Finite Group Databasebeta

← #164993

Subgroups — #164993

Order 512.

955 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36276C211yesmaximal, characteristic
2#22785C211yesmaximal, characteristic
3#22698C211yesmaximal, characteristic
4#7727C211yesmaximal, characteristic, complemented
5#7722C211yesmaximal, characteristic, complemented
6#7687C211yesmaximal, characteristic, complemented
7#7682C211yesmaximal, characteristic, complemented
8#7667C211yesmaximal, characteristic, complemented
9#7667C211yesmaximal, characteristic, complemented
10#7643C211yesmaximal, characteristic, complemented
11#7624C211yesmaximal, characteristic, complemented
12#7609C211yesmaximal, characteristic, complemented
13#7590C211yesmaximal, characteristic, complemented
14#7575C211yesmaximal, characteristic, complemented
15C2×#1272C211yesmaximal, characteristic, complemented
16C2×#554C411yescharacteristic
17C2×D4×Q8C411yescharacteristic
18C2×#435—21no—
19#2862C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2567C411yescharacteristic
23#2550—21no—
24#2550—21no—
25#2549—21no—
26#2549—21no—
27#2549—21no—
28#2549—21no—
29#2546C411yescharacteristic
30#2502C411yescharacteristic
31#2502C411yescharacteristic
32#2434C411yescharacteristic
33#2434C411yescharacteristic
34#2349—21no—
35#2349—21no—
36C4×C4⋊Q8C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1940—21no—
41#1923—21no—
42#1916—21no—
43#1869—21no—
44#1851—21no—
45#1850—21no—
46#1839C2211yescharacteristic
47#1831C2211yescharacteristic
48#1817C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1767C2211yescharacteristic
53C8×C22⋊C4—21no—
54C4×C22⋊C8C2211yescharacteristic
55#1744C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1272C2211yescharacteristic
68#1272C2211yescharacteristic
69#1272C2211yescharacteristic
70#1272C2211yescharacteristic
71#1272C2211yescharacteristic
72#1272C2211yescharacteristic
73#1272C2211yescharacteristic
74#1272C2211yescharacteristic
75C23×Q8C2×C411yescharacteristic
76C23×Q8C2×C411yescharacteristic
77C22×M16—21no—
78C23×C8—21no—
79C23×C8—21no—
80#554—44no—
81D4×Q8—44no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×C4⋊Q8C2×C411yescharacteristic
85C2×C4⋊Q8C2×C411yescharacteristic
86C2×C4⋊Q8C2×C411yescharacteristic
87C2×C4⋊Q8C2×C411yescharacteristic
88C2×#125—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8C2×C411yescharacteristic
94C2×C22⋊Q8C2×C411yescharacteristic
95C2×C22⋊Q8C2×C411yescharacteristic
96C2×C22⋊Q8C2×C411yescharacteristic
97C2×C4×Q8C2×C411yescharacteristic
98C2×C4×Q8C2×C411yescharacteristic
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4C2×C411yescharacteristic
102C2×C4×D4D411yescharacteristic
103C2×C4×D4D411yescharacteristic
104C22×C4⋊C4C2×C411yescharacteristic
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#435—44no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×#97—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122#398—21no—
123#398—21no—
124#398—21no—
125#398—21no—
126#398—21no—
127#398—21no—
128#395—21no—
129#395—21no—
130#395—21no—
131#395—21no—
132#395—21no—
133#395—21no—
134#391—21no—
135#391—21no—
136#391—21no—
137#391—21no—
138#391—21no—
139#391—21no—
140#391C2×C411yescharacteristic
141#391C2×C411yescharacteristic
142#391C2×C411yescharacteristic
143#391C2×C411yescharacteristic
144#391C2×C411yescharacteristic
145#391C2×C411yescharacteristic
146#391C2311yescharacteristic
147#391C2311yescharacteristic
148#391C2311yescharacteristic
149#391C2311yescharacteristic
150#389—21no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#385—21no—
161#385—21no—
162#385—21no—
163#385—21no—
164#385—21no—
165#385—21no—
166#385—21no—
167#385—21no—
168#384C2×C411yescharacteristic
169#384C2×C411yescharacteristic
170#384C2×C411yescharacteristic
171#382—21no—
172#382—21no—
173#380—21no—
174#380—21no—
175#380—21no—
176#380—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4C2×C411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C22⋊C4C2×C411yescharacteristic
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
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×Q8—21no—
210C22×Q8—21no—
211C22×Q8—21no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—41no—
215C22×Q8—41no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8—81no—
219C22×Q8—81no—
220C22×Q8C22×C411yescharacteristic
221C22×Q8C22×C411yescharacteristic
222C22×Q8C22×C411yescharacteristic
223C22×Q8C22×C411yescharacteristic
224C22×D4C22⋊C411yescharacteristic
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4C22×C411yescharacteristic
230C23×C4C22×C411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C2×M16—42no—
236C2×M16—42no—
237C2×M16—81no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8—42no—
250C22×C8—81no—
251C22×C8—81no—
252C4⋊Q8—22no—
253C4⋊Q8—41no—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C4⋊Q8—44no—
257C4⋊Q8—81no—
258C4⋊Q8—81no—
259#125—44no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C22⋊Q8—42no—
263C22⋊Q8—42no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C22⋊Q8—42no—
267C22⋊Q8—42no—
268C4×Q8—42no—
269C4×Q8—42no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—24no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4SD1614yes—
276C4×D4Q1614yes—
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—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
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⋊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—81no—
322C2×C42—81no—
323C2×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2411yescharacteristic
336C4⋊C8—42no—
337C4⋊C8—42no—
338C4⋊C8—81no—
339C4⋊C8—81no—
340C4⋊C8—81no—
341C22⋊C8—42no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—81no—
346#97—42no—
347C4×C8—42no—
348C4×C8—81no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
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#95C22⋊C411yescharacteristic
370#95C22⋊C411yescharacteristic
371#95C22⋊C411yescharacteristic
372#95C22⋊C411yescharacteristic
373#95C22⋊C411yescharacteristic
374#95C22⋊C411yescharacteristic
375C24C2×C22⋊C411yescharacteristic
376C24C2×C22⋊C411yescharacteristic
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8—41no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×Q8—84no—
402C2×Q8—84no—
403C2×Q8—84no—
404C2×Q8—84no—
405C2×Q8#13112yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4#10314yes—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
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—41no—
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—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×SD1612yes—
458C22×C4C2×SD1612yes—
459C22×C4C2×Q1612yes—
460C22×C4C2×Q1612yes—
461C22×C4#13712yes—
462C22×C4#13712yes—
463C22×C4C23×C411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466M16—84no—
467C2×C8—22no—
468C2×C8—22no—
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—41no—
490C2×C8—41no—
491C2×C8—84no—
492C2×C8—84no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4#10314yes—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4#10014yes—
531C22⋊C4#10314yes—
532C22⋊C4#10314yes—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—22no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—42no—
548C42—42no—
549C42C2×SD1612yes—
550C42C2×Q1612yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23C2×#10012yes—
562C23C2×#10312yes—
563C23C2×#10312yes—
564C23#41912yes—
565C23C22×C22⋊C411yescharacteristic
566Q8—42no—
567Q8—42no—
568Q8—84no—
569Q8—81no—
570Q8—81no—
571Q8—81no—
572Q8—81no—
573Q8—81no—
574Q8—81no—
575Q8—84no—
576Q8—162no—
577D4—28no—
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—22no—
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—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
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—41no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—162no—
646C2×C4—162no—
647C2×C4C2×#10012yes—
648C2×C4C2×#10012yes—
649C2×C4C2×#10312yes—
650C2×C4C2×#10312yes—
651C2×C4C2×#10312yes—
652C2×C4C2×#10312yes—
653C2×C4#41912yes—
654C2×C4C2×#13111yescharacteristic
655C2×C4#56811yescharacteristic
656C2×C4C22×SD1611yescharacteristic
657C2×C4C22×Q1611yescharacteristic
658C2×C4C2×#13711yescharacteristic
659C2×C4C2×#13711yescharacteristic
660C8—42no—
661C8—42no—
662C8—81no—
663C8—81no—
664C8—81no—
665C8—81no—
666C8—81no—
667C8—81no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22#154814yes—
682C22#288011yescharacteristic
683C22#288311yescharacteristic
684C22C22×#10011yescharacteristic
685C22C22×#10311yescharacteristic
686C22C2×#41911yescharacteristic
687C22#290111yescharacteristic
688C22#290111yescharacteristic
689C4—21no—
690C4—21no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—21no—
696C4—21no—
697C4—22no—
698C4—22no—
699C4—21no—
700C4—21no—
701C4—22no—
702C4—41no—
703C4—42no—
704C4—41no—
705C4—41no—
706C4—81no—
707C4—81no—
708C4—81no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4—81no—
713C2—24no—
714C2—42no—
715C2#1742111yescharacteristic
716C2#1750911yescharacteristic
717C2#1751411yescharacteristic
718C2#1761411yescharacteristic
719C2#1762111yescharacteristic
720C2#1768511yescharacteristic
721C2C2×#154811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
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
4. #7727C2semidirect
5. #7722C2semidirect
6. #7687C2semidirect
7. #7682C2semidirect
8. #7667C2semidirect
9. #7667C2semidirect
10. #7643C2semidirect
11. #7624C2semidirect
12. #7609C2semidirect
13. #7590C2semidirect
14. #7575C2semidirect
15. C2×#1272C2semidirect

Maximal subgroups

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

subgroupindex
#362762
#227852
#226982
#77272
#77222
#76872
#76822
#76672
#76432
#76242
#76092
#75902
#75752
C2×#12722