Finite Group Databasebeta

← #496823

Subgroups — #496823

Order 512.

790 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2187C211yesmaximal, characteristic
2#21525C211yesmaximal, characteristic, complemented
3#21233C211yesmaximal, characteristic
4#13238C211yesmaximal, characteristic, complemented
5#13237C211yesmaximal, characteristic, complemented
6#13149C211yesmaximal, characteristic, complemented
7#13147C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#13013C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic, complemented
12#12994C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic, complemented
14#12731C211yesmaximal, characteristic
15#12600C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C22×SD32C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic, complemented
21C2×#427C2211yescharacteristic
22C2×#418—21no—
23C2×#405C2211yescharacteristic
24#2323—21no—
25#2309—21no—
26#2187C418yescomplemented
27C2×#367C2211yescharacteristic, complemented
28C2×#366C2211yescharacteristic, complemented
29C2×#358C2211yescharacteristic, complemented
30C2×#358C2211yescharacteristic, complemented
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×#346C2211yescharacteristic, complemented
40C2×C4×C16C2211yescharacteristic, complemented
41#1972—21no—
42#1972—21no—
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic
46#1970C2211yescharacteristic
47#1969C2211yescharacteristic, complemented
48#1942C2211yescharacteristic, complemented
49#1935—21no—
50#1894—21no—
51#1808C2211yescharacteristic
52C4×#107C2211yescharacteristic, complemented
53#1763—21no—
54#1380C2211yescharacteristic, complemented
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1374C2211yescharacteristic, complemented
59#1374C2211yescharacteristic, complemented
60C23×D4—21no—
61C22×Q16C2311yescharacteristic
62C22×D8—21no—
63C22×D8—22no—
64C22×D8—41no—
65C22×D8—41no—
66C22×D8—41no—
67C22×D8—41no—
68C22×D8C2×C414yes—
69C22×D8C2311yescharacteristic
70C22×D8C2311yescharacteristic
71C22×M16D411yescharacteristic
72C23×C8C2311yescharacteristic
73C2×C22⋊Q8—21no—
74C2×C4×Q8—21no—
75C2×#117—21no—
76C2×#117D411yescharacteristic
77C2×SD32—41no—
78C2×SD32—41no—
79C2×SD32—41no—
80C2×SD32—41no—
81C2×SD32C2×C414yes—
82C2×SD32C2×C414yes—
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87#467—22no—
88#467C2×C414yes—
89C4×Q16—41no—
90C4×Q16—41no—
91#427—41no—
92#427—41no—
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#106—21no—
99#418—44no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107#405—41no—
108#405—41no—
109C2×#97D411yescharacteristic
110C2×#97D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391—21no—
114#390—21no—
115#390—21no—
116#390—21no—
117#390—21no—
118#390—41no—
119#390—41no—
120#389—21no—
121#389—41no—
122#388—41no—
123#388—41no—
124#387—41no—
125#380—21no—
126#379—41no—
127#379—41no—
128C4×C4⋊C4—21no—
129#367—41no—
130#366—41no—
131#358—41no—
132#358—41no—
133#357—41no—
134#357—41no—
135#357—41no—
136#357—41no—
137#357C2×C414yes—
138#357C2×C414yes—
139C22⋊C16—22no—
140C22⋊C16C2×C414yes—
141#346—41no—
142C4×C16—41no—
143#340—21no—
144#340—21no—
145#340—21no—
146C25—41no—
147C22×Q8—21no—
148C22×D4—24no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—41no—
156C22×D4—42no—
157C23×C4C2×D411yescharacteristic
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16C22×C412yes—
164C2×D8—24no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—44no—
176C2×D8—44no—
177C2×D8—42no—
178C2×D8—42no—
179C2×D8—44no—
180C2×D8—44no—
181C2×D8C22×C412yes—
182C2×D8C22×C412yes—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—41no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8C22×C412yes—
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8#4112yes—
195C22×C8C2411yescharacteristic
196C22⋊Q8—24no—
197C22⋊Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200#117—41no—
201#117—41no—
202#117—42no—
203#117—42no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4D811yescharacteristic
213C2×C4⋊C4D811yescharacteristic
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4D811yescharacteristic
226C2×C22⋊C4D811yescharacteristic
227C2×C42—41no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232SD32—44no—
233SD32—44no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16C22×C412yes—
245C2×C16C22×C412yes—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107C22×C412yes—
256#106—42no—
257#103—24no—
258#103—42no—
259#103—42no—
260#102—42no—
261#102—42no—
262#102—42no—
263#102—42no—
264#97—41no—
265#97—41no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275C24—41no—
276C24—41no—
277C24—44no—
278C24—82no—
279C24—82no—
280C24—161no—
281C24—161no—
282C24—161no—
283C24—161no—
284C24C2×D811yescharacteristic
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—48no—
294C2×D4—48no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—44no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—84no—
309C2×D4—84no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4C4×D412yes—
321C22×C4C2×D811yescharacteristic
322C22×C4C2×D811yescharacteristic
323C22×C4C2×D811yescharacteristic
324C22×C4C2×D811yescharacteristic
325C22×C4C2×D811yescharacteristic
326C22×C4#13512yes—
327C22×C4C22×D411yescharacteristic
328Q16—22no—
329Q16—22no—
330Q16—41no—
331Q16—41no—
332D8—22no—
333D8—22no—
334D8—22no—
335D8—22no—
336D8—48no—
337D8—41no—
338D8—41no—
339D8—41no—
340D8—41no—
341M16—44no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8C4×D412yes—
354C2×C8AGL(1,ℤ/8)12yes—
355C2×C8C23×C411yescharacteristic
356C2×C8C2×#4111yescharacteristic
357C2×C8H(5,𝔽2)11yescharacteristic
358C2×C8H(5,𝔽2)11yescharacteristic
359C2×C8H(5,𝔽2)11yescharacteristic
360C2×C8H(5,𝔽2)11yescharacteristic
361C4⋊C4—22no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—44no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—82no—
375C4⋊C4—161no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—44no—
379C22⋊C4—82no—
380C22⋊C4—161no—
381C22⋊C4—161no—
382C22⋊C4—161no—
383C22⋊C4—161no—
384C22⋊C4—161no—
385C22⋊C4—161no—
386C22⋊C4—161no—
387C22⋊C4—161no—
388C22⋊C4—161no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—82no—
394C42—82no—
395C42—161no—
396C16—22no—
397C16—22no—
398C16—41no—
399C16—41no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—42no—
407C23—42no—
408C23—88no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—164no—
422C23—164no—
423C23—164no—
424C23—164no—
425C23—164no—
426C23—164no—
427C23C4×D812yes—
428C23C22×D811yescharacteristic
429Q8—42no—
430Q8—42no—
431Q8—81no—
432Q8—81no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—88no—
438D4—81no—
439D4—81no—
440D4—81no—
441D4—81no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4C4×D812yes—
479C2×C4C2×C4×D411yescharacteristic
480C2×C4#53411yescharacteristic
481C2×C4#53511yescharacteristic
482C2×C4C2×#13511yescharacteristic
483C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
484C2×C4#57511yescharacteristic
485C8—22no—
486C8—22no—
487C8—41no—
488C8#51811yescharacteristic
489C8#51811yescharacteristic
490C8#53811yescharacteristic
491C8#53811yescharacteristic
492C22—22no—
493C22—22no—
494C22—22no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—1616no—
506C22—161no—
507C22—161no—
508C22—161no—
509C22—161no—
510C22—161no—
511C22—161no—
512C22—161no—
513C22—161no—
514C22C2×C4×D811yescharacteristic
515C22#318611yescharacteristic
516C22#320511yescharacteristic
517C22#341511yescharacteristic
518C22#341611yescharacteristic
519C22#341611yescharacteristic
520C22#341611yescharacteristic
521C4—22no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—82no—
527C4—82no—
528C4—161no—
529C4—161no—
530C4—161no—
531C4—161no—
532C4#232111yescharacteristic
533C4C4×AGL(1,ℤ/8)11yescharacteristic
534C4#318811yescharacteristic
535C4#321111yescharacteristic
536C2—42no—
537C2—162no—
538C2—162no—
539C2#2204011yescharacteristic
540C2#3345411yescharacteristic
541C2#3345711yescharacteristic
542C2#3376411yescharacteristic
543C2#3376611yescharacteristic
544C2#3378811yescharacteristic
545C2#3379011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21525C2semidirect
4. #13238C2semidirect
5. #13237C2semidirect
6. #13149C2semidirect
7. #13147C2semidirect
8. #13131C2semidirect
9. #13130C2semidirect
10. #13013C2semidirect
11. #13013C2semidirect
12. #12994C2semidirect
13. #12921C2semidirect
15. #12600C2semidirect
20. C2×C4×Q16C22semidirect
26. #2187C4semidirect
27. C2×#367C22semidirect
28. C2×#366C22semidirect
29. C2×#358C22semidirect
30. C2×#358C22semidirect
39. C2×#346C22semidirect
40. C2×C4×C16C22semidirect
47. #1969C22semidirect
48. #1942C22semidirect
52. C4×#107C22semidirect
54. #1380C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1374C22semidirect
59. #1374C22semidirect

Maximal subgroups

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

subgroupindex
C2×#21872
#215252
#212332
#132382
#132372
#131492
#131472
#131312
#131302
#130132
#129942
#129212
#127312
#126002