Finite Group Databasebeta

← #6850695

Subgroups — #6850695

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36211C211yesmaximal, characteristic, complemented
2#36179C211yesmaximal, characteristic, complemented
3#33297C211yesmaximal, characteristic, complemented
4#32072C218yesmaximal, complemented
5#29806C212yesmaximal, complemented
6#29143C211yesmaximal, characteristic, complemented
7#28509C211yesmaximal, characteristic, complemented
8#27973C212yesmaximal, complemented
9#27890C212yesmaximal, complemented
10#24692C214yesmaximal, complemented
11#24128C214yesmaximal, complemented
12#21981C212yesmaximal, complemented
13#21690C211yesmaximal, characteristic, complemented
14#21265C211yesmaximal, characteristic, complemented
15C2×#547—22no—
16C2×#547C2212yes—
17C2×#493C2211yescharacteristic
18C2×#476C2214yes—
19C2×#442C2214yes—
20C2×#431C2212yes—
21#2868C2211yescharacteristic
22#2859C2211yescharacteristic
23#2856C2212yes—
24#2856C2212yes—
25#2542—22no—
26#2542C2212yes—
27#2535C2212yes—
28#2513—22no—
29#2501—22no—
30#2501—22no—
31#2501C2214yes—
32#2490C2214yes—
33#2420—22no—
34#2414—22no—
35#2363—22no—
36#2357C2212yes—
37C4×#127C2211yescharacteristic
38C4×#127C2211yescharacteristic
39#2285C2212yes—
40C2×#384C2212yes—
41C42×D4C2211yescharacteristic
42#2094C2214yes—
43#2071C2218yes—
44#2054C2218yes—
45#2034C2218yes—
46#2025C2214yes—
47#1999C2218yes—
48#1993C2214yes—
49#1992C2214yes—
50#1972C2214yes—
51#1957C2211yescharacteristic
52#1957C2212yes—
53#1957C2212yes—
54#1957C2211yescharacteristic
55#1955C2212yes—
56#1948C2218yes—
57#1940C2212yes—
58#1926C2218yes—
59#1924C2214yes—
60#1874C2218yes—
61#1853C2218yes—
62#1844C2212yes—
63#1844C2211yescharacteristic
64#1834C2212yes—
65#1834C2212yes—
66#1834C2212yes—
67#1804C2214yes—
68C4×C4⋊C8C2211yescharacteristic
69C4×C22⋊C8C2212yes—
70C4×#97C2211yescharacteristic
71C22×#41—22no—
72C22×D8C2314yes—
73#547—48no—
74#547—416no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8C2312yes—
77C2×#127C2311yescharacteristic
78C2×#127C2311yescharacteristic
79C2×#125C2312yes—
80C2×#123—24no—
81C2×#123—24no—
82C2×#123—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—22no—
85C2×#121—22no—
86C2×#121—22no—
87C2×#121—22no—
88C2×#121—22no—
89C2×#121C2314yes—
90C2×#121C2314yes—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4D412yes—
94C2×C4×D4C2312yes—
95C2×C4×D4C2312yes—
96C2×C4×D4C2311yescharacteristic
97C2×#117—22no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4C2314yes—
101C22×C42C2312yes—
102#493—24no—
103#476—48no—
104#442—48no—
105#431—28no—
106C2×#106C2318yes—
107C2×#106C2314yes—
108C2×C4⋊C8C2314yes—
109C2×C4⋊C8C2312yes—
110C2×#102C2318yes—
111C2×#102C2318yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×C22⋊C8C2314yes—
115C2×C22⋊C8C2314yes—
116C2×#97C2312yes—
117C2×#97C2312yes—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2312yes—
120C2×C4×C8C2311yescharacteristic
121#399—22no—
122#399—22no—
123#399—24no—
124#399—24no—
125#397—22no—
126#397—24no—
127#394—24no—
128#392—22no—
129#390—24no—
130#390—22no—
131#390—22no—
132#390—22no—
133#390—22no—
134#390—22no—
135#390—22no—
136#390C2312yes—
137#390C2312yes—
138#390C2314yes—
139#390C2314yes—
140#389—22no—
141#389—22no—
142#388—24no—
143#388—24no—
144#386—24no—
145#384—22no—
146#384—22no—
147#384—22no—
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#384C2314yes—
151#384C2312yes—
152#384C2312yes—
153#384C2312yes—
154#382—22no—
155#382C2312yes—
156#381—22no—
157#380—22no—
158#380C2314yes—
159C4×C4⋊C4—22no—
160C4×C4⋊C4D412yes—
161C4×C4⋊C4C2314yes—
162C4×C4⋊C4C2311yescharacteristic
163C4×C22⋊C4—22no—
164C4×C22⋊C4D412yes—
165C4×C22⋊C4D412yes—
166C4×C22⋊C4C2312yes—
167C43C2311yescharacteristic
168#340C2318yes—
169#340C2318yes—
170#340C2318yes—
171#336C2314yes—
172#336C2312yes—
173C2×#41—416no—
174C2×#41—48no—
175C22×Q8—22no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4C2×D411yescharacteristic
181C22×D4C2412yes—
182C22×D4C2412yes—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2412yes—
189C2×D8—28no—
190C2×D8—44no—
191C2×D8—44no—
192C22×C8C2412yes—
193C22×C8C2414yes—
194C22×C8C2412yes—
195C4⋊Q8—28no—
196C4⋊Q8—48no—
197#127—22no—
198#127—22no—
199#127—22no—
200#127—22no—
201#125—44no—
202#123—416no—
203#123—416no—
204#123—416no—
205C22⋊Q8—48no—
206C22⋊Q8—416no—
207#121—48no—
208#121—48no—
209#121—48no—
210#121—48no—
211#121—48no—
212#121—84no—
213C4×D4—28no—
214C4×D4—24no—
215C4×D4—28no—
216C4×D4—28no—
217C4×D4—48no—
218C4×D4—84no—
219#117—84no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2414yes—
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2414yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4C2×D412yes—
253C2×C22⋊C4C2×D412yes—
254C2×C22⋊C4C2×D412yes—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2411yescharacteristic
263C2×C42C2412yes—
264C2×C42C2411yescharacteristic
265C2×C42C2411yescharacteristic
266#106—44no—
267#106—48no—
268C4⋊C8—28no—
269C4⋊C8—24no—
270#102—48no—
271#102—44no—
272#102—44no—
273#102—48no—
274C22⋊C8—28no—
275C22⋊C8—44no—
276#97—24no—
277#97—42no—
278C4×C8—24no—
279C4×C8—22no—
280C4×C8#4114yescomplemented
281#95—22no—
282#95—24no—
283#95—24no—
284#95—22no—
285#95C2×D411yescharacteristic
286#95C2×D411yescharacteristic
287#95C2×D412yes—
288#95C2×D412yes—
289C24—22no—
290C24—22no—
291C24C22×D412yes—
292#41—832no—
293C2×Q8—48no—
294C2×Q8—44no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—84no—
316C2×D4—84no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—84no—
337C22×C4AGL(1,ℤ/8)18yes—
338C22×C4C22×D412yes—
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345C22×C4C2511yescharacteristic
346D8—416no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8C2×#4114yes—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—48no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—84no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—48no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—48no—
383C22⋊C4—48no—
384C22⋊C4—84no—
385C22⋊C4—84no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—24no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—84no—
396C42C2×#4112yes—
397C42H(5,𝔽2)14yes—
398C422−1+412yes—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23C2×AGL(1,ℤ/8)14yes—
416C23C23×D411yescharacteristic
417Q8—88no—
418D4—28no—
419D4—44no—
420D4—44no—
421D4—44no—
422D4—44no—
423D4—816no—
424D4—88no—
425D4—88no—
426D4—168no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—216no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—48no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—88no—
455C2×C4—816no—
456C2×C4—164no—
457C2×C4#53512yes—
458C2×C4#53512yes—
459C2×C4#54714yes—
460C2×C4C2×AGL(1,ℤ/8)14yes—
461C2×C4C2×AGL(1,ℤ/8)14yes—
462C2×C4#57714yes—
463C2×C4C22×#4111yescharacteristic
464C2×C4C2×H(5,𝔽2)12yes—
465C2×C4C2×2−1+411yescharacteristic
466C2×C4#58512yes—
467C8—24no—
468C8—44no—
469C8—42no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—84no—
478C22—84no—
479C22—82no—
480C22—82no—
481C22—84no—
482C22—84no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22C2×#53511yescharacteristic
488C22C2×#54711yescharacteristic
489C22#350011yescharacteristic
490C22C22×AGL(1,ℤ/8)12yes—
491C22C2×#57712yes—
492C4—24no—
493C4—22no—
494C4—22no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—84no—
499C4—82no—
500C4—84no—
501C4—82no—
502C4#334518yes—
503C4#350612yes—
504C4#354712yes—
505C2—44no—
506C2—162no—
507C2—162no—
508C2#3949111yescharacteristic
509C2#6079811yescharacteristic
510C2#6082011yescharacteristic
511C2C2×#334512yes—
512C2#6254112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36211C2semidirect
2. #36179C2semidirect
3. #33297C2semidirect
4. #32072C2semidirect
5. #29806C2semidirect
6. #29143C2semidirect
7. #28509C2semidirect
8. #27973C2semidirect
9. #27890C2semidirect
10. #24692C2semidirect
11. #24128C2semidirect
12. #21981C2semidirect
13. #21690C2semidirect
14. #21265C2semidirect
280. C4×C8#41semidirect

Maximal subgroups

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

subgroupindex
#362112
#361792
#332972
#320722
#298062
#291432
#285092
#279732
#278902
#246922
#241282
#219812
#216902
#212652