Finite Group Databasebeta

← #490862

Subgroups — #490862

Order 512.

447 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33661C211yesmaximal, characteristic
2#33627C211yesmaximal, characteristic
3#19648C211yesmaximal, characteristic, complemented
4#18054C211yesmaximal, characteristic
5#12501C211yesmaximal, characteristic, complemented
6#12499C211yesmaximal, characteristic, complemented
7#12489C211yesmaximal, characteristic, complemented
8#12487C211yesmaximal, characteristic, complemented
9#12449C211yesmaximal, characteristic, complemented
10#12447C211yesmaximal, characteristic, complemented
11#12422C211yesmaximal, characteristic, complemented
12#12420C211yesmaximal, characteristic, complemented
13#12018C211yesmaximal, characteristic, complemented
14#11987C211yesmaximal, characteristic, complemented
15#11973C211yesmaximal, characteristic, complemented
16#3231—21no—
17#3185—21no—
18#3170C2211yescharacteristic
19#3160C2211yescharacteristic
20#2974—21no—
21C2×C4×D8C2211yescharacteristic
22#2235—21no—
23#2228—21no—
24#2228—21no—
25#2226C2211yescharacteristic
26#2226C2211yescharacteristic
27#2226C2211yescharacteristic
28#2226C2211yescharacteristic
29#2224C2211yescharacteristic
30#2224C2211yescharacteristic
31#2208C2211yescharacteristic
32#2208C2211yescharacteristic
33#2187—21no—
34C4×SD32—21no—
35C4×SD32—21no—
36#2162C2211yescharacteristic
37#2159—21no—
38#2150C2211yescharacteristic
39#2140—21no—
40#1718—21no—
41#1716—21no—
42#1713C2211yescharacteristic, complemented
43#1710C2211yescharacteristic, complemented
44#1707C2211yescharacteristic, complemented
45#1707C2211yescharacteristic, 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#1365C2211yescharacteristic, complemented
62#1365C2211yescharacteristic, complemented
63#1364C2211yescharacteristic, complemented
64#1364C2211yescharacteristic, complemented
65#1362C2211yescharacteristic, complemented
66#1362C2211yescharacteristic, complemented
67C22×D8D411yescharacteristic
68#533—21no—
69#533—21no—
70C2×C4×D4—21no—
71C2×SD32—21no—
72C2×SD32—21no—
73C2×M32—21no—
74#500—21no—
75#500C2311yescharacteristic
76#500C2311yescharacteristic
77#500C2311yescharacteristic
78#499—21no—
79#499—21no—
80#498—21no—
81#498—21no—
82#498—21no—
83#494—21no—
84#494C2311yescharacteristic
85#487—21no—
86#487—21no—
87#487—21no—
88#487—21no—
89#487—21no—
90#485—41no—
91#485—41no—
92#479—41no—
93#479—41no—
94#470—42no—
95#467—21no—
96#467—21no—
97#441—42no—
98C4×Q16—21no—
99C4×Q16—21no—
100C4×D8—21no—
101C4×D8—21no—
102C4×D8D411yescharacteristic
103C4×D8D411yescharacteristic
104C4×D8C2311yescharacteristic
105C4×D8C2311yescharacteristic
106#436—21no—
107#436—21no—
108#435—41no—
109#435—41no—
110C8×D4—21no—
111C8×D4—21no—
112#432—21no—
113#431D411yescharacteristic
114#427—21no—
115#427—21no—
116C2×#107D411yescharacteristic
117C2×#102—21no—
118#407—41no—
119C2×C22⋊C8—41no—
120C4×M16D411yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#367—21no—
123#367—21no—
124#367—21no—
125#367—21no—
126#366—21no—
127#366—21no—
128C4⋊C16—21no—
129C4⋊C16C2311yescharacteristic
130C4⋊C16C2311yescharacteristic
131C4⋊C16C2311yescharacteristic
132C4⋊C16C2311yescharacteristic
133#358—21no—
134#358—21no—
135#357—21no—
136#357—21no—
137#357—21no—
138#357—21no—
139#357—21no—
140#357—21no—
141C22⋊C16—21no—
142C4×C16—21no—
143#335C2311yescharacteristic
144#335C2311yescharacteristic
145#334—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148D4⋊C8—21no—
149D4⋊C8—21no—
150#322D411yescharacteristic
151#322D411yescharacteristic
152C82C2311yescharacteristic
153C82C2311yescharacteristic
154C22×D4—21no—
155C23×C4—41no—
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—42no—
159C2×D8—22no—
160C2×D8—21no—
161C2×D8—41no—
162C2×D8C2×D411yescharacteristic
163C2×D8C2×D411yescharacteristic
164C2×M16—21no—
165C2×M16—41no—
166C2×M16—41no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8C2×D411yescharacteristic
170C22×C8#4111yescharacteristic
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175#125—41no—
176#125—41no—
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—41no—
190C4×D4—41no—
191#117—41no—
192#117—41no—
193C2×C4⋊C4—21no—
194C2×C22⋊C4—41no—
195C2×C42C2×D411yescharacteristic
196SD32—42no—
197SD32—42no—
198M32—42no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—41no—
211#107—41no—
212#107C2×D411yescharacteristic
213#107C2×D411yescharacteristic
214#106—42no—
215#106—41no—
216#106—41no—
217#106—41no—
218#106—41no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8D811yescharacteristic
222C4⋊C8D811yescharacteristic
223#103—42no—
224#103—42no—
225#103—41no—
226#103—41no—
227#103—41no—
228#103—41no—
229#103—41no—
230#103—41no—
231#102—21no—
232#102—21no—
233#102—21no—
234#102—21no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240C22⋊C8D811yescharacteristic
241C22⋊C8D811yescharacteristic
242#97—21no—
243#97—41no—
244C4×C8C2×D411yescharacteristic
245C4×C8C2×D411yescharacteristic
246C4×C8C2×D411yescharacteristic
247C4×C8C2×D411yescharacteristic
248C4×C8C2×D411yescharacteristic
249C4×C8C2×D411yescharacteristic
250C4×C8#4111yescharacteristic
251C4×C8C2411yescharacteristic
252C24—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—42no—
260C2×D4—41no—
261C2×D4—81no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—161no—
268C22×C4—161no—
269C22×C4#12111yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4AGL(1,ℤ/8)11yescharacteristic
272Q16—41no—
273Q16—41no—
274Q16—81no—
275Q16—84no—
276D8—21no—
277D8—21no—
278D8—21no—
279D8—42no—
280D8#13712yes—
281M16—42no—
282M16—161no—
283M16—161no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—161no—
292C2×C8—161no—
293C2×C8#12111yescharacteristic
294C2×C8#12111yescharacteristic
295C2×C8C2×D811yescharacteristic
296C2×C8C2×D811yescharacteristic
297C2×C8C2×D811yescharacteristic
298C2×C8C2×D811yescharacteristic
299C2×C8C22×D411yescharacteristic
300C2×C8C2×#4111yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C4⋊C4—81no—
315C4⋊C4—81no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—81no—
323C42—81no—
324C42#12111yescharacteristic
325C42C2×D811yescharacteristic
326C42AGL(1,ℤ/8)11yescharacteristic
327C42C22×D411yescharacteristic
328C16—41no—
329C16—41no—
330C16—41no—
331C16—41no—
332C23—41no—
333C23—41no—
334C23—82no—
335C23—82no—
336C23—161no—
337C23#45911yescharacteristic
338Q8—82no—
339Q8—82no—
340Q8—81no—
341Q8—81no—
342Q8—81no—
343Q8—81no—
344D4—22no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—82no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—161no—
365C2×C4—161no—
366C2×C4—161no—
367C2×C4#45911yescharacteristic
368C2×C4#45911yescharacteristic
369C2×C4#45911yescharacteristic
370C2×C4C2×#12111yescharacteristic
371C2×C4C22×D811yescharacteristic
372C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
373C8—22no—
374C8—21no—
375C8—21no—
376C8—21no—
377C8—21no—
378C8—21no—
379C8—21no—
380C8—21no—
381C8—41no—
382C8—81no—
383C8—81no—
384C8C2×#13711yescharacteristic
385C8#57511yescharacteristic
386C22—21no—
387C22—21no—
388C22—21no—
389C22—42no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—164no—
396C22C2×#45911yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—81no—
402C4—81no—
403C4—161no—
404C4—161no—
405C4—161no—
406C4—161no—
407C4#305611yescharacteristic
408C4#306911yescharacteristic
409C4#341611yescharacteristic
410C4#341611yescharacteristic
411C2—41no—
412C2—82no—
413C2—161no—
414C2#1916311yescharacteristic
415C2#3358611yescharacteristic
416C2#3359411yescharacteristic
subgroupG/H
center, socleC22C2×#459
commutator, Frattini, cosocleC4×C8C24
prime radicalC22×D8D4
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
3. #19648C2semidirect
5. #12501C2semidirect
6. #12499C2semidirect
7. #12489C2semidirect
8. #12487C2semidirect
9. #12449C2semidirect
10. #12447C2semidirect
11. #12422C2semidirect
12. #12420C2semidirect
13. #12018C2semidirect
14. #11987C2semidirect
15. #11973C2semidirect
42. #1713C22semidirect
43. #1710C22semidirect
44. #1707C22semidirect
45. #1707C22semidirect
55. #1370C22semidirect
56. #1370C22semidirect
57. #1367C22semidirect
58. #1367C22semidirect
59. #1365C22semidirect
60. #1365C22semidirect
61. #1365C22semidirect
62. #1365C22semidirect
63. #1364C22semidirect
64. #1364C22semidirect
65. #1362C22semidirect
66. #1362C22semidirect

Maximal subgroups

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

subgroupindex
#336612
#336272
#196482
#180542
#125012
#124992
#124892
#124872
#124492
#124472
#124222
#124202
#120182
#119872
#119732