Finite Group Databasebeta

← #189907

Subgroups — #189907

Order 512.

1025 conjugacy classes of subgroups.

The subgroup lattice has 769 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34191C211yesmaximal, characteristic
2#21379C211yesmaximal, characteristic
3#20375C211yesmaximal, characteristic
4#8722C211yesmaximal, characteristic, complemented
5#8721C211yesmaximal, characteristic, complemented
6#8720C211yesmaximal, characteristic, complemented
7#8720C211yesmaximal, characteristic, complemented
8#8720C211yesmaximal, characteristic, complemented
9#8720C211yesmaximal, characteristic, complemented
10#8719C211yesmaximal, characteristic, complemented
11#8718C211yesmaximal, characteristic, complemented
12#8036C211yesmaximal, characteristic, complemented
13#8033C211yesmaximal, characteristic, complemented
14#8029C211yesmaximal, characteristic, complemented
15C2×#1295C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#408—21no—
19C2×C4×M16C2211yescharacteristic
20C22×#97C2211yescharacteristic
21C22×#97C2211yescharacteristic
22#2355C411yescharacteristic
23#2355C411yescharacteristic
24#2352C411yescharacteristic
25#2352C411yescharacteristic
26#2352C411yescharacteristic
27#2352C411yescharacteristic
28D4×C4⋊C4C411yescharacteristic
29D4×C4⋊C4C411yescharacteristic
30#2346—21no—
31#2343—21no—
32#2343—21no—
33#2339—21no—
34#2320—21no—
35#2318—21no—
36#2318—21no—
37#2285C2211yescharacteristic
38#2283C2211yescharacteristic
39#2283C2211yescharacteristic
40#2283C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43#2275—21no—
44#1985—21no—
45#1955—21no—
46#1869—21no—
47#1869—21no—
48#1750—21no—
49#1750—21no—
50#1749C2211yescharacteristic
51#1749C2211yescharacteristic
52#1749C2211yescharacteristic
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56#1749C2211yescharacteristic
57#1749C2211yescharacteristic
58#1729—21no—
59#1295C2211yescharacteristic
60#1295C2211yescharacteristic
61#1295C2211yescharacteristic
62#1295C2211yescharacteristic
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×#41D411yescharacteristic
76C22×M16—21no—
77C23×C8—21no—
78C23×C8—21no—
79C4×#41—22no—
80C4×#41—22no—
81C4×#41—22no—
82C4×#41—22no—
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×Q8D411yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4C2×C411yescharacteristic
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4D411yescharacteristic
91C2×C4×D4D411yescharacteristic
92C2×C4×D4D411yescharacteristic
93C2×#117C2×C411yescharacteristic
94C2×#117C2×C411yescharacteristic
95C2×#117D411yescharacteristic
96C2×#117D411yescharacteristic
97C2×#117D411yescharacteristic
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C4⋊C4C2×C411yescharacteristic
100C22×C4⋊C4C2×C411yescharacteristic
101C22×C4⋊C4C2×C411yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105#432—22no—
106#432—22no—
107C2×C4⋊C8—21no—
108#408—28no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C4×M16—22no—
119C4×M16—22no—
120C2×#97—22no—
121C2×#97—22no—
122C2×#97—22no—
123C2×#97—22no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97C2311yescharacteristic
127C2×#97C2311yescharacteristic
128C2×#97C2311yescharacteristic
129C2×#97C2311yescharacteristic
130C2×#97C2311yescharacteristic
131C2×#97C2311yescharacteristic
132C2×#97C2311yescharacteristic
133C2×#97C2311yescharacteristic
134C2×C4×C8—21no—
135#391—21no—
136#391—21no—
137#389—21no—
138#389—21no—
139#388—21no—
140#388—21no—
141#388—21no—
142#388—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#386—21no—
152#386—21no—
153#386—21no—
154#386—21no—
155#385—21no—
156#385—21no—
157#385—21no—
158#385—21no—
159#384C2×C411yescharacteristic
160#384C2×C411yescharacteristic
161#383—21no—
162#383—21no—
163#383—21no—
164#382—21no—
165#382—21no—
166#382—21no—
167#382—21no—
168#382C2×C411yescharacteristic
169#382C2×C411yescharacteristic
170#382C2×C411yescharacteristic
171#382C2×C411yescharacteristic
172#382C2311yescharacteristic
173#382C2311yescharacteristic
174#382C2311yescharacteristic
175#382C2311yescharacteristic
176#381—21no—
177#381—21no—
178#381—21no—
179#380—21no—
180#380—21no—
181#380C2×C411yescharacteristic
182#380C2×C411yescharacteristic
183#380C2×C411yescharacteristic
184#380C2×C411yescharacteristic
185#380C2×C411yescharacteristic
186#380C2×C411yescharacteristic
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4C2×C411yescharacteristic
192C4×C4⋊C4C2×C411yescharacteristic
193C4×C4⋊C4C2×C411yescharacteristic
194C4×C4⋊C4C2×C411yescharacteristic
195C4×C4⋊C4C2×C411yescharacteristic
196C4×C4⋊C4C2×C411yescharacteristic
197C4×C22⋊C4—21no—
198C4×C22⋊C4—21no—
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—21no—
202C4×C22⋊C4—21no—
203C4×C22⋊C4—21no—
204#376—21no—
205#376—21no—
206#376—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214#336—21no—
215#336—21no—
216#336—21no—
217#336—21no—
218#336—21no—
219C2×#41—24no—
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—42no—
223C22×Q8C22⋊C411yescharacteristic
224C22×D4C22⋊C411yescharacteristic
225C22×D4C22⋊C411yescharacteristic
226C22×D4C22⋊C411yescharacteristic
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4C22×C411yescharacteristic
232C23×C4C22×C411yescharacteristic
233C23×C4C22×C411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C2×M16—24no—
241C2×M16—42no—
242C2×M16—42no—
243C22×C8—24no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—42no—
253C22×C8—42no—
254C22×C8—42no—
255C22×C8—42no—
256C22×C8—42no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—44no—
274C4×D4—44no—
275#117—22no—
276#117—22no—
277#117—22no—
278#117—22no—
279#117—22no—
280#117—22no—
281#117—81no—
282#117—81no—
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—81no—
292C2×C4⋊C4—81no—
293C2×C4⋊C4—81no—
294C2×C4⋊C4—81no—
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×C4⋊C4C22×C411yescharacteristic
306C2×C4⋊C4C22×C411yescharacteristic
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C4⋊C4C22×C411yescharacteristic
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⋊C4C22⋊C411yescharacteristic
318C2×C22⋊C4C22⋊C411yescharacteristic
319C2×C22⋊C4C22⋊C411yescharacteristic
320C2×C22⋊C4C22⋊C411yescharacteristic
321C2×C22⋊C4C22⋊C411yescharacteristic
322C2×C22⋊C4C22⋊C411yescharacteristic
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×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42C22×C411yescharacteristic
340C2×C42C22×C411yescharacteristic
341C2×C42C22×C411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42#4112yes—
349C2×C42#4112yes—
350C2×C42#4112yes—
351C2×C42#4112yes—
352C2×C42C2411yescharacteristic
353C4⋊C8—42no—
354C22⋊C8—24no—
355C22⋊C8—24no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358C22⋊C8—42no—
359C22⋊C8—42no—
360C22⋊C8—42no—
361C22⋊C8—42no—
362C22⋊C8—42no—
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—21no—
372#97—21no—
373#97—21no—
374#97—21no—
375#97—21no—
376#97—21no—
377#97—21no—
378#97—21no—
379#97—42no—
380#97—42no—
381C4×C8—42no—
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#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410C24C2×C22⋊C411yescharacteristic
411C24C2×C22⋊C411yescharacteristic
412C24C2×C22⋊C411yescharacteristic
413#41—48no—
414#41—48no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—41no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—41no—
426C2×D4#10414yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4C2×C22⋊C411yescharacteristic
478C22×C4C2×C22⋊C411yescharacteristic
479C22×C4C2×C22⋊C411yescharacteristic
480C22×C4C2×C22⋊C411yescharacteristic
481C22×C4C2×C22⋊C411yescharacteristic
482C22×C4C2×C22⋊C411yescharacteristic
483C22×C4C2×C22⋊C411yescharacteristic
484C22×C4C2×C22⋊C411yescharacteristic
485C22×C4C2×C22⋊C411yescharacteristic
486C22×C4#11712yes—
487C22×C4#11712yes—
488C22×C4#12312yes—
489C22×C4#12312yes—
490C22×C4#12312yes—
491C22×C4#12312yes—
492C22×C4C23×C411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495M16—48no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—48no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—48no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—41no—
516C2×C8—41no—
517C2×C8—41no—
518C2×C8—41no—
519C2×C8—41no—
520C2×C8—41no—
521C2×C8—41no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—81no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C4⋊C4—81no—
542C4⋊C4—81no—
543C4⋊C4—81no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—81no—
555C22⋊C4—81no—
556C22⋊C4—81no—
557C22⋊C4—81no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—42no—
571C42—42no—
572C42—81no—
573C42—81no—
574C42—81no—
575C42—81no—
576C42—81no—
577C42—81no—
578C42#11712yes—
579C42#11712yes—
580C42#12312yes—
581C42#12312yes—
582C42#12312yes—
583C42#12312yes—
584C42C2×#4111yescharacteristic
585C42C2×#4111yescharacteristic
586C42C2×#4111yescharacteristic
587C42C2×#4111yescharacteristic
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23#38112yes—
606C23C2×#10412yes—
607C23C2×#10412yes—
608C23#42112yes—
609C23C22×C22⋊C411yescharacteristic
610Q8—44no—
611D4—28no—
612D4—44no—
613D4—44no—
614C2×C4—24no—
615C2×C4—24no—
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—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—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
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—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—81no—
682C2×C4—81no—
683C2×C4—81no—
684C2×C4—81no—
685C2×C4—162no—
686C2×C4—162no—
687C2×C4—162no—
688C2×C4—162no—
689C2×C4#38112yes—
690C2×C4#38112yes—
691C2×C4#38112yes—
692C2×C4C2×#10412yes—
693C2×C4C2×#11711yescharacteristic
694C2×C4C2×#11711yescharacteristic
695C2×C4C2×#12311yescharacteristic
696C2×C4C2×#12311yescharacteristic
697C2×C4C2×#12311yescharacteristic
698C2×C4C2×#12311yescharacteristic
699C8—42no—
700C8—42no—
701C8—42no—
702C8—42no—
703C8—42no—
704C8—42no—
705C8—81no—
706C8—81no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—22no—
716C22—41no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—41no—
724C22—41no—
725C22—41no—
726C22—41no—
727C22#182614yes—
728C22C2×#38111yescharacteristic
729C22#288811yescharacteristic
730C22#288911yescharacteristic
731C22C22×#10411yescharacteristic
732C22C2×#42111yescharacteristic
733C22#290111yescharacteristic
734C22#290111yescharacteristic
735C4—21no—
736C4—21no—
737C4—22no—
738C4—21no—
739C4—21no—
740C4—22no—
741C4—22no—
742C4—21no—
743C4—21no—
744C4—22no—
745C4—41no—
746C4—41no—
747C4—41no—
748C4—41no—
749C4—41no—
750C4—81no—
751C4—81no—
752C4—81no—
753C4—81no—
754C4—81no—
755C4—81no—
756C4—81no—
757C4—81no—
758C2—24no—
759C2—42no—
760C2—81no—
761C2#1765211yescharacteristic
762C2#1765311yescharacteristic
763C2#1767911yescharacteristic
764C2#1768011yescharacteristic
765C2#2069511yescharacteristic
766C2C2×#182611yescharacteristic
767C2#2070611yescharacteristic
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 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8722C2semidirect
5. #8721C2semidirect
6. #8720C2semidirect
7. #8720C2semidirect
8. #8720C2semidirect
9. #8720C2semidirect
10. #8719C2semidirect
11. #8718C2semidirect
12. #8036C2semidirect
13. #8033C2semidirect
14. #8029C2semidirect
15. C2×#1295C2semidirect

Maximal subgroups

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

subgroupindex
#341912
#213792
#203752
#87222
#87212
#87202
#87192
#87182
#80362
#80332
#80292
C2×#12952