Finite Group Databasebeta

← #189917

Subgroups — #189917

Order 512.

1282 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2#20715C212yesmaximal
3#8721C214yesmaximal, complemented
4#8718C214yesmaximal, complemented
5#8031C214yesmaximal, complemented
6C22×#117C412yes—
7C23×C42C2211yescharacteristic
8C2×#432C2214yes—
9#2285—22no—
10#2285—22no—
11#2285C412yes—
12#2285C412yes—
13#2285C2212yes—
14#2285C2212yes—
15C2×#384C411yescharacteristic
16C2×#384C411yescharacteristic
17C2×#382C2212yes—
18C2×#380—22no—
19C2×C4×C4⋊C4—22no—
20#1817—22no—
21#1797—22no—
22#1797—22no—
23#1779—22no—
24#1749C2214yes—
25#1749C2214yes—
26#1295C22116yescomplemented
27C24×C4C2×C411yescharacteristic
28C24×C4D412yes—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117—48no—
32C2×#117C2×C414yes—
33C2×#117C2×C414yes—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C412yes—
38C22×C22⋊C4—22no—
39C22×C42—28no—
40C22×C42—22no—
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42D412yes—
45C22×C42D412yes—
46C22×C42D412yes—
47C22×C42C2311yescharacteristic
48C22×C42C2311yescharacteristic
49C22×C42C2311yescharacteristic
50#432—216no—
51C2×C4⋊C8—24no—
52C2×C22⋊C8—24no—
53C2×C22⋊C8—24no—
54C2×C22⋊C8—24no—
55C2×#97C2318yes—
56#384—22no—
57#384—22no—
58#384C2×C412yes—
59#384C2×C412yes—
60#384C2×C412yes—
61#384C2×C412yes—
62#382—24no—
63#382C2×C414yes—
64#382C2314yes—
65#380—24no—
66#380—24no—
67#380—24no—
68#380—24no—
69C4×C4⋊C4—24no—
70C4×C4⋊C4—24no—
71C4×C4⋊C4—24no—
72C4×C4⋊C4—24no—
73C2×#95—24no—
74#336—24no—
75#336—24no—
76C25C22⋊C411yescharacteristic
77C23×C4—24no—
78C23×C4—24no—
79C23×C4—24no—
80C23×C4—28no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—44no—
84C23×C4—44no—
85C23×C4C22⋊C412yes—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C412yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C22×C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C22×C8—28no—
98#117—48no—
99#117—48no—
100#117—48no—
101#117—48no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—48no—
111C2×C4⋊C4—48no—
112C2×C4⋊C4C22×C414yes—
113C2×C4⋊C4C22×C414yes—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—48no—
117C2×C42—24no—
118C2×C42—28no—
119C2×C42—24no—
120C2×C42—24no—
121C2×C42—28no—
122C2×C42—28no—
123C2×C42—24no—
124C2×C42—24no—
125C2×C42—24no—
126C2×C42—24no—
127C2×C42—48no—
128C2×C42C22⋊C412yes—
129C2×C42C22⋊C412yes—
130C2×C42C22⋊C412yes—
131C2×C42C22×C411yescharacteristic
132C2×C42C22×C411yescharacteristic
133C2×C42C22×C411yescharacteristic
134C2×C42C2×D412yes—
135C2×C42C2×D412yes—
136C2×C42C2×D412yes—
137C2×C42#4118yes—
138C2×C42C2411yescharacteristic
139C4⋊C8—48no—
140C22⋊C8—48no—
141C22⋊C8—48no—
142C22⋊C8—48no—
143#97—216no—
144#95—28no—
145#95—28no—
146C24—22no—
147C24—22no—
148C24—22no—
149C24—24no—
150C24—42no—
151C24C2×C22⋊C411yescharacteristic
152C24C2×C22⋊C411yescharacteristic
153C24C2×C22⋊C411yescharacteristic
154C22×C4—232no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—432no—
190C22×C4—48no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—48no—
194C22×C4#10014yes—
195C22×C4#10114yes—
196C22×C4C2×C22⋊C412yes—
197C22×C4C2×C22⋊C412yes—
198C22×C4C2×C22⋊C411yescharacteristic
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C2×C22⋊C412yes—
201C22×C4C2×C22⋊C411yescharacteristic
202C22×C4#11714yes—
203C22×C4#12318yes—
204C22×C4C23×C411yescharacteristic
205C22×C4C22×D412yes—
206C2×C8—48no—
207C2×C8—48no—
208C2×C8—48no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—48no—
212C4⋊C4—48no—
213C4⋊C4—48no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—48no—
217C22⋊C4—48no—
218C22⋊C4—48no—
219C42—22no—
220C42—22no—
221C42—24no—
222C42—24no—
223C42—24no—
224C42—24no—
225C42—24no—
226C42—48no—
227C42—48no—
228C42#11714yes—
229C42#12318yes—
230C42C2×#4114yes—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—28no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—21no—
243C23—48no—
244C23—48no—
245C23—44no—
246C23—44no—
247C23#38112yes—
248C23C2×#10012yes—
249C23C2×#10112yes—
250C23#42114yes—
251C23C22×C22⋊C411yescharacteristic
252C2×C4—28no—
253C2×C4—28no—
254C2×C4—28no—
255C2×C4—216no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—28no—
269C2×C4—28no—
270C2×C4—28no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—48no—
275C2×C4—48no—
276C2×C4—48no—
277C2×C4—48no—
278C2×C4—48no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—48no—
282C2×C4—48no—
283C2×C4—88no—
284C2×C4—88no—
285C2×C4#38114yes—
286C2×C4#38112yes—
287C2×C4C2×#10012yes—
288C2×C4C2×#10012yes—
289C2×C4C2×#10112yes—
290C2×C4C2×#10112yes—
291C2×C4#42114yes—
292C2×C4C2×#11712yes—
293C2×C4C2×#12314yes—
294C8—88no—
295C22—24no—
296C22—22no—
297C22—24no—
298C22—22no—
299C22—24no—
300C22—22no—
301C22—22no—
302C22—24no—
303C22—416no—
304C22—42no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22#153614yes—
311C22C2×#38111yescharacteristic
312C22C22×#10011yescharacteristic
313C22C22×#10111yescharacteristic
314C22C2×#42112yes—
315C22C2×#42112yes—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—24no—
320C4—24no—
321C4—24no—
322C4—42no—
323C4—84no—
324C4—84no—
325C4#182114yes—
326C2—24no—
327C2—42no—
328C2—42no—
329C2C2×#153611yescharacteristic
330C2#1766311yescharacteristic
331C2#1767212yes—
332C2C2×#182111yescharacteristic
333C2#2070512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8721C2semidirect
4. #8718C2semidirect
5. #8031C2semidirect
26. #1295C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
#207152
#87212
#87182
#80312