Finite Group Databasebeta

← #190661

Subgroups — #190661

Order 512.

851 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34227C211yesmaximal, characteristic
2#21435C211yesmaximal, characteristic, complemented
3#21435C211yesmaximal, characteristic, complemented
4#8720C211yesmaximal, characteristic, complemented
5#8720C211yesmaximal, characteristic, complemented
6#8584C211yesmaximal, characteristic, complemented
7#8581C211yesmaximal, characteristic, complemented
8#8581C211yesmaximal, characteristic, complemented
9#8580C211yesmaximal, characteristic, complemented
10#8336C211yesmaximal, characteristic
11#8336C211yesmaximal, characteristic
12#8226C211yesmaximal, characteristic, complemented
13#8226C211yesmaximal, characteristic, complemented
14#8225C211yesmaximal, characteristic, complemented
15C4×#330C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2374—21no—
18#2374—21no—
19#2373—21no—
20#2373—21no—
21#2357—21no—
22#2353—21no—
23#2347—21no—
24#2343—21no—
25#2324C411yescharacteristic
26#2324C411yescharacteristic
27#2323C411yescharacteristic
28#2323C411yescharacteristic
29#2320C411yescharacteristic
30#2319C411yescharacteristic
31C4×#125C2211yescharacteristic
32C4×#125C2211yescharacteristic
33C4×#124C2211yescharacteristic
34C4×#124C2211yescharacteristic
35#2283C2211yescharacteristic
36#2276C411yescharacteristic
37#2275C411yescharacteristic
38C4×#117C2211yescharacteristic
39#1924—21no—
40#1924—21no—
41#1923—21no—
42#1923—21no—
43#1917—21no—
44#1917—21no—
45#1916—21no—
46#1916—21no—
47#1767C2211yescharacteristic
48#1767C2211yescharacteristic
49#1767C2211yescharacteristic
50#1767C2211yescharacteristic
51#1749C2211yescharacteristic
52#1749C2211yescharacteristic
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55#1745C2211yescharacteristic
56#1745C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C4×#97C2211yescharacteristic
59C2×#330C2211yescharacteristic
60C2×#330C2211yescharacteristic
61C2×#330C2211yescharacteristic
62C2×#330C2211yescharacteristic
63C2×#329C2211yescharacteristic
64C2×#329C2211yescharacteristic
65C2×#329C2211yescharacteristic
66C2×#329C2211yescharacteristic
67#1295C2211yescharacteristic
68#1295C2211yescharacteristic
69#1295C2211yescharacteristic
70#1295C2211yescharacteristic
71#1295C2211yescharacteristic
72#1295C2211yescharacteristic
73#1295C2211yescharacteristic
74#1295C2211yescharacteristic
75#529—22no—
76#529—22no—
77#529—22no—
78#529—22no—
79C2×#126—21no—
80C2×#125C2311yescharacteristic
81C2×#124C2311yescharacteristic
82C2×#123—21no—
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×D4—21no—
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4C2×C411yescharacteristic
92C2×#117—21no—
93C2×#117C2×C411yescharacteristic
94C2×#117C2×C411yescharacteristic
95C22×C42C2311yescharacteristic
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97C2311yescharacteristic
107C2×#97C2311yescharacteristic
108C2×#97C2311yescharacteristic
109C2×#97C2311yescharacteristic
110C2×#97C2311yescharacteristic
111C2×#97C2311yescharacteristic
112C2×#97C2311yescharacteristic
113C2×#97C2311yescharacteristic
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#390—21no—
119#390—21no—
120#390C2×C411yescharacteristic
121#390C2×C411yescharacteristic
122#389—21no—
123#389—21no—
124#389C2×C411yescharacteristic
125#389C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#388—21no—
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#388—21no—
139#388—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#386—21no—
149#386—21no—
150#385—21no—
151#385—21no—
152#383—21no—
153#383—21no—
154#382—21no—
155#382C2311yescharacteristic
156#382C2311yescharacteristic
157#381—21no—
158#381—21no—
159#380—21no—
160#380C2×C411yescharacteristic
161#380C2×C411yescharacteristic
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⋊C4C2×C411yescharacteristic
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4D411yescharacteristic
176C4×C4⋊C4D411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191#376—21no—
192#376—21no—
193#376C2×C411yescharacteristic
194#376C2×C411yescharacteristic
195C2×#95—21no—
196C43C2311yescharacteristic
197C43C2311yescharacteristic
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#329—22no—
215#329—22no—
216#329—22no—
217#329—22no—
218C22×Q8C22×C411yescharacteristic
219C22×D4—21no—
220C22×D4C22×C411yescharacteristic
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4C22×C411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235#126—24no—
236#126—24no—
237#125—21no—
238#125—21no—
239#125#4112yes—
240#125#4112yes—
241#124—21no—
242#124—21no—
243#124#4112yes—
244#124#4112yes—
245#123—24no—
246#123—24no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249#121—22no—
250#121—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—42no—
254C4×Q8—42no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—42no—
260C4×D4—42no—
261#117—24no—
262#117—24no—
263#117—42no—
264#117—42no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
276C2×C4⋊C4C22⋊C411yescharacteristic
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×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
301C2×C22⋊C4C22⋊C411yescharacteristic
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
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—42no—
335C22⋊C8—42no—
336C22⋊C8—42no—
337C22⋊C8—42no—
338C22⋊C8—42no—
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—21no—
350#97—21no—
351#97—21no—
352#97—21no—
353#97—21no—
354#97—21no—
355#97—42no—
356#97—42no—
357C4×C8—42no—
358C4×C8—42no—
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#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95C22⋊C411yescharacteristic
380#95C22⋊C411yescharacteristic
381#95C22⋊C411yescharacteristic
382#95C22⋊C411yescharacteristic
383#95C22⋊C411yescharacteristic
384#95C22⋊C411yescharacteristic
385#95C22⋊C411yescharacteristic
386#95C22⋊C411yescharacteristic
387C24—21no—
388C24C2×C22⋊C411yescharacteristic
389C2×Q8—21no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8#11712yes—
393C2×D4—24no—
394C2×D4—21no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—81no—
398C2×D4—81no—
399C2×D4#11712yes—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
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×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
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—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4#11712yes—
489C4⋊C4#11712yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C42—24no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—22no—
517C42—22no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—42no—
529C42—42no—
530C42—81no—
531C42—81no—
532C42—81no—
533C42—81no—
534C42—81no—
535C42—81no—
536C42C2×#4111yescharacteristic
537C42C2×#4111yescharacteristic
538C42C2×#4111yescharacteristic
539C42C2×#4111yescharacteristic
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—21no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C22×C22⋊C411yescharacteristic
555Q8—42no—
556Q8—42no—
557D4—42no—
558D4—42no—
559D4—84no—
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—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—21no—
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—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—84no—
625C2×C4—162no—
626C2×C4—162no—
627C2×C4#41312yes—
628C2×C4#41312yes—
629C2×C4C2×#11711yescharacteristic
630C2×C4C2×#11711yescharacteristic
631C2×C4#52511yescharacteristic
632C2×C4#52511yescharacteristic
633C2×C4#52511yescharacteristic
634C2×C4#52511yescharacteristic
635C8—42no—
636C8—42no—
637C8—42no—
638C8—42no—
639C8—81no—
640C8—81no—
641C8—81no—
642C8—81no—
643C22—41no—
644C22—41no—
645C22—42no—
646C22—42no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—42no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22#228011yescharacteristic
658C22C2×#41311yescharacteristic
659C22C2×#41311yescharacteristic
660C22#290111yescharacteristic
661C22#290111yescharacteristic
662C22#290111yescharacteristic
663C22#290111yescharacteristic
664C4—21no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—22no—
669C4—22no—
670C4—21no—
671C4—21no—
672C4—41no—
673C4—41no—
674C4—42no—
675C4—42no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—42no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C2—81no—
689C2—82no—
690C2#1769011yescharacteristic
691C2#1769011yescharacteristic
692C2#1769111yescharacteristic
693C2#1769111yescharacteristic
694C2#2072811yescharacteristic
695C2#2074911yescharacteristic
696C2#2074911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×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
2. #21435C2semidirect
3. #21435C2semidirect
4. #8720C2semidirect
5. #8720C2semidirect
6. #8584C2semidirect
7. #8581C2semidirect
8. #8581C2semidirect
9. #8580C2semidirect
12. #8226C2semidirect
13. #8226C2semidirect
14. #8225C2semidirect
15. C4×#330C2semidirect

Maximal subgroups

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

subgroupindex
#342272
#214352
#87202
#85842
#85812
#85802
#83362
#82262
#82252
C4×#3302