Finite Group Databasebeta

← #508306

Subgroups — #508306

Order 512.

811 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2220C211yesmaximal, characteristic, complemented
2C2×#2112C211yesmaximal, characteristic, complemented
3#32931C211yesmaximal, characteristic, complemented
4#32930C211yesmaximal, characteristic
5#13493C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13383C211yesmaximal, characteristic, complemented
8#13382C211yesmaximal, characteristic, complemented
9#13381C211yesmaximal, characteristic, complemented
10#13380C211yesmaximal, characteristic, complemented
11#13305C211yesmaximal, characteristic, complemented
12#13305C211yesmaximal, characteristic, complemented
13#13062C211yesmaximal, characteristic, complemented
14#13061C211yesmaximal, characteristic, complemented
15#12836C211yesmaximal, characteristic, complemented
16C24×C8C2211yescharacteristic
17C22×SD32C2211yescharacteristic, complemented
18C22×M32C2211yescharacteristic
19C22×M32C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×#467C2211yescharacteristic
22C2×#467C2211yescharacteristic
23C2×#466C2211yescharacteristic
24C2×#466C2211yescharacteristic
25C2×#466C2211yescharacteristic
26#2848—21no—
27#2847—21no—
28#2220—24no—
29C2×#367C2211yescharacteristic, complemented
30C2×#366C2211yescharacteristic, complemented
31C2×#366C2211yescharacteristic, complemented
32C2×#358C2211yescharacteristic, complemented
33C2×#358C2211yescharacteristic, complemented
34C2×#358C2211yescharacteristic, complemented
35C2×#357C2211yescharacteristic, complemented
36C2×#357C2211yescharacteristic, complemented
37C2×#357C2211yescharacteristic, complemented
38#2112—24no—
39C2×C22⋊C16C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×C22⋊C16C2211yescharacteristic
42C2×C22⋊C16C2211yescharacteristic
43#1893—21no—
44#1893—21no—
45#1893—21no—
46#1892—21no—
47#1892—21no—
48#1892—21no—
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1808C2211yescharacteristic
54#1808C2211yescharacteristic
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59#1380C2211yescharacteristic, complemented
60#1380C2211yescharacteristic, complemented
61C24×C4D411yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C23×C8—22no—
65C23×C8—22no—
66C23×C8—41no—
67C23×C8—41no—
68C23×C8—41no—
69C23×C8—41no—
70C23×C8D411yescharacteristic
71C23×C8D411yescharacteristic
72C23×C8D411yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×#121—21no—
80C2×#121—21no—
81C2×#121—21no—
82C2×#120—41no—
83C2×SD32—22no—
84C2×SD32—41no—
85C2×SD32—41no—
86C2×M32—22no—
87C2×M32—41no—
88C2×M32—41no—
89C2×M32—41no—
90C2×M32—41no—
91C2×M32D414yescomplemented
92C22×C16C2311yescharacteristic
93C22×C16C2311yescharacteristic
94C22×C16C2311yescharacteristic
95C22×C16C2311yescharacteristic
96#467—22no—
97#467—22no—
98#467—41no—
99#467—41no—
100#467—41no—
101#467—41no—
102#466—22no—
103#466—22no—
104#466—41no—
105#466—41no—
106#466—41no—
107#466—41no—
108C2×#107C2311yescharacteristic
109C2×#107C2311yescharacteristic
110C2×#107C2311yescharacteristic
111C2×#107C2311yescharacteristic
112C2×#107C2311yescharacteristic
113C2×#107C2311yescharacteristic
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#102—21no—
120#386—41no—
121#386—41no—
122#386—41no—
123#385—41no—
124#385—41no—
125#385—41no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132#379—41no—
133#367—41no—
134#366—22no—
135#366—41no—
136#358—22no—
137#358—41no—
138#358—41no—
139#357—22no—
140#357—41no—
141#357—41no—
142C22⋊C16—22no—
143C22⋊C16—22no—
144C22⋊C16—22no—
145C22⋊C16—22no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152C25D811yescharacteristic
153C22×Q8—21no—
154C22×D4—21no—
155C22×D4—41no—
156C22×D4—41no—
157C22×D4—41no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—41no—
161C23×C4—41no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4D811yescharacteristic
165C23×C4D811yescharacteristic
166C23×C4D811yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—21no—
173C2×Q16—21no—
174C2×Q16—21no—
175C2×Q16—21no—
176C2×D8—21no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8—21no—
181C2×D8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—41no—
202C22×C8—41no—
203C22×C8—41no—
204C22×C8—41no—
205C22×C8—41no—
206C22×C8—41no—
207C22×C8—44no—
208C22×C8—44no—
209C22×C8—44no—
210C22×C8—44no—
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D412yes—
215C22×C8#4112yes—
216C22×C8C2411yescharacteristic
217C22⋊Q8—42no—
218C22⋊Q8—42no—
219C22⋊Q8—42no—
220C22⋊Q8—42no—
221C22⋊Q8—42no—
222C22⋊Q8—42no—
223#121—42no—
224#121—42no—
225#121—42no—
226#121—42no—
227#121—42no—
228#121—42no—
229#120—84no—
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—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251SD32—44no—
252M32—44no—
253M32—44no—
254C2×C16—21no—
255C2×C16—21no—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—21no—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16—21no—
262C2×C16—21no—
263C2×C16—21no—
264C2×C16C2×D412yes—
265C2×C16C2×D412yes—
266#107—21no—
267#107—21no—
268#107—21no—
269#107—21no—
270#107—21no—
271#107—21no—
272#107—21no—
273#107—21no—
274#107—21no—
275#107—21no—
276#107—21no—
277#107—21no—
278#103—42no—
279#103—42no—
280#103—42no—
281#102—42no—
282#102—42no—
283#102—42no—
284#95—41no—
285#95—41no—
286#95—41no—
287#95—41no—
288#95—41no—
289#95—41no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24C2×D811yescharacteristic
299C24C2×D811yescharacteristic
300C24C2×D811yescharacteristic
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—82no—
314C2×D4—82no—
315C2×D4—82no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—82no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4#12112yes—
359C22×C4C2×D811yescharacteristic
360C22×C4C2×D811yescharacteristic
361C22×C4C2×D811yescharacteristic
362C22×C4#13512yes—
363C22×C4C22×D411yescharacteristic
364Q16—41no—
365Q16—41no—
366Q16—41no—
367Q16—41no—
368D8—41no—
369D8—41no—
370D8—41no—
371D8—41no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—44no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8#12112yes—
396C2×C8C22×D411yescharacteristic
397C2×C8C2×#4111yescharacteristic
398C2×C8H(5,𝔽2)11yescharacteristic
399C2×C8H(5,𝔽2)11yescharacteristic
400C2×C8H(5,𝔽2)11yescharacteristic
401C2×C8H(5,𝔽2)11yescharacteristic
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C22⋊C4—82no—
422C22⋊C4—82no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C16—22no—
430C16—22no—
431C16—41no—
432C16—41no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23#46612yes—
466C23#50912yes—
467C23#51012yes—
468C23C22×D811yescharacteristic
469Q8—81no—
470Q8—81no—
471Q8—81no—
472Q8—81no—
473D4—81no—
474D4—81no—
475D4—81no—
476D4—81no—
477D4—164no—
478D4—164no—
479D4—164no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
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—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4#46612yes—
525C2×C4C2×#12111yescharacteristic
526C2×C4#53411yescharacteristic
527C2×C4#53411yescharacteristic
528C2×C4C2×#13511yescharacteristic
529C2×C4#57511yescharacteristic
530C2×C4#57511yescharacteristic
531C8—22no—
532C8—41no—
533C8—41no—
534C8#53411yescharacteristic
535C8#53511yescharacteristic
536C8#53811yescharacteristic
537C8#53811yescharacteristic
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—416no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—161no—
561C22—161no—
562C22—161no—
563C22—161no—
564C22—161no—
565C22—161no—
566C22—161no—
567C22C2×#46611yescharacteristic
568C22#318611yescharacteristic
569C22#318611yescharacteristic
570C22C2×#50911yescharacteristic
571C22C2×#51011yescharacteristic
572C22#341411yescharacteristic
573C22#341411yescharacteristic
574C4—22no—
575C4—41no—
576C4—41no—
577C4—161no—
578C4—161no—
579C4—161no—
580C4—161no—
581C4—161no—
582C4—161no—
583C4—161no—
584C4#261711yescharacteristic
585C4#305411yescharacteristic
586C4#319111yescharacteristic
587C4#319111yescharacteristic
588C2—42no—
589C2—42no—
590C2—42no—
591C2—321no—
592C2#2973011yescharacteristic
593C2#3360511yescharacteristic
594C2#3360611yescharacteristic
595C2#3375911yescharacteristic
596C2#3376011yescharacteristic
597C2#3376211yescharacteristic
598C2#3376311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#32930C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ 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
1. C2×#2220C2semidirect
2. C2×#2112C2semidirect
3. #32931C2semidirect
5. #13493C2semidirect
6. #13492C2semidirect
7. #13383C2semidirect
8. #13382C2semidirect
9. #13381C2semidirect
10. #13380C2semidirect
11. #13305C2semidirect
12. #13305C2semidirect
13. #13062C2semidirect
14. #13061C2semidirect
15. #12836C2semidirect
17. C22×SD32C22semidirect
29. C2×#367C22semidirect
30. C2×#366C22semidirect
31. C2×#366C22semidirect
32. C2×#358C22semidirect
33. C2×#358C22semidirect
34. C2×#358C22semidirect
35. C2×#357C22semidirect
36. C2×#357C22semidirect
37. C2×#357C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1380C22semidirect
60. #1380C22semidirect
91. C2×M32D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22202
C2×#21122
#329312
#329302
#134932
#134922
#133832
#133822
#133812
#133802
#133052
#130622
#130612
#128362