Finite Group Databasebeta

← #181625

Subgroups — #181625

Order 512.

1643 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2#34126C211yesmaximal, characteristic
3#34057C211yesmaximal, characteristic
4#8052C212yesmaximal, complemented
5#8051C212yesmaximal, complemented
6#8051C212yesmaximal, complemented
7#8050C212yesmaximal, complemented
8#8049C212yesmaximal, complemented
9C2×#1297C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11#2975—21no—
12#2974—24no—
13#2973—21no—
14#2973—21no—
15#2972—21no—
16D4×M16C418yescomplemented
17#2920C2212yes—
18C2×#407C2212yes—
19C2×#407C2212yes—
20#2357C412yescomplemented
21#2353C412yescomplemented
22#2353C412yescomplemented
23#2349C412yescomplemented
24#2345—21no—
25#2342—21no—
26#2342—21no—
27D4×C22⋊C4—21no—
28#2318C418yescomplemented
29#2315—21no—
30#2313—21no—
31#2312—21no—
32#2308—21no—
33#2293—21no—
34#2290—24no—
35#2290—21no—
36#2290—21no—
37#2283C2212yes—
38#2283C2211yescharacteristic
39#2280C2211yescharacteristic
40#2280C2212yes—
41#2280C2211yescharacteristic
42C42×D4—21no—
43C4×#117C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47#1297C2214yes—
48#1297C2214yes—
49#1297C2214yes—
50#1297C2214yes—
51C23×D4D411yescharacteristic
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16C2×C414yescomplemented
55C22×M16C2×C414yescomplemented
56C2×C4×D4—22no—
57C2×C4×D4C2×C414yes—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C414yes—
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D414yes—
67C2×C4×D4D411yescharacteristic
68C2×C4×D4Q814yes—
69C2×#117—21no—
70C2×#117—22no—
71C2×#117—21no—
72C2×#117—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42—21no—
81C22×C42—22no—
82C22×C42C2311yescharacteristic
83#436—28no—
84#435—28no—
85#435—28no—
86C8×D4—28no—
87#432—22no—
88#432—22no—
89#431—22no—
90#431—22no—
91#423C2×C414yescomplemented
92#407—22no—
93#407—22no—
94#407—22no—
95#407—22no—
96#407C2×C414yescomplemented
97#407C2×C414yescomplemented
98#407C2314yes—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C4×M16C2×C414yescomplemented
104#390—22no—
105#389—22no—
106#388—28no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#387—28no—
112#387—22no—
113#387—22no—
114#386—28no—
115#386—22no—
116#386—22no—
117#385—28no—
118#385—22no—
119#385—22no—
120#383C2×C414yescomplemented
121#383C2×C412yescomplemented
122#382C2×C414yescomplemented
123#382C2×C412yescomplemented
124#381—22no—
125#381—21no—
126#381C2×C412yes—
127#381C2×C414yescomplemented
128#381C2×C412yescomplemented
129#380—22no—
130#380—22no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380C2×C412yes—
135#380C2×C414yescomplemented
136#380C2×C412yescomplemented
137#379—22no—
138#379—21no—
139#379—21no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4C2×C412yescomplemented
145C4×C22⋊C4—22no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—22no—
155C4×C22⋊C4—22no—
156C4×C22⋊C4—22no—
157C4×C22⋊C4C2×C412yes—
158C4×C22⋊C4C2×C414yescomplemented
159C4×C22⋊C4C2×C412yescomplemented
160C4×C22⋊C4C2311yescharacteristic
161C4×C22⋊C4C2311yescharacteristic
162C4×C22⋊C4C2311yescharacteristic
163C4×C22⋊C4C2314yes—
164C4×C22⋊C4C2311yescharacteristic
165#376—22no—
166#376—21no—
167#376C2×C412yescomplemented
168C2×#95C2×C412yes—
169C2×#95C2×C414yescomplemented
170C2×#95C2×C412yescomplemented
171C43—21no—
172C25C2×D411yescharacteristic
173C25C2×D411yescharacteristic
174C22×D4—44no—
175C22×D4C22⋊C411yescharacteristic
176C22×D4C22⋊C411yescharacteristic
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C414yes—
179C22×D4C22⋊C411yescharacteristic
180C22×D4C4⋊C414yes—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—82no—
185C23×C4—82no—
186C23×C4C22×C412yes—
187C23×C4C22×C412yes—
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C412yes—
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D412yes—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×Q812yes—
205C23×C4C2×Q812yes—
206C23×C4C2411yescharacteristic
207C2×M16—24no—
208C2×M16—24no—
209C2×M16—84no—
210C2×M16—84no—
211C2×M16C22×C418yes—
212C22×C8—24no—
213C22×C8—24no—
214C4×D4—48no—
215C4×D4C4214yes—
216C4×D4C4214yes—
217C4×D4C4214yes—
218C4×D4C4214yes—
219C4×D4C22⋊C414yes—
220C4×D4C22⋊C414yes—
221C4×D4C22⋊C414yes—
222C4×D4C22⋊C414yes—
223C4×D4C4⋊C414yes—
224C4×D4C4⋊C414yes—
225C4×D4C4⋊C414yes—
226C4×D4C4⋊C414yes—
227C4×D4C4⋊C414yes—
228C4×D4C4⋊C414yes—
229C4×D4C4⋊C414yes—
230C4×D4C4⋊C414yes—
231#117—44no—
232#117—84no—
233#117—82no—
234#117—82no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C414yes—
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C4⋊C414yes—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C22⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4C22⋊C414yes—
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C414yes—
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C4⋊C414yes—
275C2×C22⋊C4C4⋊C414yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—82no—
287C2×C42—82no—
288C2×C42—84no—
289C2×C42C22×C412yes—
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C412yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D412yes—
296C2×C42C2×Q812yes—
297C4⋊C8—24no—
298C4⋊C8—24no—
299C22⋊C8—24no—
300C22⋊C8—24no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C22⋊C8—24no—
304C22⋊C8—24no—
305#97—24no—
306C4×C8—24no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95C22×C412yes—
321#95C22×C412yes—
322#95C22×C412yes—
323#95C22×C418yes—
324#95C22×C412yes—
325C24—24no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—81no—
330C24—81no—
331C24C2×C22⋊C412yes—
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334C24C2×C22⋊C412yes—
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C4⋊C412yes—
338C24C2×C4⋊C412yes—
339C24C22×D411yescharacteristic
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4#9514yes—
347C2×D4#9514yes—
348C2×D4#9514yes—
349C2×D4#9514yes—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—48no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—82no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—82no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—84no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—84no—
385C22×C4C2×C4212yes—
386C22×C4C2×C4212yes—
387C22×C4C2×C4212yes—
388C22×C4C2×C4212yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C4⋊C412yes—
406C22×C4C2×C4⋊C412yes—
407C22×C4C2×C4⋊C412yes—
408C22×C4C2×C4⋊C412yes—
409C22×C4C2×C4⋊C412yes—
410C22×C4C2×C4⋊C412yes—
411C22×C4C2×C4⋊C412yes—
412C22×C4C2×C4⋊C412yes—
413C22×C4C2×C4⋊C412yes—
414C22×C4C2×C4⋊C412yes—
415C22×C4C2×C4⋊C412yes—
416C22×C4C2×C4⋊C412yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C23×C411yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×Q811yescharacteristic
423M16—48no—
424M16—84no—
425M16—84no—
426C2×C8—28no—
427C2×C8—84no—
428C2×C8—84no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4#9514yes—
436C4⋊C4#9514yes—
437C4⋊C4#9514yes—
438C4⋊C4#9514yes—
439C22⋊C4—24no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—48no—
446C22⋊C4—42no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C22⋊C4—82no—
452C22⋊C4—82no—
453C22⋊C4#9514yes—
454C22⋊C4#9514yes—
455C22⋊C4#9514yes—
456C22⋊C4#9514yes—
457C22⋊C4#9514yes—
458C22⋊C4#9514yes—
459C22⋊C4#9514yes—
460C22⋊C4#9514yes—
461C42—42no—
462C42—42no—
463C42—42no—
464C42—48no—
465C42—44no—
466C42—42no—
467C42—44no—
468C42—82no—
469C42—82no—
470C42—84no—
471C42C2×C4212yes—
472C42C2×C4212yes—
473C42C2×C22⋊C412yes—
474C42C2×C22⋊C412yes—
475C42C2×C4⋊C412yes—
476C42C2×C4⋊C412yes—
477C42C2×C4⋊C412yes—
478C42C2×C4⋊C412yes—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—88no—
506C23—88no—
507C23C2×#9512yes—
508C23C2×#9512yes—
509C23C2×#9512yes—
510C23C2×#9512yes—
511C23#42114yes—
512C23C22×C22⋊C411yescharacteristic
513C23C22×C22⋊C411yescharacteristic
514C23C22×C4⋊C411yescharacteristic
515D4—28no—
516D4—28no—
517D4—44no—
518D4—44no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—48no—
572C2×C4—88no—
573C2×C4—88no—
574C2×C4—816no—
575C2×C4C2×#9512yes—
576C2×C4C2×#9512yes—
577C2×C4C2×#9512yes—
578C2×C4C2×#9512yes—
579C2×C4C2×#9512yes—
580C2×C4C2×#9512yes—
581C2×C4C2×#9512yes—
582C2×C4C2×#9512yes—
583C2×C4C22×C4211yescharacteristic
584C2×C4C22×C22⋊C411yescharacteristic
585C2×C4C22×C4⋊C411yescharacteristic
586C2×C4C22×C4⋊C411yescharacteristic
587C8—48no—
588C22—21no—
589C22—21no—
590C22—22no—
591C22—21no—
592C22—21no—
593C22—22no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—21no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—44no—
607C22—48no—
608C22—44no—
609C22—44no—
610C22—82no—
611C22—82no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22#173314yes—
616C22C22×#9511yescharacteristic
617C22C2×#42112yes—
618C22#290112yes—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—44no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—88no—
635C4#288312yes—
636C2—22no—
637C2—22no—
638C2—24no—
639C2—42no—
640C2—42no—
641C2—41no—
642C2—42no—
643C2C2×#173311yescharacteristic
644C2#2034311yescharacteristic
645C2#2035411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC22C22×#95
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8052C2semidirect
5. #8051C2semidirect
6. #8051C2semidirect
7. #8050C2semidirect
8. #8049C2semidirect
9. C2×#1297C2semidirect
16. D4×M16C4semidirect
20. #2357C4semidirect
21. #2353C4semidirect
22. #2353C4semidirect
23. #2349C4semidirect
28. #2318C4semidirect
54. C22×M16C2×C4semidirect
55. C22×M16C2×C4semidirect
91. #423C2×C4semidirect
96. #407C2×C4semidirect
97. #407C2×C4semidirect
103. C4×M16C2×C4semidirect
120. #383C2×C4semidirect
121. #383C2×C4semidirect
122. #382C2×C4semidirect
123. #382C2×C4semidirect
127. #381C2×C4semidirect
128. #381C2×C4semidirect
135. #380C2×C4semidirect
136. #380C2×C4semidirect
144. C4×C4⋊C4C2×C4semidirect
158. C4×C22⋊C4C2×C4semidirect
159. C4×C22⋊C4C2×C4semidirect
167. #376C2×C4semidirect
169. C2×#95C2×C4semidirect
170. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602302
#341262
#340572
#80522
#80512
#80502
#80492
C2×#12972