Finite Group Databasebeta

← #161320

Subgroups — #161320

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34327C211yesmaximal, characteristic
2#22698C211yesmaximal, characteristic, complemented
3#22014C211yesmaximal, characteristic, complemented
4#7647C211yesmaximal, characteristic, complemented
5#7643C211yesmaximal, characteristic, complemented
6#7635C211yesmaximal, characteristic, complemented
7#7630C211yesmaximal, characteristic, complemented
8#7618C211yesmaximal, characteristic, complemented
9#7616C211yesmaximal, characteristic, complemented
10#7615C211yesmaximal, characteristic, complemented
11#7609C211yesmaximal, characteristic, complemented
12#7600C211yesmaximal, characteristic, complemented
13#7597C211yesmaximal, characteristic, complemented
14#7589C211yesmaximal, characteristic, complemented
15#7578C211yesmaximal, characteristic, complemented
16C2×#548C411yescharacteristic
17C2×#435—21no—
18#2372—21no—
19#2368—21no—
20#2367C411yescharacteristic
21#2364—21no—
22#2362C411yescharacteristic
23#2361C411yescharacteristic
24#2356C411yescharacteristic
25#2353—21no—
26#2352C411yescharacteristic
27Q8×C22⋊C4C411yescharacteristic
28#2339C411yescharacteristic
29#2337—21no—
30#2332—21no—
31#2325—21no—
32C4×#124C2211yescharacteristic
33C4×C22⋊Q8C2211yescharacteristic
34C4×#121C2211yescharacteristic
35#2297C2211yescharacteristic
36#2295C2211yescharacteristic
37C2×#391C2211yescharacteristic
38#2275—21no—
39C42×D4C2211yescharacteristic
40#1955—21no—
41#1940—21no—
42#1940—21no—
43#1924—21no—
44#1924—21no—
45#1851—21no—
46#1850—21no—
47#1831C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1797C2211yescharacteristic
54#1767C2211yescharacteristic
55#1767C2211yescharacteristic
56#1745C2211yescharacteristic
57#1728C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×#41C2×C411yescharacteristic
76C23×Q8C2×C411yescharacteristic
77C22×M16—21no—
78C23×C8—21no—
79#548—24no—
80#548—24no—
81C2×#124—21no—
82C2×#124C2×C411yescharacteristic
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×#121—21no—
86C2×#121C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×Q8C2×C411yescharacteristic
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×C4×D4C2×C411yescharacteristic
94C2×C4×D4C2×C411yescharacteristic
95C2×C4×D4C2×C411yescharacteristic
96C2×C4×D4D411yescharacteristic
97C2×C4×D4D411yescharacteristic
98C2×#117C2×C411yescharacteristic
99C22×C4⋊C4C2×C411yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102#435—44no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
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×#97—21no—
118C2×#97—21no—
119#391—21no—
120#391—21no—
121#391—21no—
122#391—21no—
123#391C2×C411yescharacteristic
124#391C2×C411yescharacteristic
125#391C2×C411yescharacteristic
126#391C2×C411yescharacteristic
127#391C2311yescharacteristic
128#391C2311yescharacteristic
129#390C2×C411yescharacteristic
130#390C2311yescharacteristic
131#389C2×C411yescharacteristic
132#389C2311yescharacteristic
133#388—21no—
134#388—21no—
135#388—21no—
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#388—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#386—21no—
150#386—21no—
151#386—21no—
152#386—21no—
153#386—21no—
154#386—21no—
155#386—21no—
156#385—21no—
157#385—21no—
158#385—21no—
159#383—21no—
160#382C2×C411yescharacteristic
161#381—21no—
162#380—21no—
163#380C2×C411yescharacteristic
164#380C2×C411yescharacteristic
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4C2×C411yescharacteristic
169C4×C4⋊C4C2×C411yescharacteristic
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C4⋊C4D411yescharacteristic
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189#376—21no—
190#376—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C43C2311yescharacteristic
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209C2×#41—22no—
210C2×#41—81no—
211C2×#41—81no—
212C2×#41M1614yescomplemented
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—21no—
216C22×Q8—81no—
217C22×Q8—81no—
218C22×Q8C22×C411yescharacteristic
219C22×Q8C22×C411yescharacteristic
220C22×D4—21no—
221C22×D4—21no—
222C22×D4C22⋊C411yescharacteristic
223C22×D4C22×C411yescharacteristic
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4C22×C411yescharacteristic
230C23×C4C22×C411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C2×M16—42no—
236C2×M16—42no—
237C2×M16—81no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8—81no—
249#124—28no—
250#124—22no—
251#124—22no—
252C22⋊Q8—28no—
253C22⋊Q8—22no—
254C22⋊Q8M1614yescomplemented
255#121—28no—
256#121—22no—
257#121—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—81no—
261C4×Q8—81no—
262C4×D4—28no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—42no—
268C4×D4—44no—
269C4×D4—42no—
270C4×D4—42no—
271C4×D4—81no—
272C4×D4—81no—
273#117—81no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—81no—
282C2×C4⋊C4C22⋊C411yescharacteristic
283C2×C4⋊C4C22⋊C411yescharacteristic
284C2×C4⋊C4C22⋊C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4C22⋊C411yescharacteristic
304C2×C22⋊C4C22⋊C411yescharacteristic
305C2×C22⋊C4C22⋊C411yescharacteristic
306C2×C22⋊C4C22⋊C411yescharacteristic
307C2×C22⋊C4C22⋊C411yescharacteristic
308C2×C22⋊C4C22⋊C411yescharacteristic
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—81no—
317C2×C42—81no—
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321C2×C42C22×C411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2411yescharacteristic
331C4⋊C8—42no—
332C4⋊C8—42no—
333C4⋊C8—42no—
334C4⋊C8—81no—
335C4⋊C8—81no—
336C4⋊C8—81no—
337C4⋊C8—81no—
338C22⋊C8—42no—
339C22⋊C8—42no—
340C22⋊C8—81no—
341C22⋊C8—81no—
342C22⋊C8—81no—
343C22⋊C8—81no—
344#97—42no—
345#97—42no—
346#97—81no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95C22⋊C411yescharacteristic
369#95C22⋊C411yescharacteristic
370#95C22⋊C411yescharacteristic
371#95C22⋊C411yescharacteristic
372#95C22⋊C411yescharacteristic
373#95C22⋊C411yescharacteristic
374C24—21no—
375C24C2×C22⋊C411yescharacteristic
376C24C2×C22⋊C411yescharacteristic
377#41—84no—
378#41—84no—
379C2×Q8—24no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8—84no—
389C2×Q8—84no—
390C2×Q8C2×M1612yes—
391C2×Q8C2×M1612yes—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4C2×M1612yes—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×M1612yes—
458C22×C4C2×M1612yes—
459C22×C4AGL(1,ℤ/8)12yes—
460C22×C4#13712yes—
461C22×C4C23×C411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464M16—84no—
465C2×C8—22no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—84no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—42no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4C2×M1612yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—81no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—81no—
528C42—24no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—22no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—42no—
546C42—42no—
547C42—81no—
548C42—81no—
549C42—81no—
550C42—81no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23#40712yes—
571C23#41712yes—
572C23C22×C22⋊C411yescharacteristic
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—84no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—48no—
582D4—84no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
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—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—162no—
652C2×C4—162no—
653C2×C4—162no—
654C2×C4#40712yes—
655C2×C4#41712yes—
656C2×C4#42112yes—
657C2×C4C22×M1611yescharacteristic
658C2×C4C22×M1611yescharacteristic
659C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
660C2×C4C2×#13711yescharacteristic
661C2×C4#57711yescharacteristic
662C2×C4#57811yescharacteristic
663C8—42no—
664C8—81no—
665C8—81no—
666C8—81no—
667C8—81no—
668C8—81no—
669C8—81no—
670C8—81no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22C2×#40711yescharacteristic
693C22#288311yescharacteristic
694C22#288811yescharacteristic
695C22C2×#41711yescharacteristic
696C22#289811yescharacteristic
697C22C2×#42111yescharacteristic
698C22#290111yescharacteristic
699C4—22no—
700C4—21no—
701C4—21no—
702C4—21no—
703C4—21no—
704C4—22no—
705C4—21no—
706C4—21no—
707C4—22no—
708C4—41no—
709C4—42no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—42no—
717C4—42no—
718C4—81no—
719C4—81no—
720C4—81no—
721C4—81no—
722C2—42no—
723C2—42no—
724C2—82no—
725C2#1736711yescharacteristic
726C2#1745111yescharacteristic
727C2#1745411yescharacteristic
728C2#1760011yescharacteristic
729C2#1762211yescharacteristic
730C2#1767911yescharacteristic
731C2#1776411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#548C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22698C2semidirect
3. #22014C2semidirect
4. #7647C2semidirect
5. #7643C2semidirect
6. #7635C2semidirect
7. #7630C2semidirect
8. #7618C2semidirect
9. #7616C2semidirect
10. #7615C2semidirect
11. #7609C2semidirect
12. #7600C2semidirect
13. #7597C2semidirect
14. #7589C2semidirect
15. #7578C2semidirect
212. C2×#41M16semidirect
254. C22⋊Q8M16semidirect

Maximal subgroups

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

subgroupindex
#343272
#226982
#220142
#76472
#76432
#76352
#76302
#76182
#76162
#76152
#76092
#76002
#75972
#75892
#75782