Finite Group Databasebeta

← #190995

Subgroups — #190995

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3#13756C211yesmaximal, characteristic, complemented
4#8715C212yesmaximal, complemented
5#8578C212yesmaximal, complemented
6#8577C212yesmaximal, complemented
7#8331C212yesmaximal
8C4×#328C212yesmaximal, complemented
9C4×#327C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C2×C4×#41C411yescharacteristic, complemented
12C22×#117—21no—
13C23×C42C2211yescharacteristic
14C2×#431—21no—
15C4×C22⋊Q8C411yescharacteristic, complemented
16C4×C22⋊Q8C411yescharacteristic, complemented
17C4×C22⋊Q8C2211yescharacteristic
18C4×C22⋊Q8C2211yescharacteristic
19C4×#123—21no—
20C4×#123—21no—
21C4×#121C411yescharacteristic, complemented
22C4×#121C411yescharacteristic, complemented
23C4×#121C2211yescharacteristic
24C4×#121C2211yescharacteristic
25C4×#120—21no—
26C4×#120—21no—
27#2297C411yescharacteristic, complemented
28#2296C411yescharacteristic, complemented
29#2290—21no—
30#2290—21no—
31#2280C2211yescharacteristic
32C42×D4—21no—
33C42×D4—21no—
34C4×#117C411yescharacteristic, complemented
35#1817—21no—
36#1797—21no—
37#1797C2212yes—
38#1779C2212yes—
39C4×C22⋊C8—21no—
40C4×C22⋊C8C2212yes—
41C2×#328C2214yes—
42C2×#327C2214yes—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C22⋊#95—21no—
46C22⋊#95C2212yes—
47C4×#95—21no—
48C4×#95C2212yes—
49#1294C2214yes—
50#1294C2214yes—
51C22×#41C2×C411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4C2311yescharacteristic
55#525—24no—
56#525—24no—
57C2×#123—21no—
58C2×C22⋊Q8C2×C411yescharacteristic
59C2×C22⋊Q8C2311yescharacteristic
60C2×#121C2×C411yescharacteristic
61C2×#121C2311yescharacteristic
62C2×#120—21no—
63C4×#41—44no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—28no—
77C2×#117—21no—
78C2×#117—44no—
79C2×#117—44no—
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C22×C4⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C42—22no—
86C22×C42—21no—
87C22×C42—44no—
88C22×C42C2×C411yescharacteristic
89C22×C42C2×C411yescharacteristic
90C22×C42C2×C411yescharacteristic
91C22×C42C2×C411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42D411yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#431—44no—
101C2×C4⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8C2314yes—
105C2×C4×C8—22no—
106#391C2×C411yescharacteristic
107#391C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#390C2×C411yescharacteristic
110#389C2×C411yescharacteristic
111#389C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#388—21no—
115#388—21no—
116#388—21no—
117#388—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#386—21no—
123#386—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#386—21no—
128#385—21no—
129#385—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#381—21no—
135#380C2×C411yescharacteristic
136#380C2×C411yescharacteristic
137#380C2311yescharacteristic
138#380C2311yescharacteristic
139#379—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4C2×C411yescharacteristic
148C4×C4⋊C4C2×C411yescharacteristic
149C4×C4⋊C4C2×C411yescharacteristic
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164#376—21no—
165C2×#95—22no—
166C2×#95—22no—
167C2×#95—22no—
168C2×#95C2314yes—
169C43—21no—
170#336—22no—
171#336—22no—
172#328—28no—
173#327—28no—
174C25C2×D411yescharacteristic
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—42no—
178C2×#41—42no—
179C22×Q8C22×C411yescharacteristic
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4C22×C411yescharacteristic
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C22⋊C411yescharacteristic
205C23×C4C22⋊C411yescharacteristic
206C23×C4C22⋊C411yescharacteristic
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2411yescharacteristic
217C22×C8—24no—
218#123—28no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8#4114yes—
223#121—22no—
224#121—22no—
225#121—22no—
226#121#4114yes—
227#120—28no—
228C4×Q8—42no—
229C4×Q8—42no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—42no—
233C4×D4—42no—
234C4×D4—44no—
235C4×D4—44no—
236C4×D4—44no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—42no—
240C4×D4—42no—
241#117—22no—
242#117—22no—
243#117—44no—
244#117—432no—
245#117—42no—
246#117—42no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—21no—
287C2×C42—28no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—44no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42C22⋊C411yescharacteristic
310C2×C42C22⋊C411yescharacteristic
311C2×C42C22⋊C411yescharacteristic
312C2×C42C22⋊C411yescharacteristic
313C2×C42C22⋊C411yescharacteristic
314C2×C42C22⋊C411yescharacteristic
315C2×C42C22⋊C411yescharacteristic
316C2×C42C22⋊C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C4⋊C8—44no—
326C22⋊C8—24no—
327C22⋊C8—24no—
328C22⋊C8—44no—
329C22⋊C8—44no—
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—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—24no—
348#95—24no—
349#95—44no—
350#95—44no—
351#95—44no—
352C24—21no—
353C24—21no—
354C24—22no—
355C24—22no—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—41no—
360C24—41no—
361C24C2×C22⋊C411yescharacteristic
362C24C2×C22⋊C411yescharacteristic
363C24C22×D411yescharacteristic
364#41—48no—
365#41—48no—
366C2×Q8—21no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8#11712yescomplemented
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4#11712yescomplemented
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4#9914yes—
465C22×C4#10414yes—
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C2×C22⋊C411yescharacteristic
469C22×C4C2×C22⋊C411yescharacteristic
470C22×C4C2×C22⋊C411yescharacteristic
471C22×C4C2×C22⋊C411yescharacteristic
472C22×C4C2×C22⋊C411yescharacteristic
473C22×C4C2×C22⋊C411yescharacteristic
474C22×C4C2×C22⋊C411yescharacteristic
475C22×C4C2×C22⋊C411yescharacteristic
476C22×C4#13514yes—
477C22×C4AGL(1,ℤ/8)12yes—
478C22×C4#13712yes—
479C22×C4C23×C411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C2×C8—44no—
484C2×C8—44no—
485C2×C8—44no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—24no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—416no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4#11712yescomplemented
502C4⋊C4#11712yescomplemented
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—416no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—42no—
540C42—416no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—42no—
547C42—42no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—24no—
553C23—24no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23C2×#9912yes—
582C23C2×#10412yes—
583C23C22×C22⋊C411yescharacteristic
584C23#52512yes—
585C23C2×#13512yes—
586C23C2×AGL(1,ℤ/8)11yescharacteristic
587C23C2×#13711yescharacteristic
588Q8—42no—
589Q8—42no—
590D4—42no—
591D4—42no—
592D4—48no—
593D4—48no—
594D4—44no—
595D4—44no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
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—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—48no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—48no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—41no—
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—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4—84no—
674C2×C4C2×#9912yes—
675C2×C4C2×#9912yes—
676C2×C4#41012yes—
677C2×C4#41612yes—
678C2×C4#41612yes—
679C2×C4#41712yes—
680C2×C4#41712yes—
681C2×C4C2×#10412yes—
682C2×C4C2×#10412yes—
683C2×C4#42112yes—
684C2×C4C2×#11711yescharacteristic
685C2×C4C2×#11711yescharacteristic
686C2×C4#52512yes—
687C8—84no—
688C22—22no—
689C22—22no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—48no—
703C22—48no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—81no—
716C22—81no—
717C22—81no—
718C22—81no—
719C22#228011yescharacteristic
720C22C22×#9911yescharacteristic
721C22C2×#41011yescharacteristic
722C22C2×#41611yescharacteristic
723C22C2×#41711yescharacteristic
724C22C22×#10411yescharacteristic
725C22C2×#42111yescharacteristic
726C22#310612yes—
727C22#310712yes—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—21no—
737C4—21no—
738C4—42no—
739C4—42no—
740C4—41no—
741C4—41no—
742C4—42no—
743C4—88no—
744C4—81no—
745C4—81no—
746C4—81no—
747C4—81no—
748C4#150414yes—
749C2—22no—
750C2—22no—
751C2—42no—
752C2—42no—
753C2—82no—
754C2#1753311yescharacteristic
755C2C2×#150411yescharacteristic
756C2#1756311yescharacteristic
757C2#1757211yescharacteristic
758C2#2072211yescharacteristic
759C2#2073711yescharacteristic
760C2#2074711yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. #13756C2semidirect
4. #8715C2semidirect
5. #8578C2semidirect
6. #8577C2semidirect
8. C4×#328C2semidirect
9. C4×#327C2semidirect
11. C2×C4×#41C4semidirect
15. C4×C22⋊Q8C4semidirect
16. C4×C22⋊Q8C4semidirect
21. C4×#121C4semidirect
22. C4×#121C4semidirect
27. #2297C4semidirect
28. #2296C4semidirect
34. C4×#117C4semidirect
369. C2×Q8#117semidirect
383. C2×D4#117semidirect
501. C4⋊C4#117semidirect
502. C4⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
#207142
#137562
#87152
#85782
#85772
#83312
C4×#3282
C4×#3272