Finite Group Databasebeta

← #193198

Subgroups — #193198

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34141C211yesmaximal, characteristic
2#21859C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic, complemented
4C2×#1948C211yesmaximal, characteristic, complemented
5C2×#1547C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8925C212yesmaximal, complemented
8#8924C212yesmaximal, complemented
9#8921C212yesmaximal, complemented
10#7901C211yesmaximal, characteristic, complemented
11#7891C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×C4×#41—21no—
14C2×#431C2211yescharacteristic
15C2×#428—21no—
16C22×#106—21no—
17C2×#423C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19#2367—21no—
20#2367—21no—
21#2366C2211yescharacteristic
22#2366C2211yescharacteristic
23#2365C2211yescharacteristic
24#2365C2211yescharacteristic
25#2364—22no—
26#2361—21no—
27#2361—21no—
28#2297—21no—
29#2293—21no—
30#2290—22no—
31C2×#384C2211yescharacteristic
32C42×D4—21no—
33C4×#117C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1990C2212yes—
36#1957C2212yes—
37#1948—21no—
38#1948—21no—
39#1948—21no—
40#1948C2212yes—
41#1948C2212yes—
42#1948C2212yes—
43#1947—21no—
44#1947C2212yes—
45#1874—22no—
46#1802—22no—
47#1736—21no—
48C2×#340—21no—
49#1547C418yescomplemented
50C2×#332C2212yes—
51C2×#331C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×#41—21no—
57C22×M16—21no—
58C23×C8—21no—
59#532—22no—
60#532—22no—
61#532C2×C414yes—
62#532Q814yes—
63C2×C4⋊Q8C2311yescharacteristic
64C2×#127C2311yescharacteristic
65C2×#124—21no—
66C2×#121—21no—
67C2×#121D412yes—
68C4×#41—44no—
69C4×#41—44no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4D412yes—
80C2×#117—22no—
81C2×#117D411yescharacteristic
82C2×#117D411yescharacteristic
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4D411yescharacteristic
88C22×C42—22no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#431—22no—
92#431—22no—
93#428—44no—
94C2×#107—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106—44no—
99#423—22no—
100#423C2×C414yescomplemented
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2×C414yescomplemented
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2312yes—
108C2×#102—24no—
109C2×C22⋊C8—21no—
110C2×C4×C8C2312yes—
111#391—21no—
112#391—21no—
113#390—21no—
114#390—21no—
115#390—21no—
116#390—21no—
117#389—21no—
118#389—21no—
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#387—22no—
124#387—22no—
125#386—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#385—22no—
130#385—22no—
131#384—21no—
132#384—21no—
133#384—21no—
134#384C2312yes—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4D411yescharacteristic
142C4×C4⋊C4D411yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C4×C4⋊C4C2311yescharacteristic
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153#376—21no—
154C2×#95—21no—
155C43—21no—
156#340—24no—
157#340—22no—
158#340—22no—
159#340—22no—
160#340—22no—
161#332C2×C418yescomplemented
162#331C2×C418yescomplemented
163C2×#41—24no—
164C2×#41—24no—
165C2×#41—42no—
166C2×#41—42no—
167C2×#41—42no—
168C2×#41—42no—
169C22×Q8—21no—
170C22×D4—22no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C2×D412yes—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—41no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C2×M16—24no—
184C2×M16—42no—
185C2×M16—42no—
186C22×C8—24no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—42no—
191C22×C8—42no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8C22×C412yes—
195C4⋊Q8C2×Q812yes—
196#127—21no—
197#127—21no—
198#127C22×C412yes—
199#127C2×Q812yes—
200#124—24no—
201#124—24no—
202#121—24no—
203#121—24no—
204#121—24no—
205#121C4⋊C418yes—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×Q8—42no—
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—48no—
214C4×D4—48no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4C4⋊C418yes—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—48no—
228#117SD1614yes—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—48no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C22×C412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×Q812yes—
288C2×C42#4112yes—
289C2×C42#4112yes—
290C2×C42C2411yescharacteristic
291#107—44no—
292#106—44no—
293#106—44no—
294#106—44no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—22no—
298C4⋊C8—22no—
299C4⋊C8—42no—
300C4⋊C8C22×C414yes—
301C4⋊C8C22×C414yes—
302#102—84no—
303C22⋊C8—42no—
304C4×C8—22no—
305C4×C8—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318C24—22no—
319C24#12011yescharacteristic
320#41—48no—
321#41—48no—
322#41—48no—
323#41—48no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×D4—28no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4C2×C4⋊C414yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—28no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4C2×C4⋊C412yes—
384C22×C4C4×D412yes—
385C22×C4C4×D412yes—
386C22×C4#12011yescharacteristic
387C22×C4#12011yescharacteristic
388C22×C4#12011yescharacteristic
389C22×C4C22⋊Q812yes—
390C22×C4C22⋊Q812yes—
391C22×C4#12312yes—
392C22×C4C2×SD1612yes—
393C22×C4#13512yes—
394C22×C4AGL(1,ℤ/8)12yes—
395C22×C4AGL(1,ℤ/8)12yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399M16—48no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—48no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4C2×C4⋊C414yes—
438C4⋊C4C2×SD1612yes—
439C4⋊C4C2×SD1612yes—
440C4⋊C4AGL(1,ℤ/8)12yes—
441C4⋊C4#13712yes—
442C22⋊C4—24no—
443C22⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—42no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C42—24no—
453C42—44no—
454C42—44no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—81no—
467C42—81no—
468C42C2×C4⋊C412yes—
469C42C4×D412yes—
470C42C4×D412yes—
471C42C22⋊Q812yes—
472C42C22⋊Q812yes—
473C42#12312yes—
474C42C23×C411yescharacteristic
475C42C22×Q811yescharacteristic
476C42C2×#4111yescharacteristic
477C42C2×#4111yescharacteristic
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23#38512yes—
490C23#44912yes—
491C23#45012yes—
492C23C2×#12011yescharacteristic
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497D4—44no—
498D4—416no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—82no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—416no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—164no—
556C2×C4#38512yes—
557C2×C4#38512yes—
558C2×C4#38512yes—
559C2×C4C4×SD1614yes—
560C2×C4#44214yes—
561C2×C4#44912yes—
562C2×C4#45012yes—
563C2×C4#47614yes—
564C2×C4#47814yes—
565C2×C4C22×C4⋊C411yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C2×C22⋊Q811yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C2×C4C2×#12311yescharacteristic
571C2×C4C22×SD1611yescharacteristic
572C2×C4C2×#13511yescharacteristic
573C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
574C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
575C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
576C2×C4C2×#13711yescharacteristic
577C2×C4#57511yescharacteristic
578C2×C4#57511yescharacteristic
579C8—44no—
580C8—44no—
581C8—82no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22#186414yes—
596C22C2×#38511yescharacteristic
597C22C2×#45011yescharacteristic
598C22C2×#44911yescharacteristic
599C22#300611yescharacteristic
600C22#300711yescharacteristic
601C22#302111yescharacteristic
602C22#302211yescharacteristic
603C4—21no—
604C4—21no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—44no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—162no—
617C4#186414yes—
618C4C2×C4×SD1611yescharacteristic
619C4C2×#44211yescharacteristic
620C4C4×AGL(1,ℤ/8)11yescharacteristic
621C4C4×#13711yescharacteristic
622C4C2×#47611yescharacteristic
623C4C2×#47811yescharacteristic
624C4#307911yescharacteristic
625C4#308011yescharacteristic
626C2—24no—
627C2—84no—
628C2#1878711yescharacteristic
629C2#1879011yescharacteristic
630C2#1879911yescharacteristic
631C2#1880211yescharacteristic
632C2#2085811yescharacteristic
633C2C2×#186411yescharacteristic
634C2#2087511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21171C2semidirect
4. C2×#1948C2semidirect
6. #8926C2semidirect
7. #8925C2semidirect
8. #8924C2semidirect
9. #8921C2semidirect
10. #7901C2semidirect
11. #7891C2semidirect
49. #1547C4semidirect
100. #423C2×C4semidirect
105. C2×C4⋊C8C2×C4semidirect
161. #332C2×C4semidirect
162. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341412
#218592
#211712
C2×#19482
C2×#15472
#89262
#89252
#89242
#89212
#79012
#78912