Finite Group Databasebeta

← #484106

Subgroups — #484106

Order 512.

474 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#509C211yesmaximal, characteristic
2#18065C211yesmaximal, characteristic, complemented
3#18044C211yesmaximal, characteristic, complemented
4#12087C212yesmaximal, complemented
5#12078C212yesmaximal, complemented
6#12077C212yesmaximal, complemented
7#12067C212yesmaximal, complemented
8#12020C211yesmaximal, characteristic, complemented
9#11666C211yesmaximal, characteristic, complemented
10#11663C211yesmaximal, characteristic, complemented
11#11414C211yesmaximal, characteristic, complemented
12C2×#509C411yescharacteristic
13#2990—21no—
14C4×#135C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic
16D4×M16—21no—
17#2177C2211yescharacteristic
18#2177C2211yescharacteristic
19#2175C2211yescharacteristic
20#2175C2211yescharacteristic
21C4×SD32C411yescharacteristic
22C4×SD32C411yescharacteristic
23C4×D16C411yescharacteristic
24C4×D16C411yescharacteristic
25#2163C2211yescharacteristic
26#2159C411yescharacteristic
27#2140C411yescharacteristic
28#2139C411yescharacteristic
29C4×M32C2211yescharacteristic
30#1594C2212yes—
31#1593C2212yescomplemented
32#1586—22no—
33#1582—22no—
34#1580C2212yes—
35#1580C2212yescomplemented
36#1566C2211yescharacteristic
37#1566C2211yescharacteristic
38#1489—21no—
39#1486—21no—
40#1451C2211yescharacteristic
41#1368C2212yescomplemented
42#1368C2212yescomplemented
43#1334C2212yescomplemented
44#1333C2212yes—
45#1333C2212yes—
46#1333C2212yescomplemented
47#1313C2212yescomplemented
48#1313C2212yescomplemented
49C2×#135C2×C411yescharacteristic
50C22×D8C2×C411yescharacteristic
51C22×M16—41no—
52C4×#41—21no—
53C2×C4×D4—21no—
54#509—24no—
55C2×SD32C2×C411yescharacteristic
56C2×SD32C2×C411yescharacteristic
57C2×D16C2×C411yescharacteristic
58C2×D16C2×C411yescharacteristic
59C2×M32C2×C411yescharacteristic
60C2×M32C2×C411yescharacteristic
61#446—22no—
62C4×Q16C2×C411yescharacteristic
63C4×Q16C2311yescharacteristic
64C4×SD16—21no—
65C4×SD16—21no—
66C4×D8—21no—
67C4×D8—21no—
68C4×D8C2×C411yescharacteristic
69C4×D8C2×C411yescharacteristic
70C4×D8C2×C411yescharacteristic
71C4×D8C2311yescharacteristic
72C4×D8C2311yescharacteristic
73C4×D8C2311yescharacteristic
74#436—22no—
75#435—42no—
76#435—42no—
77C8×D4—22no—
78C8×D4—22no—
79#432—41no—
80#431D411yescharacteristic
81#427C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83#423—21no—
84#423—21no—
85C2×C4⋊C8—41no—
86#416—21no—
87C2×#102—21no—
88#407—41no—
89C4×M16D411yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#367C2×C411yescharacteristic
92#367C2×C411yescharacteristic
93#366C2×C411yescharacteristic
94#366C2×C411yescharacteristic
95C4⋊C16—21no—
96#358C2×C411yescharacteristic
97#358C2×C411yescharacteristic
98#357C2×C411yescharacteristic
99#357C2×C411yescharacteristic
100#357C2×C411yescharacteristic
101#357C2×C411yescharacteristic
102#357C2×C411yescharacteristic
103#357C2×C411yescharacteristic
104C22⋊C16—21no—
105#346C2312yes—
106#346C2311yescharacteristic
107#346C2311yescharacteristic
108C4×C16C2×C411yescharacteristic
109C4×C16C2×C411yescharacteristic
110#334—21no—
111#334—21no—
112#334C2312yes—
113#334C2312yes—
114Q8⋊C8—22no—
115D4⋊C8—22no—
116D4⋊C8—22no—
117D4⋊C8—22no—
118#322D412yes—
119#322C2312yes—
120C2×#41—21no—
121C22×D4—21no—
122C23×C4—41no—
123#135—22no—
124#135—41no—
125C2×Q16C22×C411yescharacteristic
126C2×SD16—21no—
127C2×D8—22no—
128C2×D8—21no—
129C2×D8—41no—
130C2×D8C22×C411yescharacteristic
131C2×D8C22×C411yescharacteristic
132C2×D8C22×C411yescharacteristic
133C2×M16—21no—
134C2×M16—42no—
135C2×M16—41no—
136C2×M16—41no—
137C2×M16—84no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8C22×C411yescharacteristic
141C22×C8#4111yescharacteristic
142C4×Q8—21no—
143C4×Q8—21no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—21no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153C4×D4—41no—
154#117—21no—
155#117—41no—
156C2×C4⋊C4—21no—
157C2×C22⋊C4—41no—
158C2×C42—41no—
159C2×C42C2×D411yescharacteristic
160SD32—22no—
161SD32—22no—
162D16—22no—
163D16M1614yescomplemented
164M32—22no—
165M32—41no—
166C2×C16—22no—
167C2×C16C22×C411yescharacteristic
168C2×C16C22×C411yescharacteristic
169C2×C16C22×C411yescharacteristic
170C2×C16C22×C411yescharacteristic
171#107—21no—
172#107C22×C411yescharacteristic
173#107C22×C411yescharacteristic
174#107C22×C411yescharacteristic
175#107C22×C411yescharacteristic
176#106—21no—
177C4⋊C8—21no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—42no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183#103—21no—
184#103—21no—
185#102—21no—
186#102—21no—
187#102—21no—
188#102—21no—
189#102—21no—
190#102—21no—
191C22⋊C8—21no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195#97—21no—
196#97—42no—
197C4×C8C22×C411yescharacteristic
198C4×C8C2×D412yes—
199C4×C8C2×D412yes—
200C4×C8C2×D411yescharacteristic
201C4×C8#4111yescharacteristic
202C4×C8C2411yescharacteristic
203C24—41no—
204#41—42no—
205#41—81no—
206C2×Q8—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—81no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4C4×D411yescharacteristic
225C22×C4AGL(1,ℤ/8)11yescharacteristic
226C22×C4AGL(1,ℤ/8)11yescharacteristic
227Q16—21no—
228Q16—21no—
229SD16—42no—
230D8—21no—
231D8—21no—
232D8—21no—
233D8—21no—
234D8—42no—
235D8C2×M1612yes—
236D8C2×M1612yescomplemented
237M16—42no—
238M16—84no—
239M16—82no—
240M16—161no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—84no—
249C2×C8—162no—
250C2×C8C4×D411yescharacteristic
251C2×C8C4×D411yescharacteristic
252C2×C8C23×C411yescharacteristic
253C2×C8C2×#4111yescharacteristic
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42—41no—
269C42—41no—
270C42—41no—
271C42C4×D411yescharacteristic
272C42AGL(1,ℤ/8)11yescharacteristic
273C42AGL(1,ℤ/8)11yescharacteristic
274C42C22×D411yescharacteristic
275C16—21no—
276C16—21no—
277C16—21no—
278C16—42no—
279C16C2×M1612yescomplemented
280C23—41no—
281C23—41no—
282C23—41no—
283C23—82no—
284C23—82no—
285C23—161no—
286C23#44211yescharacteristic
287Q8—41no—
288Q8—41no—
289D4—22no—
290D4—22no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—82no—
296D4—82no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4#44211yescharacteristic
318C2×C4#44211yescharacteristic
319C2×C4#44211yescharacteristic
320C2×C4C2×C4×D411yescharacteristic
321C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
322C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
323C8—21no—
324C8—21no—
325C8—21no—
326C8—42no—
327C8—42no—
328C8—82no—
329C8—82no—
330C8C22×M1611yescharacteristic
331C8#56811yescharacteristic
332C22—21no—
333C22—21no—
334C22—21no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—164no—
345C22C2×#44211yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4AGL(1,ℤ/16)12yes—
356C4D4×M1611yescharacteristic
357C4#295711yescharacteristic
358C2—41no—
359C2—82no—
360C2—82no—
361C2—161no—
362C2#1804411yescharacteristic
363C2C2×AGL(1,ℤ/16)11yescharacteristic
364C2#3346911yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radicalC2×#509C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18065C2semidirect
3. #18044C2semidirect
4. #12087C2semidirect
5. #12078C2semidirect
6. #12077C2semidirect
7. #12067C2semidirect
8. #12020C2semidirect
9. #11666C2semidirect
10. #11663C2semidirect
11. #11414C2semidirect
31. #1593C22semidirect
35. #1580C22semidirect
41. #1368C22semidirect
42. #1368C22semidirect
43. #1334C22semidirect
46. #1333C22semidirect
47. #1313C22semidirect
48. #1313C22semidirect
163. D16M16semidirect
236. D8C2×M16semidirect
279. C16C2×M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5092
#180652
#180442
#120872
#120782
#120772
#120672
#120202
#116662
#116632
#114142