Finite Group Databasebeta

← #120557

Subgroups — #120557

Order 512.

422 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21994C211yesmaximal, characteristic
2#17822C211yesmaximal, characteristic
3#8109C211yesmaximal, characteristic
4#8041C211yesmaximal, characteristic
5#8003C211yesmaximal, characteristic
6#7990C211yesmaximal, characteristic
7#7952C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2285C2211yescharacteristic
15#1950—22no—
16#1855—22no—
17#1849C412yescomplemented
18#1838C412yescomplemented
19#1573—22no—
20#1571—21no—
21#1570—21no—
22#1569—22no—
23#1305—22no—
24#1296—22no—
25#1292—21no—
26#1292—21no—
27#1290—22no—
28#1285—22no—
29C22×M16—21no—
30C22×M16—21no—
31C2×#117Q812yes—
32C22×C42C2311yescharacteristic
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432D412yes—
43#432Q812yes—
44#432Q812yes—
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49C2×#108—24no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423—21no—
56#423—21no—
57#423—21no—
58#423C2×C412yes—
59#423C2×C411yescharacteristic
60#423C2×C411yescharacteristic
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8D411yescharacteristic
66C2×C4⋊C8D411yescharacteristic
67C2×C4⋊C8D412yes—
68#407—21no—
69#407D412yes—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C4×M16—21no—
73C4×M16D412yes—
74C2×#97—21no—
75C2×#97C2×C412yes—
76#384—21no—
77#382—21no—
78#380—21no—
79#380—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82#341—24no—
83#335—24no—
84#334—24no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×Q811yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—22no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—82no—
94C2×M16—82no—
95C2×M16C22⋊C412yes—
96C2×M16C22⋊C412yes—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—41no—
102#117—82no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C22⋊C4—22no—
107C2×C42—22no—
108C2×C42C22×C411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×Q811yescharacteristic
114C2×C42C2×Q811yescharacteristic
115C2×C42#4111yescharacteristic
116C2×C42#4111yescharacteristic
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120#108—84no—
121C4⋊C8—21no—
122C4⋊C8—21no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—21no—
128C4⋊C8—22no—
129C4⋊C8—21no—
130C4⋊C8—22no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149#97—22no—
150#97—22no—
151#97—22no—
152#97—41no—
153#97—41no—
154#97C4⋊C414yes—
155C4×C8—22no—
156C4×C8—41no—
157#95—41no—
158#95—41no—
159C24C22⋊Q811yescharacteristic
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C4×Q811yescharacteristic
169C22×C4C4×Q811yescharacteristic
170C22×C4#12111yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4C22⋊Q811yescharacteristic
175C22×C4C22⋊Q811yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4#12311yescharacteristic
179C22×C4#12411yescharacteristic
180C22×C4#12511yescharacteristic
181C22×C4#12511yescharacteristic
182C22×C4#12611yescharacteristic
183C22×C4C4⋊Q811yescharacteristic
184C22×C4C4⋊Q811yescharacteristic
185M16—42no—
186M16—42no—
187M16—44no—
188M16—81no—
189M16—81no—
190M16—82no—
191M16—82no—
192M16—82no—
193C2×C8—24no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—82no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—82no—
207C4⋊C4—82no—
208C4⋊C4—81no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C22⋊C4—82no—
213C42—82no—
214C42C2×C4⋊C411yescharacteristic
215C42#11711yescharacteristic
216C42C4×D411yescharacteristic
217C42C4×D411yescharacteristic
218C42#12011yescharacteristic
219C42#12111yescharacteristic
220C42#12111yescharacteristic
221C42C22⋊Q811yescharacteristic
222C42C22⋊Q811yescharacteristic
223C42C22⋊Q811yescharacteristic
224C42#12311yescharacteristic
225C42#12311yescharacteristic
226C42#12511yescharacteristic
227C42#12511yescharacteristic
228C42#12611yescharacteristic
229C42C4⋊Q811yescharacteristic
230C23—21no—
231C23—21no—
232C23—41no—
233C23—41no—
234C23#39111yescharacteristic
235C23#39611yescharacteristic
236C23#39811yescharacteristic
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#38011yescharacteristic
267C2×C4#38611yescharacteristic
268C2×C4#38711yescharacteristic
269C2×C4#38711yescharacteristic
270C2×C4#38911yescharacteristic
271C2×C4#38911yescharacteristic
272C2×C4#39411yescharacteristic
273C2×C4#39511yescharacteristic
274C2×C4#39611yescharacteristic
275C2×C4#39811yescharacteristic
276C2×C4#39811yescharacteristic
277C2×C4#40011yescharacteristic
278C8—44no—
279C8—42no—
280C8—42no—
281C8—82no—
282C8—81no—
283C8—82no—
284C8—81no—
285C8—82no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—81no—
295C22—81no—
296C22#144111yescharacteristic
297C22#206911yescharacteristic
298C22#207011yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—162no—
310C4#190911yescharacteristic
311C4#191011yescharacteristic
312C4#204811yescharacteristic
313C4#204911yescharacteristic
314C2—21no—
315C2—21no—
316C2—41no—
317C2—41no—
318C2#1122511yescharacteristic
319C2#1125211yescharacteristic
320C2#1125311yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1849C4semidirect
18. #1838C4semidirect

Maximal subgroups

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

subgroupindex
#219942
#178222
#81092
#80412
#80032
#79902
#79522