Finite Group Databasebeta

← #496276

Subgroups — #496276

Order 512.

1156 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3050C211yesmaximal, characteristic
2C2×#2112C211yesmaximal, characteristic, complemented
3C2×#1807C211yesmaximal, characteristic, complemented
4#13065C214yesmaximal, complemented
5#13001C214yesmaximal, complemented
6#12870C212yesmaximal, complemented
7#12735C212yesmaximal, complemented
8C22×#135C2211yescharacteristic
9C24×C8C2211yescharacteristic
10C2×#525—21no—
11C22×M32C2212yescomplemented
12#3050—21no—
13#3050—21no—
14#3050—21no—
15#3050—21no—
16#3050C412yescomplemented
17#3050C412yescomplemented
18#3050C2212yes—
19#3050C2212yes—
20C2×#467C411yescharacteristic, complemented
21C2×#467C411yescharacteristic, complemented
22C2×#466C411yescharacteristic, complemented
23C2×#466C411yescharacteristic, complemented
24C2×#465—21no—
25C2×#465—21no—
26C2×#427C2211yescharacteristic
27C22×#106C2212yescomplemented
28C2×#416—21no—
29C2×#380—21no—
30C2×#360C2218yescomplemented
31#2112—21no—
32#2112—21no—
33#2112C2212yes—
34#2112C2212yescomplemented
35C2×C22⋊C16—21no—
36C2×C22⋊C16—21no—
37#1807—21no—
38#1807—21no—
39#1807C2212yes—
40#1807C2212yescomplemented
41C2×#340—21no—
42#1386C2218yescomplemented
43C22×#41—21no—
44C24×C4D411yescharacteristic
45C2×#135—21no—
46C2×#135—21no—
47C2×#135—44no—
48C2×#135C2×C412yescomplemented
49C2×#135C2312yes—
50C22×Q16C2×C411yescharacteristic, complemented
51C22×SD16—21no—
52C22×D8C2×C411yescharacteristic, complemented
53C23×C8—44no—
54C23×C8C2×C411yescharacteristic
55C23×C8C2×C411yescharacteristic
56C23×C8C2×C411yescharacteristic
57C23×C8C2×C411yescharacteristic
58C23×C8D411yescharacteristic
59C23×C8D411yescharacteristic
60C23×C8D411yescharacteristic
61C23×C8D411yescharacteristic
62C23×C8D411yescharacteristic
63C23×C8D411yescharacteristic
64C23×C8D411yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68#525—22no—
69#525—22no—
70#525—22no—
71#525—22no—
72#525—41no—
73#525—41no—
74#525—41no—
75#525—41no—
76C2×#123—41no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×#121—21no—
80C2×#121—21no—
81C2×#120—41no—
82C2×C4×D4—41no—
83C2×C4×D4—41no—
84C2×#117—21no—
85C22×C4⋊C4—22no—
86C22×C22⋊C4—41no—
87C2×M32—22no—
88C2×M32—44no—
89C2×M32C2314yes—
90C22×C16—22no—
91#467—21no—
92#467—21no—
93#467—21no—
94#467—21no—
95#467C2×C412yes—
96#467C2×C412yes—
97#467C2×C412yescomplemented
98#467C2×C412yescomplemented
99#466—21no—
100#466—21no—
101#466—21no—
102#466—21no—
103#466C2×C412yes—
104#466C2×C412yes—
105#466C2×C412yescomplemented
106#466C2×C412yescomplemented
107#465—22no—
108#465—22no—
109#465—22no—
110#465—22no—
111#465—22no—
112#465—22no—
113#465—22no—
114#465—22no—
115#427—21no—
116#427—21no—
117#427C2×C412yescomplemented
118#427C2312yes—
119C2×#107C2×C411yescharacteristic, complemented
120C2×#107C2×C411yescharacteristic, complemented
121C2×#106—21no—
122C2×#106—22no—
123C2×#106—44no—
124C2×#106C2314yes—
125#416—22no—
126#416—22no—
127#416—22no—
128#416—22no—
129C2×#103—21no—
130C2×#103—21no—
131C2×#102—21no—
132C2×#102—21no—
133#380—22no—
134#380—22no—
135#380—41no—
136#380—41no—
137C2×#95—41no—
138#360—48no—
139C22⋊C16—22no—
140C22⋊C16—22no—
141C22⋊C16—22no—
142C22⋊C16—22no—
143#340—22no—
144#340—22no—
145C25D811yescharacteristic
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—44no—
149C2×#41—44no—
150C2×#41—41no—
151C2×#41—41no—
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4—41no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—41no—
160C23×C4—42no—
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4D811yescharacteristic
166C23×C4D811yescharacteristic
167C23×C4D811yescharacteristic
168C23×C4SD1611yescharacteristic
169C23×C4SD1611yescharacteristic
170C23×C4SD1611yescharacteristic
171C23×C4SD1611yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175#135—28no—
176#135—44no—
177#135—44no—
178#135—44no—
179C2×Q16—21no—
180C2×Q16—42no—
181C2×Q16C22×C412yes—
182C2×SD16—22no—
183C2×SD16—22no—
184C2×SD16—44no—
185C2×D8—21no—
186C2×D8—42no—
187C2×D8C22×C412yes—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—24no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8—42no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8—42no—
202C22×C8C22⋊C411yescharacteristic
203C22×C8C22⋊C411yescharacteristic
204C22×C8C22⋊C411yescharacteristic
205C22×C8C22⋊C411yescharacteristic
206C22×C8C22⋊C411yescharacteristic
207C22×C8C22⋊C411yescharacteristic
208C22×C8C22⋊C411yescharacteristic
209C22×C8C22⋊C411yescharacteristic
210C22×C8C22⋊C411yescharacteristic
211C22×C8C22⋊C411yescharacteristic
212C22×C8C22⋊C411yescharacteristic
213C22×C8C22⋊C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8C2×D411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22×C8C2411yescharacteristic
230#123—42no—
231#123—42no—
232#123—42no—
233#123—42no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C22⋊Q8—41no—
239C22⋊Q8—41no—
240C22⋊Q8—41no—
241C22⋊Q8—41no—
242#121—22no—
243#121—22no—
244#121—22no—
245#121—22no—
246#121—41no—
247#121—41no—
248#121—41no—
249#121—41no—
250#120—42no—
251#120—42no—
252#120—42no—
253#120—42no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—42no—
262#117—22no—
263#117—22no—
264#117—41no—
265#117—41no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—84no—
282C2×C42—41no—
283M32—28no—
284M32—44no—
285C2×C16—24no—
286C2×C16—44no—
287#107—21no—
288#107—21no—
289#107C22×C412yes—
290#107C22×C412yes—
291#106—22no—
292#106—22no—
293#106—28no—
294#106—44no—
295#103—22no—
296#103—22no—
297#103—22no—
298#103—22no—
299#102—22no—
300#102—22no—
301#102—22no—
302#102—22no—
303#95—42no—
304#95—42no—
305C24—24no—
306C24—42no—
307C24—42no—
308C24—41no—
309C24#10211yescharacteristic
310C24#10211yescharacteristic
311C24#10211yescharacteristic
312C24#10211yescharacteristic
313C24C2×D811yescharacteristic
314C24C2×D811yescharacteristic
315C24C2×D811yescharacteristic
316#41—44no—
317#41—44no—
318#41—44no—
319#41—44no—
320#41—84no—
321#41—84no—
322C2×Q8—22no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—41no—
326C2×D4—22no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—84no—
337C2×D4—84no—
338C2×D4—84no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4#10211yescharacteristic
371C22×C4#10211yescharacteristic
372C22×C4#10211yescharacteristic
373C22×C4#10211yescharacteristic
374C22×C4#10211yescharacteristic
375C22×C4#10211yescharacteristic
376C22×C4#10211yescharacteristic
377C22×C4#10211yescharacteristic
378C22×C4#10211yescharacteristic
379C22×C4#10211yescharacteristic
380C22×C4#10211yescharacteristic
381C22×C4#10211yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×D811yescharacteristic
389C22×C4C2×D811yescharacteristic
390C22×C4C2×D811yescharacteristic
391C22×C4C2×SD1611yescharacteristic
392C22×C4C2×SD1611yescharacteristic
393C22×C4C2×SD1611yescharacteristic
394C22×C4C2×SD1611yescharacteristic
395C22×C4C2×SD1611yescharacteristic
396C22×C4C2×SD1611yescharacteristic
397C22×C4C22×D411yescharacteristic
398Q16—24no—
399Q16—42no—
400SD16—44no—
401SD16—44no—
402D8—24no—
403D8—42no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8C2×C22⋊C411yescharacteristic
432C2×C8C2×C22⋊C411yescharacteristic
433C2×C8C2×C22⋊C411yescharacteristic
434C2×C8C2×C22⋊C411yescharacteristic
435C2×C8C2×C22⋊C411yescharacteristic
436C2×C8C2×C22⋊C411yescharacteristic
437C2×C8C23×C411yescharacteristic
438C2×C8C22×D411yescharacteristic
439C2×C8H(5,𝔽2)14yes—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—84no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C42—42no—
464C42—42no—
465C16—44no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—416no—
487C23—42no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23#36014yescomplemented
492C23C2×#10211yescharacteristic
493C23C2×#10211yescharacteristic
494C23C2×#10211yescharacteristic
495C23C2×#10211yescharacteristic
496C23C2×#10211yescharacteristic
497C23C2×#10211yescharacteristic
498C23C22×D811yescharacteristic
499Q8—42no—
500Q8—42no—
501Q8—82no—
502D4—42no—
503D4—42no—
504D4—82no—
505D4—84no—
506D4—84no—
507D4—84no—
508D4—84no—
509D4—84no—
510D4—84no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—84no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—168no—
556C2×C4C2×#10211yescharacteristic
557C2×C4C2×#10211yescharacteristic
558C2×C4C2×#10211yescharacteristic
559C2×C4C2×#10211yescharacteristic
560C2×C4C2×#10211yescharacteristic
561C2×C4C2×#10211yescharacteristic
562C2×C4C22×C22⋊C411yescharacteristic
563C2×C4#53412yes—
564C2×C4C22×SD1611yescharacteristic
565C2×C4#57512yes—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—21no—
570C8—21no—
571C8—21no—
572C8#51812yes—
573C8#53412yes—
574C22—28no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—164no—
601C22C2×#36012yes—
602C22C2×#36012yes—
603C22C2×#36012yes—
604C22#214512yes—
605C22C22×#10211yescharacteristic
606C22#318612yes—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—164no—
617C4#230811yescharacteristic
618C4#289611yescharacteristic
619C4#318712yes—
620C2—22no—
621C2—22no—
622C2—22no—
623C2—22no—
624C2—22no—
625C2—41no—
626C2—41no—
627C2—162no—
628C2#2175811yescharacteristic
629C2C22×#36011yescharacteristic
630C2C2×#214511yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#3050C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2112C2semidirect
3. C2×#1807C2semidirect
4. #13065C2semidirect
5. #13001C2semidirect
6. #12870C2semidirect
7. #12735C2semidirect
11. C22×M32C22semidirect
16. #3050C4semidirect
17. #3050C4semidirect
20. C2×#467C4semidirect
21. C2×#467C4semidirect
22. C2×#466C4semidirect
23. C2×#466C4semidirect
27. C22×#106C22semidirect
30. C2×#360C22semidirect
34. #2112C22semidirect
40. #1807C22semidirect
42. #1386C22semidirect
48. C2×#135C2×C4semidirect
50. C22×Q16C2×C4semidirect
52. C22×D8C2×C4semidirect
97. #467C2×C4semidirect
98. #467C2×C4semidirect
105. #466C2×C4semidirect
106. #466C2×C4semidirect
117. #427C2×C4semidirect
119. C2×#107C2×C4semidirect
120. C2×#107C2×C4semidirect
491. C23#360semidirect

Maximal subgroups

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

subgroupindex
C2×#30502
C2×#21122
C2×#18072
#130652
#130012
#128702
#127352