Finite Group Databasebeta

← #206757

Subgroups — #206757

Order 512.

938 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#544C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic
3#20549C211yesmaximal, 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#7903C211yesmaximal, characteristic
14#7902C211yesmaximal, characteristic
15#7901C211yesmaximal, characteristic
16C2×#544C411yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2331—21no—
22#2331C411yescharacteristic
23#2331C411yescharacteristic
24#2330—21no—
25#2330C411yescharacteristic
26#2330C411yescharacteristic
27#2329—21no—
28#2329—21no—
29#2317C411yescharacteristic
30#2317C411yescharacteristic
31#2316C411yescharacteristic
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×#125—21no—
36C4×#125C2211yescharacteristic
37C4×#125C2211yescharacteristic
38C4×C22⋊Q8—21no—
39C4×C22⋊Q8—21no—
40C4×#117C2211yescharacteristic
41C4×#117C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43#1940—21no—
44#1940—21no—
45#1851—21no—
46#1851—21no—
47#1771—21no—
48#1771—21no—
49#1767C2211yescharacteristic
50#1767C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56C4×C4⋊C8C2211yescharacteristic
57#1753—21no—
58#1753—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—
76C22×M16—21no—
77#544—24no—
78#544—24no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×C4⋊Q8C2×C411yescharacteristic
82C2×#125—21no—
83C2×#125C2×C411yescharacteristic
84C2×#125C2×C411yescharacteristic
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×#117C2×C411yescharacteristic
90C2×#117C2×C411yescharacteristic
91C2×#117C2×C411yescharacteristic
92C2×#117C2×C411yescharacteristic
93C22×C4⋊C4C2×C411yescharacteristic
94C22×C4⋊C4C2×C411yescharacteristic
95C22×C42—21no—
96C22×C42C2311yescharacteristic
97#432—22no—
98#432Q814yescomplemented
99#431—22no—
100#431—22no—
101#423—22no—
102#423Q814yescomplemented
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C4×M16—22no—
116C4×M16—22no—
117C2×#97—21no—
118C2×#97—21no—
119C2×#97—21no—
120C2×#97—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#391—21no—
128#391—21no—
129#391—21no—
130#391—21no—
131#391—21no—
132#391—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—21no—
143#389C2×C411yescharacteristic
144#389C2×C411yescharacteristic
145#389C2×C411yescharacteristic
146#389C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#389C2×C411yescharacteristic
151#387—21no—
152#387—21no—
153#387—21no—
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#387—21no—
163#385—21no—
164#385—21no—
165#385—21no—
166#385—21no—
167#384—21no—
168#384C2×C411yescharacteristic
169#384C2×C411yescharacteristic
170#382—21no—
171#382C2×C411yescharacteristic
172#382C2×C411yescharacteristic
173#380—21no—
174#380—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4C2×C411yescharacteristic
184C4×C4⋊C4C2×C411yescharacteristic
185C4×C4⋊C4C2×C411yescharacteristic
186C4×C4⋊C4C2×C411yescharacteristic
187C4×C4⋊C4D411yescharacteristic
188C4×C4⋊C4D411yescharacteristic
189C4×C4⋊C4D411yescharacteristic
190C4×C4⋊C4D411yescharacteristic
191C4×C4⋊C4C2311yescharacteristic
192C4×C4⋊C4C2311yescharacteristic
193C4×C4⋊C4C2311yescharacteristic
194C4×C4⋊C4C2311yescharacteristic
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4D411yescharacteristic
198C4×C22⋊C4D411yescharacteristic
199C4×C22⋊C4D411yescharacteristic
200C4×C22⋊C4D411yescharacteristic
201#376—21no—
202#376C2×C411yescharacteristic
203#376C2×C411yescharacteristic
204C2×#95—21no—
205C43—21no—
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208#336—21no—
209#336—21no—
210#336—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214#336—21no—
215#336—21no—
216C22×Q8—21no—
217C22×Q8—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4C22×C411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C2×M16—24no—
224C2×M16—42no—
225C2×M16—42no—
226C2×M16—42no—
227C2×M16—81no—
228C2×M16—81no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C4⋊Q8—28no—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8M1614yescomplemented
242#125—28no—
243#125—22no—
244#125—22no—
245#125—22no—
246#125M1614yescomplemented
247C22⋊Q8—28no—
248C22⋊Q8—28no—
249C4×Q8—81no—
250C4×Q8—81no—
251C4×Q8—81no—
252C4×Q8—81no—
253#117—22no—
254#117—22no—
255#117—22no—
256#117—22no—
257#117—41no—
258#117—41no—
259#117—41no—
260#117—41no—
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—22no—
269C2×C4⋊C4—22no—
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—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4—41no—
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×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
301C2×C22⋊C4C22⋊C411yescharacteristic
302C2×C22⋊C4C22⋊C411yescharacteristic
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—81no—
314C2×C42—81no—
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×Q812yes—
331C2×C42C2×Q812yes—
332C2×C42#4112yes—
333C2×C42#4112yes—
334C2×C42C2411yescharacteristic
335C4⋊C8—24no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—42no—
341C4⋊C8—42no—
342C4⋊C8—42no—
343C4⋊C8—42no—
344C4⋊C8—42no—
345C4⋊C8C2×Q812yes—
346C4⋊C8C2×Q812yes—
347C4⋊C8C2×Q812yes—
348C4⋊C8C2×Q812yes—
349C22⋊C8—24no—
350C22⋊C8—42no—
351#97—24no—
352#97—42no—
353#97—81no—
354#97—81no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—81no—
364C4×C8—81no—
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—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95C22⋊C411yescharacteristic
386#95C22⋊C411yescharacteristic
387#95C22⋊C411yescharacteristic
388#95C22⋊C411yescharacteristic
389#95C22⋊C411yescharacteristic
390#95C22⋊C411yescharacteristic
391#95C22⋊C411yescharacteristic
392#95C22⋊C411yescharacteristic
393C24C2×C22⋊C411yescharacteristic
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—81no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×Q8—81no—
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—21no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C4×Q812yes—
436C22×C4C4×Q812yes—
437C22×C4C22⋊Q812yes—
438C22×C4C22⋊Q812yes—
439C22×C4#12412yes—
440C22×C4C4⋊Q812yes—
441C22×C4C23×C411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444M16—48no—
445M16—84no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
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—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4C2×M1612yes—
507C4⋊C4C2×M1612yes—
508C4⋊C4C2×M1612yes—
509C4⋊C4C2×M1612yes—
510C22⋊C4—24no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—42no—
516C42—24no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—42no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—81no—
545C42—81no—
546C42—81no—
547C42—81no—
548C42C4×Q812yes—
549C42C4×Q812yes—
550C42C22⋊Q812yes—
551C42C22⋊Q812yes—
552C42#12412yes—
553C42C4⋊Q812yes—
554C42C2×M1612yes—
555C42C2×M1612yes—
556C42C22×Q811yescharacteristic
557C42C22×Q811yescharacteristic
558C42C2×#4111yescharacteristic
559C42C2×#4111yescharacteristic
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#39112yes—
567C23C22×C22⋊C411yescharacteristic
568Q8—84no—
569Q8—84no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
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—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—42no—
613C2×C4—42no—
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—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—84no—
635C2×C4—84no—
636C2×C4#39112yes—
637C2×C4#39112yes—
638C2×C4#39112yes—
639C2×C4#40712yes—
640C2×C4#40712yes—
641C2×C4C2×C4×Q811yescharacteristic
642C2×C4C2×C4×Q811yescharacteristic
643C2×C4C2×C22⋊Q811yescharacteristic
644C2×C4C2×C22⋊Q811yescharacteristic
645C2×C4C2×#12411yescharacteristic
646C2×C4C2×C4⋊Q811yescharacteristic
647C2×C4C22×M1611yescharacteristic
648C2×C4C22×M1611yescharacteristic
649C2×C4#56811yescharacteristic
650C2×C4#56811yescharacteristic
651C2×C4#57611yescharacteristic
652C2×C4#57711yescharacteristic
653C2×C4#57711yescharacteristic
654C2×C4#57811yescharacteristic
655C8—42no—
656C8—42no—
657C8—42no—
658C8—42no—
659C8—42no—
660C8—42no—
661C8—81no—
662C8—81no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22C2×#39111yescharacteristic
671C22C2×#40711yescharacteristic
672C22#287911yescharacteristic
673C22#289711yescharacteristic
674C22#289811yescharacteristic
675C22#290111yescharacteristic
676C22#290111yescharacteristic
677C4—22no—
678C4—22no—
679C4—22no—
680C4—21no—
681C4—21no—
682C4—22no—
683C4—22no—
684C4—41no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—41no—
689C4—42no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4Q8×M1611yescharacteristic
699C4Q8×M1611yescharacteristic
700C4Q8×M1611yescharacteristic
701C4Q8×M1611yescharacteristic
702C4#308111yescharacteristic
703C4#308211yescharacteristic
704C4#308311yescharacteristic
705C4#308411yescharacteristic
706C2—42no—
707C2#1745311yescharacteristic
708C2#1745411yescharacteristic
709C2#1750511yescharacteristic
710C2#1750611yescharacteristic
711C2#2130311yescharacteristic
712C2#2130711yescharacteristic
713C2#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
98. #432Q8semidirect
102. #423Q8semidirect
241. C4⋊Q8M16semidirect
246. #125M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5442
#213502
#205492
#98692
#98682
#98672
#98662
#98652
#98642
#79042
#79032
#79022
#79012