Finite Group Databasebeta

← #162634

Subgroups — #162634

Order 512.

1546 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3C2×#1728C211yesmaximal, characteristic, complemented
4#7678C214yesmaximal, complemented
5#7668C212yesmaximal, complemented
6#7664C212yesmaximal, complemented
7#7616C214yesmaximal, complemented
8C23×M16—21no—
9C2×#532C411yescharacteristic
10C2×#525—21no—
11C23×C42C2211yescharacteristic
12C22×C4⋊C8—21no—
13C22×C22⋊C8—21no—
14#2857C411yescharacteristic
15#2857C2211yescharacteristic
16#2856C411yescharacteristic
17#2856C2211yescharacteristic
18#2854—21no—
19#2854C2212yes—
20#2850—21no—
21#2850—21no—
22#2847—21no—
23#2847C412yes—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121C412yes—
27C4×#121C2212yes—
28#2297—21no—
29#2285C411yescharacteristic
30#1797—21no—
31#1797—21no—
32#1797C2212yes—
33#1797C2212yes—
34#1728—21no—
35#1728—21no—
36#1728C2212yes—
37#1728C2212yes—
38C2×#336—21no—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1271C2218yescomplemented
42#1270C2218yescomplemented
43C22×#41—21no—
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C22×M16—28no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—44no—
51C22×M16—44no—
52C23×C8—22no—
53C23×C8—22no—
54#532—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#127C2×C411yescharacteristic
57C2×#124—21no—
58#525—44no—
59#525—44no—
60C2×#123—22no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121C2×C412yes—
69C2×#121C2×C412yes—
70C2×#121C2×C412yes—
71C2×#120—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C412yes—
76C2×#117—21no—
77C22×C42—44no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—44no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—44no—
96#391—21no—
97#391—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390C2×C412yes—
103#390C2312yes—
104#389—21no—
105#389—21no—
106#389C2×C412yes—
107#389C2312yes—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#382—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380C2×C412yes—
125#380C2×C412yes—
126#380C2×C412yes—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C412yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336C2314yes—
136#336C2314yes—
137C25C22⋊C411yescharacteristic
138C2×#41—44no—
139C2×#41—44no—
140C2×#41—44no—
141C22×Q8—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C22×C412yes—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4D814yes—
168C23×C4SD1614yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C2×M16—28no—
179C2×M16—28no—
180C2×M16—44no—
181C2×M16—432no—
182C2×M16—44no—
183C2×M16—44no—
184C2×M16—44no—
185C22×C8—28no—
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—44no—
189C22×C8—44no—
190C22×C8—44no—
191C22×C8—44no—
192C22×C8—44no—
193C4⋊Q8—42no—
194#127—42no—
195#124—44no—
196#123—44no—
197#123—44no—
198C22⋊Q8—44no—
199C22⋊Q8—42no—
200C22⋊Q8—42no—
201#121—44no—
202#121—44no—
203#121—44no—
204#121—44no—
205#121—44no—
206#121—44no—
207#121—42no—
208#121—42no—
209#121—42no—
210#121—42no—
211#120—48no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—48no—
215C4×D4—44no—
216C4×D4—44no—
217#117—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2411yescharacteristic
268C4⋊C8—44no—
269C4⋊C8—44no—
270C22⋊C8—44no—
271C22⋊C8—44no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—41no—
284C24—41no—
285C24#10214yes—
286C24C2×C22⋊C411yescharacteristic
287C24C2×C22⋊C411yescharacteristic
288C24C2×C22⋊C411yescharacteristic
289#41—816no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×D4—24no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—216no—
317C22×C4—216no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—416no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4#9914yes—
370C22×C4#10214yes—
371C22×C4#10214yes—
372C22×C4#10214yes—
373C22×C4#10414yes—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×D812yes—
384C22×C4C2×D812yes—
385C22×C4C2×D812yes—
386C22×C4C2×SD1612yes—
387C22×C4C2×SD1612yes—
388C22×C4C2×SD1612yes—
389C22×C4AGL(1,ℤ/8)12yes—
390C22×C4#13712yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394M16—416no—
395M16—416no—
396C2×C8—28no—
397C2×C8—44no—
398C2×C8—416no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—88no—
404C4⋊C4—24no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—44no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—28no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—28no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23C2×#9912yes—
474C23#41312yes—
475C23C2×#10212yes—
476C23C2×#10212yes—
477C23C2×#10212yes—
478C23#41712yes—
479C23C2×#10412yes—
480C23C22×C22⋊C411yescharacteristic
481Q8—84no—
482D4—84no—
483D4—84no—
484D4—88no—
485D4—88no—
486D4—88no—
487D4—84no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
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—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—88no—
545C2×C4#327116yes—
546C2×C4C2×#9912yes—
547C2×C4C2×#9912yes—
548C2×C4C2×#10212yes—
549C2×C4C2×#10212yes—
550C2×C4C2×#10212yes—
551C2×C4#41712yes—
552C2×C4C2×#10412yes—
553C2×C4C2×#10412yes—
554C2×C4#42112yes—
555C2×C4#56812yes—
556C2×C4C22×D811yescharacteristic
557C2×C4C22×SD1611yescharacteristic
558C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
559C2×C4C2×#13711yescharacteristic
560C8—48no—
561C8—84no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—416no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22C2×#32714yes—
590C22#154714yes—
591C22#287911yescharacteristic
592C22C22×#9911yescharacteristic
593C22C2×#41311yescharacteristic
594C22C22×#10211yescharacteristic
595C22C2×#41711yescharacteristic
596C22C22×#10411yescharacteristic
597C22C2×#42111yescharacteristic
598C4—21no—
599C4—21no—
600C4—22no—
601C4—21no—
602C4—21no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4C2×#32714yes—
614C4C2×#32714yes—
615C2—24no—
616C2—24no—
617C2—42no—
618C2—162no—
619C2#1740011yescharacteristic
620C2#1748511yescharacteristic
621C2#1750411yescharacteristic
622C2C22×#32711yescharacteristic
623C2#1759211yescharacteristic
624C2#1765111yescharacteristic
625C2C2×#154711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1797C2semidirect
3. C2×#1728C2semidirect
4. #7678C2semidirect
5. #7668C2semidirect
6. #7664C2semidirect
7. #7616C2semidirect
41. #1271C22semidirect
42. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#17972
C2×#17282
#76782
#76682
#76642
#76162