Finite Group Databasebeta

← #165363

Subgroups — #165363

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36375C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic
3#20630C211yesmaximal, characteristic
4#7715C211yesmaximal, characteristic, complemented
5#7715C211yesmaximal, characteristic, complemented
6#7711C211yesmaximal, characteristic, complemented
7#7701C211yesmaximal, characteristic, complemented
8#7691C211yesmaximal, characteristic, complemented
9#7661C211yesmaximal, characteristic, complemented
10#7641C211yesmaximal, characteristic, complemented
11#7634C211yesmaximal, characteristic, complemented
12#7613C211yesmaximal, characteristic, complemented
13#7598C211yesmaximal, characteristic, complemented
14#7589C211yesmaximal, characteristic, complemented
15#7573C211yesmaximal, characteristic, complemented
16C2×#552C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#446—21no—
19C2×#424—21no—
20C2×#408—21no—
21#2855C2211yescharacteristic
22#2855C2211yescharacteristic
23#2851C2211yescharacteristic
24#2584C411yescharacteristic
25#2581C411yescharacteristic
26#2486C411yescharacteristic
27#2485C411yescharacteristic
28#2478—21no—
29#2475—21no—
30#2466—21no—
31#2462—21no—
32#2419—21no—
33#2417C411yescharacteristic
34#2415C411yescharacteristic
35#2406—21no—
36#2359—21no—
37#2352C411yescharacteristic
38#2344—21no—
39C4×C22⋊Q8C2211yescharacteristic
40#2296C2211yescharacteristic
41C2×#389C2211yescharacteristic
42#1955—21no—
43#1955—21no—
44#1954—21no—
45#1798—21no—
46#1798—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1797C2211yescharacteristic
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56C4×C22⋊C8C2211yescharacteristic
57#1728C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C22×#41D411yescharacteristic
76C22×M16—21no—
77C23×C8—21no—
78#552—44no—
79C2×#126—21no—
80C2×#125—21no—
81C2×#125C2×C411yescharacteristic
82C2×#123—21no—
83C2×#123—21no—
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×C22⋊Q8C2×C411yescharacteristic
87C4×#41—44no—
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×Q8C2×C411yescharacteristic
90C2×C4×Q8D411yescharacteristic
91C2×C4×D4C2×C411yescharacteristic
92C2×C4×D4C2×C411yescharacteristic
93C2×C4×D4C2×C411yescharacteristic
94C2×C4×D4D411yescharacteristic
95C2×C4×D4D411yescharacteristic
96C2×C4×D4D411yescharacteristic
97C2×#117—21no—
98C2×#117C2×C411yescharacteristic
99C2×#117C2×C411yescharacteristic
100C2×#117D411yescharacteristic
101C2×#117D411yescharacteristic
102C2×#117D411yescharacteristic
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2×C411yescharacteristic
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#446—44no—
110#424—44no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118#408—44no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×#97—21no—
130C2×#97—21no—
131C2×C4×C8—21no—
132#400—21no—
133#398—21no—
134#398—21no—
135#398—21no—
136#397—21no—
137#397—21no—
138#397—21no—
139#396—21no—
140#395—21no—
141#395—21no—
142#394—21no—
143#394—21no—
144#391C2×C411yescharacteristic
145#391C2×C411yescharacteristic
146#391C2311yescharacteristic
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389C2×C411yescharacteristic
153#389C2×C411yescharacteristic
154#389C2×C411yescharacteristic
155#389C2×C411yescharacteristic
156#389C2×C411yescharacteristic
157#389C2×C411yescharacteristic
158#389C2311yescharacteristic
159#389C2311yescharacteristic
160#389C2311yescharacteristic
161#388—21no—
162#388—21no—
163#388—21no—
164#387—21no—
165#387—21no—
166#387—21no—
167#387—21no—
168#387—21no—
169#387—21no—
170#387—21no—
171#387—21no—
172#387—21no—
173#387—21no—
174#387—21no—
175#386—21no—
176#386—21no—
177#385—21no—
178#385—21no—
179#385—21no—
180#385—21no—
181#385—21no—
182#385—21no—
183#385—21no—
184#382C2×C411yescharacteristic
185#381—21no—
186#381—21no—
187#381—21no—
188#380—21no—
189#380C2×C411yescharacteristic
190#380C2×C411yescharacteristic
191#380C2×C411yescharacteristic
192#380C2×C411yescharacteristic
193#380C2×C411yescharacteristic
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2×C411yescharacteristic
198C4×C22⋊C4—21no—
199C4×C22⋊C4—21no—
200C2×#95—21no—
201#336—21no—
202#336—21no—
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209#336C2311yescharacteristic
210#336C2311yescharacteristic
211C2×#41—22no—
212C2×#41—22no—
213C2×#41—42no—
214C2×#41—42no—
215C22×Q8C22⋊C411yescharacteristic
216C22×Q8C22×C411yescharacteristic
217C22×D4C22⋊C411yescharacteristic
218C22×D4C22⋊C411yescharacteristic
219C22×D4C22⋊C411yescharacteristic
220C23×C4—21no—
221C23×C4—21no—
222C23×C4C22×C411yescharacteristic
223C23×C4C22×C411yescharacteristic
224C23×C4C22×C411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C2×M16—42no—
232C2×M16—42no—
233C2×M16—42no—
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—
245#126—44no—
246#125—42no—
247#125—44no—
248#123—44no—
249#123—44no—
250C22⋊Q8—22no—
251C22⋊Q8—41no—
252C22⋊Q8—42no—
253C22⋊Q8—42no—
254C4×Q8—24no—
255C4×Q8—42no—
256C4×Q8—42no—
257C4×D4—24no—
258C4×D4—42no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—42no—
263#117—24no—
264#117—22no—
265#117—42no—
266#117—42no—
267#117—44no—
268#117—41no—
269#117—42no—
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—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⋊C4—21no—
288C2×C4⋊C4—41no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4C22⋊C411yescharacteristic
293C2×C4⋊C4C22⋊C411yescharacteristic
294C2×C4⋊C4C22⋊C411yescharacteristic
295C2×C4⋊C4C22⋊C411yescharacteristic
296C2×C4⋊C4C22⋊C411yescharacteristic
297C2×C4⋊C4C22⋊C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4C22⋊C411yescharacteristic
316C2×C22⋊C4C22⋊C411yescharacteristic
317C2×C22⋊C4C22⋊C411yescharacteristic
318C2×C22⋊C4C22⋊C411yescharacteristic
319C2×C22⋊C4C22⋊C411yescharacteristic
320C2×C22⋊C4C22⋊C411yescharacteristic
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
330C2×C42C22×C411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2411yescharacteristic
338C4⋊C8—42no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341C4⋊C8—42no—
342C4⋊C8—42no—
343C4⋊C8—42no—
344C4⋊C8—42no—
345C22⋊C8—42no—
346C22⋊C8—42no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351C22⋊C8—42no—
352C22⋊C8—81no—
353C22⋊C8—81no—
354C22⋊C8—81no—
355#97—42no—
356#97—42no—
357C4×C8—42no—
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#95—21no—
379#95—21no—
380#95—21no—
381C24C2×C22⋊C411yescharacteristic
382C24C2×C22⋊C411yescharacteristic
383C24C2×C22⋊C411yescharacteristic
384#41—48no—
385#41—48no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
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—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—81no—
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4AGL(1,ℤ/8)12yes—
460C22×C4#13712yes—
461C22×C4C23×C411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464M16—84no—
465C2×C8—22no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
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—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—84no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—81no—
524C4⋊C4#9914yes—
525C4⋊C4#13112yes—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—42no—
553C42—42no—
554C42—42no—
555C42—81no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23#41712yes—
576C23#42112yes—
577C23C22×C22⋊C411yescharacteristic
578Q8—44no—
579Q8—81no—
580Q8—81no—
581D4—44no—
582D4—44no—
583D4—44no—
584C2×C4—24no—
585C2×C4—24no—
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—22no—
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—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
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—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4C2×#9912yes—
658C2×C4C2×#9912yes—
659C2×C4C2×#9912yes—
660C2×C4#41712yes—
661C2×C4C2×#13111yescharacteristic
662C2×C4#56811yescharacteristic
663C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
664C2×C4C2×#13711yescharacteristic
665C2×C4#57711yescharacteristic
666C2×C4#57811yescharacteristic
667C8—42no—
668C8—81no—
669C8—81no—
670C8—81no—
671C8—81no—
672C8—81no—
673C8—81no—
674C8—81no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22#288011yescharacteristic
697C22C22×#9911yescharacteristic
698C22#288911yescharacteristic
699C22C2×#41711yescharacteristic
700C22#289811yescharacteristic
701C22C2×#42111yescharacteristic
702C22#290111yescharacteristic
703C4—21no—
704C4—21no—
705C4—22no—
706C4—22no—
707C4—21no—
708C4—21no—
709C4—22no—
710C4—22no—
711C4—21no—
712C4—21no—
713C4—41no—
714C4—42no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—81no—
720C4—81no—
721C4—81no—
722C4—81no—
723C4—81no—
724C4—81no—
725C4—81no—
726C2—42no—
727C2—42no—
728C2—81no—
729C2#1741611yescharacteristic
730C2#1751211yescharacteristic
731C2#1752411yescharacteristic
732C2#1759211yescharacteristic
733C2#1760611yescharacteristic
734C2#1768011yescharacteristic
735C2#1776311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7715C2semidirect
5. #7715C2semidirect
6. #7711C2semidirect
7. #7701C2semidirect
8. #7691C2semidirect
9. #7661C2semidirect
10. #7641C2semidirect
11. #7634C2semidirect
12. #7613C2semidirect
13. #7598C2semidirect
14. #7589C2semidirect
15. #7573C2semidirect

Maximal subgroups

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

subgroupindex
#363752
#225322
#206302
#77152
#77112
#77012
#76912
#76612
#76412
#76342
#76132
#75982
#75892
#75732