Finite Group Databasebeta

← #182633

Subgroups — #182633

Order 512.

1236 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#529C211yesmaximal, characteristic
2C4×#432C211yesmaximal, characteristic, complemented
3C2×#1512C212yesmaximal
4C4×#382C211yesmaximal, characteristic, complemented
5#8221C212yesmaximal, complemented
6#8220C212yesmaximal
7#8219C212yesmaximal, complemented
8#8218C212yesmaximal
9#8017C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C2×#432C2212yes—
12C4×#125C411yescharacteristic, complemented
13C4×#125C411yescharacteristic, complemented
14C4×#126—21no—
15C4×#126—21no—
16C4×#124C411yescharacteristic, complemented
17C4×#124C411yescharacteristic, complemented
18C4×C22⋊Q8C2211yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20C4×#123—21no—
21C4×#123—21no—
22C4×#121C2211yescharacteristic
23C4×#121C2211yescharacteristic
24#2298—21no—
25#2295C411yescharacteristic, complemented
26#2293C411yescharacteristic, complemented
27#2290—21no—
28#2283C2211yescharacteristic
29C2×#382C2212yes—
30C42×Q8C411yescharacteristic, complemented
31C42×D4—21no—
32C42×D4C411yescharacteristic, complemented
33C4×#117—21no—
34C2×C4×C4⋊C4—21no—
35C2×C43C2211yescharacteristic
36#1767—21no—
37C4×C4⋊C8—21no—
38#1749C2212yes—
39C4×C22⋊C8C2212yes—
40#1745—21no—
41C4×#97—21no—
42#1512C4116yescomplemented
43C2×#328C2214yes—
44C2×#327C2214yes—
45C4⋊#95—21no—
46#1434—21no—
47#1434C2212yes—
48C4×#95—21no—
49C4×#95C2212yes—
50#1294C2214yes—
51#1294C2214yes—
52#529—24no—
53#529C2×C418yes—
54C2×#126—21no—
55C2×#125C2×C411yescharacteristic
56C2×#124C2×C411yescharacteristic
57C2×#123—21no—
58C2×C22⋊Q8C2311yescharacteristic
59C2×#121C2311yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4—22no—
72C22×C42—21no—
73C22×C42—22no—
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#432—24no—
82#432C2×C418yescomplemented
83C2×C4⋊C8—22no—
84C2×C22⋊C8C2314yes—
85C2×#97—22no—
86C2×#97—22no—
87C2×C4×C8—22no—
88#391C2×C411yescharacteristic
89#391C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#389C2×C411yescharacteristic
93#389C2×C411yescharacteristic
94#389C2×C411yescharacteristic
95#389C2×C411yescharacteristic
96#388—21no—
97#388—21no—
98#388—21no—
99#388—21no—
100#388—21no—
101#388—21no—
102#388—21no—
103#388—21no—
104#387—21no—
105#387—21no—
106#387—21no—
107#387—21no—
108#387—21no—
109#387—21no—
110#387—21no—
111#387—21no—
112#386—21no—
113#386—21no—
114#385—21no—
115#385—21no—
116#383—21no—
117#382—24no—
118#382C2×C411yescharacteristic
119#382C2×C411yescharacteristic
120#382C2×C418yescomplemented
121#381—21no—
122#380C2311yescharacteristic
123#380C2311yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—44no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154#376—21no—
155#376—21no—
156C2×#95—22no—
157C2×#95—22no—
158C2×#95—22no—
159C2×#95—22no—
160C2×#95C2314yes—
161C43—21no—
162C43C2×C411yescharacteristic
163C43C2×C411yescharacteristic
164C43C2×C411yescharacteristic
165C43C2×C411yescharacteristic
166C43D411yescharacteristic
167C43D411yescharacteristic
168C43D411yescharacteristic
169C43D411yescharacteristic
170#336—22no—
171#336—22no—
172#328C2×C4116yescomplemented
173#327C2×C4116yes—
174C22×Q8C22×C411yescharacteristic
175C22×D4—21no—
176C22×D4C22×C411yescharacteristic
177C23×C4—21no—
178C23×C4—24no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2411yescharacteristic
188C22×C8—24no—
189#126—28no—
190#125—22no—
191#125C4214yescomplemented
192#124—22no—
193#124C4214yescomplemented
194#123—28no—
195C22⋊Q8—22no—
196C22⋊Q8C22×C414yes—
197#121—22no—
198#121C22×C414yes—
199C4×Q8—22no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8C4214yescomplemented
203C4×D4—28no—
204C4×D4—22no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—44no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4C4214yescomplemented
211#117—28no—
212#117—44no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—44no—
261C2×C42—42no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C414yes—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42#4114yes—
295C4⋊C8—28no—
296C22⋊C8—24no—
297C22⋊C8C22×C418yes—
298#97—28no—
299#97—44no—
300C4×C8—44no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—28no—
320#95—21no—
321#95—21no—
322#95—24no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95C22×C418yes—
327C24—21no—
328C24C22×D411yescharacteristic
329C2×Q8—21no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8C2×C4212yes—
333C2×D4—24no—
334C2×D4—21no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4C2×C4212yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—28no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C4×D414yes—
396C22×C4C4×D414yes—
397C22×C4AGL(1,ℤ/8)12yes—
398C22×C4#13712yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C23×C412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4112yes—
403C2×C8—28no—
404C2×C8—44no—
405C2×C8—44no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—416no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4C2×C4212yes—
424C4⋊C4C2×C4212yes—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C42—24no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—88no—
473C42C2×C4212yes—
474C42C2×C4212yes—
475C42#11712yes—
476C42#11712yes—
477C42C4×D414yes—
478C42C4×D414yes—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C22×C22⋊C411yescharacteristic
491C23C2×C4×D412yes—
492C23C2×AGL(1,ℤ/8)11yescharacteristic
493C23C2×#13711yescharacteristic
494C23#57512yes—
495Q8—42no—
496Q8—42no—
497D4—42no—
498D4—42no—
499D4—44no—
500D4—44no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—48no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4C4×C22⋊C412yes—
574C2×C4C4×C22⋊C412yes—
575C2×C4C4×C22⋊C412yes—
576C2×C4C4×C22⋊C412yes—
577C2×C4#41012yes—
578C2×C4#41012yes—
579C2×C4#41712yes—
580C2×C4#41712yes—
581C2×C4#42112yes—
582C2×C4#42112yes—
583C2×C4C22×C4211yescharacteristic
584C2×C4C2×#11711yescharacteristic
585C2×C4C2×C4×D412yes—
586C8—48no—
587C22—22no—
588C22—22no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—22no—
594C22—22no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C2×C4×C22⋊C411yescharacteristic
603C22C2×#41011yescharacteristic
604C22C2×#41011yescharacteristic
605C22C2×#41711yescharacteristic
606C22#289611yescharacteristic
607C22C2×#42111yescharacteristic
608C22C2×#42111yescharacteristic
609C22C4×AGL(1,ℤ/8)12yes—
610C22C4×#13712yes—
611C4—22no—
612C4—21no—
613C4—21no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—42no—
629C4—88no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C2—22no—
635C2—22no—
636C2—82no—
637C2#1757911yescharacteristic
638C2#1758111yescharacteristic
639C2#1758811yescharacteristic
640C2#1759111yescharacteristic
641C2C4×#41011yescharacteristic
642C2C4×#42111yescharacteristic
643C2C4×#41711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#432C2semidirect
4. C4×#382C2semidirect
5. #8221C2semidirect
7. #8219C2semidirect
9. #8017C2semidirect
12. C4×#125C4semidirect
13. C4×#125C4semidirect
16. C4×#124C4semidirect
17. C4×#124C4semidirect
25. #2295C4semidirect
26. #2293C4semidirect
30. C42×Q8C4semidirect
32. C42×D4C4semidirect
42. #1512C4semidirect
82. #432C2×C4semidirect
120. #382C2×C4semidirect
172. #328C2×C4semidirect
191. #125C42semidirect
193. #124C42semidirect
202. C4×Q8C42semidirect
210. C4×D4C42semidirect

Maximal subgroups

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

subgroupindex
C4×#5292
C4×#4322
C2×#15122
C4×#3822
#82212
#82202
#82192
#82182
#80172