Finite Group Databasebeta

← #484103

Subgroups — #484103

Order 512.

455 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#509C211yesmaximal, characteristic
2#18064C211yesmaximal, characteristic, complemented
3#18043C211yesmaximal, characteristic, complemented
4#12089C212yesmaximal, complemented
5#12080C212yesmaximal, complemented
6#12078C212yesmaximal, complemented
7#12065C212yesmaximal, complemented
8#12021C211yesmaximal, characteristic, complemented
9#11667C211yesmaximal, characteristic, complemented
10#11665C211yesmaximal, characteristic, complemented
11#11414C211yesmaximal, characteristic, complemented
12C2×#509C411yescharacteristic
13#2974—21no—
14#2971—21no—
15C4×#135C2211yescharacteristic
16C2×C4×D8C2211yescharacteristic
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#1594C2212yescomplemented
32#1588—22no—
33#1581—22no—
34#1580C2212yes—
35#1579C2212yescomplemented
36#1568C2211yescharacteristic
37#1568C2211yescharacteristic
38#1490—21no—
39#1488—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
51C4×#41—21no—
52C2×C4×D4—21no—
53#509—24no—
54C2×SD32C2×C411yescharacteristic
55C2×SD32C2×C411yescharacteristic
56C2×D16C2×C411yescharacteristic
57C2×D16C2×C411yescharacteristic
58C2×M32C2×C411yescharacteristic
59C2×M32C2×C411yescharacteristic
60C8×Q8—22no—
61C4×Q16C2×C411yescharacteristic
62C4×Q16C2311yescharacteristic
63C4×SD16—21no—
64C4×SD16—21no—
65C4×D8—21no—
66C4×D8—21no—
67C4×D8C2×C411yescharacteristic
68C4×D8C2×C411yescharacteristic
69C4×D8C2×C411yescharacteristic
70C4×D8C2311yescharacteristic
71C4×D8C2311yescharacteristic
72C4×D8C2311yescharacteristic
73#436—22no—
74#436—22no—
75#436—42no—
76#435—42no—
77C8×D4—22no—
78#432—21no—
79#432—21no—
80#432—41no—
81#431D411yescharacteristic
82#427C2×C411yescharacteristic
83C2×#107C2×C411yescharacteristic
84#416—21no—
85C2×#102—21no—
86#407—41no—
87C2×C22⋊C8—41no—
88C4×M16—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#335—21no—
111#335—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—42no—
136C22×C8—42no—
137C22×C8C22×C411yescharacteristic
138C22×C8#4111yescharacteristic
139C4×Q8—21no—
140C4×Q8—21no—
141C4×D4—21no—
142C4×D4—21no—
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151#117—21no—
152#117—41no—
153C2×C4⋊C4—21no—
154C2×C22⋊C4—41no—
155C2×C42—41no—
156C2×C42C2×D411yescharacteristic
157SD32—22no—
158SD32M1614yescomplemented
159D16—22no—
160D16—22no—
161M32—22no—
162M32—41no—
163C2×C16—22no—
164C2×C16C22×C411yescharacteristic
165C2×C16C22×C411yescharacteristic
166C2×C16C22×C411yescharacteristic
167C2×C16C22×C411yescharacteristic
168#107—21no—
169#107C22×C411yescharacteristic
170#107C22×C411yescharacteristic
171#107C22×C411yescharacteristic
172#107C22×C411yescharacteristic
173#106—21no—
174C4⋊C8—21no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—42no—
178#103—21no—
179#103—21no—
180#102—21no—
181#102—21no—
182#102—21no—
183#102—21no—
184#102—21no—
185#102—21no—
186C22⋊C8—21no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192#97—21no—
193#97—41no—
194#97—41no—
195C4×C8—42no—
196C4×C8C22×C411yescharacteristic
197C4×C8C2×D412yes—
198C4×C8C2×D412yes—
199C4×C8C2×D411yescharacteristic
200C4×C8#4111yescharacteristic
201C4×C8C2411yescharacteristic
202C24—41no—
203#41—42no—
204#41—81no—
205C2×Q8—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—42no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—81no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—162no—
222C22×C4C4×D411yescharacteristic
223C22×C4AGL(1,ℤ/8)11yescharacteristic
224C22×C4AGL(1,ℤ/8)11yescharacteristic
225Q16—21no—
226Q16C2×M1612yes—
227SD16—42no—
228D8—21no—
229D8—21no—
230D8—21no—
231D8—21no—
232D8—21no—
233D8—42no—
234D8C2×M1612yes—
235M16—81no—
236M16—84no—
237M16—162no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—162no—
246C2×C8C4×D411yescharacteristic
247C2×C8C4×D411yescharacteristic
248C2×C8C23×C411yescharacteristic
249C2×C8C2×#4111yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C42—41no—
265C42—41no—
266C42—41no—
267C42C4×D411yescharacteristic
268C42AGL(1,ℤ/8)11yescharacteristic
269C42AGL(1,ℤ/8)11yescharacteristic
270C42C22×D411yescharacteristic
271C16—21no—
272C16—21no—
273C16—21no—
274C16—42no—
275C16C2×M1612yes—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—82no—
280C23—82no—
281C23—161no—
282C23#44211yescharacteristic
283Q8—22no—
284Q8—41no—
285D4—22no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—82no—
292D4—82no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—161no—
307C2×C4—161no—
308C2×C4—161no—
309C2×C4—161no—
310C2×C4#44211yescharacteristic
311C2×C4#44211yescharacteristic
312C2×C4#44211yescharacteristic
313C2×C4C2×C4×D411yescharacteristic
314C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
315C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8—42no—
320C8—42no—
321C8—82no—
322C8—82no—
323C8C22×M1611yescharacteristic
324C8#56811yescharacteristic
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—164no—
338C22C2×#44211yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4D4×M1611yescharacteristic
350C4#295711yescharacteristic
351C2—41no—
352C2—82no—
353C2—161no—
354C2—161no—
355C2#1804411yescharacteristic
356C2#3346811yescharacteristic
357C2#3347111yescharacteristic
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. #18064C2semidirect
3. #18043C2semidirect
4. #12089C2semidirect
5. #12080C2semidirect
6. #12078C2semidirect
7. #12065C2semidirect
8. #12021C2semidirect
9. #11667C2semidirect
10. #11665C2semidirect
11. #11414C2semidirect
31. #1594C22semidirect
35. #1579C22semidirect
41. #1368C22semidirect
42. #1368C22semidirect
43. #1334C22semidirect
46. #1333C22semidirect
47. #1313C22semidirect
48. #1313C22semidirect
158. SD32M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5092
#180642
#180432
#120892
#120802
#120782
#120652
#120212
#116672
#116652
#114142