Finite Group Databasebeta

← #190824

Subgroups — #190824

Order 512.

1364 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C42×#41C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic, complemented
3#20777C211yesmaximal, characteristic, complemented
4#8708C212yesmaximal, complemented
5#8566C212yesmaximal, complemented
6#8565C212yesmaximal, complemented
7#8312C212yesmaximal
8#8171C212yesmaximal, complemented
9#8170C212yesmaximal, complemented
10C2×C4×#41C411yescharacteristic
11C2×C4×#41C411yescharacteristic
12C2×C4×#41C2211yescharacteristic
13C42×Q8C411yescharacteristic
14C42×Q8C411yescharacteristic
15C42×Q8C2211yescharacteristic
16C42×Q8C2211yescharacteristic
17C42×D4—21no—
18C42×D4—21no—
19C42×D4—21no—
20C42×D4—21no—
21C42×D4C411yescharacteristic
22C42×D4C411yescharacteristic
23C42×D4C2211yescharacteristic
24C42×D4C2211yescharacteristic
25C4×#117—21no—
26C4×#117—21no—
27C4×#117—21no—
28C4×#117C411yescharacteristic
29C4×#117C411yescharacteristic
30C4×#117C2211yescharacteristic
31C2×C43—21no—
32C2×C43C2211yescharacteristic
33#1839—21no—
34#1839—21no—
35#1831—21no—
36#1831C2212yes—
37#1817—21no—
38#1817—21no—
39#1797—21no—
40#1797C2212yes—
41C4×C4⋊C8C2212yes—
42C4×C4⋊C8C2212yes—
43C4×C22⋊C8—21no—
44C4×C22⋊C8C2212yes—
45#1744—21no—
46#1744C2212yes—
47C2×Q8⋊C8C2214yes—
48C2×D4⋊C8C2214yes—
49#1279C2214yes—
50#1279C2214yes—
51C22×#41C2×C411yescharacteristic
52C4×#41—24no—
53C4×#41—24no—
54C4×#41—24no—
55C4×#41—24no—
56C4×#41—24no—
57C4×#41C818yescomplemented
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117—21no—
77C2×#117—21no—
78C2×#117—21no—
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42—21no—
89C22×C42C2×C411yescharacteristic
90C22×C42C2×C411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42C2311yescharacteristic
96C2×C4⋊C8—22no—
97C2×C4⋊C8C2314yes—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8C2314yes—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144#376—21no—
145#376—21no—
146#376—21no—
147C43—21no—
148C43—21no—
149C43—21no—
150C43—21no—
151C43—21no—
152C43—21no—
153C43—21no—
154C43C2×C411yescharacteristic
155C43C2×C411yescharacteristic
156C43D411yescharacteristic
157C43D411yescharacteristic
158C43D411yescharacteristic
159C43D411yescharacteristic
160C43C2311yescharacteristic
161C43C2311yescharacteristic
162#336—22no—
163#336—22no—
164#336—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168#336—22no—
169#336—22no—
170Q8⋊C8—28no—
171D4⋊C8—28no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—22no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41C2×C814yes—
178C22×Q8C22×C411yescharacteristic
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C22×C8—24no—
196C22×C8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8C2×C814yes—
206C4×Q8C2×C814yes—
207C4×Q8M1614yes—
208C4×Q8#4114yes—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—28no—
212C4×D4—28no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—28no—
216C4×D4—28no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4C2×C814yes—
224C4×D4C2×C814yes—
225C4×D4M1614yes—
226C4×D4#4114yes—
227#117—28no—
228#117—28no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—22no—
234#117—28no—
235#117C2×C814yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—28no—
258C2×C42—28no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—28no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22⋊C411yescharacteristic
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22⋊C411yescharacteristic
299C2×C42C22⋊C411yescharacteristic
300C2×C42C22⋊C411yescharacteristic
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22⋊C411yescharacteristic
303C2×C42C22⋊C411yescharacteristic
304C2×C42C22⋊C411yescharacteristic
305C2×C42C2×C814yes—
306C2×C42M1614yes—
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
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×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2411yescharacteristic
323C4⋊C8—24no—
324C4⋊C8—24no—
325C4⋊C8—44no—
326C22⋊C8—44no—
327C22⋊C8—44no—
328C4×C8—24no—
329C4×C8—24no—
330C4×C8—44no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343C24—21no—
344C24C2×C22⋊C411yescharacteristic
345#41—28no—
346#41—28no—
347#41—28no—
348#41—28no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8#11712yes—
353C2×Q8C22×C812yes—
354C2×Q8C2×M1612yes—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4#11712yes—
362C2×D4C22×C812yes—
363C2×D4C2×M1612yes—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4C22⋊C814yes—
413C22×C4C22⋊C814yes—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
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×C4C22×C812yes—
426C22×C4C2×M1612yes—
427C22×C4C23×C411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C2×C8—44no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8—44no—
434C2×C8—44no—
435C2×C8—44no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—24no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4#11712yes—
449C4⋊C4#11712yes—
450C4⋊C4C22×C812yes—
451C4⋊C4C22×C812yes—
452C4⋊C4C2×M1612yes—
453C4⋊C4C2×M1612yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C42—24no—
464C42—24no—
465C42—24no—
466C42—24no—
467C42—21no—
468C42—21no—
469C42—22no—
470C42—24no—
471C42—22no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—24no—
479C42—24no—
480C42—24no—
481C42—22no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42C22⋊C814yes—
509C42C22⋊C814yes—
510C42C22×C812yes—
511C42C22×C812yes—
512C42C2×M1612yes—
513C42C2×M1612yes—
514C42C2×M1612yes—
515C42C2×M1612yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C23—22no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×C22⋊C812yes—
532C23C22×C22⋊C411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8#43114yes—
537D4—216no—
538D4—22no—
539D4—22no—
540D4—22no—
541D4#43114yes—
542C2×C4—216no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4C2×C22⋊C812yes—
607C2×C4C2×C22⋊C812yes—
608C2×C4C2×C22⋊C812yes—
609C2×C4C2×C22⋊C812yes—
610C2×C4C2×C22⋊C812yes—
611C2×C4#40712yes—
612C2×C4#40712yes—
613C2×C4#42112yes—
614C2×C4#42112yes—
615C2×C4C2×#11711yescharacteristic
616C2×C4C2×#11711yescharacteristic
617C2×C4#52512yes—
618C2×C4#52512yes—
619C2×C4C23×C811yescharacteristic
620C2×C4C22×M1611yescharacteristic
621C2×C4C22×M1611yescharacteristic
622C2×C4C22×M1611yescharacteristic
623C2×C4#57512yes—
624C2×C4#57512yes—
625C8—84no—
626C8—84no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—41no—
640C22—42no—
641C22#228011yescharacteristic
642C22C22×C22⋊C811yescharacteristic
643C22C2×#40711yescharacteristic
644C22#289611yescharacteristic
645C22#289611yescharacteristic
646C22C2×#42111yescharacteristic
647C22C2×#42111yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—24no—
659C4—41no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—41no—
665C4—42no—
666C4—42no—
667C4C2×#43111yescharacteristic
668C4C2×#43111yescharacteristic
669C4#292011yescharacteristic
670C4#292011yescharacteristic
671C4#311114yes—
672C2—42no—
673C2—44no—
674C2#1743711yescharacteristic
675C2#1743711yescharacteristic
676C2#1745011yescharacteristic
677C2#1745211yescharacteristic
678C2#2071411yescharacteristic
679C2#2074211yescharacteristic
680C2#2074811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20777C2semidirect
3. #20777C2semidirect
4. #8708C2semidirect
5. #8566C2semidirect
6. #8565C2semidirect
8. #8171C2semidirect
9. #8170C2semidirect
57. C4×#41C8semidirect

Maximal subgroups

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

subgroupindex
C42×#412
#207772
#87082
#85662
#85652
#83122
#81712
#81702