Finite Group Databasebeta

← #223124

Subgroups — #223124

Order 512.

838 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36172C211yesmaximal, characteristic
2#21108C211yesmaximal, characteristic, complemented
3#21107C211yesmaximal, characteristic, complemented
4#9893C211yesmaximal, characteristic, complemented
5#9893C211yesmaximal, characteristic, complemented
6#9123C211yesmaximal, characteristic, complemented
7#9123C211yesmaximal, characteristic, complemented
8#9120C211yesmaximal, characteristic, complemented
9#9119C211yesmaximal, characteristic, complemented
10#8720C211yesmaximal, characteristic, complemented
11#8718C211yesmaximal, characteristic, complemented
12#8581C211yesmaximal, characteristic, complemented
13#8224C211yesmaximal, characteristic, complemented
14#8036C211yesmaximal, characteristic, complemented
15#8033C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2864C2211yescharacteristic
21#2850C2211yescharacteristic
22#2850C2211yescharacteristic
23#2576—21no—
24#2529—21no—
25#2518—21no—
26#2515—21no—
27#2514C411yescharacteristic
28#2513C411yescharacteristic
29#2510—21no—
30#2444—21no—
31#2421C411yescharacteristic
32#2420C411yescharacteristic
33#2417C411yescharacteristic
34#2416C411yescharacteristic
35#2364—21no—
36#2361C411yescharacteristic
37#2360C411yescharacteristic
38#2353—21no—
39C4×#124C2211yescharacteristic
40#2283C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C42×D4C2211yescharacteristic
43#1924—21no—
44#1923—21no—
45#1917—21no—
46#1916—21no—
47#1870—21no—
48#1869—21no—
49#1851—21no—
50#1850—21no—
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56#1749C2211yescharacteristic
57#1745C2211yescharacteristic
58#1745C2211yescharacteristic
59C2×#329C2211yescharacteristic
60C2×#329C2211yescharacteristic
61C2×#329C2211yescharacteristic
62C2×#329C2211yescharacteristic
63#1295C2211yescharacteristic
64#1295C2211yescharacteristic
65#1295C2211yescharacteristic
66#1295C2211yescharacteristic
67#1295C2211yescharacteristic
68#1295C2211yescharacteristic
69#1295C2211yescharacteristic
70#1295C2211yescharacteristic
71#1295C2211yescharacteristic
72#1295C2211yescharacteristic
73#1295C2211yescharacteristic
74#1295C2211yescharacteristic
75C22×M16—21no—
76C23×C8—21no—
77C2×#124C2311yescharacteristic
78C2×C22⋊Q8C2×C411yescharacteristic
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×#121—21no—
81C2×#121C2×C411yescharacteristic
82C2×#121C2×C411yescharacteristic
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4D411yescharacteristic
88C2×C4×D4D411yescharacteristic
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#432—22no—
94#432—22no—
95#432—41no—
96#432—41no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C4×M16—22no—
108C4×M16—41no—
109C2×#97—22no—
110C2×#97—21no—
111C2×#97—41no—
112C2×#97C2311yescharacteristic
113C2×#97C2311yescharacteristic
114C2×#97C2311yescharacteristic
115C2×#97C2311yescharacteristic
116C2×#97C2311yescharacteristic
117C2×#97C2311yescharacteristic
118C2×#97C2311yescharacteristic
119C2×#97C2311yescharacteristic
120C2×C4×C8—21no—
121#401—21no—
122#400—21no—
123#400—21no—
124#398—21no—
125#397—21no—
126#397—21no—
127#397—21no—
128#397—21no—
129#396—21no—
130#396—21no—
131#394—21no—
132#394—21no—
133#391C2×C411yescharacteristic
134#391C2×C411yescharacteristic
135#391C2×C411yescharacteristic
136#390C2×C411yescharacteristic
137#390C2×C411yescharacteristic
138#390C2×C411yescharacteristic
139#389—21no—
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#388—21no—
145#388—21no—
146#388—21no—
147#388—21no—
148#388—21no—
149#388—21no—
150#388—21no—
151#388—21no—
152#388—21no—
153#388—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#387—21no—
161#386—21no—
162#386—21no—
163#386—21no—
164#386—21no—
165#386—21no—
166#386—21no—
167#386—21no—
168#386—21no—
169#385—21no—
170#385—21no—
171#383—21no—
172#383—21no—
173#382—21no—
174#382—21no—
175#382C2×C411yescharacteristic
176#382C2×C411yescharacteristic
177#382C2311yescharacteristic
178#382C2311yescharacteristic
179#382C2311yescharacteristic
180#381—21no—
181#381—21no—
182#380—21no—
183#380C2×C411yescharacteristic
184#380C2×C411yescharacteristic
185C4×C4⋊C4C2×C411yescharacteristic
186C4×C4⋊C4C2×C411yescharacteristic
187C4×C4⋊C4C2×C411yescharacteristic
188C4×C4⋊C4D411yescharacteristic
189C4×C4⋊C4D411yescharacteristic
190C4×C22⋊C4—21no—
191C4×C22⋊C4—21no—
192C4×C22⋊C4—21no—
193C4×C22⋊C4—21no—
194C4×C22⋊C4C2×C411yescharacteristic
195C4×C22⋊C4C2×C411yescharacteristic
196C4×C22⋊C4D411yescharacteristic
197C4×C22⋊C4D411yescharacteristic
198C4×C22⋊C4D411yescharacteristic
199C4×C22⋊C4D411yescharacteristic
200#376—21no—
201C2×#95—21no—
202C2×#95—21no—
203C43C2311yescharacteristic
204#336—21no—
205#336—21no—
206#336—21no—
207#336—21no—
208#336—21no—
209#336—21no—
210#336—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214#329—22no—
215#329—22no—
216#329—41no—
217#329—41no—
218C22×Q8C22×C411yescharacteristic
219C22×D4—21no—
220C22×D4—21no—
221C22×D4C22⋊C411yescharacteristic
222C22×D4C22×C411yescharacteristic
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4C22×C411yescharacteristic
227C23×C4C22×C411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C2×M16—42no—
233C2×M16—42no—
234C2×M16—42no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—42no—
244C22×C8—42no—
245C22×C8—42no—
246#124—21no—
247#124—21no—
248#124—21no—
249#124—21no—
250C22⋊Q8—42no—
251C22⋊Q8—81no—
252#121—44no—
253#121—42no—
254#121—81no—
255C4×Q8—81no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4—44no—
260C4×D4—81no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—81no—
271C2×C4⋊C4—81no—
272C2×C4⋊C4C22⋊C411yescharacteristic
273C2×C4⋊C4C22⋊C411yescharacteristic
274C2×C4⋊C4C22⋊C411yescharacteristic
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×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
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×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2411yescharacteristic
326C4⋊C8—42no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329C4⋊C8—42no—
330C22⋊C8—42no—
331C22⋊C8—42no—
332C22⋊C8—42no—
333C22⋊C8—42no—
334C22⋊C8—42no—
335C22⋊C8—42no—
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—21no—
350#97—42no—
351#97#4112yes—
352#97#4112yes—
353C4×C8—42no—
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#95—21no—
376#95—21no—
377#95—21no—
378#95C22⋊C411yescharacteristic
379#95C22⋊C411yescharacteristic
380#95C22⋊C411yescharacteristic
381#95C22⋊C411yescharacteristic
382#95C22⋊C411yescharacteristic
383#95C22⋊C411yescharacteristic
384C24—21no—
385C24C2×C22⋊C411yescharacteristic
386C24C2×C22⋊C411yescharacteristic
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—81no—
403C2×D4—81no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
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—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457M16—48no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—48no—
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—
483C4⋊C4—22no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—42no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—42no—
543C42—81no—
544C42—81no—
545C42C2×#4111yescharacteristic
546C42C2×#4111yescharacteristic
547C42H(5,𝔽2)11yescharacteristic
548C422−1+411yescharacteristic
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23#42112yes—
569C23#42112yes—
570C23C22×C22⋊C411yescharacteristic
571Q8—81no—
572Q8—81no—
573D4—44no—
574D4—81no—
575D4—81no—
576D4—84no—
577D4—162no—
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—21no—
598C2×C4—21no—
599C2×C4—21no—
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—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
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—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—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—162no—
647C2×C4—162no—
648C2×C4—162no—
649C2×C4#41312yes—
650C2×C4C4×#4111yescharacteristic
651C2×C4#52011yescharacteristic
652C2×C4#54611yescharacteristic
653C2×C4#54611yescharacteristic
654C2×C4#54711yescharacteristic
655C2×C4#54711yescharacteristic
656C8—42no—
657C8—42no—
658C8—42no—
659C8—42no—
660C8—81no—
661C8—81no—
662C8—81no—
663C8—81no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22#233911yescharacteristic
686C22C2×#41311yescharacteristic
687C22#288811yescharacteristic
688C22C2×#42111yescharacteristic
689C22C2×#42111yescharacteristic
690C22#290111yescharacteristic
691C22#290111yescharacteristic
692C4—21no—
693C4—21no—
694C4—22no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—81no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4—81no—
713C4—81no—
714C4—81no—
715C2—42no—
716C2—42no—
717C2—161no—
718C2#1766811yescharacteristic
719C2#1767911yescharacteristic
720C2#1767911yescharacteristic
721C2#1768711yescharacteristic
722C2#2248211yescharacteristic
723C2#2250611yescharacteristic
724C2#2250611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2416C4
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
2. #21108C2semidirect
3. #21107C2semidirect
4. #9893C2semidirect
5. #9893C2semidirect
6. #9123C2semidirect
7. #9123C2semidirect
8. #9120C2semidirect
9. #9119C2semidirect
10. #8720C2semidirect
11. #8718C2semidirect
12. #8581C2semidirect
13. #8224C2semidirect
14. #8036C2semidirect
15. #8033C2semidirect

Maximal subgroups

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

subgroupindex
#361722
#211082
#211072
#98932
#91232
#91202
#91192
#87202
#87182
#85812
#82242
#80362
#80332