Finite Group Databasebeta

← #484086

Subgroups — #484086

Order 512.

454 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#508C211yesmaximal, characteristic
2#18059C211yesmaximal, characteristic, complemented
3#18055C211yesmaximal, characteristic, complemented
4#12082C212yesmaximal, complemented
5#12076C212yesmaximal, complemented
6#12073C212yesmaximal, complemented
7C8×SD32C212yesmaximal, complemented
8#12023C211yesmaximal, characteristic, complemented
9#11660C211yesmaximal, characteristic, complemented
10#11660C211yesmaximal, characteristic, complemented
11C16×M16C211yesmaximal, characteristic, complemented
12C2×#508C411yescharacteristic
13#2991—21no—
14#2990—21no—
15C4×#135C2211yescharacteristic
16C4×#135C2211yescharacteristic
17C4×Q32C411yescharacteristic
18C4×Q32C2211yescharacteristic
19C4×SD32C411yescharacteristic
20C4×SD32C411yescharacteristic
21C4×SD32C2211yescharacteristic
22C4×SD32C2211yescharacteristic
23C4×D16C411yescharacteristic
24C4×D16C2211yescharacteristic
25#2162C2211yescharacteristic
26#2158C411yescharacteristic
27#2138C411yescharacteristic
28#2138C411yescharacteristic
29C2×C4×C16C2211yescharacteristic
30#1591C2212yes—
31#1589C2212yes—
32#1588—22no—
33#1585—22no—
34C8×Q16C2212yes—
35C8×D8C2212yes—
36#1567C2211yescharacteristic
37#1567C2211yescharacteristic
38#1489—21no—
39#1489—21no—
40C8×M16C2211yescharacteristic
41#1372C2212yes—
42#1372C2212yes—
43#1332C2212yes—
44#1332C2212yes—
45#1331C2212yes—
46#1331C2212yes—
47#1312C2212yes—
48C8×C16C2212yes—
49C2×#135C2×C411yescharacteristic
50C2×#135C2×C411yescharacteristic
51C4×#41—21no—
52C4×#41—21no—
53#508—24no—
54C2×Q32C2×C411yescharacteristic
55C2×SD32C2×C411yescharacteristic
56C2×SD32C2×C411yescharacteristic
57C2×D16C2×C411yescharacteristic
58C22×C16C2×C411yescharacteristic
59C22×C16C2×C411yescharacteristic
60#446—22no—
61C8×Q8—22no—
62C4×Q16C2×C411yescharacteristic
63C4×Q16C2×C411yescharacteristic
64C4×Q16C2311yescharacteristic
65C4×Q16C2311yescharacteristic
66C4×SD16—21no—
67C4×SD16—21no—
68C4×SD16—21no—
69C4×SD16—21no—
70C4×D8C2×C411yescharacteristic
71C4×D8C2×C411yescharacteristic
72C4×D8C2311yescharacteristic
73C4×D8C2311yescharacteristic
74#436—22no—
75#436—42no—
76#435—42no—
77C8×D4—22no—
78#432—41no—
79#431—41no—
80#431D411yescharacteristic
81#427C2×C411yescharacteristic
82#427C2×C411yescharacteristic
83#423—21no—
84#423—21no—
85#423—41no—
86C2×C4⋊C8—41no—
87#416—21no—
88#416—21no—
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#358C2×C411yescharacteristic
99#358C2×C411yescharacteristic
100#357C2×C411yescharacteristic
101#357C2×C411yescharacteristic
102#357C2×C411yescharacteristic
103#357C2×C411yescharacteristic
104C22⋊C16—21no—
105C4×C16C2×C411yescharacteristic
106C4×C16C2×C411yescharacteristic
107C4×C16C2312yes—
108C4×C16C2311yescharacteristic
109C4×C16C2311yescharacteristic
110#335—21no—
111#335—21no—
112#335C2312yes—
113#335C2312yes—
114Q8⋊C8—22no—
115Q8⋊C8—22no—
116D4⋊C8—22no—
117D4⋊C8—22no—
118#322D412yes—
119C82C2312yes—
120C2×#41—21no—
121C2×#41—21no—
122#135—22no—
123#135—22no—
124#135—41no—
125#135—41no—
126C2×Q16C22×C411yescharacteristic
127C2×Q16C22×C411yescharacteristic
128C2×SD16—21no—
129C2×SD16—21no—
130C2×D8C22×C411yescharacteristic
131C2×D8C22×C411yescharacteristic
132C2×M16—21no—
133C2×M16—41no—
134C2×M16—41no—
135C2×M16—42no—
136C22×C8—42no—
137C22×C8C22×C411yescharacteristic
138C22×C8#4111yescharacteristic
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—21no—
142C4×Q8—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—21no—
153#117—41no—
154#117—41no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42C2×D411yescharacteristic
158Q32—22no—
159SD32—22no—
160SD32M1614yescomplemented
161D16—22no—
162C2×C16—22no—
163C2×C16—22no—
164C2×C16—41no—
165C2×C16C22×C411yescharacteristic
166C2×C16C22×C411yescharacteristic
167C2×C16C22×C411yescharacteristic
168C2×C16C22×C411yescharacteristic
169#107C22×C411yescharacteristic
170#107C22×C411yescharacteristic
171#107C22×C411yescharacteristic
172#107C22×C411yescharacteristic
173#106—21no—
174#106—21no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—42no—
178C4⋊C8—42no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8D811yescharacteristic
182C4⋊C8D811yescharacteristic
183#103—21no—
184#103—21no—
185#103—21no—
186#103—21no—
187#102—21no—
188#102—21no—
189#102—21no—
190#102—21no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193C22⋊C8D812yes—
194#97—21no—
195#97—42no—
196C4×C8—42no—
197C4×C8C22×C411yescharacteristic
198C4×C8C2×D412yes—
199C4×C8C2×D411yescharacteristic
200C4×C8C2×D412yes—
201C4×C8#4111yescharacteristic
202C4×C8C2411yescharacteristic
203#41—42no—
204#41—42no—
205#41—81no—
206#41—81no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—41no—
212C2×D4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4C4×D411yescharacteristic
220C22×C4C2×D811yescharacteristic
221C22×C4#13511yescharacteristic
222Q16—21no—
223Q16—21no—
224Q16—21no—
225Q16C2×M1612yes—
226SD16—42no—
227SD16—42no—
228D8—21no—
229D8—21no—
230D8—21no—
231D8C2×M1612yes—
232M16—24no—
233M16—162no—
234M16—161no—
235M16—161no—
236C2×C8—22no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—162no—
243C2×C8C4×D411yescharacteristic
244C2×C8C4×D411yescharacteristic
245C2×C8C2×D812yes—
246C2×C8C2×D812yes—
247C2×C8C23×C411yescharacteristic
248C2×C8C2×#4111yescharacteristic
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—41no—
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—
263C42—41no—
264C42—41no—
265C42—41no—
266C42—41no—
267C42—41no—
268C42—41no—
269C42C4×D411yescharacteristic
270C42C2×D811yescharacteristic
271C42#13511yescharacteristic
272C42C22×D411yescharacteristic
273C16—21no—
274C16—21no—
275C16—21no—
276C16—42no—
277C16C2×M1612yes—
278C23—41no—
279C23—41no—
280C23C4×D811yescharacteristic
281Q8—22no—
282Q8—41no—
283Q8—41no—
284Q8—41no—
285D4—22no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—82no—
290D4—82no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
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—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4C4×D811yescharacteristic
311C2×C4C4×D811yescharacteristic
312C2×C4C4×D811yescharacteristic
313C2×C4C2×C4×D411yescharacteristic
314C2×C4C22×D811yescharacteristic
315C2×C4C2×#13511yescharacteristic
316C8—24no—
317C8—22no—
318C8—22no—
319C8—21no—
320C8—21no—
321C8—21no—
322C8—82no—
323C8—82no—
324C8C22×M1611yescharacteristic
325C8#56811yescharacteristic
326C22—21no—
327C22—21no—
328C22—21no—
329C22—42no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22C2×C4×D811yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—42no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4#217812yes—
346C4D4×M1611yescharacteristic
347C4#295711yescharacteristic
348C4#341611yescharacteristic
349C4#341611yescharacteristic
350C2—41no—
351C2—82no—
352C2—161no—
353C2M16×D811yescharacteristic
354C2#3346011yescharacteristic
355C2C2×#217811yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radicalC2×#508C4
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. #18059C2semidirect
3. #18055C2semidirect
4. #12082C2semidirect
5. #12076C2semidirect
6. #12073C2semidirect
7. C8×SD32C2semidirect
8. #12023C2semidirect
9. #11660C2semidirect
10. #11660C2semidirect
11. C16×M16C2semidirect
160. SD32M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5082
#180592
#180552
#120822
#120762
#120732
C8×SD322
#120232
#116602
C16×M162