Finite Group Databasebeta

← #206259

Subgroups — #206259

Order 512.

1102 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3#21863C211yesmaximal, characteristic
4#21860C211yesmaximal, characteristic, complemented
5#21409C211yesmaximal, characteristic, complemented
6#21200C211yesmaximal, characteristic
7C2×#1453C211yesmaximal, characteristic, complemented
8#9838C211yesmaximal, characteristic, complemented
9#9836C211yesmaximal, characteristic, complemented
10#9831C211yesmaximal, characteristic, complemented
11#9828C211yesmaximal, characteristic, complemented
12#9827C211yesmaximal, characteristic, complemented
13#9827C211yesmaximal, characteristic, complemented
14#9822C211yesmaximal, characteristic, complemented
15#9819C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3228—24no—
19#3228C418yescomplemented
20C2×#500C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#482—22no—
25C2×#478—22no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#417—22no—
29C2×C4×M16C2211yescharacteristic
30#2285C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1959—22no—
39#1957C2211yescharacteristic
40#1957C2211yescharacteristic
41#1957C2211yescharacteristic
42#1957C2211yescharacteristic
43#1956C2211yescharacteristic
44#1956C2211yescharacteristic
45#1956C2211yescharacteristic
46#1956C2211yescharacteristic
47#1956C2211yescharacteristic
48#1956C2211yescharacteristic
49#1948—22no—
50#1894—22no—
51#1875—22no—
52#1872—22no—
53#1853—22no—
54#1844C2211yescharacteristic
55#1844C2211yescharacteristic
56#1834C2211yescharacteristic
57#1834C2211yescharacteristic
58#1804—22no—
59#1803—22no—
60#1763—22no—
61#1453—24no—
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65C2×C82C2211yescharacteristic
66C22×#41—21no—
67C22×M16D412yes—
68#533—22no—
69#533—22no—
70#533—22no—
71#533C2×C414yes—
72#532—22no—
73#532—22no—
74#532—22no—
75#532C2×C414yes—
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2311yescharacteristic
79C2×#127C2311yescharacteristic
80C2×#125—21no—
81C2×#124—21no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×#121—21no—
85C2×#121—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×#117—21no—
91C2×#117—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C42C2311yescharacteristic
96#500—22no—
97#500C2×C414yescomplemented
98#498—22no—
99#498C2×C414yescomplemented
100#486—22no—
101#486—22no—
102#486C2×C414yescomplemented
103#486C2×C414yescomplemented
104#482—216no—
105#478—216no—
106#431—22no—
107#431—22no—
108#431C2×C414yes—
109#431D414yes—
110C2×#107—22no—
111C2×#106—22no—
112C2×#106—22no—
113C2×#106—22no—
114C2×C4⋊C8D412yes—
115C2×C4⋊C8D412yes—
116#417—48no—
117C2×#103—22no—
118C2×#103—22no—
119C2×#103—22no—
120C2×#103—22no—
121C2×#103—22no—
122C2×#103—22no—
123C2×#102—22no—
124C2×#102—22no—
125C2×#102—22no—
126C2×#102—22no—
127C2×#102—22no—
128C2×#102—22no—
129C2×C22⋊C8D412yes—
130C2×C22⋊C8D412yes—
131C4×M16—22no—
132C4×M16—22no—
133C2×#97D412yes—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#384—21no—
141#384—21no—
142#384—21no—
143#384C2311yescharacteristic
144#384C2311yescharacteristic
145#384C2311yescharacteristic
146#384C2311yescharacteristic
147#382—21no—
148#380—21no—
149#380—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#340—22no—
160#322—22no—
161#322—22no—
162#322—22no—
163C82—22no—
164C2×#41—24no—
165C2×#41—44no—
166C2×#41—44no—
167C2×#41—42no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—41no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C2×M16—24no—
181C2×M16—24no—
182C2×M16—42no—
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8C22×C412yes—
199C4⋊Q8C22×C412yes—
200C4⋊Q8C22×C412yes—
201#127—21no—
202#127—21no—
203#127—21no—
204#127C22×C412yes—
205#125—24no—
206#125—42no—
207#124—24no—
208#124—42no—
209C22⋊Q8—24no—
210C22⋊Q8—24no—
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213#121—24no—
214#121—24no—
215#121—42no—
216#121—42no—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—42no—
220C4×Q8—42no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—42no—
224C4×D4—42no—
225#117—24no—
226#117—42no—
227#117—82no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—82no—
253C2×C4⋊C4—82no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42C22×C412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42#4112yes—
266C2×C42#4112yes—
267C2×C42C2411yescharacteristic
268#107—28no—
269#106—28no—
270#106—82no—
271#106—82no—
272C4⋊C8—24no—
273C4⋊C8C22⋊C418yes—
274#103—44no—
275#103—44no—
276#103—44no—
277#103—44no—
278#103—82no—
279#103—82no—
280#102—28no—
281#102—28no—
282#102—44no—
283#102—44no—
284#102—82no—
285#102—82no—
286C22⋊C8—24no—
287C22⋊C8C22⋊C418yes—
288#97—24no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8C22×C412yes—
298C4×C8C22×C412yes—
299C4×C8C2×D412yes—
300C4×C8C2×D412yes—
301#95—41no—
302#95—41no—
303C24—41no—
304C24#12711yescharacteristic
305#41—416no—
306#41—88no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—44no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—82no—
333C2×D4—82no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—44no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4C2×C22⋊C412yes—
354C22×C4C4×D412yes—
355C22×C4C4×D412yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12412yes—
359C22×C4#12711yescharacteristic
360C22×C4#12711yescharacteristic
361C22×C4#12711yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365M16—48no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8C2×C22⋊C414yes—
382C2×C8C2×C22⋊C414yes—
383C2×C8AGL(1,ℤ/8)14yes—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—82no—
417C42—44no—
418C42—82no—
419C42C2×C22⋊C412yes—
420C42C4×D412yes—
421C42C4×D412yes—
422C42#12112yes—
423C42#12112yes—
424C42#12412yes—
425C42C23×C411yescharacteristic
426C42C22×D411yescharacteristic
427C42C2×#4111yescharacteristic
428C42C2×#4111yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—42no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23#39012yes—
443C23C2×#12711yescharacteristic
444Q8—44no—
445Q8—44no—
446Q8—44no—
447Q8—82no—
448Q8—82no—
449Q8—82no—
450D4—44no—
451D4—44no—
452D4—44no—
453D4—88no—
454D4—82no—
455D4—82no—
456D4—82no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—88no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—164no—
492C2×C4#39012yes—
493C2×C4#39012yes—
494C2×C4#39012yes—
495C2×C4C22×C22⋊C411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12411yescharacteristic
501C2×C4C2×AGL(1,ℤ/8)12yes—
502C2×C4#57512yes—
503C2×C4#57712yes—
504C2×C4#57712yes—
505C2×C4#57712yes—
506C2×C4#57812yes—
507C8—24no—
508C8—24no—
509C8—24no—
510C8—42no—
511C8—42no—
512C8—42no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—161no—
524C22—161no—
525C22—161no—
526C22—161no—
527C22C2×#39011yescharacteristic
528C22#315111yescharacteristic
529C22#315311yescharacteristic
530C22#315411yescharacteristic
531C22#315411yescharacteristic
532C22#315411yescharacteristic
533C22#315611yescharacteristic
534C4—22no—
535C4—22no—
536C4—22no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—161no—
541C4—161no—
542C4—161no—
543C4—161no—
544C4#289812yes—
545C4C4×AGL(1,ℤ/8)12yes—
546C4#295212yes—
547C4#305512yes—
548C4#306012yes—
549C4#313712yes—
550C2—42no—
551C2—162no—
552C2#1971311yescharacteristic
553C2#1971611yescharacteristic
554C2#1971811yescharacteristic
555C2#1972511yescharacteristic
556C2#2120511yescharacteristic
557C2#2122911yescharacteristic
558C2#2123611yescharacteristic
subgroupG/H
center, socleC23C2×#127
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
2. #21981C2semidirect
4. #21860C2semidirect
5. #21409C2semidirect
7. C2×#1453C2semidirect
8. #9838C2semidirect
9. #9836C2semidirect
10. #9831C2semidirect
11. #9828C2semidirect
12. #9827C2semidirect
13. #9827C2semidirect
14. #9822C2semidirect
15. #9819C2semidirect
19. #3228C4semidirect
97. #500C2×C4semidirect
99. #498C2×C4semidirect
102. #486C2×C4semidirect
103. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
#219812
#218632
#218602
#214092
#212002
C2×#14532
#98382
#98362
#98312
#98282
#98272
#98222
#98192