Finite Group Databasebeta

← #360167

Subgroups — #360167

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34065C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic, complemented
3C2×#1897C211yesmaximal, characteristic
4C2×#1896C211yesmaximal, characteristic
5C2×#1806C212yesmaximal, complemented
6#20627C211yesmaximal, characteristic, complemented
7#11226C212yesmaximal, complemented
8#11075C212yesmaximal, complemented
9#10852C212yesmaximal, complemented
10#10720C211yesmaximal, characteristic, complemented
11#10719C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#423C2212yes—
18C22×#104C2212yes—
19C22×#104C2212yes—
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22#2324C414yescomplemented
23#2323C414yescomplemented
24#2322—22no—
25#2321—22no—
26#2297C2211yescharacteristic
27#2296—28no—
28#2296C2211yescharacteristic
29#2293—22no—
30C2×#389—22no—
31#2290—22no—
32C2×#385—22no—
33#2280C2211yescharacteristic
34C4×#117C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37#1897—24no—
38#1896—24no—
39#1806—28no—
40#1798—22no—
41#1798—22no—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48#1728C2212yes—
49C22×#41C2311yescharacteristic
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2312yes—
54#525—22no—
55#525—22no—
56#525D414yes—
57#525Q814yes—
58C2×#123—21no—
59C2×C22⋊Q8C2×C412yes—
60C2×#121C2×C412yes—
61C2×#120—21no—
62C4×#41—24no—
63C4×#41—24no—
64C2×C4×Q8C2×C412yescomplemented
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4C2×C412yescomplemented
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—24no—
71C2×#117—24no—
72C2×#117—21no—
73C2×#117—24no—
74C2×#117—42no—
75C2×#117C2×C414yes—
76C2×#117D412yes—
77C2×#117D412yes—
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4D412yes—
85C22×C22⋊C4—21no—
86C22×C22⋊C4D412yes—
87C22×C42—21no—
88C22×C42—24no—
89C22×C42—21no—
90C22×C42—22no—
91C22×C42—22no—
92C22×C42—42no—
93C22×C42C2312yes—
94C22×C42C2312yes—
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#423—24no—
98#423—24no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×#104—24no—
102C2×#104—24no—
103C2×#104—24no—
104C2×#104—24no—
105C2×#104—42no—
106C2×#104—42no—
107#407—22no—
108#407—22no—
109#407—22no—
110#407—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113#391C2×C412yescomplemented
114#390—24no—
115#390C2×C412yescomplemented
116#389—216no—
117#389—24no—
118#389C2×C414yescomplemented
119#388—24no—
120#388—24no—
121#388—24no—
122#387—24no—
123#386—22no—
124#386—24no—
125#385—216no—
126#385—22no—
127#385—24no—
128#381—21no—
129#380C2×C414yes—
130#379—21no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2×C414yescomplemented
136C4×C4⋊C4D418yescomplemented
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140#376—21no—
141C2×#95—22no—
142C2×#95—22no—
143C43—21no—
144#339—24no—
145#339—24no—
146#339—24no—
147#339—24no—
148#339—42no—
149#339—42no—
150#336—24no—
151C25C2×D411yescharacteristic
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—22no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41C2×D412yes—
159C2×#41C2×Q812yes—
160C22×Q8C22×C411yescharacteristic
161C22×D4—21no—
162C22×D4—22no—
163C22×D4C22×C411yescharacteristic
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—48no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4C22×C412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×Q812yes—
183C23×C4#4112yes—
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C22×C8—22no—
199C22×C8—22no—
200#123—24no—
201#123—24no—
202C22⋊Q8—24no—
203C22⋊Q8C4⋊C418yescomplemented
204#121—24no—
205#121C4⋊C418yescomplemented
206#120—24no—
207#120—24no—
208C4×Q8—28no—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—28no—
212C4×D4—44no—
213C4×D4—44no—
214#117—21no—
215#117—21no—
216#117—216no—
217#117—28no—
218#117—28no—
219#117—22no—
220#117—22no—
221#117—22no—
222#117—22no—
223#117—48no—
224#117—48no—
225#117—44no—
226#117C2×D412yes—
227#117C2×Q812yes—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C22⋊C414yes—
241C2×C4⋊C4C22⋊C414yes—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C412yes—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4C22⋊C414yes—
259C2×C22⋊C4C22⋊C414yes—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—24no—
278C2×C42—48no—
279C2×C42—48no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—44no—
283C2×C42C22×C412yes—
284C2×C42C22×C412yes—
285C2×C42C2×D414yes—
286C2×C42C2×D414yes—
287C2×C42C2×D412yes—
288C2×C42C2×D414yes—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42#4114yes—
292C4⋊C8—44no—
293C4⋊C8—44no—
294#104—48no—
295#104—48no—
296C22⋊C8—44no—
297C22⋊C8—44no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—28no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—44no—
310C24C2×C22⋊C412yes—
311C24#12112yes—
312C24C22⋊Q812yes—
313C24C22×D411yescharacteristic
314#41—24no—
315#41—24no—
316#41—24no—
317#41—24no—
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322C2×Q8—21no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8C2×C4⋊C412yes—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—28no—
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—84no—
337C2×D4C2×C4⋊C412yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—416no—
368C22×C4—48no—
369C22×C4—44no—
370C22×C4—416no—
371C22×C4—416no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—84no—
382C22×C4C2×C22⋊C414yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C4⋊C414yes—
387C22×C4#11714yes—
388C22×C4#12014yes—
389C22×C4#12114yes—
390C22×C4#12112yes—
391C22×C4C22⋊Q814yes—
392C22×C4C22⋊Q812yes—
393C22×C4#12314yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404M16—42no—
405M16—42no—
406M16—42no—
407M16—42no—
408M16—42no—
409M16—42no—
410M16—42no—
411M16—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C4⋊C4—22no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—28no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—416no—
425C4⋊C4—416no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—84no—
429C4⋊C4C2×C4⋊C414yes—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—28no—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—48no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—44no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C42—24no—
445C42—24no—
446C42—22no—
447C42—22no—
448C42—24no—
449C42—24no—
450C42—24no—
451C42—24no—
452C42—48no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—44no—
460C42—84no—
461C42C4×D418yes—
462C42#12114yes—
463C42#12114yes—
464C23—24no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—41no—
481C23—44no—
482C23—41no—
483C23#38014yes—
484C23C22×C22⋊C411yescharacteristic
485C23C2×#12011yescharacteristic
486C23C2×#12111yescharacteristic
487C23C2×C22⋊Q811yescharacteristic
488C23C2×#12311yescharacteristic
489C23#53511yescharacteristic
490C23#53611yescharacteristic
491Q8—24no—
492Q8—42no—
493D4—24no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—88no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—432no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—88no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4#38014yes—
543C2×C4#38614yes—
544C2×C4#38614yes—
545C2×C4C22×C4⋊C411yescharacteristic
546C2×C4C2×#11711yescharacteristic
547C2×C4C2×C4×D412yes—
548C2×C4#52012yes—
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×#12112yes—
551C2×C4C2×C22⋊Q811yescharacteristic
552C2×C4#53712yes—
553C2×C4D4212yes—
554C2×C4#54612yes—
555C2×C4#54712yes—
556C2×C4#54812yes—
557C2×C4D4×Q812yes—
558C2×C4#55212yes—
559C2×C4#55312yes—
560C2×C4#55412yes—
561C8—82no—
562C8—82no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—44no—
580C22—82no—
581C22—82no—
582C22C2×#38011yescharacteristic
583C22C2×#38611yescharacteristic
584C22#231111yescharacteristic
585C22#240811yescharacteristic
586C22#240911yescharacteristic
587C22#248611yescharacteristic
588C22#248711yescharacteristic
589C22#290114yes—
590C4—24no—
591C4—24no—
592C4—24no—
593C4—48no—
594C4—42no—
595C4—44no—
596C4—44no—
597C4—84no—
598C4—82no—
599C4—82no—
600C4D4×C4⋊C412yes—
601C4#235212yes—
602C4#241612yes—
603C4#241712yes—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—44no—
609C2—82no—
610C2#1387211yescharacteristic
611C2#2065512yes—
612C2#2098012yes—
613C2#2190112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2296C2semidirect
5. C2×#1806C2semidirect
6. #20627C2semidirect
7. #11226C2semidirect
8. #11075C2semidirect
9. #10852C2semidirect
10. #10720C2semidirect
11. #10719C2semidirect
22. #2324C4semidirect
23. #2323C4semidirect
64. C2×C4×Q8C2×C4semidirect
67. C2×C4×D4C2×C4semidirect
113. #391C2×C4semidirect
115. #390C2×C4semidirect
118. #389C2×C4semidirect
135. C4×C4⋊C4C2×C4semidirect
136. C4×C4⋊C4D4semidirect
203. C22⋊Q8C4⋊C4semidirect
205. #121C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#340652
C2×#22962
C2×#18972
C2×#18962
C2×#18062
#206272
#112262
#110752
#108522
#107202
#107192