Finite Group Databasebeta

← #484033

Subgroups — #484033

Order 512.

434 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33661C211yesmaximal, characteristic
2#18086C211yesmaximal, characteristic, complemented
3#18054C211yesmaximal, characteristic
4#12084C211yesmaximal, characteristic, complemented
5#12082C211yesmaximal, characteristic, complemented
6#12075C211yesmaximal, characteristic, complemented
7#12074C211yesmaximal, characteristic, complemented
8#12071C211yesmaximal, characteristic, complemented
9#12070C211yesmaximal, characteristic, complemented
10#12043C211yesmaximal, characteristic, complemented
11C8×SD32C211yesmaximal, characteristic, complemented
12#12034C211yesmaximal, characteristic, complemented
13#11676C211yesmaximal, characteristic, complemented
14#11593C211yesmaximal, characteristic
15#11409C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#2986—21no—
18#2974—21no—
19C2×C4×D8C2211yescharacteristic
20#2235C411yescharacteristic
21#2235C411yescharacteristic
22#2226C411yescharacteristic
23#2226C411yescharacteristic
24#2208C411yescharacteristic
25#2208C411yescharacteristic
26#2187C411yescharacteristic
27#2187C411yescharacteristic
28C4×SD32C2211yescharacteristic
29C4×SD32C2211yescharacteristic
30C4×SD32C2211yescharacteristic
31C4×SD32C2211yescharacteristic
32#2162C2211yescharacteristic
33C4×M32C2211yescharacteristic
34#1591C2211yescharacteristic, complemented
35#1591C2211yescharacteristic, complemented
36#1589C2211yescharacteristic
37#1589C2211yescharacteristic
38#1587—21no—
39#1587—21no—
40#1586—21no—
41#1586—21no—
42C8×Q16C2211yescharacteristic, complemented
43C8×Q16C2211yescharacteristic, complemented
44C8×D8C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1568C2211yescharacteristic
47#1564C2211yescharacteristic
48#1490—21no—
49#1485—21no—
50C8×M16C2211yescharacteristic
51#1372C2211yescharacteristic, complemented
52#1372C2211yescharacteristic, complemented
53#1371C2211yescharacteristic, complemented
54#1371C2211yescharacteristic, complemented
55#1332C2211yescharacteristic, complemented
56#1332C2211yescharacteristic, complemented
57#1332C2211yescharacteristic, complemented
58#1332C2211yescharacteristic, complemented
59#1331C2211yescharacteristic
60#1331C2211yescharacteristic
61#1331C2211yescharacteristic
62#1331C2211yescharacteristic
63#1312C2211yescharacteristic, complemented
64#1312C2211yescharacteristic, complemented
65#1311C2211yescharacteristic, complemented
66C8×C16C2211yescharacteristic, complemented
67C22×D8C2×C411yescharacteristic
68#533—21no—
69C2×C4×D4—21no—
70C2×SD32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×SD32C2×C411yescharacteristic
73C2×SD32C2×C411yescharacteristic
74C2×M32—21no—
75#500C2×C411yescharacteristic
76#494C2×C411yescharacteristic
77#487—21no—
78#467—21no—
79#467C2×C411yescharacteristic
80#467C2×C411yescharacteristic
81#446—21no—
82#446—21no—
83#446—41no—
84C8×Q8—21no—
85C8×Q8—21no—
86C8×Q8—41no—
87C4×Q16—21no—
88C4×Q16C2311yescharacteristic
89C4×Q16C2311yescharacteristic
90C4×D8—21no—
91C4×D8—21no—
92C4×D8C2×C411yescharacteristic
93C4×D8C2×C411yescharacteristic
94C4×D8C2311yescharacteristic
95C4×D8C2311yescharacteristic
96#436—21no—
97#436—21no—
98#435—41no—
99#435—41no—
100C8×D4—21no—
101C8×D4—21no—
102#433—41no—
103#433—41no—
104#432—21no—
105#431—21no—
106#431D411yescharacteristic
107C2×#107C2×C411yescharacteristic
108C2×#102—21no—
109#407—41no—
110C2×C22⋊C8—41no—
111C4×M16D411yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#367C2×C411yescharacteristic
114#367C2×C411yescharacteristic
115#367C2×C411yescharacteristic
116#367C2×C411yescharacteristic
117C4⋊C16C2×C411yescharacteristic
118C4⋊C16C2×C411yescharacteristic
119#358C2×C411yescharacteristic
120#358C2×C411yescharacteristic
121#358C2×C411yescharacteristic
122#358C2×C411yescharacteristic
123#357C2×C411yescharacteristic
124#357C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127C22⋊C16C2×C411yescharacteristic
128C22⋊C16C2×C411yescharacteristic
129#346—21no—
130C4×C16C2311yescharacteristic
131C4×C16C2311yescharacteristic
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134#335C2311yescharacteristic
135#335C2311yescharacteristic
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#334—21no—
139#334—21no—
140Q8⋊C8—21no—
141Q8⋊C8—21no—
142Q8⋊C8—21no—
143Q8⋊C8—21no—
144D4⋊C8—21no—
145D4⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150C82C2311yescharacteristic
151C82C2311yescharacteristic
152C22×D4—21no—
153C23×C4—41no—
154C2×Q16—21no—
155C2×Q16C22×C411yescharacteristic
156C2×Q16C22×C411yescharacteristic
157C2×D8—22no—
158C2×D8—21no—
159C2×D8—41no—
160C2×D8C22×C411yescharacteristic
161C2×D8C22×C411yescharacteristic
162C2×M16—21no—
163C2×M16—41no—
164C2×M16—41no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8C22×C411yescharacteristic
168C22×C8#4111yescharacteristic
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171#125—41no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8—41no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—41no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—41no—
183C4×D4—41no—
184#117—41no—
185C2×C4⋊C4—21no—
186C2×C22⋊C4—41no—
187C2×C42C2×D411yescharacteristic
188SD32—22no—
189SD32—22no—
190SD32—41no—
191SD32—41no—
192M32—42no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16C22×C411yescharacteristic
196C2×C16C22×C411yescharacteristic
197C2×C16C22×C411yescharacteristic
198C2×C16C22×C411yescharacteristic
199#107—21no—
200#107—21no—
201#107C22×C411yescharacteristic
202#107C22×C411yescharacteristic
203#107C22×C411yescharacteristic
204#107C22×C411yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8D811yescharacteristic
214C4⋊C8D811yescharacteristic
215#103—21no—
216#103—21no—
217#103—21no—
218#103—21no—
219#102—21no—
220#102—21no—
221#102—21no—
222#102—21no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8D811yescharacteristic
230C22⋊C8D811yescharacteristic
231#97—21no—
232#97—41no—
233#97—41no—
234#97—41no—
235C4×C8—41no—
236C4×C8—41no—
237C4×C8—41no—
238C4×C8C22×C411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8#4111yescharacteristic
245C4×C8C2411yescharacteristic
246C24—41no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—41no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—42no—
253C2×D4—41no—
254C2×D4—81no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—161no—
261C22×C4—161no—
262C22×C4C4×D411yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4#13511yescharacteristic
265Q16—21no—
266Q16—21no—
267Q16—21no—
268Q16—21no—
269Q16—81no—
270D8—21no—
271D8—21no—
272D8—21no—
273D8—42no—
274D8#13112yes—
275M16—42no—
276M16—161no—
277M16—161no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—161no—
288C2×C8—161no—
289C2×C8C4×D411yescharacteristic
290C2×C8C4×D411yescharacteristic
291C2×C8C2×D811yescharacteristic
292C2×C8C2×D811yescharacteristic
293C2×C8C2×D811yescharacteristic
294C2×C8C2×D811yescharacteristic
295C2×C8C23×C411yescharacteristic
296C2×C8C2×#4111yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C42—41no—
316C42—41no—
317C42C4×D411yescharacteristic
318C42C2×D811yescharacteristic
319C42#13511yescharacteristic
320C42C22×D411yescharacteristic
321C16—22no—
322C16—21no—
323C16—21no—
324C16—21no—
325C16—21no—
326C16—41no—
327C23—41no—
328C23—41no—
329C23—82no—
330C23—82no—
331C23—161no—
332C23C4×D811yescharacteristic
333Q8—41no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—82no—
338D4—22no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—82no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—161no—
359C2×C4—161no—
360C2×C4—161no—
361C2×C4C4×D811yescharacteristic
362C2×C4C4×D811yescharacteristic
363C2×C4C4×D811yescharacteristic
364C2×C4C2×C4×D411yescharacteristic
365C2×C4C22×D811yescharacteristic
366C2×C4C2×#13511yescharacteristic
367C8—22no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—21no—
375C8—41no—
376C8—81no—
377C8—81no—
378C8—81no—
379C8—81no—
380C8C2×#13111yescharacteristic
381C8#56811yescharacteristic
382C22—21no—
383C22—21no—
384C22—21no—
385C22—42no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—164no—
392C22C2×C4×D811yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4#293311yescharacteristic
404C4#295611yescharacteristic
405C4#341611yescharacteristic
406C4#341611yescharacteristic
407C2—41no—
408C2—82no—
409C2—161no—
410C2#1799411yescharacteristic
411C2#3345711yescharacteristic
412C2#3346311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radicalC22×D8C2×C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #431 ◁ 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. #18086C2semidirect
4. #12084C2semidirect
5. #12082C2semidirect
6. #12075C2semidirect
7. #12074C2semidirect
8. #12071C2semidirect
9. #12070C2semidirect
10. #12043C2semidirect
11. C8×SD32C2semidirect
12. #12034C2semidirect
13. #11676C2semidirect
15. #11409C2semidirect
34. #1591C22semidirect
35. #1591C22semidirect
42. C8×Q16C22semidirect
43. C8×Q16C22semidirect
51. #1372C22semidirect
52. #1372C22semidirect
53. #1371C22semidirect
54. #1371C22semidirect
55. #1332C22semidirect
56. #1332C22semidirect
57. #1332C22semidirect
58. #1332C22semidirect
63. #1312C22semidirect
64. #1312C22semidirect
65. #1311C22semidirect
66. C8×C16C22semidirect

Maximal subgroups

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

subgroupindex
#336612
#180862
#180542
#120842
#120822
#120752
#120742
#120712
#120702
#120432
C8×SD322
#120342
#116762
#115932
#114092