Finite Group Databasebeta

← #161121

Subgroups — #161121

Order 512.

918 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
1#34323C211yesmaximal, characteristic
2#22788C211yesmaximal, characteristic, complemented
3#21996C211yesmaximal, characteristic, complemented
4#7650C211yesmaximal, characteristic, complemented
5#7647C211yesmaximal, characteristic, complemented
6#7642C211yesmaximal, characteristic, complemented
7#7641C211yesmaximal, characteristic, complemented
8#7619C211yesmaximal, characteristic, complemented
9#7616C211yesmaximal, characteristic, complemented
10#7608C211yesmaximal, characteristic, complemented
11#7607C211yesmaximal, characteristic, complemented
12#7582C211yesmaximal, characteristic, complemented
13#7579C211yesmaximal, characteristic, complemented
14#7565C211yesmaximal, characteristic, complemented
15#7564C211yesmaximal, characteristic, complemented
16C2×#553C411yescharacteristic
17#2373—21no—
18#2370C411yescharacteristic
19#2363C411yescharacteristic
20#2361—21no—
21#2361C411yescharacteristic
22#2359—21no—
23#2352—21no—
24#2352C411yescharacteristic
25#2349C411yescharacteristic
26#2346C411yescharacteristic
27#2345C411yescharacteristic
28#2335—21no—
29#2334—21no—
30#2325—21no—
31C4×#125C2211yescharacteristic
32C4×#121C2211yescharacteristic
33C4×#121C2211yescharacteristic
34#2296C2211yescharacteristic
35#2293C2211yescharacteristic
36C2×#389C2211yescharacteristic
37#2275—21no—
38C42×D4C2211yescharacteristic
39#1924—21no—
40#1923—21no—
41#1916—21no—
42#1916—21no—
43#1870—21no—
44#1869—21no—
45#1850—21no—
46#1839C2211yescharacteristic
47#1839C2211yescharacteristic
48#1831C2211yescharacteristic
49#1817C2211yescharacteristic
50#1817C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1753—21no—
54C4×C22⋊C8C2211yescharacteristic
55C4×C22⋊C8C2211yescharacteristic
56#1745C2211yescharacteristic
57#1728C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1271C2211yescharacteristic
60#1271C2211yescharacteristic
61#1271C2211yescharacteristic
62#1271C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×M16—21no—
76C23×C8—21no—
77#553—24no—
78#553—24no—
79C2×#126—21no—
80C2×#125C2×C411yescharacteristic
81C2×#123—21no—
82C2×#121—21no—
83C2×#121C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4C2×C411yescharacteristic
92C2×C4×D4C2×C411yescharacteristic
93C2×C4×D4D411yescharacteristic
94C2×C4×D4D411yescharacteristic
95C2×#117—21no—
96C2×#117C2×C411yescharacteristic
97C2×#117C2×C411yescharacteristic
98C22×C4⋊C4—21no—
99C22×C4⋊C4C2×C411yescharacteristic
100C22×C4⋊C4C2×C411yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115#390—21no—
116#390C2×C411yescharacteristic
117#390C2311yescharacteristic
118#389—21no—
119#389—21no—
120#389—21no—
121#389C2×C411yescharacteristic
122#389C2×C411yescharacteristic
123#389C2×C411yescharacteristic
124#389C2×C411yescharacteristic
125#389C2×C411yescharacteristic
126#389C2311yescharacteristic
127#389C2311yescharacteristic
128#389C2311yescharacteristic
129#388—21no—
130#388—21no—
131#388—21no—
132#388—21no—
133#388—21no—
134#388—21no—
135#388—21no—
136#388—21no—
137#388—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#386—21no—
146#386—21no—
147#386—21no—
148#386—21no—
149#386—21no—
150#385—21no—
151#385—21no—
152#385—21no—
153#385—21no—
154#385—21no—
155#382—21no—
156#382C2×C411yescharacteristic
157#380—21no—
158#380—21no—
159#380C2×C411yescharacteristic
160#380C2×C411yescharacteristic
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4C2×C411yescharacteristic
167C4×C4⋊C4C2×C411yescharacteristic
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⋊C4C2×C411yescharacteristic
181C4×C22⋊C4C2×C411yescharacteristic
182C4×C22⋊C4D411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186#376—21no—
187C2×#95—21no—
188C2×#95—21no—
189C43C2311yescharacteristic
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C22⋊C411yescharacteristic
211C22×D4C22×C411yescharacteristic
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C22×C411yescharacteristic
218C23×C4C22×C411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C2×M16—42no—
224C2×M16—42no—
225C2×M16—81no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—42no—
235C22×C8—81no—
236C22×C8—81no—
237#126—28no—
238#125—22no—
239#125—22no—
240#123—28no—
241#121—28no—
242#121—22no—
243#121—22no—
244#121—22no—
245#121M1614yescomplemented
246C4×D4—28no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—42no—
254C4×D4—44no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—81no—
258C4×D4—81no—
259C4×D4—81no—
260C4×D4—81no—
261#117—22no—
262#117—81no—
263#117—81no—
264#117—81no—
265#117M1614yescomplemented
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
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—81no—
281C2×C4⋊C4—81no—
282C2×C4⋊C4—81no—
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×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4C22⋊C411yescharacteristic
307C2×C22⋊C4C22⋊C411yescharacteristic
308C2×C22⋊C4C22⋊C411yescharacteristic
309C2×C22⋊C4C22⋊C411yescharacteristic
310C2×C22⋊C4C22⋊C411yescharacteristic
311C2×C22⋊C4C22⋊C411yescharacteristic
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—81no—
320C2×C42—81no—
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C411yescharacteristic
323C2×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2411yescharacteristic
334C4⋊C8—81no—
335C4⋊C8—81no—
336C4⋊C8—81no—
337C22⋊C8—42no—
338C22⋊C8—42no—
339C22⋊C8—42no—
340C22⋊C8—81no—
341C22⋊C8—81no—
342C22⋊C8—81no—
343#97—42no—
344C4×C8—81no—
345C4×C8—81no—
346#95—21no—
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#95C22⋊C411yescharacteristic
364#95C22⋊C411yescharacteristic
365#95C22⋊C411yescharacteristic
366#95C22⋊C411yescharacteristic
367#95C22⋊C411yescharacteristic
368#95C22⋊C411yescharacteristic
369C24—21no—
370C24C2×C22⋊C411yescharacteristic
371C24C2×C22⋊C411yescharacteristic
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—81no—
383C2×D4—81no—
384C2×D4C2×M1612yes—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×M1612yes—
438C22×C4C2×M1612yes—
439C22×C4#13512yes—
440C22×C4#13512yes—
441C22×C4C23×C411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444M16—84no—
445C2×C8—22no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—84no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4C2×M1612yes—
496C4⋊C4C2×M1612yes—
497C4⋊C4C2×M1612yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—41no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—81no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—24no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—42no—
537C42—42no—
538C42—81no—
539C42—81no—
540C42—81no—
541C42—81no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23#40712yes—
562C23#41612yes—
563C23C22×C22⋊C411yescharacteristic
564D4—42no—
565D4—42no—
566D4—44no—
567D4—48no—
568D4—84no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
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—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
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—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—162no—
638C2×C4—162no—
639C2×C4—162no—
640C2×C4#40712yes—
641C2×C4#41612yes—
642C2×C4#42112yes—
643C2×C4C22×M1611yescharacteristic
644C2×C4C22×M1611yescharacteristic
645C2×C4C2×#13511yescharacteristic
646C2×C4C2×#13511yescharacteristic
647C2×C4#57611yescharacteristic
648C2×C4#57711yescharacteristic
649C8—42no—
650C8—81no—
651C8—81no—
652C8—81no—
653C8—81no—
654C8—81no—
655C8—81no—
656C8—81no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22C2×#40711yescharacteristic
679C22#288311yescharacteristic
680C22#288911yescharacteristic
681C22C2×#41611yescharacteristic
682C22#289711yescharacteristic
683C22C2×#42111yescharacteristic
684C22#290111yescharacteristic
685C4—21no—
686C4—21no—
687C4—22no—
688C4—21no—
689C4—21no—
690C4—22no—
691C4—21no—
692C4—21no—
693C4—22no—
694C4—41no—
695C4—41no—
696C4—42no—
697C4—41no—
698C4—42no—
699C4—42no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—81no—
705C4—81no—
706C4—81no—
707C4—81no—
708C2—42no—
709C2—42no—
710C2—82no—
711C2#1736811yescharacteristic
712C2#1744311yescharacteristic
713C2#1745311yescharacteristic
714C2#1759711yescharacteristic
715C2#1761511yescharacteristic
716C2#1768211yescharacteristic
717C2#1772711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#553C4
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. #22788C2semidirect
3. #21996C2semidirect
4. #7650C2semidirect
5. #7647C2semidirect
6. #7642C2semidirect
7. #7641C2semidirect
8. #7619C2semidirect
9. #7616C2semidirect
10. #7608C2semidirect
11. #7607C2semidirect
12. #7582C2semidirect
13. #7579C2semidirect
14. #7565C2semidirect
15. #7564C2semidirect
245. #121M16semidirect
265. #117M16semidirect

Maximal subgroups

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

subgroupindex
#343232
#227882
#219962
#76502
#76472
#76422
#76412
#76192
#76162
#76082
#76072
#75822
#75792
#75652
#75642