Finite Group Databasebeta

← #355459

Subgroups — #355459

Order 512.

1740 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34065C211yesmaximal, characteristic
2C2×#1897C211yesmaximal, characteristic
3C2×#1896C211yesmaximal, characteristic
4#20377C211yesmaximal, characteristic, complemented
5#13756C211yesmaximal, characteristic, complemented
6#10933C212yesmaximal, complemented
7#10886C212yesmaximal, complemented
8#10808C212yesmaximal, complemented
9#10790C212yesmaximal, complemented
10#10720C211yesmaximal, characteristic, complemented
11#10719C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×#104C2212yes—
18C22×#104C2212yes—
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C2×C4×M16C2212yes—
22#2324C414yescomplemented
23#2323C414yescomplemented
24#2322—22no—
25#2321—22no—
26#2297C2211yescharacteristic
27#2296C2211yescharacteristic
28#2293—22no—
29#2290—22no—
30#2280C2211yescharacteristic
31C4×#117C2211yescharacteristic
32#1897C418yescomplemented
33#1896C418yescomplemented
34#1750—22no—
35#1750—22no—
36C2×#339C2212yes—
37C2×#339C2212yes—
38C2×#339C2212yes—
39C2×#339C2212yes—
40C2×#339C2212yes—
41C2×#339C2212yes—
42#1728C2212yes—
43C4⋊#95—22no—
44C22⋊#95—22no—
45C4×#95C2212yes—
46C4×#95C2212yes—
47C22×#41C2311yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2312yes—
52#525—22no—
53#525—22no—
54#525C2×C414yes—
55#525C2×C414yes—
56C2×#123—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×#121C2×C412yes—
59C2×#120—21no—
60C4×#41—42no—
61C4×#41—42no—
62C2×C4×Q8C2×C412yescomplemented
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C412yescomplemented
66C2×#117—22no—
67C2×#117—21no—
68C2×#117—42no—
69C2×#117—42no—
70C2×#117—42no—
71C2×#117—42no—
72C2×#117C2×C414yes—
73C2×#117C2×C414yescomplemented
74C2×#117D412yes—
75C2×#117D412yes—
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4D412yes—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D412yes—
83C22×C42—21no—
84C22×C42—22no—
85C22×C42—21no—
86C22×C42—42no—
87C22×C42—42no—
88C22×C42C2×C414yescomplemented
89C22×C42C2312yes—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×#104—42no—
94C2×#104—42no—
95C2×#104—42no—
96C2×#104—42no—
97C2×#104C2×C418yes—
98C2×#104C2×C418yescomplemented
99#407—22no—
100#407—22no—
101#407C2×C414yescomplemented
102#407C2×C414yescomplemented
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C4×M16—42no—
106C4×M16—42no—
107C2×#97—22no—
108C2×C4×C8—22no—
109#391C2×C412yescomplemented
110#390—24no—
111#390C2×C412yescomplemented
112#389—24no—
113#389C2×C414yescomplemented
114#388—24no—
115#388—24no—
116#388—24no—
117#387—24no—
118#386—22no—
119#386—24no—
120#385—22no—
121#385—24no—
122#381—21no—
123#380C2×C414yes—
124#379—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4C2×C414yescomplemented
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130#376—21no—
131C2×#95—24no—
132C2×#95—24no—
133C43—21no—
134#339—42no—
135#339—42no—
136#339—42no—
137#339—42no—
138#339C2×C418yescomplemented
139#339C2×C418yescomplemented
140#336—22no—
141#336—22no—
142C25C2×D411yescharacteristic
143C2×#41—21no—
144C2×#41—21no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41C22×C412yes—
150C2×#41C22×C412yes—
151C22×Q8C22×C411yescharacteristic
152C22×D4—21no—
153C22×D4—22no—
154C22×D4C22×C411yescharacteristic
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—48no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4C22×C412yes—
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4C2411yescharacteristic
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16C22×C414yes—
188C2×M16C22×C414yes—
189C22×C8—22no—
190C22×C8—22no—
191#123—24no—
192#123—24no—
193C22⋊Q8—24no—
194C22⋊Q8C4218yes—
195#121—24no—
196#121C4218yes—
197#120—24no—
198#120—24no—
199C4×Q8—44no—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205#117—21no—
206#117—21no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—48no—
212#117—48no—
213#117—42no—
214#117—42no—
215#117—42no—
216#117—42no—
217#117—44no—
218#117—48no—
219#117C22×C412yes—
220#117C22×C412yes—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C22⋊C414yes—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4C22⋊C414yes—
248C2×C22⋊C4C22⋊C414yes—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—48no—
270C2×C42—44no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—48no—
274C2×C42C22×C414yes—
275C2×C42C22×C414yes—
276C2×C42C22×C412yes—
277C2×C42C22×C412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D412yes—
281#104—48no—
282#104—48no—
283C22⋊C8—28no—
284C22⋊C8—28no—
285#97—44no—
286C4×C8—44no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—48no—
292#95—48no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—44no—
299C24C2×C22⋊C412yes—
300C24C4×D412yes—
301C24C4×D412yes—
302C24C22×D411yescharacteristic
303#41—24no—
304#41—24no—
305#41—24no—
306#41—24no—
307#41—42no—
308#41—42no—
309#41—42no—
310#41—42no—
311C2×Q8—21no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8C2×C4212yes—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—28no—
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—84no—
326C2×D4C2×C4212yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—416no—
361C22×C4—44no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—416no—
365C22×C4—416no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—84no—
375C22×C4C2×C4214yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C414yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4#11714yes—
381C22×C4C4×D414yes—
382C22×C4C4×D414yes—
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C23×C411yescharacteristic
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C4H(5,𝔽2)12yes—
394C22×C42−1+412yes—
395M16—24no—
396M16—24no—
397M16—24no—
398M16—24no—
399M16—42no—
400M16—42no—
401M16—42no—
402M16—42no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C4⋊C4—22no—
410C4⋊C4—28no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—48no—
416C4⋊C4—44no—
417C4⋊C4—84no—
418C4⋊C4C2×C4214yes—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—48no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C42—22no—
434C42—22no—
435C42—24no—
436C42—24no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—48no—
446C42—44no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—84no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—41no—
471C23—41no—
472C23—44no—
473C23C4×C22⋊C414yes—
474C23C22×C22⋊C411yescharacteristic
475C23C2×C4×D411yescharacteristic
476C23C2×C4×D411yescharacteristic
477C23#53411yescharacteristic
478C23#53411yescharacteristic
479C23#53511yescharacteristic
480C23#53611yescharacteristic
481Q8—24no—
482Q8—42no—
483D4—24no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—88no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—432no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—88no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4C4×C22⋊C414yes—
537C2×C4C22×C4211yescharacteristic
538C2×C4C2×#11711yescharacteristic
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C2×C4×D411yescharacteristic
541C2×C4#51812yes—
542C2×C4#51812yes—
543C2×C4#51812yes—
544C2×C4#52012yes—
545C2×C4#52012yes—
546C2×C4#52012yes—
547C2×C4#53412yes—
548C2×C4#53712yes—
549C2×C4#53812yes—
550C2×C4#53812yes—
551C2×C4#53912yes—
552C2×C4#53912yes—
553C8—44no—
554C8—44no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—42no—
572C22—82no—
573C22—82no—
574C22C2×C4×C22⋊C411yescharacteristic
575C22#230811yescharacteristic
576C22#231111yescharacteristic
577C22#232111yescharacteristic
578C22#232111yescharacteristic
579C22#232211yescharacteristic
580C22#232211yescharacteristic
581C22#290114yes—
582C4—24no—
583C4—48no—
584C4—42no—
585C4—44no—
586C4—42no—
587C4—44no—
588C4—42no—
589C4—84no—
590C4—82no—
591C4—82no—
592C4#227512yes—
593C4#231812yes—
594C4#232512yes—
595C4#232512yes—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—44no—
601C2—82no—
602C2#1378411yescharacteristic
603C2#2040512yes—
604C2#2177512yes—
605C2#2190112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C2×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
4. #20377C2semidirect
5. #13756C2semidirect
6. #10933C2semidirect
7. #10886C2semidirect
8. #10808C2semidirect
9. #10790C2semidirect
10. #10720C2semidirect
11. #10719C2semidirect
22. #2324C4semidirect
23. #2323C4semidirect
32. #1897C4semidirect
33. #1896C4semidirect
62. C2×C4×Q8C2×C4semidirect
65. C2×C4×D4C2×C4semidirect
73. C2×#117C2×C4semidirect
88. C22×C42C2×C4semidirect
98. C2×#104C2×C4semidirect
101. #407C2×C4semidirect
102. #407C2×C4semidirect
109. #391C2×C4semidirect
111. #390C2×C4semidirect
113. #389C2×C4semidirect
126. C4×C4⋊C4C2×C4semidirect
138. #339C2×C4semidirect
139. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340652
C2×#18972
C2×#18962
#203772
#137562
#109332
#108862
#108082
#107902
#107202
#107192