Finite Group Databasebeta

← #223854

Subgroups — #223854

Order 512.

923 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36185C211yesmaximal, characteristic
2#21271C211yesmaximal, characteristic, complemented
3#21107C211yesmaximal, characteristic, complemented
4#9895C211yesmaximal, characteristic, complemented
5#9890C211yesmaximal, characteristic, complemented
6#9124C211yesmaximal, characteristic, complemented
7#9123C211yesmaximal, characteristic, complemented
8#9123C211yesmaximal, characteristic, complemented
9#9119C211yesmaximal, characteristic, complemented
10#8722C211yesmaximal, characteristic, complemented
11#8721C211yesmaximal, characteristic, complemented
12#8581C211yesmaximal, characteristic, complemented
13#8226C211yesmaximal, characteristic, complemented
14#8034C211yesmaximal, characteristic, complemented
15#8031C211yesmaximal, characteristic, complemented
16C2×#556C411yescharacteristic
17C2×#555C411yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C22×#97C2211yescharacteristic
22#2862C2211yescharacteristic
23#2850C2211yescharacteristic
24#2850C2211yescharacteristic
25#2567C411yescharacteristic
26#2542C411yescharacteristic
27#2521—21no—
28#2521—21no—
29#2521—21no—
30#2521—21no—
31#2500C411yescharacteristic
32#2500C411yescharacteristic
33#2499C411yescharacteristic
34#2499C411yescharacteristic
35#2427—21no—
36#2427—21no—
37#2362—21no—
38#2362—21no—
39C4×#124C2211yescharacteristic
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42C42×D4C2211yescharacteristic
43#1955—21no—
44#1940—21no—
45#1924—21no—
46#1923—21no—
47#1923—21no—
48#1917—21no—
49#1869—21no—
50#1850—21no—
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56#1749C2211yescharacteristic
57#1745C2211yescharacteristic
58C4×#97C2211yescharacteristic
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
75C23×C8—21no—
76#556—44no—
77#555—44no—
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×#126—21no—
81C2×#126—21no—
82C2×#124—21no—
83C2×#124—21no—
84C2×#124—21no—
85C2×#124C2311yescharacteristic
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8C2×C411yescharacteristic
89C2×C22⋊Q8C2×C411yescharacteristic
90C2×#121—21no—
91C2×#121C2×C411yescharacteristic
92C2×#121C2×C411yescharacteristic
93C2×C4×Q8—21no—
94C2×C4×Q8C2×C411yescharacteristic
95C2×C4×D4C2×C411yescharacteristic
96C2×C4×D4C2×C411yescharacteristic
97C2×C4×D4D411yescharacteristic
98C2×C4×D4D411yescharacteristic
99C2×#117—21no—
100C2×#117C2×C411yescharacteristic
101C2×#117C2×C411yescharacteristic
102C2×#117C2×C411yescharacteristic
103C2×#117C2×C411yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106#432—22no—
107#432—22no—
108#432—22no—
109#432—41no—
110#432—41no—
111#432—41no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×#97—22no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97—41no—
128C2×#97C2311yescharacteristic
129C2×#97C2311yescharacteristic
130C2×#97C2311yescharacteristic
131C2×#97C2311yescharacteristic
132C2×#97C2311yescharacteristic
133C2×#97C2311yescharacteristic
134C2×#97C2311yescharacteristic
135C2×#97C2311yescharacteristic
136C2×C4×C8—21no—
137#399—21no—
138#399—21no—
139#397—21no—
140#397—21no—
141#397—21no—
142#397—21no—
143#395—21no—
144#395—21no—
145#394—21no—
146#394—21no—
147#394—21no—
148#394—21no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391C2×C411yescharacteristic
154#391C2×C411yescharacteristic
155#391C2×C411yescharacteristic
156#390C2×C411yescharacteristic
157#390C2×C411yescharacteristic
158#390C2×C411yescharacteristic
159#389—21no—
160#389—21no—
161#388—21no—
162#388—21no—
163#388—21no—
164#388—21no—
165#388—21no—
166#388—21no—
167#388—21no—
168#388—21no—
169#388—21no—
170#388—21no—
171#387—21no—
172#387—21no—
173#387—21no—
174#387—21no—
175#386—21no—
176#386—21no—
177#386—21no—
178#386—21no—
179#384C2×C411yescharacteristic
180#384C2×C411yescharacteristic
181#384C2×C411yescharacteristic
182#384C2×C411yescharacteristic
183#382C2×C411yescharacteristic
184#382C2×C411yescharacteristic
185#382C2311yescharacteristic
186#382C2311yescharacteristic
187#382C2311yescharacteristic
188#380—21no—
189#380—21no—
190#380—21no—
191#380—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4C2×C411yescharacteristic
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D411yescharacteristic
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4C2×C411yescharacteristic
202C4×C22⋊C4C2×C411yescharacteristic
203C4×C22⋊C4D411yescharacteristic
204C4×C22⋊C4D411yescharacteristic
205C4×C22⋊C4D411yescharacteristic
206C4×C22⋊C4D411yescharacteristic
207C43C2311yescharacteristic
208#336—21no—
209#336—21no—
210#336—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214#336—21no—
215#336—21no—
216#329—22no—
217#329—22no—
218#329—41no—
219#329—41no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8C22×C411yescharacteristic
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C22⋊C411yescharacteristic
226C22×D4C22×C411yescharacteristic
227C23×C4—21no—
228C23×C4C22×C411yescharacteristic
229C23×C4C22×C411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—42no—
243C22×C8—42no—
244C22×C8—42no—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247#126—44no—
248#126—44no—
249#124—21no—
250#124—21no—
251#124—21no—
252#124—21no—
253#124—44no—
254#124—44no—
255#124—44no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258C22⋊Q8—42no—
259C22⋊Q8—42no—
260#121—42no—
261#121—44no—
262#121—42no—
263C4×Q8—42no—
264C4×Q8—44no—
265C4×D4—24no—
266C4×D4—42no—
267C4×D4—42no—
268C4×D4—42no—
269#117—44no—
270#117—42no—
271#117—42no—
272#117—42no—
273#117—42no—
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—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
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×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4C22⋊C411yescharacteristic
314C2×C22⋊C4C22⋊C411yescharacteristic
315C2×C22⋊C4C22⋊C411yescharacteristic
316C2×C22⋊C4C22⋊C411yescharacteristic
317C2×C22⋊C4C22⋊C411yescharacteristic
318C2×C22⋊C4C22⋊C411yescharacteristic
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42C22×C411yescharacteristic
333C2×C42C22×C411yescharacteristic
334C2×C42C22×C411yescharacteristic
335C2×C42C22×C411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2411yescharacteristic
345C4⋊C8—42no—
346C4⋊C8—42no—
347C4⋊C8—42no—
348C4⋊C8—42no—
349C4⋊C8—42no—
350C22⋊C8—42no—
351C22⋊C8—42no—
352C22⋊C8—42no—
353C22⋊C8—42no—
354C22⋊C8—42no—
355C22⋊C8—42no—
356C22⋊C8—42no—
357#97—21no—
358#97—21no—
359#97—21no—
360#97—21no—
361#97—21no—
362#97—21no—
363#97—21no—
364#97—21no—
365#97—21no—
366#97—21no—
367#97—21no—
368#97—21no—
369#97—21no—
370#97—21no—
371#97—42no—
372#97—42no—
373#97#4112yes—
374#97#4112yes—
375C4×C8—42no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95C22⋊C411yescharacteristic
393#95C22⋊C411yescharacteristic
394#95C22⋊C411yescharacteristic
395#95C22⋊C411yescharacteristic
396#95C22⋊C411yescharacteristic
397#95C22⋊C411yescharacteristic
398C24—21no—
399C24C2×C22⋊C411yescharacteristic
400C24C2×C22⋊C411yescharacteristic
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—41no—
406C2×Q8—41no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—81no—
410C2×Q8—81no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—81no—
423C2×D4—81no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C23×C411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—48no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—81no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C22⋊C4#10014yes—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—22no—
556C42—22no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—42no—
562C42—42no—
563C42—42no—
564C42—42no—
565C42—42no—
566C42C2×#4111yescharacteristic
567C42C2×#4111yescharacteristic
568C42H(5,𝔽2)11yescharacteristic
569C422−1+411yescharacteristic
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23C2×#10012yes—
591C23C22×C22⋊C411yescharacteristic
592Q8—81no—
593Q8—81no—
594Q8—84no—
595Q8—162no—
596D4—44no—
597D4—81no—
598D4—81no—
599D4—162no—
600D4—162no—
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—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—84no—
667C2×C4C2×#10012yes—
668C2×C4C2×#10012yes—
669C2×C4#42112yes—
670C2×C4#42112yes—
671C2×C4C4×#4111yescharacteristic
672C2×C4#52011yescharacteristic
673C2×C4#54611yescharacteristic
674C2×C4#54611yescharacteristic
675C2×C4#54711yescharacteristic
676C2×C4#54711yescharacteristic
677C8—42no—
678C8—42no—
679C8—42no—
680C8—42no—
681C8—81no—
682C8—81no—
683C8—81no—
684C8—81no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22#233911yescharacteristic
707C22C22×#10011yescharacteristic
708C22#288911yescharacteristic
709C22C2×#42111yescharacteristic
710C22C2×#42111yescharacteristic
711C22#290111yescharacteristic
712C22#290111yescharacteristic
713C4—22no—
714C4—22no—
715C4—21no—
716C4—21no—
717C4—22no—
718C4—21no—
719C4—21no—
720C4—22no—
721C4—21no—
722C4—21no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—41no—
729C4—81no—
730C4—81no—
731C4—81no—
732C4—81no—
733C4—81no—
734C4—81no—
735C4—81no—
736C2—42no—
737C2—81no—
738C2—161no—
739C2#1767111yescharacteristic
740C2#1768311yescharacteristic
741C2#1768311yescharacteristic
742C2#1768511yescharacteristic
743C2#2247911yescharacteristic
744C2#2250611yescharacteristic
745C2#2250711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#555C4
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. #21271C2semidirect
3. #21107C2semidirect
4. #9895C2semidirect
5. #9890C2semidirect
6. #9124C2semidirect
7. #9123C2semidirect
8. #9123C2semidirect
9. #9119C2semidirect
10. #8722C2semidirect
11. #8721C2semidirect
12. #8581C2semidirect
13. #8226C2semidirect
14. #8034C2semidirect
15. #8031C2semidirect

Maximal subgroups

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

subgroupindex
#361852
#212712
#211072
#98952
#98902
#91242
#91232
#91192
#87222
#87212
#85812
#82262
#80342
#80312