
640C2×C4C2×H(5,𝔽2)12yes—
641C2×C4#58512yes—
642C8—28no—
643C8—28no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—416no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—88no—
666C22#291114yes—
667C22C23×C4⋊C411yescharacteristic
668C22#348411yescharacteristic
669C22#348511yescharacteristic
670C22C2×#57511yescharacteristic
671C22C2×#57511yescharacteristic
672C22#358511yescharacteristic
673C22#358611yescharacteristic
674C4—22no—
675C4—24no—
676C4—24no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—44no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4C4×H(5,𝔽2)12yes—
690C4#350112yes—
691C2—24no—
692C2—44no—
693C2—44no—
694C2#3424811yescharacteristic
695C2C2×#291111yescharacteristic
696C2#6010911yescharacteristic
697C2#6157611yescharacteristic
698C2#6157811yescharacteristic
699C2#6171111yescharacteristic
700C2#6171311yescharacteristic
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. #22712C2semidirect
5. #22710C2semidirect
6. #22662C2semidirect
7. #22660C2semidirect
8. #21986C2semidirect
9. #21986C2semidirect
10. #21944C2semidirect
11. #21941C2semidirect
12. #21365C2semidirect
13. #21365C2semidirect
14. #20844C2semidirect
15. #20843C2semidirect
19. #2973C4semidirect
24. C2×#428C22semidirect
25. C2×#427C22semidirect
54. #1949C22semidirect
55. #1949C22semidirect
56. #1948C22semidirect
57. #1948C22semidirect
58. #1947C22semidirect
59. #1947C22semidirect
60. #1854C22semidirect
61. #1854C22semidirect
62. #1854C22semidirect
63. #1854C22semidirect
64. #1854C22semidirect
65. #1854C22semidirect
70. #1853C22semidirect
75. #1852C22semidirect
76. #1847C22semidirect
77. #1847C22semidirect
78. #1836C22semidirect
79. #1836C22semidirect
80. #1809C22semidirect
81. #1809C22semidirect
82. #1808C22semidirect
83. #1807C22semidirect
84. #1776C22semidirect
85. #1776C22semidirect
86. #1736C22semidirect
87. #1735C22semidirect
126. #435C2×C4semidirect
127. #435C2×C4semidirect
128. #432C2×C4semidirect
129. #432C2×C4semidirect
143. C2×C22⋊C8C2×C4semidirect
262. C22×C8C22×C4semidirect
359. C4⋊C8C22×C4semidirect
360. C4⋊C8C22×C4semidirect
364. C22⋊C8C22×C4semidirect
365. #97C22×C4semidirect
366. #97C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29732
#341862
C2×#23522
#227122
#227102
#226622
#226602
#219862
#219442
#219412
#213652
#208442
#208432