Finite Group Databasebeta

← #165090

Subgroups — #165090

Order 512.

950 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36263C211yesmaximal, characteristic
2#22787C211yesmaximal, characteristic, complemented
3#22698C211yesmaximal, characteristic, complemented
4#7712C211yesmaximal, characteristic, complemented
5#7706C211yesmaximal, characteristic, complemented
6#7698C211yesmaximal, characteristic, complemented
7#7694C211yesmaximal, characteristic, complemented
8#7664C211yesmaximal, characteristic, complemented
9#7664C211yesmaximal, characteristic, complemented
10#7641C211yesmaximal, characteristic, complemented
11#7633C211yesmaximal, characteristic, complemented
12#7628C211yesmaximal, characteristic, complemented
13#7599C211yesmaximal, characteristic, complemented
14#7594C211yesmaximal, characteristic, complemented
15C2×#1271C211yesmaximal, characteristic, complemented
16C2×#553—21no—
17C2×#435—21no—
18#2860C2211yescharacteristic
19#2856C2211yescharacteristic
20#2854C2211yescharacteristic
21#2602—21no—
22#2601—21no—
23#2591—21no—
24#2559—21no—
25#2558—21no—
26#2543—21no—
27#2506C411yescharacteristic
28#2496C411yescharacteristic
29#2445C411yescharacteristic
30#2428C411yescharacteristic
31#2416C411yescharacteristic
32#2416C411yescharacteristic
33#2364—21no—
34#2361C411yescharacteristic
35#2361C411yescharacteristic
36C4×#121C2211yescharacteristic
37#2293C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1940—21no—
41#1924—21no—
42#1917—21no—
43#1869—21no—
44#1851—21no—
45#1850—21no—
46#1839C2211yescharacteristic
47#1831C2211yescharacteristic
48#1817C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1767C2211yescharacteristic
53#1752—21no—
54#1749C2211yescharacteristic
55#1745C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1271C2211yescharacteristic
60#1271C2211yescharacteristic
61#1271C2211yescharacteristic
62#1271C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1270C2211yescharacteristic
68#1270C2211yescharacteristic
69#1270C2211yescharacteristic
70#1270C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×M16—21no—
76C23×C8—21no—
77C23×C8—21no—
78#553—44no—
79#553—44no—
80C2×#127C2×C411yescharacteristic
81C2×#126—21no—
82C2×#126—21no—
83C2×#125—21no—
84C2×#123—21no—
85C2×#123—21no—
86C2×#121—21no—
87C2×#121—21no—
88C2×#121—21no—
89C2×#121—21no—
90C2×#121—21no—
91C2×#121C2×C411yescharacteristic
92C2×#121C2×C411yescharacteristic
93C2×#121C2×C411yescharacteristic
94C2×#121C2×C411yescharacteristic
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4C2×C411yescharacteristic
99C2×C4×D4C2×C411yescharacteristic
100C2×C4×D4C2×C411yescharacteristic
101C2×C4×D4D411yescharacteristic
102C2×C4×D4D411yescharacteristic
103C2×#117—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#435—44no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×#97—21no—
121C2×#97—21no—
122C2×#97—21no—
123#400—21no—
124#400—21no—
125#398—21no—
126#397—21no—
127#396—21no—
128#396—21no—
129#394—21no—
130#394—21no—
131#394—21no—
132#394—21no—
133#394—21no—
134#392—21no—
135#390—21no—
136#390—21no—
137#390C2×C411yescharacteristic
138#390C2×C411yescharacteristic
139#390C2×C411yescharacteristic
140#390C2×C411yescharacteristic
141#390C2311yescharacteristic
142#390C2311yescharacteristic
143#389—21no—
144#389—21no—
145#389C2×C411yescharacteristic
146#389C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#389C2×C411yescharacteristic
151#389C2311yescharacteristic
152#389C2311yescharacteristic
153#388—21no—
154#388—21no—
155#388—21no—
156#388—21no—
157#388—21no—
158#388—21no—
159#388—21no—
160#388—21no—
161#388—21no—
162#388—21no—
163#387—21no—
164#387—21no—
165#386—21no—
166#386—21no—
167#386—21no—
168#386—21no—
169#386—21no—
170#386—21no—
171#386—21no—
172#386—21no—
173#385—21no—
174#385—21no—
175#384C2×C411yescharacteristic
176#380C2×C411yescharacteristic
177#380C2×C411yescharacteristic
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4C2×C411yescharacteristic
180C4×C4⋊C4C2×C411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C4⋊C4D411yescharacteristic
183C4×C22⋊C4—21no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4C2×C411yescharacteristic
186C4×C22⋊C4C2×C411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191C2×#95—21no—
192C43C2311yescharacteristic
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4C22⋊C411yescharacteristic
216C22×D4C22×C411yescharacteristic
217C22×D4C22×C411yescharacteristic
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4C22×C411yescharacteristic
223C23×C4C22×C411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C2×M16—42no—
229C2×M16—42no—
230C2×M16—81no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—42no—
240C22×C8—42no—
241C22×C8—42no—
242C22×C8—42no—
243C22×C8—81no—
244C22×C8—81no—
245#127—81no—
246#126—42no—
247#126—42no—
248#126—44no—
249#125—44no—
250#123—44no—
251#123—42no—
252#123—42no—
253#121—22no—
254#121—44no—
255#121—41no—
256#121—42no—
257#121—42no—
258#121—44no—
259#121—44no—
260#121—42no—
261#121—42no—
262#121—42no—
263#121—42no—
264#121—81no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—41no—
271C4×D4—42no—
272C4×D4—44no—
273C4×D4—44no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—81no—
277#117—42no—
278#117—42no—
279C2×C4⋊C4—22no—
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—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—81no—
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⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4C22⋊C411yescharacteristic
326C2×C22⋊C4C22⋊C411yescharacteristic
327C2×C22⋊C4C22⋊C411yescharacteristic
328C2×C22⋊C4C22⋊C411yescharacteristic
329C2×C22⋊C4C22⋊C411yescharacteristic
330C2×C22⋊C4C22⋊C411yescharacteristic
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—81no—
334C2×C42—81no—
335C2×C42C22×C411yescharacteristic
336C2×C42C22×C411yescharacteristic
337C2×C42C22×C411yescharacteristic
338C2×C42C22×C411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C4⋊C8—42no—
349C4⋊C8—42no—
350C4⋊C8—81no—
351C4⋊C8—81no—
352C4⋊C8—81no—
353C22⋊C8—42no—
354C22⋊C8—42no—
355C22⋊C8—42no—
356C22⋊C8—42no—
357C22⋊C8—81no—
358#97—42no—
359#97—42no—
360#97—81no—
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#95C22⋊C411yescharacteristic
374#95C22⋊C411yescharacteristic
375#95C22⋊C411yescharacteristic
376#95C22⋊C411yescharacteristic
377#95C22⋊C411yescharacteristic
378#95C22⋊C411yescharacteristic
379C24—21no—
380C24—21no—
381C24C2×C22⋊C411yescharacteristic
382C24C2×C22⋊C411yescharacteristic
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—81no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4—81no—
412C22×C4—21no—
413C22×C4—22no—
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—21no—
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—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
460C22×C4AGL(1,ℤ/8)12yes—
461C22×C4AGL(1,ℤ/8)12yes—
462C22×C4AGL(1,ℤ/8)12yes—
463C22×C4#13712yes—
464C22×C4C23×C411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467M16—84no—
468C2×C8—22no—
469C2×C8—22no—
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—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—84no—
493C2×C8—84no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—41no—
500C4⋊C4—42no—
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—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4#13112yes—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—41no—
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—42no—
534C22⋊C4—42no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C22⋊C4—81no—
546C22⋊C4—81no—
547C22⋊C4—81no—
548C22⋊C4—81no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—42no—
566C42—42no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23#41312yes—
593C23#41712yes—
594C23#41812yes—
595C23C22×C22⋊C411yescharacteristic
596D4—44no—
597D4—81no—
598D4—81no—
599D4—81no—
600D4—81no—
601D4—84no—
602D4—84no—
603D4—84no—
604D4—162no—
605D4—162no—
606D4—162no—
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—22no—
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—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
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—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—81no—
668C2×C4—81no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4—162no—
674C2×C4—162no—
675C2×C4#41712yes—
676C2×C4#41812yes—
677C2×C4C2×#13111yescharacteristic
678C2×C4#56811yescharacteristic
679C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
680C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
681C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
682C2×C4C2×#13711yescharacteristic
683C8—42no—
684C8—42no—
685C8—81no—
686C8—81no—
687C8—81no—
688C8—81no—
689C8—81no—
690C8—81no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22—81no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22—81no—
718C22#155714yes—
719C22#288011yescharacteristic
720C22#288311yescharacteristic
721C22C2×#41311yescharacteristic
722C22C2×#41811yescharacteristic
723C22C2×#41711yescharacteristic
724C22#290111yescharacteristic
725C22#290111yescharacteristic
726C4—21no—
727C4—21no—
728C4—21no—
729C4—21no—
730C4—22no—
731C4—21no—
732C4—21no—
733C4—22no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—42no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—81no—
744C4—81no—
745C4—81no—
746C4—81no—
747C4—81no—
748C2—24no—
749C2—81no—
750C2—161no—
751C2—161no—
752C2#1742711yescharacteristic
753C2#1751111yescharacteristic
754C2#1751511yescharacteristic
755C2#1762011yescharacteristic
756C2#1762211yescharacteristic
757C2#1769011yescharacteristic
758C2C2×#155711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36263C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22787C2semidirect
3. #22698C2semidirect
4. #7712C2semidirect
5. #7706C2semidirect
6. #7698C2semidirect
7. #7694C2semidirect
8. #7664C2semidirect
9. #7664C2semidirect
10. #7641C2semidirect
11. #7633C2semidirect
12. #7628C2semidirect
13. #7599C2semidirect
14. #7594C2semidirect
15. C2×#1271C2semidirect

Maximal subgroups

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

subgroupindex
#362632
#227872
#226982
#77122
#77062
#76982
#76942
#76642
#76412
#76332
#76282
#75992
#75942
C2×#12712