Finite Group Databasebeta

← #206146

Subgroups — #206146

Order 512.

1176 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3207C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21863C211yesmaximal, characteristic, complemented
4#21805C211yesmaximal, characteristic, complemented
5#21401C211yesmaximal, characteristic, complemented
6#21401C211yesmaximal, characteristic, complemented
7C2×#1450C211yesmaximal, characteristic, complemented
8#9833C211yesmaximal, characteristic, complemented
9#9832C211yesmaximal, characteristic, complemented
10#9829C211yesmaximal, characteristic, complemented
11#9827C211yesmaximal, characteristic, complemented
12#9824C211yesmaximal, characteristic, complemented
13#9822C211yesmaximal, characteristic, complemented
14#9821C211yesmaximal, characteristic, complemented
15#9820C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3207—24no—
19#3207C418yescomplemented
20C2×#493C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#463—22no—
25C2×#449—22no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1983—22no—
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1957C2211yescharacteristic
41#1957C2211yescharacteristic
42#1957C2211yescharacteristic
43#1957C2211yescharacteristic
44#1957C2211yescharacteristic
45#1957C2211yescharacteristic
46#1956C2211yescharacteristic
47#1956C2211yescharacteristic
48#1949—22no—
49#1893—22no—
50#1893—22no—
51#1875—22no—
52#1854—22no—
53#1845C2211yescharacteristic
54#1844C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57#1804—22no—
58#1788—22no—
59C4×#102—22no—
60C4×#102—22no—
61#1450—24no—
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65C2×C82C2211yescharacteristic
66C22×#41—21no—
67C22×#41—21no—
68C22×SD16—22no—
69C22×D8—22no—
70C23×C8D412yes—
71#532—22no—
72#532—22no—
73#532—22no—
74#532—22no—
75#532—22no—
76#532—22no—
77#532C2×C414yes—
78#532C2×C414yes—
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×#127C2311yescharacteristic
82C2×#127C2311yescharacteristic
83C2×#124—21no—
84C2×#124—21no—
85C2×#121—21no—
86C2×#121—21no—
87C2×#121—21no—
88C2×#121—21no—
89C2×C4×D4—21no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×#117—21no—
94C2×#117—21no—
95C22×C42C2311yescharacteristic
96#493—22no—
97#493C2×C414yes—
98#492—22no—
99#492C2×C414yes—
100#487—22no—
101#487C2×C414yes—
102#486—22no—
103#486C2×C414yes—
104#463—216no—
105#449—216no—
106#431—22no—
107#431—22no—
108#431C2×C414yes—
109#431D414yes—
110C2×#107—22no—
111C2×#106—22no—
112C2×C4⋊C8D412yes—
113C2×C4⋊C8D412yes—
114C2×#103—22no—
115C2×#103—22no—
116C2×#103—22no—
117C2×#102—22no—
118C2×#102—22no—
119C2×#102—22no—
120C2×#102—22no—
121C2×#102—22no—
122C2×#102—22no—
123C2×#102—22no—
124C2×#102—22no—
125C2×#102—22no—
126C2×C22⋊C8D412yes—
127C2×C22⋊C8D412yes—
128C2×C4×C8—22no—
129C2×C4×C8—22no—
130C2×C4×C8D412yes—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#382—21no—
142#382—21no—
143#380—21no—
144#380—21no—
145#380—21no—
146#380—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
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—
161C82—22no—
162C82—22no—
163C2×#41—24no—
164C2×#41—24no—
165C2×#41—42no—
166C2×#41—42no—
167C2×#41—82no—
168C2×#41—82no—
169C2×#41—82no—
170C2×#41—82no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—41no—
180C22×D4—41no—
181C23×C4—41no—
182C23×C4—41no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C2×SD16—28no—
187C2×SD16—82no—
188C2×SD16—82no—
189C2×D8—28no—
190C2×D8—82no—
191C2×D8—82no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—24no—
195C22×C8C2×D412yes—
196C22×C8C2×D412yes—
197C22×C8C2×D412yes—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8C22×C412yes—
208C4⋊Q8C22×C412yes—
209#127—21no—
210#127—21no—
211#127—21no—
212#127—21no—
213#127—21no—
214#127—21no—
215#127C22×C412yes—
216#127C22×C412yes—
217#124—24no—
218#124—24no—
219#124—42no—
220#124—42no—
221#121—24no—
222#121—24no—
223#121—24no—
224#121—24no—
225#121—42no—
226#121—42no—
227#121—42no—
228#121—42no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—42no—
234C4×D4—42no—
235C4×D4—42no—
236C4×D4—42no—
237#117—82no—
238#117—82no—
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—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C42—41no—
262C2×C42—41no—
263C2×C42C22×C412yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42#4112yes—
269C2×C42#4112yes—
270C2×C42C2411yescharacteristic
271#107—82no—
272#106—82no—
273C4⋊C8—24no—
274C4⋊C8C22⋊C418yes—
275#103—28no—
276#103—82no—
277#103—82no—
278#102—28no—
279#102—44no—
280#102—44no—
281#102—44no—
282#102—44no—
283#102—82no—
284#102—82no—
285#102—82no—
286#102—82no—
287C22⋊C8—24no—
288C22⋊C8C22⋊C418yes—
289C4×C8—24no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8C22×C412yes—
299C4×C8C22×C412yes—
300C4×C8C2×D412yes—
301C4×C8C2×D412yes—
302#95—41no—
303#95—41no—
304#95—41no—
305#95—41no—
306C24—41no—
307C24—41no—
308C24#12711yescharacteristic
309#41—88no—
310#41—88no—
311#41—88no—
312#41—88no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—82no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4C2×C22⋊C412yes—
370C22×C4C4×D412yes—
371C22×C4C4×D412yes—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4#12412yes—
375C22×C4#12711yescharacteristic
376C22×C4#12711yescharacteristic
377C22×C4#12711yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381SD16—88no—
382D8—88no—
383C2×C8—216no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8C2×C22⋊C414yes—
399C2×C8C2×C22⋊C414yes—
400C2×C8#13514yes—
401C2×C8#13514yes—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C42—82no—
429C42—82no—
430C42C2×C22⋊C412yes—
431C42C4×D412yes—
432C42C4×D412yes—
433C42#12112yes—
434C42#12112yes—
435C42#12412yes—
436C42C23×C411yescharacteristic
437C42C22×D411yescharacteristic
438C42C2×#4111yescharacteristic
439C42C2×#4111yescharacteristic
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—42no—
447C23—42no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23#39012yes—
461C23C2×#12711yescharacteristic
462Q8—44no—
463Q8—44no—
464Q8—82no—
465Q8—82no—
466D4—44no—
467D4—44no—
468D4—44no—
469D4—44no—
470D4—44no—
471D4—44no—
472D4—88no—
473D4—88no—
474D4—82no—
475D4—82no—
476D4—82no—
477D4—82no—
478D4—82no—
479D4—82no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—88no—
495C2×C4—88no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4#39012yes—
513C2×C4#39012yes—
514C2×C4#39012yes—
515C2×C4C22×C22⋊C411yescharacteristic
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×#12411yescharacteristic
521C2×C4C2×#13512yes—
522C2×C4C2×#13512yes—
523C2×C4#57612yes—
524C2×C4#57612yes—
525C2×C4#57712yes—
526C2×C4#57712yes—
527C8—24no—
528C8—24no—
529C8—24no—
530C8—24no—
531C8—42no—
532C8—42no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—161no—
547C22—161no—
548C22—161no—
549C22—161no—
550C22—161no—
551C22—161no—
552C22—161no—
553C22—161no—
554C22C2×#39011yescharacteristic
555C22#314711yescharacteristic
556C22#314711yescharacteristic
557C22#315411yescharacteristic
558C22#315411yescharacteristic
559C22#315511yescharacteristic
560C22#315511yescharacteristic
561C4—22no—
562C4—22no—
563C4—22no—
564C4—82no—
565C4—82no—
566C4—161no—
567C4—161no—
568C4—161no—
569C4—161no—
570C4#289712yes—
571C4C4×#13512yes—
572C4#294612yes—
573C4#305012yes—
574C4#306212yes—
575C4#313612yes—
576C2—42no—
577C2—162no—
578C2—162no—
579C2#1966511yescharacteristic
580C2#1966711yescharacteristic
581C2#1967011yescharacteristic
582C2#1967011yescharacteristic
583C2#2120611yescharacteristic
584C2#2122411yescharacteristic
585C2#2123511yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3207C2
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. #21984C2semidirect
3. #21863C2semidirect
4. #21805C2semidirect
5. #21401C2semidirect
6. #21401C2semidirect
7. C2×#1450C2semidirect
8. #9833C2semidirect
9. #9832C2semidirect
10. #9829C2semidirect
11. #9827C2semidirect
12. #9824C2semidirect
13. #9822C2semidirect
14. #9821C2semidirect
15. #9820C2semidirect
19. #3207C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32072
#219842
#218632
#218052
#214012
C2×#14502
#98332
#98322
#98292
#98272
#98242
#98222
#98212
#98202