Finite Group Databasebeta

← #206758

Subgroups — #206758

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34966C211yesmaximal, characteristic
2#20546C211yesmaximal, characteristic
3C4×#424C211yesmaximal, characteristic
4#9869C211yesmaximal, characteristic, complemented
5#9868C211yesmaximal, characteristic, complemented
6#9867C211yesmaximal, characteristic, complemented
7#9867C211yesmaximal, characteristic, complemented
8#9866C211yesmaximal, characteristic, complemented
9#9866C211yesmaximal, characteristic, complemented
10#9865C211yesmaximal, characteristic, complemented
11#9864C211yesmaximal, characteristic, complemented
12#7904C211yesmaximal, characteristic
13#7898C211yesmaximal, characteristic
14#7895C211yesmaximal, characteristic
15C2×#1279C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#424—21no—
18C22×C4⋊C8C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2606—21no—
22#2605—21no—
23#2570C411yescharacteristic
24#2570C411yescharacteristic
25#2550C411yescharacteristic
26#2550C411yescharacteristic
27#2442C411yescharacteristic
28#2442C411yescharacteristic
29#2440C411yescharacteristic
30#2440C411yescharacteristic
31#2438—21no—
32#2438—21no—
33#2402—21no—
34#2399—21no—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×#125C2211yescharacteristic
38C4×#125C2211yescharacteristic
39C4×C22⋊Q8—21no—
40C4×C22⋊Q8—21no—
41#2283C2211yescharacteristic
42C4×#117C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44#1771—21no—
45#1771—21no—
46#1768—21no—
47#1768—21no—
48#1767C2211yescharacteristic
49#1767C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56#1752—21no—
57C8×C22⋊C4—21no—
58#1729—21no—
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C22×M16—21no—
76C23×C8—21no—
77C23×C8—21no—
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×#125C2×C411yescharacteristic
81C2×#125C2×C411yescharacteristic
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×C4×Q8—21no—
85C2×C4×Q8—21no—
86C2×#117D411yescharacteristic
87C2×#117D411yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92#432—22no—
93#432Q814yescomplemented
94#424—28no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8Q814yescomplemented
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C22⋊C8—21no—
107C4×M16—22no—
108C4×M16—22no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#400—21no—
123#400—21no—
124#400—21no—
125#400—21no—
126#398—21no—
127#398—21no—
128#398—21no—
129#398—21no—
130#398—21no—
131#398—21no—
132#398—21no—
133#398—21no—
134#395—21no—
135#395—21no—
136#395—21no—
137#395—21no—
138#391—21no—
139#391—21no—
140#391—21no—
141#391—21no—
142#389—21no—
143#389—21no—
144#389—21no—
145#389—21no—
146#389C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#389C2×C411yescharacteristic
151#389C2×C411yescharacteristic
152#389C2×C411yescharacteristic
153#389C2×C411yescharacteristic
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#387—21no—
161#387—21no—
162#385—21no—
163#385—21no—
164#385—21no—
165#385—21no—
166#384C2×C411yescharacteristic
167#384C2×C411yescharacteristic
168#383—21no—
169#382C2×C411yescharacteristic
170#382C2×C411yescharacteristic
171#382C2×C411yescharacteristic
172#382C2×C411yescharacteristic
173#381—21no—
174#380—21no—
175#380—21no—
176#380C2×C411yescharacteristic
177#380C2×C411yescharacteristic
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2×C411yescharacteristic
187C4×C4⋊C4C2×C411yescharacteristic
188C4×C4⋊C4C2×C411yescharacteristic
189C4×C4⋊C4C2×C411yescharacteristic
190C4×C4⋊C4D411yescharacteristic
191C4×C4⋊C4D411yescharacteristic
192C4×C4⋊C4C2311yescharacteristic
193C4×C4⋊C4C2311yescharacteristic
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2311yescharacteristic
196C4×C22⋊C4—21no—
197C4×C22⋊C4C2×C411yescharacteristic
198C4×C22⋊C4C2×C411yescharacteristic
199C4×C22⋊C4D411yescharacteristic
200C4×C22⋊C4D411yescharacteristic
201#376—21no—
202#376D411yescharacteristic
203#376D411yescharacteristic
204C2×#95—21no—
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#336—21no—
208#336—21no—
209#336—21no—
210#336—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214#336—21no—
215C22×Q8—21no—
216C22×Q8—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C22×C411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C2×M16—24no—
223C2×M16—42no—
224C2×M16—42no—
225C22×C8—24no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—24no—
235C22×C8—42no—
236C22×C8—42no—
237C22×C8—81no—
238C22×C8—81no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—41no—
242C4⋊Q8—41no—
243#125—22no—
244#125—22no—
245#125—41no—
246#125—41no—
247C22⋊Q8—44no—
248C22⋊Q8—44no—
249C4×Q8—81no—
250C4×Q8—81no—
251C4×Q8—81no—
252C4×Q8—81no—
253#117—24no—
254#117—42no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
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—41no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4C22⋊C411yescharacteristic
280C2×C4⋊C4C22⋊C411yescharacteristic
281C2×C4⋊C4C22⋊C411yescharacteristic
282C2×C4⋊C4C22⋊C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4C22⋊C411yescharacteristic
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—81no—
304C2×C42—81no—
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×Q812yes—
321C2×C42C2×Q812yes—
322C2×C42#4112yes—
323C2×C42#4112yes—
324C2×C42C2411yescharacteristic
325C4⋊C8—24no—
326C4⋊C8—24no—
327C4⋊C8—24no—
328C4⋊C8—24no—
329C4⋊C8—24no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8C2×Q812yes—
335C4⋊C8C2×Q812yes—
336C4⋊C8C2×Q812yes—
337C4⋊C8C2×Q812yes—
338C22⋊C8—24no—
339#97—24no—
340#97—42no—
341#97—81no—
342#97—81no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—42no—
352C4×C8—81no—
353C4×C8—81no—
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#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#95C22⋊C411yescharacteristic
379#95C22⋊C411yescharacteristic
380#95C22⋊C411yescharacteristic
381#95C22⋊C411yescharacteristic
382#95C22⋊C411yescharacteristic
383#95C22⋊C411yescharacteristic
384#95C22⋊C411yescharacteristic
385#95C22⋊C411yescharacteristic
386C24C2×C22⋊C411yescharacteristic
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—81no—
392C2×Q8—81no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C4×Q812yes—
429C22×C4C4×Q812yes—
430C22×C4C22⋊Q812yes—
431C22×C4C22⋊Q812yes—
432C22×C4#12412yes—
433C22×C4C4⋊Q812yes—
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437M16—48no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—48no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—84no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4#13112yes—
503C4⋊C4#13112yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4#10414yes—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—22no—
527C42—22no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—81no—
539C42—81no—
540C42C4×Q812yes—
541C42C4×Q812yes—
542C42C22⋊Q812yes—
543C42C22⋊Q812yes—
544C42#12412yes—
545C42C4⋊Q812yes—
546C42C22×Q811yescharacteristic
547C42C22×Q811yescharacteristic
548C42C2×#4111yescharacteristic
549C42C2×#4111yescharacteristic
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23#39112yes—
555C23C2×#10412yes—
556C23#42112yes—
557C23C22×C22⋊C411yescharacteristic
558Q8—84no—
559Q8—84no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
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—21no—
601C2×C4—21no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4#39112yes—
627C2×C4#39112yes—
628C2×C4#39112yes—
629C2×C4C2×#10412yes—
630C2×C4C2×#10412yes—
631C2×C4C2×C4×Q811yescharacteristic
632C2×C4C2×C4×Q811yescharacteristic
633C2×C4C2×C22⋊Q811yescharacteristic
634C2×C4C2×C22⋊Q811yescharacteristic
635C2×C4C2×#12411yescharacteristic
636C2×C4C2×C4⋊Q811yescharacteristic
637C2×C4C2×#13111yescharacteristic
638C2×C4C2×#13111yescharacteristic
639C2×C4#56811yescharacteristic
640C2×C4#56811yescharacteristic
641C2×C4#57611yescharacteristic
642C2×C4#57711yescharacteristic
643C2×C4#57711yescharacteristic
644C2×C4#57811yescharacteristic
645C8—42no—
646C8—42no—
647C8—42no—
648C8—42no—
649C8—42no—
650C8—42no—
651C8—42no—
652C8—42no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22#198614yes—
660C22C2×#39111yescharacteristic
661C22#288011yescharacteristic
662C22#288011yescharacteristic
663C22#289711yescharacteristic
664C22#289811yescharacteristic
665C22C22×#10411yescharacteristic
666C22C2×#42111yescharacteristic
667C4—22no—
668C4—21no—
669C4—21no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—41no—
675C4—42no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—42no—
680C4—41no—
681C4—41no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4#296011yescharacteristic
689C4#296111yescharacteristic
690C4#296211yescharacteristic
691C4#296211yescharacteristic
692C4#308111yescharacteristic
693C4#308211yescharacteristic
694C4#308311yescharacteristic
695C4#308411yescharacteristic
696C2—24no—
697C2#1751711yescharacteristic
698C2#1751911yescharacteristic
699C2#1752111yescharacteristic
700C2#1752411yescharacteristic
701C2#2130411yescharacteristic
702C2C2×#198611yescharacteristic
703C2#2131511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9869C2semidirect
5. #9868C2semidirect
6. #9867C2semidirect
7. #9867C2semidirect
8. #9866C2semidirect
9. #9866C2semidirect
10. #9865C2semidirect
11. #9864C2semidirect
93. #432Q8semidirect
101. C2×C4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#349662
#205462
C4×#4242
#98692
#98682
#98672
#98662
#98652
#98642
#79042
#78982
#78952
C2×#12792