Finite Group Databasebeta

← #490789

Subgroups — #490789

Order 512.

506 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33670C211yesmaximal, characteristic, complemented
2#33580C211yesmaximal, characteristic
3#19645C211yesmaximal, characteristic, complemented
4M16×D8C211yesmaximal, characteristic, complemented
5#12495C211yesmaximal, characteristic, complemented
6#12493C211yesmaximal, characteristic, complemented
7#12489C211yesmaximal, characteristic, complemented
8#12483C211yesmaximal, characteristic, complemented
9#12428C211yesmaximal, characteristic, complemented
10#12426C211yesmaximal, characteristic, complemented
11#12422C211yesmaximal, characteristic, complemented
12#12416C211yesmaximal, characteristic, complemented
13#12017C211yesmaximal, characteristic, complemented
14#11988C211yesmaximal, characteristic, complemented
15#11966C211yesmaximal, characteristic, complemented
16C2×#509—21no—
17#3228—21no—
18#3182—21no—
19#3170C2211yescharacteristic
20#3147C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22D4×M16—21no—
23#2236—21no—
24#2224C2211yescharacteristic
25#2224C2211yescharacteristic
26#2210—21no—
27#2210—21no—
28#2208C2211yescharacteristic
29#2208C2211yescharacteristic
30#2208C2211yescharacteristic
31#2208C2211yescharacteristic
32#2206C2211yescharacteristic
33#2206C2211yescharacteristic
34#2187—21no—
35#2175—21no—
36#2175—21no—
37#2163C2211yescharacteristic
38#2150C2211yescharacteristic
39#2139—21no—
40#1719—21no—
41#1714—21no—
42#1713C2211yescharacteristic, complemented
43#1709C2211yescharacteristic, complemented
44#1707C2211yescharacteristic, complemented
45#1706C2211yescharacteristic, complemented
46#1589C2211yescharacteristic
47#1589C2211yescharacteristic
48#1581—21no—
49#1581—21no—
50C8×D8C2211yescharacteristic
51C8×D8C2211yescharacteristic
52#1567C2211yescharacteristic
53#1486—21no—
54C8×M16C2211yescharacteristic
55#1370C2211yescharacteristic, complemented
56#1370C2211yescharacteristic, complemented
57#1367C2211yescharacteristic, complemented
58#1367C2211yescharacteristic, complemented
59#1364C2211yescharacteristic, complemented
60#1364C2211yescharacteristic, complemented
61#1362C2211yescharacteristic, complemented
62#1362C2211yescharacteristic, complemented
63#1362C2211yescharacteristic, complemented
64#1362C2211yescharacteristic, complemented
65#1361C2211yescharacteristic, complemented
66#1361C2211yescharacteristic, complemented
67C2×#135—21no—
68C2×#135—21no—
69C22×D8D411yescharacteristic
70C22×M16—41no—
71#533—21no—
72#532—21no—
73C2×C4×D4—21no—
74#509—44no—
75C2×SD32—21no—
76C2×SD32—21no—
77C2×SD32—21no—
78C2×SD32—21no—
79C2×D16—21no—
80C2×D16—21no—
81C2×M32—21no—
82#500—21no—
83#500—21no—
84#500C2311yescharacteristic
85#498—21no—
86#495—21no—
87#495—21no—
88#494C2311yescharacteristic
89#494C2311yescharacteristic
90#493C2311yescharacteristic
91#492—21no—
92#492—21no—
93#487—21no—
94#487—21no—
95#486—21no—
96#486—21no—
97#486—21no—
98#482—42no—
99#478—42no—
100#468—41no—
101#468—41no—
102#467—21no—
103#467—21no—
104#440—41no—
105#440—41no—
106C4×Q16—21no—
107C4×Q16—21no—
108C4×D8—21no—
109C4×D8—21no—
110C4×D8D411yescharacteristic
111C4×D8D411yescharacteristic
112C4×D8C2311yescharacteristic
113C4×D8C2311yescharacteristic
114#436—21no—
115#436—21no—
116#435—41no—
117#435—41no—
118C8×D4—21no—
119C8×D4—21no—
120#431D411yescharacteristic
121C2×#107D411yescharacteristic
122#423—21no—
123C2×#102—21no—
124#407—41no—
125C4×M16D411yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#366—21no—
128#366—21no—
129C4⋊C16—21no—
130C4⋊C16C2311yescharacteristic
131C4⋊C16C2311yescharacteristic
132C4⋊C16C2311yescharacteristic
133C4⋊C16C2311yescharacteristic
134#358—21no—
135#358—21no—
136#358—21no—
137#358—21no—
138#357—21no—
139#357—21no—
140#357—21no—
141#357—21no—
142C22⋊C16—21no—
143#346—21no—
144#335—21no—
145#335C2311yescharacteristic
146#335C2311yescharacteristic
147D4⋊C8—21no—
148D4⋊C8—21no—
149D4⋊C8—21no—
150D4⋊C8—21no—
151#322D411yescharacteristic
152#322D411yescharacteristic
153C82C2311yescharacteristic
154C82C2311yescharacteristic
155C2×#41—41no—
156C22×D4—21no—
157C23×C4—41no—
158#135—42no—
159#135—42no—
160#135—82no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×SD16—41no—
166C2×SD16—41no—
167C2×SD16—41no—
168C2×SD16—41no—
169C2×D8—22no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—41no—
174C2×D8C2×D411yescharacteristic
175C2×D8C2×D411yescharacteristic
176C2×M16—41no—
177C2×M16—41no—
178C2×M16—41no—
179C2×M16—84no—
180C2×M16D811yescharacteristic
181C2×M16D811yescharacteristic
182C22×C8—41no—
183C22×C8—41no—
184C22×C8C2×D411yescharacteristic
185C22×C8#4111yescharacteristic
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189#127—21no—
190#125—41no—
191#124—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194#121—41no—
195#121—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206#117—41no—
207C2×C4⋊C4—21no—
208C2×C22⋊C4—41no—
209C2×C42C2×D411yescharacteristic
210SD32—42no—
211SD32—42no—
212SD32—42no—
213SD32—42no—
214D16—42no—
215D16—42no—
216M32—42no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—42no—
225#107—41no—
226#107—41no—
227#107C2×D411yescharacteristic
228#107C2×D411yescharacteristic
229#106—42no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—41no—
234#103—41no—
235#103—41no—
236#103—41no—
237#103—41no—
238#102—21no—
239#102—21no—
240#102—21no—
241#102—21no—
242#102—42no—
243#102—42no—
244#102—41no—
245#102—41no—
246C22⋊C8—21no—
247C22⋊C8—41no—
248C22⋊C8—41no—
249C22⋊C8—41no—
250C22⋊C8—41no—
251#97D811yescharacteristic
252#97D811yescharacteristic
253C4×C8C2×D411yescharacteristic
254C4×C8C2×D411yescharacteristic
255C4×C8C2×D411yescharacteristic
256C4×C8C2×D411yescharacteristic
257C4×C8C2×D411yescharacteristic
258C4×C8C2×D411yescharacteristic
259C4×C8#4111yescharacteristic
260C4×C8C2411yescharacteristic
261C24—41no—
262#41—82no—
263#41—82no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—81no—
275C2×D4—81no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—81no—
284C22×C4#12111yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4AGL(1,ℤ/8)11yescharacteristic
287Q16—41no—
288Q16—41no—
289Q16—41no—
290Q16—41no—
291Q16—81no—
292Q16—81no—
293SD16—82no—
294SD16—82no—
295SD16—82no—
296SD16—82no—
297D8—21no—
298D8—21no—
299D8—21no—
300D8—42no—
301D8—41no—
302D8—41no—
303D8—81no—
304D8AGL(1,ℤ/8)12yes—
305M16—22no—
306M16—41no—
307M16—82no—
308M16—82no—
309M16—82no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—82no—
318C2×C8—82no—
319C2×C8#12111yescharacteristic
320C2×C8#12111yescharacteristic
321C2×C8C2×D811yescharacteristic
322C2×C8C2×D811yescharacteristic
323C2×C8C2×D811yescharacteristic
324C2×C8C2×D811yescharacteristic
325C2×C8C22×D411yescharacteristic
326C2×C8C2×#4111yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—81no—
343C22⋊C4—81no—
344C22⋊C4—81no—
345C42—81no—
346C42#12111yescharacteristic
347C42C2×D811yescharacteristic
348C42AGL(1,ℤ/8)11yescharacteristic
349C42C22×D411yescharacteristic
350C16—41no—
351C16—41no—
352C16—41no—
353C16—41no—
354C23—41no—
355C23—41no—
356C23—82no—
357C23—82no—
358C23—81no—
359C23—161no—
360C23#45911yescharacteristic
361Q8—81no—
362Q8—81no—
363Q8—81no—
364Q8—81no—
365Q8—81no—
366Q8—81no—
367D4—22no—
368D4—41no—
369D4—41no—
370D4—41no—
371D4—82no—
372D4—82no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—162no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—162no—
396C2×C4#45911yescharacteristic
397C2×C4#45911yescharacteristic
398C2×C4#45911yescharacteristic
399C2×C4C2×#12111yescharacteristic
400C2×C4C22×D811yescharacteristic
401C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
402C8—21no—
403C8—21no—
404C8—21no—
405C8—21no—
406C8—21no—
407C8—21no—
408C8—41no—
409C8—41no—
410C8—81no—
411C8—81no—
412C8#50912yescomplemented
413C8C2×AGL(1,ℤ/8)11yescharacteristic
414C8#57511yescharacteristic
415C22—21no—
416C22—21no—
417C22—21no—
418C22—42no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—164no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22C2×#45911yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—81no—
434C4—81no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4#305511yescharacteristic
439C4#306811yescharacteristic
440C4C2×#50911yescharacteristic
441C4#341411yescharacteristic
442C2—41no—
443C2—82no—
444C2—161no—
445C2—321no—
446C2#1916211yescharacteristic
447C2#3358011yescharacteristic
448C2#3359611yescharacteristic
subgroupG/H
center, socleC22C2×#459
commutator, Frattini, cosocleC4×C8C24
prime radical#33580C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33670C2semidirect
3. #19645C2semidirect
4. M16×D8C2semidirect
5. #12495C2semidirect
6. #12493C2semidirect
7. #12489C2semidirect
8. #12483C2semidirect
9. #12428C2semidirect
10. #12426C2semidirect
11. #12422C2semidirect
12. #12416C2semidirect
13. #12017C2semidirect
14. #11988C2semidirect
15. #11966C2semidirect
42. #1713C22semidirect
43. #1709C22semidirect
44. #1707C22semidirect
45. #1706C22semidirect
55. #1370C22semidirect
56. #1370C22semidirect
57. #1367C22semidirect
58. #1367C22semidirect
59. #1364C22semidirect
60. #1364C22semidirect
61. #1362C22semidirect
62. #1362C22semidirect
63. #1362C22semidirect
64. #1362C22semidirect
65. #1361C22semidirect
66. #1361C22semidirect
412. C8#509semidirect

Maximal subgroups

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

subgroupindex
#336702
#335802
#196452
M16×D82
#124952
#124932
#124892
#124832
#124282
#124262
#124222
#124162
#120172
#119882
#119662