Finite Group Databasebeta

← #243199

Subgroups — #243199

Order 512.

642 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21964C212yesmaximal
2#21143C211yesmaximal, characteristic
3#20817C211yesmaximal, characteristic
4#10657C214yesmaximal, complemented
5#10651C212yesmaximal, complemented
6#10650C212yesmaximal, complemented
7#8041C212yesmaximal
8#7874C211yesmaximal, characteristic
9C2×#424—21no—
10C2×#423C2211yescharacteristic
11C2×#423C2212yes—
12C22×C4×C8C2211yescharacteristic
13#2285C2212yes—
14#2285C2211yescharacteristic
15#1948—22no—
16#1948C2214yes—
17#1947—22no—
18#1947C2214yes—
19#1939—21no—
20#1939—21no—
21#1939C2212yes—
22#1939C2212yes—
23#1845C2211yescharacteristic
24#1845C2211yescharacteristic
25#1834C2212yes—
26#1809—24no—
27#1808—21no—
28#1808—21no—
29C4×#107—21no—
30C4×#107—21no—
31#1729—21no—
32#1296C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35#1277C2212yes—
36C22×M16—22no—
37C22×M16—21no—
38C23×C8D411yescharacteristic
39C23×C8D411yescharacteristic
40C2×#117—21no—
41C2×#117—22no—
42C22×C42C2311yescharacteristic
43C2×#107—21no—
44C2×#107—21no—
45C2×#107—21no—
46C2×#107—21no—
47C2×#107—24no—
48C2×#106—24no—
49#424—44no—
50#423—24no—
51#423—24no—
52#423Q818yes—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2314yes—
59C2×C4×C8—24no—
60C2×C4×C8D411yescharacteristic
61C2×C4×C8D411yescharacteristic
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#384D411yescharacteristic
65#384D411yescharacteristic
66#384C2312yes—
67#384C2312yes—
68#382—21no—
69#382—22no—
70#380—21no—
71#380—22no—
72#380—22no—
73#380D412yes—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4C2312yes—
78#340—22no—
79#340—22no—
80#340—24no—
81#340—24no—
82#336—22no—
83#336—22no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4#4111yescharacteristic
87C2×M16—28no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—82no—
92C2×M16—82no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—24no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105#117—42no—
106#117—42no—
107#117—84no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4C2×D412yes—
119C2×C4⋊C4C2×D412yes—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—42no—
122C2×C42—22no—
123C2×C42—42no—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×Q814yes—
127C2×C42#4114yes—
128C2×C42#4111yescharacteristic
129C2×C42C2411yescharacteristic
130#107—42no—
131#107—42no—
132#107—42no—
133#107—42no—
134#107—84no—
135#106—84no—
136C4⋊C8—28no—
137C4⋊C8—24no—
138C4⋊C8—24no—
139C4⋊C8—44no—
140C4⋊C8—44no—
141C4⋊C8—42no—
142C4⋊C8C2×Q818yes—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—22no—
147#95—22no—
148#95—22no—
149#95—42no—
150#95—42no—
151C24#12111yescharacteristic
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4#12111yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4C22⋊Q814yes—
169C22×C4#12312yes—
170C22×C4#12514yes—
171C22×C4C4⋊Q812yes—
172C22×C4C22×D411yescharacteristic
173C22×C4C22×D411yescharacteristic
174C22×C4C2×#4111yescharacteristic
175M16—88no—
176M16—84no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—28no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—44no—
191C2×C8—44no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8AGL(1,ℤ/8)12yes—
196C2×C8#13712yes—
197C4⋊C4—22no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C22⋊C4—44no—
215C22⋊C4—84no—
216C42—42no—
217C42—42no—
218C42—84no—
219C42C22⋊Q814yes—
220C42#12312yes—
221C42#12514yes—
222C42C4⋊Q812yes—
223C42C22×Q812yes—
224C42C2×#4112yes—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23#39812yes—
232C23C2×#12111yescharacteristic
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4#39812yes—
259C2×C4#39812yes—
260C2×C4#39812yes—
261C2×C4#47212yes—
262C2×C4#47312yes—
263C2×C4C2×C22⋊Q812yes—
264C2×C4C2×#12311yescharacteristic
265C2×C4C2×#12512yes—
266C2×C4C2×C4⋊Q811yescharacteristic
267C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
268C2×C4C2×#13711yescharacteristic
269C2×C4#57511yescharacteristic
270C2×C4#57511yescharacteristic
271C2×C4#57611yescharacteristic
272C2×C4#57712yes—
273C2×C4#57811yescharacteristic
274C8—22no—
275C8—22no—
276C8—48no—
277C8—41no—
278C8—41no—
279C8—82no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22C2×#39811yescharacteristic
288C22#303611yescharacteristic
289C22#303711yescharacteristic
290C22#306011yescharacteristic
291C22#306111yescharacteristic
292C22C2×#47211yescharacteristic
293C22C2×#47311yescharacteristic
294C4—22no—
295C4—22no—
296C4—22no—
297C4—82no—
298C4—82no—
299C4—162no—
300C4—162no—
301C4#208914yes—
302C4#308112yes—
303C4#308212yes—
304C4#309411yescharacteristic
305C4#309511yescharacteristic
306C4#316511yescharacteristic
307C4#316611yescharacteristic
308C2—42no—
309C2#1933311yescharacteristic
310C2#1933711yescharacteristic
311C2#1934411yescharacteristic
312C2#1935011yescharacteristic
313C2#2365711yescharacteristic
314C2#2366911yescharacteristic
315C2C2×#208911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10657C2semidirect
5. #10651C2semidirect
6. #10650C2semidirect

Maximal subgroups

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

subgroupindex
#219642
#211432
#208172
#106572
#106512
#106502
#80412
#78742