Finite Group Databasebeta

← #6529796

Subgroups — #6529796

Order 512.

2308 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2C2×#2982C211yesmaximal, characteristic, complemented
3#35178C211yesmaximal, characteristic, complemented
4C2×#2538C211yesmaximal, characteristic, complemented
5#28263C214yesmaximal, complemented
6#28185C214yesmaximal, complemented
7#25469C214yesmaximal, complemented
8#25292C214yesmaximal, complemented
9#24600C214yesmaximal, complemented
10#24500C214yesmaximal, complemented
11#21982C211yesmaximal, characteristic, complemented
12#21859C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic, complemented
14C2×#536—21no—
15C2×#532C2211yescharacteristic
16C22×C22⋊Q8—21no—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3100—28no—
20C2×#480C2212yes—
21C2×#474C2212yes—
22C2×#459C2212yes—
23C2×#447C2212yes—
24#2982—28no—
25C2×C8×Q8C2214yescomplemented
26C2×C8×Q8C2214yescomplemented
27C2×C4×D8C2214yes—
28C2×#431C2214yes—
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2854C2214yes—
32#2618—22no—
33#2548C2212yescomplemented
34#2546C2212yescomplemented
35#2545—22no—
36#2538—22no—
37#2538—22no—
38#2538C2214yes—
39#2538C2214yes—
40#2538C2212yescomplemented
41#2538C2212yescomplemented
42#2536—22no—
43C2×#396—22no—
44#2311—21no—
45C2×#390C2212yes—
46C2×#389C2212yes—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50#2086C2218yescomplemented
51#2067C2218yes—
52#2067C2218yescomplemented
53#2054C2218yes—
54#1990C2212yescomplemented
55#1970C2218yes—
56#1957C2212yescomplemented
57#1947C2214yescomplemented
58#1942C2218yescomplemented
59#1934C2218yes—
60#1934C2218yescomplemented
61#1874C2214yes—
62#1852C2214yes—
63#1845C2212yescomplemented
64#1834C2212yescomplemented
65#1823C2212yes—
66#1802C2212yes—
67#1802C2212yes—
68#1802C2214yescomplemented
69#1786C2212yes—
70C4×#102C2214yes—
71C22×#41—21no—
72C23×Q8D411yescharacteristic
73C23×D4C2311yescharacteristic
74C24×C4—21no—
75C22×D8C2314yes—
76#536—48no—
77#532—24no—
78#532—24no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8C2311yescharacteristic
81C2×#127C2311yescharacteristic
82C2×#124—22no—
83C2×#124—21no—
84C2×#123—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—48no—
90C2×#121—21no—
91C2×#121C2312yes—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8D412yes—
95C2×C4×Q8D412yes—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×#117—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D412yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119#480—28no—
120#474—28no—
121#459—28no—
122#447—28no—
123C8×Q8—216no—
124C8×Q8—216no—
125C4×D8—216no—
126#431—24no—
127#431—216no—
128#431D418yes—
129C2×#107C2314yes—
130C2×#107C2318yes—
131C2×C4⋊C8C2318yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2318yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C4×C8C2318yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#398—24no—
145#396—24no—
146#396—24no—
147#396—28no—
148#396—28no—
149#394—24no—
150#391—22no—
151#390—22no—
152#390—24no—
153#390C2314yes—
154#390C2318yes—
155#390C2314yes—
156#389—24no—
157#389C2314yes—
158#389C2318yes—
159#389C2314yes—
160#388—24no—
161#388—24no—
162#386—24no—
163#384—21no—
164#384—21no—
165#384C2311yescharacteristic
166#384C2311yescharacteristic
167#384C2312yes—
168#384C2312yes—
169#382—21no—
170#380—21no—
171#380—22no—
172#380—22no—
173#380—24no—
174#380—24no—
175#380C2312yes—
176C4×C4⋊C4—21no—
177C4×C4⋊C4C2312yes—
178C2×#95—22no—
179#340C2318yes—
180#340C2314yes—
181#340C2314yes—
182#340C2318yes—
183C25—21no—
184C2×#41—44no—
185C2×#41—44no—
186C2×#41—44no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—21no—
190C22×Q8D818yescomplemented
191C22×Q8C2×D412yes—
192C22×Q8C2×D412yes—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—21no—
196C22×D4—22no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—42no—
201C22×D4C2412yes—
202C22×D4C2412yes—
203C23×C4—22no—
204C23×C4—21no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2411yescharacteristic
216C2×D8—28no—
217C2×D8—28no—
218C2×D8—44no—
219C22×C8C2414yes—
220C22×C8C2414yes—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—48no—
224#127—22no—
225#127—22no—
226#124—48no—
227#124—44no—
228#123—416no—
229C22⋊Q8—416no—
230C22⋊Q8—416no—
231C22⋊Q8—48no—
232C22⋊Q8—48no—
233#121—24no—
234#121—24no—
235#121—44no—
236C4×Q8—28no—
237C4×Q8—28no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4—44no—
251#117—44no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—82no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—84no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42C2×D414yes—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D414yes—
304C2×C42#4114yes—
305C2×C42C2414yes—
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#107—28no—
309#107—48no—
310C4⋊C8—216no—
311C4⋊C8—24no—
312C4⋊C8C2×D418yes—
313#102—28no—
314#102—28no—
315#102—28no—
316#102—44no—
317#102—48no—
318C22⋊C8—24no—
319C22⋊C8C2×D418yes—
320C4×C8—216no—
321C4×C8—24no—
322C4×C8C2×D418yes—
323#95—28no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—42no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24C22×D411yescharacteristic
338#41—816no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—232no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8C2×D818yes—
348C2×Q8C2×D818yes—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—42no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—81no—
371C2×D4—81no—
372C2×D42−1+414yes—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—816no—
403C22×C4#12114yes—
404C22×C4#12114yes—
405C22×C4C2×D818yes—
406C22×C4C2×D814yes—
407C22×C4#13514yes—
408C22×C4C22×D414yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D414yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4112yes—
417C22×C4C2511yescharacteristic
418D8—416no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8C22×D418yes—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—28no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—416no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—88no—
461C42—28no—
462C42—24no—
463C42—24no—
464C42—44no—
465C42#12114yes—
466C42#12114yes—
467C42C22×D412yes—
468C42C22×D412yes—
469C42C2×#4112yes—
470C42C2×#4112yes—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—24no—
476C23—48no—
477C23—48no—
478C23—48no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23C2×#12112yes—
497C23C22×D812yes—
498C23C2×#13512yes—
499C23C23×D411yescharacteristic
500Q8—216no—
501Q8—216no—
502Q8—84no—
503D4—24no—
504D4—24no—
505D4—48no—
506D4—42no—
507D4—44no—
508D4—42no—
509D4—44no—
510D4—84no—
511D4—88no—
512D4—84no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—24no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—216no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—84no—
544C2×C4—88no—
545C2×C4—84no—
546C2×C4—88no—
547C2×C4—88no—
548C2×C4—88no—
549C2×C4#466116yes—
550C2×C4C2×#12112yes—
551C2×C4C2×#12112yes—
552C2×C4C2×#12112yes—
553C2×C4C2×#12112yes—
554C2×C4C2×#12112yes—
555C2×C4C22×D818yes—
556C2×C4C22×D812yes—
557C2×C4C22×D812yes—
558C2×C4C2×#13512yes—
559C2×C4C2×#13512yes—
560C2×C4C23×D411yescharacteristic
561C2×C4C22×#4111yescharacteristic
562C2×C4C2×2−1+412yes—
563C2×C4#58512yes—
564C8—28no—
565C8—44no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—816no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22C2×#46614yes—
588C22C22×#12111yescharacteristic
589C22#348611yescharacteristic
590C22#348711yescharacteristic
591C22C23×D811yescharacteristic
592C22C22×#13511yescharacteristic
593C22#358511yescharacteristic
594C22#358611yescharacteristic
595C4—28no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—44no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—84no—
605C4C2×#46614yes—
606C4C2×#46614yes—
607C4#348912yes—
608C4#349512yes—
609C2—24no—
610C2—84no—
611C2—162no—
612C2#3566111yescharacteristic
613C2C22×#46611yescharacteristic
614C2#6045611yescharacteristic
615C2#6181711yescharacteristic
616C2#6181911yescharacteristic
617C2#6188011yescharacteristic
618C2#6188211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2982C2semidirect
3. #35178C2semidirect
4. C2×#2538C2semidirect
5. #28263C2semidirect
6. #28185C2semidirect
7. #25469C2semidirect
8. #25292C2semidirect
9. #24600C2semidirect
10. #24500C2semidirect
11. #21982C2semidirect
12. #21859C2semidirect
13. #20732C2semidirect
25. C2×C8×Q8C22semidirect
26. C2×C8×Q8C22semidirect
33. #2548C22semidirect
34. #2546C22semidirect
40. #2538C22semidirect
41. #2538C22semidirect
50. #2086C22semidirect
52. #2067C22semidirect
54. #1990C22semidirect
56. #1957C22semidirect
57. #1947C22semidirect
58. #1942C22semidirect
60. #1934C22semidirect
63. #1845C22semidirect
64. #1834C22semidirect
68. #1802C22semidirect
190. C22×Q8D8semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
C2×#29822
#351782
C2×#25382
#282632
#281852
#254692
#252922
#246002
#245002
#219822
#218592
#207322