Finite Group Databasebeta

← #225529

Subgroups — #225529

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22763C212yesmaximal
2#22739C211yesmaximal, characteristic
3#22738C211yesmaximal, characteristic
4C2×C8×Q16C211yesmaximal, characteristic
5#9619C212yesmaximal
6#8884C212yesmaximal
7#8883C212yesmaximal
8#8816C211yesmaximal, characteristic
9#8647C211yesmaximal, characteristic
10#8647C211yesmaximal, characteristic
11#8283C211yesmaximal, characteristic
12C2×C8×Q8C2211yescharacteristic
13C2×C8×Q8C2211yescharacteristic
14C2×C4×Q16C2211yescharacteristic
15#2355C2212yes—
16Q8×C4⋊C4C2212yes—
17#1949—22no—
18#1948—22no—
19#1948C2212yes—
20#1947C2212yes—
21#1935—22no—
22#1918—22no—
23#1878—22no—
24#1845C2212yes—
25#1835—21no—
26#1834—21no—
27#1834C2212yes—
28#1834C2211yescharacteristic
29#1834C2211yescharacteristic
30#1803C2212yes—
31#1803C2212yes—
32C4×#107—21no—
33C4×#106—21no—
34C4×#106C2211yescharacteristic
35#1770C2212yes—
36#1763—22no—
37C8×Q16C418yescomplemented
38C2×#335C2211yescharacteristic
39C2×#334C2211yescharacteristic
40C2×Q8⋊C8C2211yescharacteristic
41C2×Q8⋊C8C2211yescharacteristic
42C2×C82C2211yescharacteristic
43#1296C2212yes—
44#1296C2212yes—
45#1279C2212yes—
46C22×Q16D411yescharacteristic
47C2×C4×Q8—22no—
48C2×C4×Q8C2311yescharacteristic
49C2×C4×Q8C2311yescharacteristic
50C8×Q8—22no—
51C8×Q8—22no—
52C8×Q8C2×C414yes—
53C8×Q8C2×C414yes—
54C4×Q16C2×C414yes—
55C4×Q16Q814yes—
56C2×#107—22no—
57C2×#107—42no—
58C2×#107D411yescharacteristic
59C2×#106—22no—
60C2×#106D412yes—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×#103—22no—
67C2×#103—22no—
68C2×#103D411yescharacteristic
69C2×#103D411yescharacteristic
70C2×#97—22no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#389—22no—
78#389—42no—
79#387—42no—
80#384—22no—
81#384C2312yes—
82#384C2312yes—
83#382—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2312yes—
88#340—21no—
89#340—21no—
90#340—22no—
91#340—22no—
92#340—21no—
93#340—21no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340D411yescharacteristic
99#340D411yescharacteristic
100#336—22no—
101#335C2×C414yes—
102#334—22no—
103Q8⋊C8C2×C414yes—
104Q8⋊C8C2×C414yes—
105C82C2×C414yes—
106C22×Q8C2×D411yescharacteristic
107C22×Q8C2×D411yescharacteristic
108C2×Q16—24no—
109C2×Q16C4⋊C414yes—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C4×Q8—21no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—44no—
123C4×Q8—82no—
124C4×Q8C22×C412yes—
125C4×Q8C22×C412yes—
126C4×Q8C2×Q812yes—
127C4×Q8C2×Q812yes—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4C2×D412yes—
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D411yescharacteristic
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—22no—
145C2×C42—42no—
146C2×C42—42no—
147C2×C42C2411yescharacteristic
148#107—44no—
149#107—84no—
150#107C4⋊C414yes—
151#106—42no—
152#106—44no—
153C4⋊C8—24no—
154C4⋊C8—24no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8C22×C412yes—
160C4⋊C8C22×C412yes—
161#103—44no—
162#103—82no—
163#103C4⋊C414yes—
164#103C4⋊C414yes—
165#97—82no—
166C4×C8—24no—
167C4×C8—24no—
168C4×C8—21no—
169C4×C8C22×C412yes—
170C4×C8C22×C412yes—
171C4×C8C22×C412yes—
172C4×C8C2×Q812yes—
173C4×C8#4112yes—
174#95—22no—
175#95—42no—
176#95—42no—
177#95—42no—
178#95—42no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8C2×C4⋊C412yes—
185C2×Q8C2×C4⋊C412yes—
186C2×Q8#13514yes—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4C22×D411yescharacteristic
196C22×C4C22×D411yescharacteristic
197C22×C4H(5,𝔽2)11yescharacteristic
198Q16—28no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8C2×C4⋊C412yes—
214C2×C8C2×C4⋊C412yes—
215C2×C8C4×D412yes—
216C2×C8C4×D412yes—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—84no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4—162no—
239C4⋊C4C2×C4⋊C412yes—
240C4⋊C4C2×C4⋊C412yes—
241C42—22no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—42no—
246C42—42no—
247C42—84no—
248C42—162no—
249C42C23×C411yescharacteristic
250C42C22×Q811yescharacteristic
251C42C2×#4111yescharacteristic
252C422−1+411yescharacteristic
253C23D4211yescharacteristic
254Q8—24no—
255Q8—22no—
256Q8—22no—
257Q8—22no—
258Q8—42no—
259Q8#42714yes—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4C22×C4⋊C411yescharacteristic
285C2×C4C2×C4×D411yescharacteristic
286C2×C4#52011yescharacteristic
287C2×C4#54711yescharacteristic
288C2×C4D4×Q811yescharacteristic
289C2×C4#55411yescharacteristic
290C2×C4C2×#13512yes—
291C2×C4#57512yes—
292C2×C4#57712yes—
293C2×C4#57812yes—
294C8—24no—
295C8—22no—
296C8—22no—
297C8—22no—
298C8—22no—
299C8—42no—
300C8—42no—
301C8—82no—
302C22D4×C4⋊C411yescharacteristic
303C22#327712yes—
304C22#328212yes—
305C22#329211yescharacteristic
306C22#329311yescharacteristic
307C4—24no—
308C4—21no—
309C4—21no—
310C4—42no—
311C4—42no—
312C4—42no—
313C4—82no—
314C4—82no—
315C4—162no—
316C4—162no—
317C4C2×#42711yescharacteristic
318C4#291211yescharacteristic
319C4#294311yescharacteristic
320C4#295011yescharacteristic
321C4#333712yes—
322C4#339012yes—
323C2#1992012yes—
324C2#1992512yes—
325C2#2265411yescharacteristic
326C2#2267511yescharacteristic
327C2#2269211yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
37. C8×Q16C4semidirect

Maximal subgroups

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

subgroupindex
#227632
#227392
#227382
C2×C8×Q162
#96192
#88842
#88832
#88162
#86472
#82832