Finite Group Databasebeta

← #190001

Subgroups — #190001

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2298C211yesmaximal, characteristic
2#17357C212yesmaximal
3#8733C214yesmaximal, complemented
4#8732C214yesmaximal, complemented
5#7867C212yesmaximal, complemented
6#7855C212yesmaximal, complemented
7C22×#117C2211yescharacteristic
8#2920C2214yes—
9#2298—22no—
10#2298—22no—
11#2298C412yes—
12#2298C412yes—
13#2298C2212yes—
14#2298C2212yes—
15C2×#388—22no—
16C2×#387—22no—
17C2×C4×C4⋊C4C412yes—
18C2×C4×C22⋊C4C412yes—
19C2×#376C2211yescharacteristic
20C2×C43C2211yescharacteristic
21#1464—22no—
22#1464—22no—
23#1463—22no—
24#1462—22no—
25#1458C2214yes—
26#1458C2214yes—
27#1276C2218yescomplemented
28#1274C2218yescomplemented
29C24×C4C2×C411yescharacteristic
30C2×#117—24no—
31C2×#117—24no—
32C2×#117C2×C411yescharacteristic
33C2×#117C2×C411yescharacteristic
34C2×#117D412yes—
35C2×#117D412yes—
36C2×#117C2311yescharacteristic
37C2×#117C2311yescharacteristic
38C22×C4⋊C4—22no—
39C22×C4⋊C4D412yes—
40C22×C22⋊C4—22no—
41C22×C22⋊C4D412yes—
42C22×C42—22no—
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C412yes—
45C22×C42C2×C412yes—
46C22×C42C2311yescharacteristic
47#432—24no—
48#432—24no—
49#431—24no—
50#431C2318yes—
51#423—24no—
52#407—24no—
53C4×M16—24no—
54#388—24no—
55#388—24no—
56#388—24no—
57#388—24no—
58#387—24no—
59#387—24no—
60#387—24no—
61#387—24no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4C2×C414yes—
65C4×C4⋊C4C2×C414yes—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4C2×C414yes—
69C4×C22⋊C4C2×C414yes—
70#376—22no—
71#376C2×C412yes—
72#376C2312yes—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C43—22no—
77C43C2×C412yes—
78C43C2312yes—
79#324—24no—
80#324—24no—
81#323—24no—
82#323—24no—
83C25C22⋊C411yescharacteristic
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—44no—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D412yes—
97C2×M16—44no—
98#117—22no—
99#117—22no—
100#117—22no—
101#117—22no—
102#117—24no—
103#117—24no—
104#117—24no—
105#117—24no—
106#117—22no—
107#117—22no—
108#117—22no—
109#117—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—48no—
115C2×C4⋊C4C22⋊C412yes—
116C2×C4⋊C4C22⋊C412yes—
117C2×C4⋊C4C2×D412yes—
118C2×C4⋊C4C2×D412yes—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—48no—
124C2×C22⋊C4C22⋊C412yes—
125C2×C22⋊C4C22⋊C412yes—
126C2×C22⋊C4C2×D412yes—
127C2×C22⋊C4C2×D412yes—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—24no—
132C2×C42—22no—
133C2×C42—24no—
134C2×C42—44no—
135C2×C42—44no—
136C2×C42—48no—
137C2×C42C22⋊C412yes—
138C2×C42C22×C414yes—
139C2×C42C22×C414yes—
140C2×C42C22×C411yescharacteristic
141C2×C42C22×C411yescharacteristic
142C2×C42C22×C411yescharacteristic
143C2×C42C2×D412yes—
144C2×C42#4114yes—
145C2×C42#4114yes—
146C2×C42C2411yescharacteristic
147C4⋊C8—28no—
148C4⋊C8—44no—
149C22⋊C8—28no—
150C22⋊C8—44no—
151#97—44no—
152C4×C8—28no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159C24—22no—
160C24—42no—
161C24—44no—
162C24C2×C22⋊C411yescharacteristic
163C24C2×C22⋊C411yescharacteristic
164C24C2×C22⋊C411yescharacteristic
165C22×C4—24no—
166C22×C4—28no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—48no—
185C22×C4—48no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C2×C22⋊C412yes—
196C22×C4C2×C22⋊C412yes—
197C22×C4C2×C22⋊C412yes—
198C22×C4#11712yes—
199C22×C4#11712yes—
200C22×C4#12314yes—
201C22×C4#12314yes—
202C22×C4C23×C411yescharacteristic
203C22×C4C22×D412yes—
204M16—88no—
205C2×C8—48no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—48no—
211C4⋊C4—48no—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—48no—
217C22⋊C4—48no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—22no—
223C42—22no—
224C42—28no—
225C42—28no—
226C42—44no—
227C42—44no—
228C42—48no—
229C42—48no—
230C42#11712yes—
231C42#11712yes—
232C42#12314yes—
233C42#12314yes—
234C42C2×#4112yes—
235C42C2×#4112yes—
236C23—24no—
237C23—22no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—44no—
243C23—44no—
244C23—42no—
245C23—42no—
246C23—44no—
247C23—88no—
248C23#38112yes—
249C23C22×C22⋊C411yescharacteristic
250C2×C4—28no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—28no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—416no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—88no—
282C2×C4#38112yes—
283C2×C4#38114yes—
284C2×C4C2×#11711yescharacteristic
285C2×C4C2×#11711yescharacteristic
286C2×C4C2×#12314yes—
287C2×C4#56814yes—
288C8—88no—
289C22—28no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—41no—
294C22—41no—
295C22—42no—
296C22—42no—
297C22—82no—
298C22—82no—
299C22—88no—
300C22#182714yes—
301C22C2×#38111yescharacteristic
302C22#287912yes—
303C4—22no—
304C4—24no—
305C4—24no—
306C4—48no—
307C4—41no—
308C4—41no—
309C4—42no—
310C4—42no—
311C4—84no—
312C4#292112yes—
313C4#292112yes—
314C2—22no—
315C2—24no—
316C2—81no—
317C2—81no—
318C2#2068311yescharacteristic
319C2C2×#182712yes—
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#381
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#376 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8733C2semidirect
4. #8732C2semidirect
5. #7867C2semidirect
6. #7855C2semidirect
27. #1276C22semidirect
28. #1274C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22982
#173572
#87332
#87322
#78672
#78552