Finite Group Databasebeta

← #121603

Subgroups — #121603

Order 512.

830 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21948C211yesmaximal, characteristic
2#20834C212yesmaximal
3#10714C211yesmaximal, characteristic
4#8051C212yesmaximal
5#7982C211yesmaximal, characteristic
6C24×C8C2211yescharacteristic
7C2×#424—22no—
8C2×#407C2212yes—
9C22×C22⋊C8C2211yescharacteristic
10#2280C2212yes—
11C2×C4×C22⋊C4C2211yescharacteristic
12#1855—22no—
13#1855—22no—
14#1850—24no—
15#1850—24no—
16#1810—22no—
17#1729—22no—
18C2×#336—22no—
19#1297—24no—
20#1289C414yescomplemented
21C24×C4C2311yescharacteristic
22C22×M16D412yes—
23C22×M16D412yes—
24C23×C8—22no—
25C23×C8—21no—
26C23×C8—22no—
27C23×C8—21no—
28C23×C8—44no—
29C23×C8D412yes—
30C23×C8D412yes—
31C2×#117—22no—
32C22×C22⋊C4—21no—
33C22×C42—21no—
34C2×#108—22no—
35C2×#108—22no—
36#424—48no—
37C2×C4⋊C8—28no—
38#407—22no—
39#407—22no—
40#407—22no—
41#407—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8C2×C412yes—
47C2×C22⋊C8Q812yes—
48#381Q814yes—
49#380—22no—
50#380C2×C414yes—
51#379—22no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4—22no—
54C4×C22⋊C4—22no—
55C4×C22⋊C4D414yes—
56C2×#95Q812yes—
57#341—24no—
58#336—28no—
59#336—44no—
60C25#4111yescharacteristic
61C23×C4—22no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D412yes—
67C23×C4C2×D412yes—
68C23×C4C2×D411yescharacteristic
69C23×C4C2×Q811yescharacteristic
70C23×C4C2×Q812yes—
71C23×C4C2×Q811yescharacteristic
72C23×C4#4112yes—
73C23×C4#4111yescharacteristic
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—44no—
79C2×M16—44no—
80C22×C8—24no—
81C22×C8—28no—
82C22×C8—24no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—44no—
89C22×C8—48no—
90C22×C8—44no—
91C22×C8—42no—
92C22×C8—44no—
93C22×C8—41no—
94C22×C8—42no—
95C22×C8—41no—
96C22×C8—42no—
97C22×C8—42no—
98C22×C8C4⋊C414yes—
99#117—44no—
100#117—44no—
101C2×C4⋊C4—24no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C42—24no—
109C2×C42—42no—
110C2×C42—42no—
111C2×C42—41no—
112C2×C42—41no—
113#108—82no—
114#108—82no—
115C4⋊C8—416no—
116C22⋊C8—24no—
117C22⋊C8—24no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122#95—24no—
123#95—24no—
124#95C4⋊C418yes—
125C24—22no—
126C24—22no—
127C24—42no—
128C24—42no—
129C24C4×D411yescharacteristic
130C24C22⋊Q811yescharacteristic
131C24C22⋊Q812yes—
132C24#12312yes—
133C24#12511yescharacteristic
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—28no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—41no—
149C22×C4—44no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4C2×C4⋊C411yescharacteristic
156C22×C4C2×C4⋊C412yes—
157C22×C4C4×D412yes—
158C22×C4C4×Q811yescharacteristic
159C22×C4#12012yes—
160C22×C4#12011yescharacteristic
161C22×C4#12112yes—
162C22×C4#12111yescharacteristic
163C22×C4C22⋊Q812yes—
164C22×C4C22⋊Q812yes—
165C22×C4C22⋊Q812yes—
166C22×C4#12311yescharacteristic
167C22×C4#12512yes—
168C22×C4#12711yescharacteristic
169C22×C4C4⋊Q811yescharacteristic
170C22×C4C4⋊Q812yes—
171C22×C4C4⋊Q812yes—
172C22×C4C4⋊Q811yescharacteristic
173M16—44no—
174M16—44no—
175M16—82no—
176M16—82no—
177C2×C8—28no—
178C2×C8—24no—
179C2×C8—24no—
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—42no—
183C2×C8—48no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—44no—
187C2×C8—42no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—44no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—82no—
194C4⋊C4—48no—
195C22⋊C4—48no—
196C22⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—44no—
199C22⋊C4—44no—
200C42—44no—
201C42—44no—
202C42—82no—
203C42—81no—
204C42—81no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—24no—
213C23—21no—
214C23—21no—
215C23—41no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23—42no—
220C23—41no—
221C23—48no—
222C23—48no—
223C23#38512yes—
224C23#38911yescharacteristic
225C23#39511yescharacteristic
226C23#39812yes—
227C23#39911yescharacteristic
228C23#44312yes—
229C23#44412yes—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—48no—
252C2×C4—44no—
253C2×C4—48no—
254C2×C4—48no—
255C2×C4—48no—
256C2×C4—82no—
257C2×C4—84no—
258C2×C4—82no—
259C2×C4#38411yescharacteristic
260C2×C4#38511yescharacteristic
261C2×C4#38912yes—
262C2×C4#39211yescharacteristic
263C2×C4#39512yes—
264C2×C4#39811yescharacteristic
265C2×C4B(4,𝔽2)12yes—
266C2×C4#45812yes—
267C8—44no—
268C8—44no—
269C8—82no—
270C8—82no—
271C22—24no—
272C22—24no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—48no—
287C22—48no—
288C22—42no—
289C22—42no—
290C22#144411yescharacteristic
291C22#186614yes—
292C22#194511yescharacteristic
293C22#194611yescharacteristic
294C22#206212yes—
295C22#209812yes—
296C4—21no—
297C4—22no—
298C4—22no—
299C4—21no—
300C4—88no—
301C4—84no—
302C4#186712yes—
303C4#202111yescharacteristic
304C4#202211yescharacteristic
305C2—22no—
306C2—24no—
307C2—22no—
308C2—21no—
309C2—21no—
310C2—42no—
311C2#1132911yescharacteristic
312C2#1133611yescharacteristic
313C2#1134411yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1444
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1289C4semidirect

Maximal subgroups

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

subgroupindex
#219482
#208342
#107142
#80512
#79822