Finite Group Databasebeta

← #194889

Subgroups — #194889

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#21804C212yesmaximal
4C2×#1461C211yesmaximal, characteristic
5#9014C218yesmaximal
6#8014C212yesmaximal
7C22×C4⋊Q8C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2212yes—
10#2567—24no—
11#2567C418yescomplemented
12C2×#395C2214yes—
13#2356—24no—
14#2355—24no—
15#2313C2214yes—
16#2310—21no—
17C2×#391—21no—
18C2×#389—22no—
19#2285C2212yes—
20C2×#384—21no—
21#2283—22no—
22#2274—21no—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1981—24no—
25#1956—24no—
26#1787C2218yes—
27#1461—24no—
28C2×#324C2214yes—
29#1294C2218yes—
30C23×Q8D411yescharacteristic
31C23×Q8C2312yes—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—24no—
36C2×C4⋊Q8C2×C414yes—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—24no—
40C2×C4×Q8C2×C414yes—
41C2×C4×Q8D414yes—
42C2×#117—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C42—21no—
50C22×C42D412yes—
51C22×C42C2311yescharacteristic
52#431—24no—
53#431—24no—
54C2×C4⋊C8—22no—
55C2×#103—216no—
56C2×C22⋊C8C2314yes—
57C2×C4×C8—22no—
58#395—28no—
59#395C2×C4116yes—
60#391—28no—
61#391—28no—
62#389—216no—
63#387—216no—
64#384—28no—
65#384D414yes—
66#383—24no—
67#382—24no—
68#382—22no—
69#381—24no—
70#380—22no—
71#380C2314yes—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—28no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2×C414yes—
77C4×C4⋊C4D414yes—
78C4×C22⋊C4—24no—
79#376—24no—
80C2×#95—24no—
81C2×#95C2312yes—
82#324—28no—
83C22×Q8—22no—
84C22×Q8—22no—
85C22×Q8—24no—
86C22×Q8—24no—
87C22×Q8—22no—
88C22×Q8—24no—
89C22×Q8—24no—
90C22×Q8—84no—
91C22×Q8C22⋊C414yes—
92C22×Q8C22×C414yes—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4C2×D412yes—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2411yescharacteristic
100C22×C8—24no—
101C4⋊Q8—28no—
102C4⋊Q8—28no—
103C4⋊Q8—44no—
104C4⋊Q8—44no—
105C4×Q8—28no—
106C4×Q8—28no—
107C4×Q8—28no—
108C4×Q8—28no—
109#117—44no—
110#117—44no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—216no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—88no—
128C2×C4⋊C4C22⋊C414yes—
129C2×C4⋊C4C22×C414yes—
130C2×C4⋊C4C2×D414yes—
131C2×C4⋊C4C2×D418yes—
132C2×C22⋊C4—24no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—24no—
136C2×C42—24no—
137C2×C42—24no—
138C2×C42—28no—
139C2×C42—44no—
140C2×C42C22⋊C418yes—
141C2×C42C22×C412yes—
142C2×C42C2×D412yes—
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C2×C42#4114yes—
146C4⋊C8—44no—
147#103—816no—
148C22⋊C8—24no—
149C22⋊C8—24no—
150C4×C8—44no—
151#95—28no—
152#95—28no—
153#95—28no—
154#95—216no—
155#95—24no—
156#95—22no—
157#95—24no—
158#95C22×C414yes—
159C24C22×D411yescharacteristic
160C2×Q8—28no—
161C2×Q8—28no—
162C2×Q8—28no—
163C2×Q8—28no—
164C2×Q8—416no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—48no—
169C2×Q8—84no—
170C2×Q8—84no—
171C2×Q8—88no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—28no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—48no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4C2×C22⋊C414yes—
191C22×C4C2×C22⋊C412yes—
192C22×C4C2×C22⋊C414yes—
193C22×C4C2×C22⋊C412yes—
194C22×C4#12011yescharacteristic
195C22×C4#12012yes—
196C22×C4#12011yescharacteristic
197C22×C4#12312yes—
198C22×C4#12312yes—
199C22×C4#13514yes—
200C22×C4#13714yes—
201C22×C4C23×C411yescharacteristic
202C22×C4C22×D412yes—
203C22×C4C22×D411yescharacteristic
204C22×C4C2×#4112yes—
205C2×C8—48no—
206C2×C8—44no—
207C4⋊C4—216no—
208C4⋊C4—28no—
209C4⋊C4—28no—
210C4⋊C4—28no—
211C4⋊C4—48no—
212C4⋊C4—44no—
213C4⋊C4—48no—
214C4⋊C4—44no—
215C4⋊C4—48no—
216C4⋊C4—48no—
217C4⋊C4—816no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C42—28no—
221C42—28no—
222C42—48no—
223C42—48no—
224C42—88no—
225C42C4×D414yes—
226C42C4×D414yes—
227C42#12114yes—
228C42#12114yes—
229C23—22no—
230C23—22no—
231C23C22×C22⋊C411yescharacteristic
232C23C2×#12011yescharacteristic
233C23C2×#12311yescharacteristic
234C23C2×#13512yes—
235C23C2×#13712yes—
236Q8—416no—
237Q8—416no—
238Q8—88no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—28no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—416no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—48no—
262C2×C4—44no—
263C2×C4—82no—
264C2×C4—84no—
265C2×C4—84no—
266C2×C4#38612yes—
267C2×C4#38614yes—
268C2×C4#38612yes—
269C2×C4#41614yes—
270C2×C4#41914yes—
271C2×C4#45214yes—
272C2×C4#45214yes—
273C2×C4#45614yes—
274C2×C4#48518yes—
275C2×C4C2×C4×D412yes—
276C2×C4C2×#12112yes—
277C8—84no—
278C22—21no—
279C22—21no—
280C22—24no—
281C22—24no—
282C22—21no—
283C22—21no—
284C22C2×#38611yescharacteristic
285C22C2×#41611yescharacteristic
286C22C2×#41911yescharacteristic
287C22C2×#45212yes—
288C22C2×#45612yes—
289C22#302712yes—
290C22C2×#48512yes—
291C4—22no—
292C4—22no—
293C4—22no—
294C4—48no—
295C4—44no—
296C4—44no—
297C4—88no—
298C4—84no—
299C4#187318yes—
300C4#188614yes—
301C2—21no—
302C2—22no—
303C2—21no—
304C2#1839214yes—
305C2C2×#187311yescharacteristic
306C2C2×#188611yescharacteristic
307C2#2096111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2567C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
#341872
#218042
C2×#14612
#90142
#80142