Finite Group Databasebeta

← #223187

Subgroups — #223187

Order 512.

923 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36181C211yesmaximal, characteristic
2#22751C211yesmaximal, characteristic
3#22748C211yesmaximal, characteristic
4#9901C211yesmaximal, characteristic, complemented
5#9901C211yesmaximal, characteristic, complemented
6#9136C211yesmaximal, characteristic, complemented
7#9135C211yesmaximal, characteristic, complemented
8#9128C211yesmaximal, characteristic, complemented
9#9127C211yesmaximal, characteristic, complemented
10#8729C211yesmaximal, characteristic, complemented
11#8724C211yesmaximal, characteristic, complemented
12#8594C211yesmaximal, characteristic, complemented
13C4×#333C211yesmaximal, characteristic, complemented
14#8044C211yesmaximal, characteristic, complemented
15#8043C211yesmaximal, characteristic, complemented
16C2×#552C412yes—
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2565C412yes—
25#2506—21no—
26#2506—21no—
27#2503C412yes—
28#2499C412yes—
29#2496—21no—
30#2496—21no—
31#2486—21no—
32#2486—21no—
33#2344—21no—
34#2344—21no—
35C4×C4⋊Q8C2211yescharacteristic
36#2285C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1923—21no—
40#1923—21no—
41#1917—21no—
42#1916—21no—
43#1870—21no—
44#1869—21no—
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53#1753—21no—
54C8×C22⋊C4—21no—
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73#552—48no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×#126—22no—
76C2×#125—22no—
77C2×#125C2×C412yes—
78C2×#123—22no—
79C2×C22⋊Q8C2×C412yes—
80C2×C22⋊Q8C2×C412yes—
81C2×C4×Q8C2×C412yes—
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×C4×D4D411yescharacteristic
85C2×#117—22no—
86C2×#117C2×C412yes—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2×C412yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#432—22no—
93#432—41no—
94#431—22no—
95#431—41no—
96#423—22no—
97#423—41no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—41no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120#400—22no—
121#400—22no—
122#398—22no—
123#398—22no—
124#397—22no—
125#397—22no—
126#391C2×C412yes—
127#391C2×C412yes—
128#391C2×C412yes—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#387—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#384—21no—
144#384—21no—
145#384C2×C412yes—
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#382C2×C412yes—
150#382C2×C412yes—
151#381—22no—
152#381—22no—
153#380—21no—
154#380—21no—
155#380—21no—
156#380—21no—
157#380—21no—
158#380—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4C2×C411yescharacteristic
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C22⋊C4C2×C411yescharacteristic
167C4×C22⋊C4C2×C411yescharacteristic
168C4×C22⋊C4D411yescharacteristic
169C4×C22⋊C4D411yescharacteristic
170C4×C22⋊C4D411yescharacteristic
171C4×C22⋊C4D411yescharacteristic
172C43C2311yescharacteristic
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—21no—
183#333—22no—
184#333—22no—
185#333—41no—
186#333—41no—
187C22×Q8C22×C412yes—
188C22×D4C22⋊C411yescharacteristic
189C23×C4—22no—
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C2×M16—42no—
197C2×M16—42no—
198C2×M16—81no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8—81no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214#126—48no—
215#125—44no—
216#125—48no—
217#123—48no—
218C22⋊Q8—44no—
219C22⋊Q8—44no—
220C4×Q8—44no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—24no—
224C4×D4—41no—
225C4×D4—41no—
226#117—48no—
227#117—44no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4C22⋊C411yescharacteristic
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2411yescharacteristic
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—42no—
302C4⋊C8—42no—
303C4⋊C8#4112yes—
304C4⋊C8#4112yes—
305C22⋊C8—42no—
306C22⋊C8—42no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309C22⋊C8—81no—
310C22⋊C8—81no—
311#97—42no—
312C4×C8—42no—
313C4×C8—81no—
314C4×C8—81no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95C22⋊C411yescharacteristic
326#95C22⋊C411yescharacteristic
327#95C22⋊C411yescharacteristic
328#95C22⋊C411yescharacteristic
329#95C22⋊C411yescharacteristic
330#95C22⋊C411yescharacteristic
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—41no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4#13512yes—
388C22×C4#13512yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392M16—84no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—84no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4#10014yes—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—22no—
478C42—41no—
479C42—44no—
480C42—44no—
481C42C2×#4111yescharacteristic
482C42C2×#4111yescharacteristic
483C42H(5,𝔽2)11yescharacteristic
484C422−1+411yescharacteristic
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23C2×#10012yes—
498C23#41612yes—
499C23C22×C22⋊C411yescharacteristic
500Q8—82no—
501Q8—82no—
502D4—44no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—88no—
565C2×C4C2×#10012yes—
566C2×C4C2×#10012yes—
567C2×C4#41612yes—
568C2×C4C4×#4111yescharacteristic
569C2×C4#52011yescharacteristic
570C2×C4#54611yescharacteristic
571C2×C4#54611yescharacteristic
572C2×C4#54711yescharacteristic
573C2×C4#54711yescharacteristic
574C2×C4C2×#13511yescharacteristic
575C2×C4C2×#13511yescharacteristic
576C2×C4#57511yescharacteristic
577C2×C4#57511yescharacteristic
578C2×C4#57711yescharacteristic
579C2×C4#57711yescharacteristic
580C2×C4#57811yescharacteristic
581C2×C4#57811yescharacteristic
582C8—42no—
583C8—42no—
584C8—81no—
585C8—81no—
586C8—81no—
587C8—81no—
588C8—81no—
589C8—81no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22#233911yescharacteristic
605C22C22×#10011yescharacteristic
606C22#288911yescharacteristic
607C22C2×#41611yescharacteristic
608C22#289611yescharacteristic
609C22#289811yescharacteristic
610C22#289811yescharacteristic
611C4—22no—
612C4—21no—
613C4—21no—
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4#329611yescharacteristic
628C4#330011yescharacteristic
629C4#332311yescharacteristic
630C4#332711yescharacteristic
631C4#333611yescharacteristic
632C4#333711yescharacteristic
633C4#335111yescharacteristic
634C4#335211yescharacteristic
635C2—42no—
636C2—42no—
637C2#1771411yescharacteristic
638C2#1772511yescharacteristic
639C2#1776711yescharacteristic
640C2#1777911yescharacteristic
641C2#2247911yescharacteristic
642C2#2248611yescharacteristic
643C2#2249911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9901C2semidirect
5. #9901C2semidirect
6. #9136C2semidirect
7. #9135C2semidirect
8. #9128C2semidirect
9. #9127C2semidirect
10. #8729C2semidirect
11. #8724C2semidirect
12. #8594C2semidirect
13. C4×#333C2semidirect
14. #8044C2semidirect
15. #8043C2semidirect

Maximal subgroups

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

subgroupindex
#361812
#227512
#227482
#99012
#91362
#91352
#91282
#91272
#87292
#87242
#85942
C4×#3332
#80442
#80432