Finite Group Databasebeta

← #490814

Subgroups — #490814

Order 512.

490 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33661C211yesmaximal, characteristic, complemented
2#33580C211yesmaximal, characteristic
3#19645C211yesmaximal, characteristic, complemented
4#18054C211yesmaximal, characteristic, complemented
5#12495C211yesmaximal, characteristic, complemented
6#12493C211yesmaximal, characteristic, complemented
7#12487C211yesmaximal, characteristic, complemented
8#12485C211yesmaximal, characteristic, complemented
9#12428C211yesmaximal, characteristic, complemented
10#12426C211yesmaximal, characteristic, complemented
11#12420C211yesmaximal, characteristic, complemented
12#12418C211yesmaximal, characteristic, complemented
13#12018C211yesmaximal, characteristic, complemented
14#11987C211yesmaximal, characteristic, complemented
15#11963C211yesmaximal, characteristic, complemented
16C2×#509—21no—
17#3228—21no—
18#3182—21no—
19#3170C2211yescharacteristic
20#3147C2211yescharacteristic
21#2974—21no—
22C2×C4×D8C2211yescharacteristic
23#2235—21no—
24#2226C2211yescharacteristic
25#2226C2211yescharacteristic
26#2210—21no—
27#2210—21no—
28#2208C2211yescharacteristic
29#2208C2211yescharacteristic
30#2208C2211yescharacteristic
31#2208C2211yescharacteristic
32#2206C2211yescharacteristic
33#2206C2211yescharacteristic
34#2187—21no—
35C4×SD32—21no—
36C4×SD32—21no—
37#2162C2211yescharacteristic
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#1586—21no—
49#1586—21no—
50C8×D8C2211yescharacteristic
51C8×D8C2211yescharacteristic
52#1568C2211yescharacteristic
53#1490—21no—
54C8×M16C2211yescharacteristic
55#1370C2211yescharacteristic, complemented
56#1370C2211yescharacteristic, complemented
57#1367C2211yescharacteristic, complemented
58#1367C2211yescharacteristic, complemented
59#1365C2211yescharacteristic, complemented
60#1365C2211yescharacteristic, 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
70#533—21no—
71#532—21no—
72C2×C4×D4—21no—
73#509—44no—
74C2×SD32—21no—
75C2×SD32—21no—
76C2×SD32—21no—
77C2×SD32—21no—
78C2×D16—21no—
79C2×D16—21no—
80C2×M32—21no—
81#500—21no—
82#500—21no—
83#500C2311yescharacteristic
84#498—21no—
85#495—21no—
86#495—21no—
87#494C2311yescharacteristic
88#494C2311yescharacteristic
89#493C2311yescharacteristic
90#492—21no—
91#492—21no—
92#487—21no—
93#487—21no—
94#486—21no—
95#486—21no—
96#486—21no—
97#482—41no—
98#482—41no—
99#478—41no—
100#478—41no—
101#468—42no—
102#467—21no—
103#467—21no—
104#440—42no—
105C4×Q16—21no—
106C4×Q16—21no—
107C4×D8—21no—
108C4×D8—21no—
109C4×D8D411yescharacteristic
110C4×D8D411yescharacteristic
111C4×D8C2311yescharacteristic
112C4×D8C2311yescharacteristic
113#436—21no—
114#436—21no—
115#435—41no—
116#435—41no—
117C8×D4—21no—
118C8×D4—21no—
119#432—21no—
120#431D411yescharacteristic
121C2×#107D411yescharacteristic
122C2×#102—21no—
123#407—41no—
124C2×C22⋊C8—41no—
125C4×M16D411yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#367—21no—
128#367—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—
143C4×C16—21no—
144#335C2311yescharacteristic
145#335C2311yescharacteristic
146#334—21no—
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—42no—
166C2×SD16—41no—
167C2×SD16—41no—
168C2×D8—22no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—41no—
173C2×D8C2×D411yescharacteristic
174C2×D8C2×D411yescharacteristic
175C2×M16—21no—
176C2×M16—41no—
177C2×M16—41no—
178C22×C8—41no—
179C22×C8—41no—
180C22×C8C2×D411yescharacteristic
181C22×C8#4111yescharacteristic
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185#127—21no—
186#125—41no—
187#124—41no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190#121—41no—
191#121—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202#117—41no—
203C2×C4⋊C4—21no—
204C2×C22⋊C4—41no—
205C2×C42C2×D411yescharacteristic
206SD32—42no—
207SD32—42no—
208SD32—42no—
209SD32—42no—
210D16—42no—
211D16—42no—
212M32—42no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—42no—
221#107—41no—
222#107—41no—
223#107C2×D411yescharacteristic
224#107C2×D411yescharacteristic
225#106—41no—
226#106—41no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8D811yescharacteristic
230C4⋊C8D811yescharacteristic
231#103—42no—
232#103—41no—
233#103—41no—
234#102—21no—
235#102—21no—
236#102—21no—
237#102—21no—
238#102—42no—
239#102—41no—
240#102—41no—
241#102—41no—
242#102—41no—
243C22⋊C8—41no—
244C22⋊C8—41no—
245C22⋊C8—41no—
246C22⋊C8—41no—
247C22⋊C8—41no—
248C22⋊C8D811yescharacteristic
249C22⋊C8D811yescharacteristic
250#97—21no—
251#97—41no—
252C4×C8C2×D411yescharacteristic
253C4×C8C2×D411yescharacteristic
254C4×C8C2×D411yescharacteristic
255C4×C8C2×D411yescharacteristic
256C4×C8C2×D411yescharacteristic
257C4×C8C2×D411yescharacteristic
258C4×C8#4111yescharacteristic
259C4×C8C2411yescharacteristic
260C24—41no—
261#41—82no—
262#41—82no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—81no—
274C2×D4—81no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—81no—
281C22×C4—161no—
282C22×C4—161no—
283C22×C4#12111yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4AGL(1,ℤ/8)11yescharacteristic
286Q16—41no—
287Q16—41no—
288Q16—41no—
289Q16—41no—
290Q16—81no—
291Q16—81no—
292SD16—84no—
293SD16—82no—
294SD16—82no—
295D8—21no—
296D8—21no—
297D8—21no—
298D8—42no—
299D8—41no—
300D8—41no—
301D8—81no—
302D8AGL(1,ℤ/8)12yes—
303M16—42no—
304M16—161no—
305M16—161no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—161no—
314C2×C8—161no—
315C2×C8#12111yescharacteristic
316C2×C8#12111yescharacteristic
317C2×C8C2×D811yescharacteristic
318C2×C8C2×D811yescharacteristic
319C2×C8C2×D811yescharacteristic
320C2×C8C2×D811yescharacteristic
321C2×C8C22×D411yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—81no—
333C4⋊C4—81no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C42—81no—
342C42#12111yescharacteristic
343C42C2×D811yescharacteristic
344C42AGL(1,ℤ/8)11yescharacteristic
345C42C22×D411yescharacteristic
346C16—41no—
347C16—41no—
348C16—41no—
349C16—41no—
350C23—41no—
351C23—41no—
352C23—82no—
353C23—82no—
354C23—81no—
355C23—161no—
356C23#45911yescharacteristic
357Q8—82no—
358Q8—81no—
359Q8—81no—
360Q8—81no—
361Q8—81no—
362D4—22no—
363D4—41no—
364D4—41no—
365D4—41no—
366D4—82no—
367D4—82no—
368D4—81no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—162no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—161no—
388C2×C4—161no—
389C2×C4—162no—
390C2×C4—161no—
391C2×C4#45911yescharacteristic
392C2×C4#45911yescharacteristic
393C2×C4#45911yescharacteristic
394C2×C4C2×#12111yescharacteristic
395C2×C4C22×D811yescharacteristic
396C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
397C8—22no—
398C8—21no—
399C8—21no—
400C8—21no—
401C8—21no—
402C8—21no—
403C8—21no—
404C8—21no—
405C8—41no—
406C8—81no—
407C8—81no—
408C8C2×AGL(1,ℤ/8)11yescharacteristic
409C8#57511yescharacteristic
410C22—21no—
411C22—21no—
412C22—21no—
413C22—42no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—164no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22C2×#45911yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—81no—
429C4—81no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4#305511yescharacteristic
434C4#306811yescharacteristic
435C4#341611yescharacteristic
436C4#341611yescharacteristic
437C2—41no—
438C2—82no—
439C2—161no—
440C2—321no—
441C2#1916211yescharacteristic
442C2#3358611yescharacteristic
443C2#3359411yescharacteristic
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. #33661C2semidirect
3. #19645C2semidirect
4. #18054C2semidirect
5. #12495C2semidirect
6. #12493C2semidirect
7. #12487C2semidirect
8. #12485C2semidirect
9. #12428C2semidirect
10. #12426C2semidirect
11. #12420C2semidirect
12. #12418C2semidirect
13. #12018C2semidirect
14. #11987C2semidirect
15. #11963C2semidirect
42. #1713C22semidirect
43. #1709C22semidirect
44. #1707C22semidirect
45. #1706C22semidirect
55. #1370C22semidirect
56. #1370C22semidirect
57. #1367C22semidirect
58. #1367C22semidirect
59. #1365C22semidirect
60. #1365C22semidirect
61. #1362C22semidirect
62. #1362C22semidirect
63. #1362C22semidirect
64. #1362C22semidirect
65. #1361C22semidirect
66. #1361C22semidirect

Maximal subgroups

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

subgroupindex
#336612
#335802
#196452
#180542
#124952
#124932
#124872
#124852
#124282
#124262
#124202
#124182
#120182
#119872
#119632