Finite Group Databasebeta

← #120378

Subgroups — #120378

Order 512.

465 conjugacy classes of subgroups.

The subgroup lattice has 299 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21359C211yesmaximal, characteristic
2#17808C211yesmaximal, characteristic
3#8109C211yesmaximal, characteristic
4#8098C211yesmaximal, characteristic
5#8066C211yesmaximal, characteristic
6#7961C211yesmaximal, characteristic
7#7950C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1948—22no—
16#1853—22no—
17C4×#106—22no—
18#1574C414yescomplemented
19#1572—22no—
20#1565—22no—
21#1305—22no—
22#1303—22no—
23#1299—22no—
24#1286—22no—
25#1285—22no—
26C2×#117—21no—
27C22×C4⋊C4—21no—
28C22×C42—21no—
29C22×C42C2311yescharacteristic
30#432—21no—
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432Q812yes—
37#432Q812yes—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431C2×C412yes—
48#431C2×C412yes—
49#431D412yes—
50C2×#106—24no—
51#423—21no—
52#423—21no—
53#423C2×C412yes—
54#423D412yes—
55C2×C4⋊C8D412yes—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60C2×C22⋊C8—21no—
61C4×M16—21no—
62C4×M16—21no—
63C4×M16—21no—
64C4×M16Q812yes—
65C2×C4×C8—21no—
66#384—21no—
67#382—21no—
68#380—21no—
69#380D412yes—
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4D412yes—
76C2×#95—21no—
77#340—24no—
78#335—24no—
79#334—24no—
80C23×C4—41no—
81C23×C4C2×D411yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—41no—
88C22×C8—22no—
89#117—82no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—44no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C22⋊C4—22no—
97C2×C42—22no—
98C2×C42—41no—
99C2×C42—41no—
100C2×C42—82no—
101C2×C42C22×C411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42C2×Q811yescharacteristic
107C2×C42C2×Q811yescharacteristic
108C2×C42C2×Q811yescharacteristic
109C2×C42#4111yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4111yescharacteristic
112C2×C42#4111yescharacteristic
113#106—48no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8C22⋊C414yes—
123C4⋊C8C4⋊C414yes—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134#97—22no—
135#97—22no—
136#97—22no—
137#97—41no—
138C4×C8—22no—
139C4×C8—22no—
140C4×C8—22no—
141C4×C8—22no—
142C4×C8—41no—
143C4×C8—41no—
144#95—22no—
145#95—22no—
146#95—41no—
147#95—41no—
148C24#12311yescharacteristic
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—81no—
156C22×C4—82no—
157C22×C4—82no—
158C22×C4#11711yescharacteristic
159C22×C4C4×D411yescharacteristic
160C22×C4C4×Q811yescharacteristic
161C22×C4#12011yescharacteristic
162C22×C4#12111yescharacteristic
163C22×C4#12111yescharacteristic
164C22×C4#12111yescharacteristic
165C22×C4#12111yescharacteristic
166C22×C4C22⋊Q811yescharacteristic
167C22×C4C22⋊Q811yescharacteristic
168C22×C4C22⋊Q811yescharacteristic
169C22×C4#12311yescharacteristic
170C22×C4#12411yescharacteristic
171C22×C4#12511yescharacteristic
172C22×C4#12511yescharacteristic
173C22×C4#12511yescharacteristic
174C22×C4#12611yescharacteristic
175C22×C4C4⋊Q811yescharacteristic
176M16—42no—
177M16—42no—
178M16—82no—
179M16—82no—
180M16—82no—
181C2×C8—24no—
182C2×C8—24no—
183C2×C8—44no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C4⋊C4—24no—
188C4⋊C4—42no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C22⋊C4—82no—
194C42—82no—
195C42—82no—
196C42—82no—
197C42C2×C22⋊C411yescharacteristic
198C42C2×C4⋊C411yescharacteristic
199C42C4×D411yescharacteristic
200C42C4×Q811yescharacteristic
201C42#12111yescharacteristic
202C42#12111yescharacteristic
203C42C22⋊Q811yescharacteristic
204C42C22⋊Q811yescharacteristic
205C42C22⋊Q811yescharacteristic
206C42C22⋊Q811yescharacteristic
207C42C22⋊Q811yescharacteristic
208C42#12311yescharacteristic
209C42#12511yescharacteristic
210C42#12611yescharacteristic
211C42C4⋊Q811yescharacteristic
212C42C4⋊Q811yescharacteristic
213C23—21no—
214C23—21no—
215C23—41no—
216C23—41no—
217C23#38711yescharacteristic
218C23#39411yescharacteristic
219C23#39811yescharacteristic
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—44no—
239C2×C4—82no—
240C2×C4—88no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4#38011yescharacteristic
246C2×C4#38611yescharacteristic
247C2×C4#38711yescharacteristic
248C2×C4#38911yescharacteristic
249C2×C4#38911yescharacteristic
250C2×C4#39111yescharacteristic
251C2×C4#39511yescharacteristic
252C2×C4#39611yescharacteristic
253C2×C4#39611yescharacteristic
254C2×C4#39811yescharacteristic
255C2×C4#39811yescharacteristic
256C2×C4#40011yescharacteristic
257C2×C4#47112yes—
258C2×C4#47112yes—
259C8—44no—
260C8—44no—
261C8—82no—
262C8—82no—
263C8—82no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22#144111yescharacteristic
275C22#204611yescharacteristic
276C22#204611yescharacteristic
277C4—21no—
278C4—21no—
279C4—21no—
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—84no—
287C4—162no—
288C4#181512yes—
289C4#190612yes—
290C4#209012yes—
291C2—21no—
292C2—21no—
293C2—41no—
294C2—41no—
295C2#1121511yescharacteristic
296C2#1123811yescharacteristic
297C2#1124711yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
18. #1574C4semidirect

Maximal subgroups

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

subgroupindex
#213592
#178082
#81092
#80982
#80662
#79612
#79502