Finite Group Databasebeta

← #484077

Subgroups — #484077

Order 512.

481 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#509C211yesmaximal, characteristic
2#18039C211yesmaximal, characteristic, complemented
3M16×D8C211yesmaximal, characteristic, complemented
4#12086C211yesmaximal, characteristic, complemented
5#12081C211yesmaximal, characteristic, complemented
6#12075C211yesmaximal, characteristic, complemented
7#12073C211yesmaximal, characteristic, complemented
8#12071C211yesmaximal, characteristic, complemented
9#12069C211yesmaximal, characteristic, complemented
10#12043C211yesmaximal, characteristic, complemented
11C8×SD32C211yesmaximal, characteristic, complemented
12#12025C211yesmaximal, characteristic, complemented
13#11661C211yesmaximal, characteristic, complemented
14#11660C211yesmaximal, characteristic, complemented
15#11409C211yesmaximal, characteristic, complemented
16C2×#509C411yescharacteristic
17#2971—21no—
18C4×#135C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20D4×M16—21no—
21#2177C411yescharacteristic
22#2177C411yescharacteristic
23#2175C411yescharacteristic
24#2175C411yescharacteristic
25C4×SD32C2211yescharacteristic
26C4×SD32C2211yescharacteristic
27C4×D16C2211yescharacteristic
28C4×D16C2211yescharacteristic
29#2162C2211yescharacteristic
30#2159C411yescharacteristic
31#2140C411yescharacteristic
32#2139C411yescharacteristic
33C4×M32C2211yescharacteristic
34#1589C2211yescharacteristic
35#1589C2211yescharacteristic
36#1589C2211yescharacteristic, complemented
37#1589C2211yescharacteristic, complemented
38#1583—21no—
39#1583—21no—
40#1581—21no—
41#1581—21no—
42C8×Q16C2211yescharacteristic, complemented
43C8×Q16C2211yescharacteristic, complemented
44C8×D8C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1568C2211yescharacteristic
47#1567C2211yescharacteristic
48#1488—21no—
49#1486—21no—
50C8×M16C2211yescharacteristic
51#1372C2211yescharacteristic, complemented
52#1372C2211yescharacteristic, complemented
53#1371C2211yescharacteristic, complemented
54#1371C2211yescharacteristic, complemented
55#1332C2211yescharacteristic, complemented
56#1332C2211yescharacteristic, complemented
57#1331C2211yescharacteristic
58#1331C2211yescharacteristic
59#1331C2211yescharacteristic
60#1331C2211yescharacteristic
61#1331C2211yescharacteristic, complemented
62#1331C2211yescharacteristic, complemented
63#1312C2211yescharacteristic, complemented
64#1312C2211yescharacteristic, complemented
65#1311C2211yescharacteristic, complemented
66C8×C16C2211yescharacteristic, complemented
67C2×#135C2×C411yescharacteristic
68C22×D8C2×C411yescharacteristic
69C22×M16—41no—
70C4×#41—21no—
71C2×C4×D4—21no—
72#509—24no—
73C2×SD32C2×C411yescharacteristic
74C2×SD32C2×C411yescharacteristic
75C2×D16C2×C411yescharacteristic
76C2×D16C2×C411yescharacteristic
77C2×M32C2×C411yescharacteristic
78C2×M32C2×C411yescharacteristic
79C8×Q8—21no—
80C8×Q8—21no—
81C4×Q16C2×C411yescharacteristic
82C4×Q16C2311yescharacteristic
83C4×SD16—21no—
84C4×SD16—21no—
85C4×D8—21no—
86C4×D8—21no—
87C4×D8C2×C411yescharacteristic
88C4×D8C2×C411yescharacteristic
89C4×D8C2×C411yescharacteristic
90C4×D8C2311yescharacteristic
91C4×D8C2311yescharacteristic
92C4×D8C2311yescharacteristic
93#436—21no—
94#436—21no—
95#436—21no—
96#436—21no—
97#436—41no—
98#436—41no—
99#435—41no—
100#435—41no—
101C8×D4—21no—
102C8×D4—21no—
103#432—21no—
104#432—41no—
105#431D411yescharacteristic
106#427C2×C411yescharacteristic
107C2×#107C2×C411yescharacteristic
108#423—21no—
109#416—21no—
110C2×#102—21no—
111#407—41no—
112C4×M16—41no—
113C4×M16D411yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#367C2×C411yescharacteristic
116#367C2×C411yescharacteristic
117#366C2×C411yescharacteristic
118#366C2×C411yescharacteristic
119C4⋊C16—21no—
120#358C2×C411yescharacteristic
121#358C2×C411yescharacteristic
122#357C2×C411yescharacteristic
123#357C2×C411yescharacteristic
124#357C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127#357C2×C411yescharacteristic
128C22⋊C16—21no—
129#346C2×C411yescharacteristic
130#346C2×C411yescharacteristic
131C4×C16C2311yescharacteristic
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134C4×C16C2311yescharacteristic
135#335—21no—
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#335C2311yescharacteristic
139#335C2311yescharacteristic
140#334—21no—
141Q8⋊C8—21no—
142Q8⋊C8—21no—
143D4⋊C8—21no—
144D4⋊C8—21no—
145D4⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148D4⋊C8—21no—
149#322D411yescharacteristic
150#322D411yescharacteristic
151C82C2311yescharacteristic
152C82C2311yescharacteristic
153C2×#41—21no—
154C22×D4—21no—
155C23×C4—41no—
156#135—22no—
157#135—41no—
158C2×Q16C22×C411yescharacteristic
159C2×SD16—21no—
160C2×D8—22no—
161C2×D8—21no—
162C2×D8—41no—
163C2×D8C22×C411yescharacteristic
164C2×D8C22×C411yescharacteristic
165C2×D8C22×C411yescharacteristic
166C2×M16—41no—
167C2×M16—41no—
168C2×M16—41no—
169C2×M16—41no—
170C2×M16—41no—
171C2×M16—84no—
172C2×M16D811yescharacteristic
173C2×M16D811yescharacteristic
174C22×C8—41no—
175C22×C8—41no—
176C22×C8C22×C411yescharacteristic
177C22×C8#4111yescharacteristic
178C4×Q8—21no—
179C4×Q8—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190#117—21no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C22⋊C4—41no—
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196SD32—22no—
197SD32M1614yescomplemented
198D16—22no—
199D16—22no—
200M32—22no—
201M32—41no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16C22×C411yescharacteristic
205C2×C16C22×C411yescharacteristic
206C2×C16C22×C411yescharacteristic
207C2×C16C22×C411yescharacteristic
208#107—21no—
209#107C22×C411yescharacteristic
210#107C22×C411yescharacteristic
211#107C22×C411yescharacteristic
212#107C22×C411yescharacteristic
213#106—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—41no—
220C4⋊C8—41no—
221C4⋊C8—41no—
222#103—21no—
223#103—21no—
224#102—21no—
225#102—21no—
226#102—21no—
227#102—21no—
228#102—21no—
229#102—21no—
230C22⋊C8—21no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233C22⋊C8—41no—
234C22⋊C8—41no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238#97—41no—
239#97D811yescharacteristic
240#97D811yescharacteristic
241C4×C8—41no—
242C4×C8—41no—
243C4×C8C22×C411yescharacteristic
244C4×C8C2×D411yescharacteristic
245C4×C8C2×D411yescharacteristic
246C4×C8C2×D411yescharacteristic
247C4×C8C2×D411yescharacteristic
248C4×C8C2×D411yescharacteristic
249C4×C8#4111yescharacteristic
250C4×C8C2411yescharacteristic
251C24—41no—
252#41—42no—
253#41—81no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—42no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—81no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4C4×D411yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4#13511yescharacteristic
275Q16—21no—
276Q16C2×M1612yescomplemented
277SD16—42no—
278D8—21no—
279D8—21no—
280D8—21no—
281D8—21no—
282D8—21no—
283D8—42no—
284D8C2×M1612yes—
285M16—22no—
286M16—41no—
287M16—82no—
288M16—82no—
289M16—82no—
290M16—82no—
291M16—82no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—82no—
302C2×C8—82no—
303C2×C8C4×D411yescharacteristic
304C2×C8C4×D411yescharacteristic
305C2×C8C2×D811yescharacteristic
306C2×C8C2×D811yescharacteristic
307C2×C8C2×D811yescharacteristic
308C2×C8C2×D811yescharacteristic
309C2×C8C23×C411yescharacteristic
310C2×C8C2×#4111yescharacteristic
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42C4×D411yescharacteristic
329C42C2×D811yescharacteristic
330C42#13511yescharacteristic
331C42C22×D411yescharacteristic
332C16—21no—
333C16—21no—
334C16—21no—
335C16—41no—
336C16—41no—
337C16C2×M1612yescomplemented
338C23—41no—
339C23—41no—
340C23—41no—
341C23—82no—
342C23—82no—
343C23—161no—
344C23C4×D811yescharacteristic
345Q8—22no—
346Q8—41no—
347D4—22no—
348D4—41no—
349D4—41no—
350D4—41no—
351D4—41no—
352D4—41no—
353D4—82no—
354D4—82no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—161no—
376C2×C4C4×D811yescharacteristic
377C2×C4C4×D811yescharacteristic
378C2×C4C4×D811yescharacteristic
379C2×C4C2×C4×D411yescharacteristic
380C2×C4C22×D811yescharacteristic
381C2×C4C2×#13511yescharacteristic
382C8—21no—
383C8—21no—
384C8—21no—
385C8—21no—
386C8—21no—
387C8—21no—
388C8—41no—
389C8—41no—
390C8—81no—
391C8—81no—
392C8—81no—
393C8—81no—
394C8#50912yescomplemented
395C8C22×M1611yescharacteristic
396C8#56811yescharacteristic
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—164no—
410C22C2×C4×D811yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4D4×M1611yescharacteristic
421C4#295711yescharacteristic
422C4C2×#50911yescharacteristic
423C4#341411yescharacteristic
424C2—41no—
425C2—82no—
426C2—161no—
427C2—161no—
428C2M16×D811yescharacteristic
429C2C4×#50911yescharacteristic
430C2#3346411yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radicalC2×#509C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18039C2semidirect
3. M16×D8C2semidirect
4. #12086C2semidirect
5. #12081C2semidirect
6. #12075C2semidirect
7. #12073C2semidirect
8. #12071C2semidirect
9. #12069C2semidirect
10. #12043C2semidirect
11. C8×SD32C2semidirect
12. #12025C2semidirect
13. #11661C2semidirect
14. #11660C2semidirect
15. #11409C2semidirect
36. #1589C22semidirect
37. #1589C22semidirect
42. C8×Q16C22semidirect
43. C8×Q16C22semidirect
51. #1372C22semidirect
52. #1372C22semidirect
53. #1371C22semidirect
54. #1371C22semidirect
55. #1332C22semidirect
56. #1332C22semidirect
61. #1331C22semidirect
62. #1331C22semidirect
63. #1312C22semidirect
64. #1312C22semidirect
65. #1311C22semidirect
66. C8×C16C22semidirect
197. SD32M16semidirect
276. Q16C2×M16semidirect
337. C16C2×M16semidirect
394. C8#509semidirect

Maximal subgroups

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

subgroupindex
C4×#5092
#180392
M16×D82
#120862
#120812
#120752
#120732
#120712
#120692
#120432
C8×SD322
#120252
#116612
#116602
#114092