Finite Group Databasebeta

← #6810330

Subgroups — #6810330

Order 512.

2061 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C212yesmaximal
2#36133C211yesmaximal, characteristic
3C2×#2546C211yesmaximal, characteristic
4#32906C211yesmaximal, characteristic
5#30880C214yesmaximal, complemented
6#30744C214yesmaximal, complemented
7#29627C212yesmaximal
8#28462C212yesmaximal, complemented
9#27769C212yesmaximal, complemented
10#25722C211yesmaximal, characteristic
11#25370C212yesmaximal, complemented
12#24732C212yesmaximal, complemented
13#24502C212yesmaximal
14#24212C211yesmaximal, characteristic
15#23934C212yesmaximal
16#22599C211yesmaximal, characteristic
17C4×#451C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#3209—216no—
23C2×#494C2214yes—
24C2×#494C2212yes—
25C2×#487C2214yes—
26C2×#462C2214yes—
27C2×#462C2214yes—
28C2×#462C2212yes—
29C2×#462C2212yes—
30C2×#451C2212yes—
31C2×#451C2212yes—
32C2×#451C2212yes—
33C2×#451C2212yes—
34C2×#451C2211yescharacteristic
35C2×C4×Q16C2212yes—
36C2×#431C2212yes—
37#2857C2211yescharacteristic
38#2855C2212yes—
39#2551C2211yescharacteristic
40#2546—21no—
41#2546—22no—
42#2546—21no—
43#2546C2214yes—
44#2546C2212yes—
45#2546C2211yescharacteristic
46#2546C2212yes—
47#2506—21no—
48#2462—22no—
49#2451C2212yes—
50#2446C2212yes—
51#2440—21no—
52#2438—22no—
53C2×#398—22no—
54#2365—21no—
55#2365C2211yescharacteristic
56Q8×C4⋊C4C2211yescharacteristic
57#2344—21no—
58C4×C22⋊Q8C2211yescharacteristic
59C2×#391C2212yes—
60C2×#389—21no—
61C2×#389—21no—
62C2×C4×C4⋊C4C2211yescharacteristic
63#2068C2214yes—
64#2056C2214yes—
65#2043C2214yes—
66#2043C2212yes—
67#2043C2214yes—
68#2024C2214yes—
69#2016C2212yes—
70#2016C2214yes—
71#2016C2214yes—
72#2016C2214yes—
73#2016C2214yes—
74#1981C2212yes—
75#1969C2214yes—
76#1956C2212yes—
77#1939C2212yes—
78#1933C2214yes—
79#1916C2214yes—
80#1878C2212yes—
81#1878C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1872C2211yescharacteristic
85#1872C2212yes—
86#1863C2212yes—
87#1852C2211yescharacteristic
88#1817C2212yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91#1787C2212yes—
92#1787C2211yescharacteristic
93C8×C4⋊C4C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8C2312yes—
97C23×Q8C2311yescharacteristic
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100C22×Q16C2314yes—
101C22×Q16C2314yes—
102C22×Q16C2312yes—
103#533—28no—
104#533—44no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—28no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8—44no—
112C2×C4⋊Q8—44no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2314yes—
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2314yes—
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2312yes—
120C2×#125—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8C2312yes—
123C2×C22⋊Q8C2311yescharacteristic
124C2×C4×Q8—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—44no—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2311yescharacteristic
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×#117—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—22no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4D411yescharacteristic
140C22×C4⋊C4D411yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C42—21no—
143C22×C42C2311yescharacteristic
144C22×C42C2312yes—
145#494—216no—
146#494—44no—
147#487—216no—
148#462—216no—
149#462—216no—
150#462—44no—
151#462—44no—
152#451—28no—
153#451—28no—
154#451—28no—
155#451—44no—
156#451D418yes—
157C4×Q16—44no—
158#431—28no—
159C2×#107C2312yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2314yes—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×C22⋊C8C2312yes—
174C2×C22⋊C8C2311yescharacteristic
175C2×C22⋊C8C2311yescharacteristic
176C2×C4×C8C2314yes—
177C2×C4×C8C2312yes—
178#400—22no—
179#398—22no—
180#398—24no—
181#398—24no—
182#398—24no—
183#398—24no—
184#398—22no—
185#398—22no—
186#395—22no—
187#395—22no—
188#391—22no—
189#391—22no—
190#391—22no—
191#391C2312yes—
192#391C2314yes—
193#391C2312yes—
194#391C2312yes—
195#391C2311yescharacteristic
196#391C2314yes—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—22no—
202#389—24no—
203#389—22no—
204#389—24no—
205#389—22no—
206#389—22no—
207#389—21no—
208#389—21no—
209#389—22no—
210#389C2312yes—
211#389C2312yes—
212#389C2311yescharacteristic
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—21no—
218#387—22no—
219#387—22no—
220#385—22no—
221#385—21no—
222#384—21no—
223#384C2311yescharacteristic
224#381—22no—
225#380—21no—
226#380—21no—
227#380C2311yescharacteristic
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4D411yescharacteristic
232C4×C4⋊C4D412yes—
233C4×C4⋊C4D411yescharacteristic
234C4×C4⋊C4C2311yescharacteristic
235C4×C4⋊C4C2311yescharacteristic
236C4×C4⋊C4C2312yes—
237C4×C4⋊C4C2311yescharacteristic
238C4×C22⋊C4—21no—
239C2×#95—22no—
240C2×#95—22no—
241C2×#95D412yes—
242#340C2312yes—
243#340C2314yes—
244#340C2312yes—
245#336C2312yes—
246#336C2314yes—
247#336C2314yes—
248#336C2312yes—
249C22×Q8—21no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—24no—
254C22×Q8—24no—
255C22×Q8—42no—
256C22×Q8—42no—
257C22×Q8—44no—
258C22×Q8—44no—
259C22×Q8C2×D414yes—
260C22×Q8C2412yes—
261C22×Q8C2414yes—
262C22×Q8C2411yescharacteristic
263C22×Q8C2412yes—
264C23×C4—21no—
265C23×C4—22no—
266C23×C4—21no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D412yes—
271C23×C4C2411yescharacteristic
272C2×Q16—28no—
273C2×Q16—28no—
274C2×Q16—28no—
275C2×Q16—42no—
276C2×Q16—42no—
277C2×Q16—44no—
278C2×Q16—44no—
279C2×Q16—44no—
280C2×Q16—44no—
281C2×Q16—44no—
282C2×Q16—44no—
283C2×Q16—44no—
284C2×Q16—44no—
285C2×Q16—44no—
286C2×Q16C2×D418yes—
287C22×C8C2412yes—
288C22×C8C2414yes—
289C22×C8C2412yes—
290C4⋊Q8—24no—
291C4⋊Q8—28no—
292C4⋊Q8—28no—
293C4⋊Q8—24no—
294C4⋊Q8—24no—
295C4⋊Q8—24no—
296C4⋊Q8—24no—
297C4⋊Q8—42no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301C4⋊Q8—42no—
302C4⋊Q8—48no—
303C4⋊Q8—48no—
304C4⋊Q8—44no—
305C4⋊Q8—44no—
306C4⋊Q8—44no—
307#125—48no—
308C22⋊Q8—22no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—48no—
312C22⋊Q8C2×D414yes—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×Q8—42no—
321C4×Q8—42no—
322C4×Q8—48no—
323C4×Q8—44no—
324C4×Q8—44no—
325#117—48no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—48no—
355C2×C4⋊C4—48no—
356C2×C4⋊C4—44no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4Q1618yescomplemented
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D412yes—
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2411yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C42—24no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—44no—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D412yes—
387C2×C42C2×D412yes—
388C2×C42C2411yescharacteristic
389C2×C42C2412yes—
390C2×C42C2412yes—
391C2×C42C2411yescharacteristic
392#107—42no—
393C4⋊C8—28no—
394C4⋊C8—24no—
395#103—28no—
396#103—28no—
397#103—28no—
398#103—42no—
399#103—44no—
400#103—42no—
401#103—42no—
402#103—44no—
403#103—44no—
404#103—44no—
405#103C2×D418yes—
406C22⋊C8—24no—
407C22⋊C8—22no—
408C22⋊C8C2×D414yes—
409C4×C8—28no—
410C4×C8—24no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—22no—
419#95—22no—
420#95—21no—
421#95—21no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95C2×D412yes—
429#95C2×D412yes—
430#95C2×D412yes—
431#95C2×D412yes—
432C24C22×D411yescharacteristic
433C2×Q8—24no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—24no—
451C2×Q8—24no—
452C2×Q8—44no—
453C2×Q8—48no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—48no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—44no—
460C2×Q8—44no—
461C2×Q8—82no—
462C2×Q8—88no—
463C2×Q8—88no—
464C2×Q8C22×D414yes—
465C2×Q8C22×D412yes—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—41no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—44no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—42no—
496C22×C4C2×Q1618yes—
497C22×C4C2×Q1614yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4H(5,𝔽2)12yes—
509C22×C4C2511yescharacteristic
510Q16—48no—
511Q16—88no—
512Q16—88no—
513Q16—88no—
514Q16—88no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—24no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—22no—
523C2×C8C22×D414yes—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—81no—
557C4⋊C4—81no—
558C4⋊C4—82no—
559C4⋊C4C2×Q1618yes—
560C4⋊C4C2×Q1614yes—
561C4⋊C4C2×Q1614yes—
562C4⋊C4#13714yes—
563C4⋊C4C22×D412yes—
564C22⋊C4—24no—
565C22⋊C4—44no—
566C22⋊C4—42no—
567C42—22no—
568C42—22no—
569C42—24no—
570C42—24no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—44no—
577C42—44no—
578C42—44no—
579C42—44no—
580C42—42no—
581C42—42no—
582C42H(5,𝔽2)14yes—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23D4212yes—
588C23C22×Q1612yes—
589C23#57812yes—
590C23C23×D411yescharacteristic
591Q8—24no—
592Q8—24no—
593Q8—24no—
594Q8—48no—
595Q8—42no—
596Q8—42no—
597Q8—42no—
598Q8—44no—
599Q8—44no—
600Q8—44no—
601Q8—44no—
602Q8—44no—
603Q8—44no—
604Q8—88no—
605Q8—88no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—216no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—48no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—41no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—84no—
659C2×C4—88no—
660C2×C4#53512yes—
661C2×C4#53512yes—
662C2×C4D4212yes—
663C2×C4C22×Q1612yes—
664C2×C4C22×Q1614yes—
665C2×C4C22×Q1614yes—
666C2×C4C22×Q1612yes—
667C2×C4C2×#13714yes—
668C2×C4C2×#13712yes—
669C2×C4C23×D411yescharacteristic
670C2×C4C2×H(5,𝔽2)12yes—
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4#58512yes—
673C8—24no—
674C8—44no—
675C8—42no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22D4×Q1614yes—
683C22C2×#53511yescharacteristic
684C22C2×D4211yescharacteristic
685C22#348711yescharacteristic
686C22C23×Q1611yescharacteristic
687C22C22×#13711yescharacteristic
688C22C2×#57811yescharacteristic
689C22#358611yescharacteristic
690C4—22no—
691C4—24no—
692C4—24no—
693C4—24no—
694C4—22no—
695C4—22no—
696C4—42no—
697C4—42no—
698C4—41no—
699C4—44no—
700C4—42no—
701C4—82no—
702C4—81no—
703C4—82no—
704C4—84no—
705C4—82no—
706C4—81no—
707C4#320914yescomplemented
708C4#349612yes—
709C4#354212yes—
710C2—24no—
711C2#3933911yescharacteristic
712C2C2×#320911yescharacteristic
713C2#6081411yescharacteristic
714C2C2×D4×Q1611yescharacteristic
715C2D4×#13711yescharacteristic
716C2#6188411yescharacteristic
717C2#6190011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #30880C2semidirect
6. #30744C2semidirect
8. #28462C2semidirect
9. #27769C2semidirect
11. #25370C2semidirect
12. #24732C2semidirect
359. C2×C4⋊C4Q16semidirect
707. C4#3209semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#361332
C2×#25462
#329062
#308802
#307442
#296272
#284622
#277692
#257222
#253702
#247322
#245022
#242122
#239342
#225992
C4×#4512