Finite Group Databasebeta

← #206281

Subgroups — #206281

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3210C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21863C211yesmaximal, characteristic
4#21860C211yesmaximal, characteristic, complemented
5#21859C211yesmaximal, characteristic
6#21835C211yesmaximal, characteristic, complemented
7C2×#1454C211yesmaximal, characteristic, complemented
8#9842C211yesmaximal, characteristic, complemented
9#9840C211yesmaximal, characteristic, complemented
10#9835C211yesmaximal, characteristic, complemented
11#9833C211yesmaximal, characteristic, complemented
12#9830C211yesmaximal, characteristic, complemented
13#9829C211yesmaximal, characteristic, complemented
14#9825C211yesmaximal, characteristic, complemented
15#9822C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3210—24no—
19#3210C418yescomplemented
20C2×#494C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#464—22no—
25C2×#452—22no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×#384C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1990C2211yescharacteristic
41#1989—22no—
42#1957C2211yescharacteristic
43#1957C2211yescharacteristic
44#1957C2211yescharacteristic
45#1957C2211yescharacteristic
46#1956C2211yescharacteristic
47#1956C2211yescharacteristic
48#1947—22no—
49#1875—22no—
50#1874—22no—
51#1872—22no—
52#1852—22no—
53#1845C2211yescharacteristic
54#1845C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57#1825—22no—
58#1804—22no—
59#1803—22no—
60#1802—22no—
61#1454—24no—
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65C2×C82C2211yescharacteristic
66C22×#41—21no—
67C22×Q16—22no—
68C22×SD16—22no—
69#533—22no—
70#533—22no—
71#533—22no—
72#533C2×C414yes—
73#532—22no—
74#532—22no—
75#532—22no—
76#532C2×C414yes—
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×#127C2311yescharacteristic
81C2×#125—21no—
82C2×#124—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×#121—21no—
86C2×#121—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×C4×D4—21no—
90C2×C4×D4—21no—
91C2×#117—21no—
92C2×#117—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C42C2311yescharacteristic
97#494—22no—
98#494C2×C414yescomplemented
99#492—22no—
100#492C2×C414yescomplemented
101#487—22no—
102#487C2×C414yescomplemented
103#486—22no—
104#486C2×C414yescomplemented
105#464—216no—
106#452—216no—
107#431—22no—
108#431—22no—
109#431—22no—
110#431—22no—
111#431C2×C414yes—
112#431D414yes—
113C2×#107—22no—
114C2×#107—22no—
115C2×C4⋊C8D412yes—
116C2×C4⋊C8D412yes—
117C2×C4⋊C8D412yes—
118C2×#103—22no—
119C2×#103—22no—
120C2×#103—22no—
121C2×#102—22no—
122C2×#102—22no—
123C2×#102—22no—
124C2×#102—22no—
125C2×#102—22no—
126C2×C22⋊C8D412yes—
127C2×C22⋊C8D412yes—
128C2×C22⋊C8D412yes—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135#384—21no—
136#384—21no—
137#384—21no—
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#384C2311yescharacteristic
142#382—21no—
143#380—21no—
144#380—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#322—22no—
160#322—22no—
161#322—22no—
162C82—22no—
163C2×#41—24no—
164C2×#41—42no—
165C2×#41—82no—
166C2×#41—82no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—41no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C2×Q16—28no—
180C2×Q16—82no—
181C2×Q16—82no—
182C2×SD16—28no—
183C2×SD16—82no—
184C2×SD16—82no—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8C22×C412yes—
201C4⋊Q8C22×C412yes—
202C4⋊Q8C22×C412yes—
203#127—21no—
204#127—21no—
205#127—21no—
206#127C22×C412yes—
207#125—24no—
208#125—42no—
209#124—24no—
210#124—42no—
211C22⋊Q8—24no—
212C22⋊Q8—24no—
213C22⋊Q8—42no—
214C22⋊Q8—42no—
215#121—24no—
216#121—24no—
217#121—42no—
218#121—42no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—42no—
226C4×D4—42no—
227#117—24no—
228#117—42no—
229#117—82no—
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—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—82no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4—82no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42C22×C412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42#4112yes—
268C2×C42#4112yes—
269C2×C42C2411yescharacteristic
270#107—82no—
271#107—82no—
272C4⋊C8—24no—
273C4⋊C8—24no—
274C4⋊C8C22⋊C418yes—
275#103—28no—
276#103—82no—
277#103—82no—
278#102—28no—
279#102—82no—
280#102—82no—
281#102—82no—
282#102—82no—
283C22⋊C8—24no—
284C22⋊C8—24no—
285C22⋊C8C22⋊C418yes—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8C22×C412yes—
295C4×C8C22×C412yes—
296C4×C8C2×D412yes—
297C4×C8C2×D412yes—
298#95—41no—
299#95—41no—
300C24—41no—
301C24#12711yescharacteristic
302#41—88no—
303#41—88no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—44no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—82no—
330C2×D4—82no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—44no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4C2×C22⋊C412yes—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4#12412yes—
356C22×C4#12711yescharacteristic
357C22×C4#12711yescharacteristic
358C22×C4#12711yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362Q16—88no—
363SD16—88no—
364C2×C8—22no—
365C2×C8—22no—
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×C8C2×C22⋊C414yes—
381C2×C8C2×C22⋊C414yes—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
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—82no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—82no—
415C42—44no—
416C42—82no—
417C42C2×C22⋊C412yes—
418C42C4×D412yes—
419C42C4×D412yes—
420C42#12112yes—
421C42#12112yes—
422C42#12412yes—
423C42C23×C411yescharacteristic
424C42C22×D411yescharacteristic
425C42C2×#4111yescharacteristic
426C42C2×#4111yescharacteristic
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—42no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23#39012yes—
441C23C2×#12711yescharacteristic
442Q8—44no—
443Q8—44no—
444Q8—44no—
445Q8—82no—
446Q8—82no—
447Q8—82no—
448D4—44no—
449D4—44no—
450D4—44no—
451D4—88no—
452D4—82no—
453D4—82no—
454D4—82no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—88no—
471C2×C4—81no—
472C2×C4—81no—
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—164no—
490C2×C4#39012yes—
491C2×C4#39012yes—
492C2×C4#39012yes—
493C2×C4C22×C22⋊C411yescharacteristic
494C2×C4C2×C4×D411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×#12411yescharacteristic
499C2×C4#57612yes—
500C2×C4#57612yes—
501C2×C4#57712yes—
502C2×C4#57712yes—
503C2×C4#57812yes—
504C2×C4#57812yes—
505C8—24no—
506C8—24no—
507C8—42no—
508C8—42no—
509C8—42no—
510C8—42no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—161no—
522C22—161no—
523C22—161no—
524C22—161no—
525C22C2×#39011yescharacteristic
526C22#315411yescharacteristic
527C22#315411yescharacteristic
528C22#315511yescharacteristic
529C22#315511yescharacteristic
530C22#315611yescharacteristic
531C22#315611yescharacteristic
532C4—22no—
533C4—22no—
534C4—22no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—161no—
539C4—161no—
540C4—161no—
541C4—161no—
542C4#289712yes—
543C4#294712yes—
544C4#295112yes—
545C4#306112yes—
546C4#306312yes—
547C4#313612yes—
548C2—42no—
549C2—162no—
550C2#1973511yescharacteristic
551C2#1973811yescharacteristic
552C2#1974111yescharacteristic
553C2#1974411yescharacteristic
554C2#2120611yescharacteristic
555C2#2123011yescharacteristic
556C2#2123411yescharacteristic
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. #21982C2semidirect
4. #21860C2semidirect
6. #21835C2semidirect
7. C2×#1454C2semidirect
8. #9842C2semidirect
9. #9840C2semidirect
10. #9835C2semidirect
11. #9833C2semidirect
12. #9830C2semidirect
13. #9829C2semidirect
14. #9825C2semidirect
15. #9822C2semidirect
19. #3210C4semidirect
98. #494C2×C4semidirect
100. #492C2×C4semidirect
102. #487C2×C4semidirect
104. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32102
#219822
#218632
#218602
#218592
#218352
C2×#14542
#98422
#98402
#98352
#98332
#98302
#98292
#98252
#98222