Finite Group Databasebeta

← #190647

Subgroups — #190647

Order 512.

818 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34239C211yesmaximal, characteristic
2#21435C211yesmaximal, characteristic
3#20783C211yesmaximal, characteristic
4#8720C211yesmaximal, characteristic
5#8720C211yesmaximal, characteristic
6#8584C211yesmaximal, characteristic, complemented
7#8582C211yesmaximal, characteristic, complemented
8#8582C211yesmaximal, characteristic, complemented
9#8580C211yesmaximal, characteristic, complemented
10#8335C211yesmaximal, characteristic
11#8335C211yesmaximal, characteristic
12#8227C211yesmaximal, characteristic, complemented
13#8227C211yesmaximal, characteristic, complemented
14#8225C211yesmaximal, characteristic, complemented
15#8225C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17#2371—21no—
18#2371—21no—
19#2370C411yescharacteristic
20#2370C411yescharacteristic
21#2370C411yescharacteristic
22#2370C411yescharacteristic
23#2345C411yescharacteristic
24#2345C411yescharacteristic
25#2344—21no—
26#2344—21no—
27#2344C411yescharacteristic
28#2344C411yescharacteristic
29#2336—21no—
30#2336—21no—
31#2313—21no—
32C4×#125C2211yescharacteristic
33C4×#125C2211yescharacteristic
34C4×#125C2211yescharacteristic
35C4×#125C2211yescharacteristic
36#2283C2211yescharacteristic
37#2274—21no—
38C4×#117C2211yescharacteristic
39#1924—21no—
40#1923—21no—
41#1917—21no—
42#1916—21no—
43#1840—21no—
44#1832—21no—
45#1818—21no—
46#1798—21no—
47#1767C2211yescharacteristic
48#1767C2211yescharacteristic
49#1767C2211yescharacteristic
50#1767C2211yescharacteristic
51#1749C2211yescharacteristic
52#1749C2211yescharacteristic
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55#1745C2211yescharacteristic
56#1745C2211yescharacteristic
57#1745C2211yescharacteristic
58C4×#97C2211yescharacteristic
59C2×#330C2211yescharacteristic
60C2×#330C2211yescharacteristic
61C2×#330C2211yescharacteristic
62C2×#330C2211yescharacteristic
63C2×#330C2211yescharacteristic
64C2×#330C2211yescharacteristic
65C2×#330C2211yescharacteristic
66C2×#330C2211yescharacteristic
67#1295C2211yescharacteristic
68#1295C2211yescharacteristic
69#1295C2211yescharacteristic
70#1295C2211yescharacteristic
71#1295C2211yescharacteristic
72#1295C2211yescharacteristic
73#1295C2211yescharacteristic
74#1295C2211yescharacteristic
75C2×#125—22no—
76C2×#125—22no—
77C2×#125—22no—
78C2×#125—22no—
79C2×#125—21no—
80C2×#125—21no—
81C2×#125C2×C411yescharacteristic
82C2×#125C2×C411yescharacteristic
83C2×#125C2311yescharacteristic
84C2×#125C2311yescharacteristic
85C2×#117—21no—
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4C2×C411yescharacteristic
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C42C2311yescharacteristic
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×#97C2311yescharacteristic
104C2×#97C2311yescharacteristic
105C2×#97C2311yescharacteristic
106C2×#97C2311yescharacteristic
107C2×#97C2311yescharacteristic
108C2×#97C2311yescharacteristic
109C2×#97C2311yescharacteristic
110C2×#97C2311yescharacteristic
111C2×C4×C8—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#389C2×C411yescharacteristic
119#389C2×C411yescharacteristic
120#389C2×C411yescharacteristic
121#389C2×C411yescharacteristic
122#389C2×C411yescharacteristic
123#389C2×C411yescharacteristic
124#387—21no—
125#387—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#383—21no—
149#382C2311yescharacteristic
150#382C2311yescharacteristic
151#381—21no—
152#381—21no—
153#381—21no—
154#380—21no—
155#380—21no—
156#380C2×C411yescharacteristic
157#380C2×C411yescharacteristic
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4C2×C411yescharacteristic
180C4×C4⋊C4C2×C411yescharacteristic
181C4×C4⋊C4C2×C411yescharacteristic
182C4×C4⋊C4D411yescharacteristic
183C4×C4⋊C4D411yescharacteristic
184C4×C22⋊C4—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4C2×C411yescharacteristic
188C4×C22⋊C4C2×C411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191#376—21no—
192#376D411yescharacteristic
193#376D411yescharacteristic
194C43C2311yescharacteristic
195C43C2311yescharacteristic
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336—21no—
202#336—21no—
203#336—21no—
204#336—21no—
205#336—21no—
206#336—21no—
207#336—21no—
208#336—21no—
209#336—21no—
210#330—22no—
211#330—22no—
212#330—22no—
213#330—22no—
214#330—22no—
215#330—22no—
216#330—22no—
217#330—22no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4C22×C411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231#125—24no—
232#125—24no—
233#125—24no—
234#125—24no—
235#125—22no—
236#125—22no—
237#125—22no—
238#125—22no—
239#125—21no—
240#125—21no—
241#125—21no—
242#125—21no—
243#125#4112yes—
244#125#4112yes—
245#125#4112yes—
246#125#4112yes—
247#117—42no—
248#117—42no—
249#117—44no—
250#117—44no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
278C2×C4⋊C4C22⋊C411yescharacteristic
279C2×C4⋊C4C22⋊C411yescharacteristic
280C2×C4⋊C4C22⋊C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4C22⋊C411yescharacteristic
292C2×C22⋊C4C22⋊C411yescharacteristic
293C2×C22⋊C4C22⋊C411yescharacteristic
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2411yescharacteristic
325C4⋊C8—42no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329C22⋊C8—42no—
330C22⋊C8—42no—
331C22⋊C8—42no—
332C22⋊C8—42no—
333#97—21no—
334#97—21no—
335#97—21no—
336#97—21no—
337#97—21no—
338#97—21no—
339#97—21no—
340#97—21no—
341#97—21no—
342#97—21no—
343#97—21no—
344#97—21no—
345#97—21no—
346#97—21no—
347#97—21no—
348#97—21no—
349#97—42no—
350C4×C8—42no—
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#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95C22⋊C411yescharacteristic
376#95C22⋊C411yescharacteristic
377#95C22⋊C411yescharacteristic
378#95C22⋊C411yescharacteristic
379#95C22⋊C411yescharacteristic
380#95C22⋊C411yescharacteristic
381#95C22⋊C411yescharacteristic
382#95C22⋊C411yescharacteristic
383C24C2×C22⋊C411yescharacteristic
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4#11712yes—
483C4⋊C4#11712yes—
484C4⋊C4#11712yes—
485C4⋊C4#11712yes—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—41no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—22no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42—81no—
515C42—81no—
516C42—81no—
517C42—81no—
518C42—81no—
519C42—81no—
520C42C2×#4111yescharacteristic
521C42C2×#4111yescharacteristic
522C42C2×#4111yescharacteristic
523C42C2×#4111yescharacteristic
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#42112yes—
531C23C22×C22⋊C411yescharacteristic
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—84no—
602C2×C4#42112yes—
603C2×C4C2×#11711yescharacteristic
604C2×C4C2×#11711yescharacteristic
605C2×C4#52511yescharacteristic
606C2×C4#52511yescharacteristic
607C2×C4#52511yescharacteristic
608C2×C4#52511yescharacteristic
609C8—42no—
610C8—42no—
611C8—81no—
612C8—81no—
613C8—81no—
614C8—81no—
615C8—81no—
616C8—81no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22#228011yescharacteristic
625C22#288911yescharacteristic
626C22#288911yescharacteristic
627C22C2×#42111yescharacteristic
628C22C2×#42111yescharacteristic
629C22#290111yescharacteristic
630C22#290111yescharacteristic
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—22no—
636C4—21no—
637C4—21no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—81no—
656C2—42no—
657C2#1768011yescharacteristic
658C2#1768011yescharacteristic
659C2#1768011yescharacteristic
660C2#1768211yescharacteristic
661C2#2073011yescharacteristic
662C2#2074811yescharacteristic
663C2#2074911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8584C2semidirect
7. #8582C2semidirect
8. #8582C2semidirect
9. #8580C2semidirect
12. #8227C2semidirect
13. #8227C2semidirect
14. #8225C2semidirect
15. #8225C2semidirect

Maximal subgroups

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

subgroupindex
#342392
#214352
#207832
#87202
#85842
#85822
#85802
#83352
#82272
#82252