Finite Group Databasebeta

← #120517

Subgroups — #120517

Order 512.

706 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 132 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#21939C211yesmaximal, characteristic
2#20836C211yesmaximal, characteristic
3#10737C212yesmaximal
4#8008C211yesmaximal, characteristic
5#7946C212yesmaximal
6C23×M16C2211yescharacteristic
7C2×#424—21no—
8C2×#407C2212yes—
9C22×C22⋊C8C2211yescharacteristic
10#2280C2211yescharacteristic
11#2280C2212yes—
12#1854—24no—
13#1851C414yescomplemented
14#1850—22no—
15#1808—21no—
16#1807—21no—
17C2×#344—24no—
18#1729—21no—
19#1293—22no—
20#1284—24no—
21C24×C4C2311yescharacteristic
22C22×M16—21no—
23C22×M16—22no—
24C22×M16—22no—
25C22×M16—22no—
26C22×M16—44no—
27C22×M16C2×C412yes—
28C23×C8—21no—
29C23×C8D411yescharacteristic
30C23×C8D411yescharacteristic
31C2×#117—21no—
32C2×#117D414yes—
33C2×#107—22no—
34C2×#106—22no—
35#424—44no—
36C2×C4⋊C8—24no—
37#407—22no—
38#407—22no—
39#407—22no—
40#407—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8C2×C412yes—
45C2×C22⋊C8D412yes—
46C2×C22⋊C8Q814yes—
47#381—22no—
48#381D412yes—
49#380—21no—
50#380—22no—
51#380—22no—
52#380Q812yes—
53#379—21no—
54#379—22no—
55C4×C22⋊C4—21no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4—22no—
58C4×C22⋊C4C2×C412yes—
59#344—48no—
60#340—22no—
61#340—22no—
62#336—24no—
63C25#4111yescharacteristic
64C23×C4—42no—
65C23×C4—42no—
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C23×C4C2×D412yes—
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×Q811yescharacteristic
72C23×C4C2×Q812yes—
73C23×C4#4112yes—
74C23×C4#4112yes—
75C23×C4#4111yescharacteristic
76C2×M16—24no—
77C2×M16—28no—
78C2×M16—24no—
79C2×M16—44no—
80C2×M16—48no—
81C2×M16—42no—
82C2×M16—44no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—44no—
86C2×M16—44no—
87C2×M16—42no—
88C2×M16—42no—
89C22×C8—22no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—24no—
95C22×C8—44no—
96C22×C8—42no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8C22⋊C412yes—
100C22×C8C22⋊C412yes—
101#117—42no—
102#117—42no—
103#117—44no—
104#117—44no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C42—22no—
114C2×C42—24no—
115#107—82no—
116#106—82no—
117C4⋊C8—48no—
118C22⋊C8—24no—
119C22⋊C8—24no—
120C22⋊C8—24no—
121C22⋊C8—24no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124#95—22no—
125#95—42no—
126#95—42no—
127#95C4⋊C414yes—
128C24—42no—
129C24—42no—
130C24—81no—
131C24#11711yescharacteristic
132C24#12111yescharacteristic
133C24C22⋊Q811yescharacteristic
134C24#12312yes—
135C24#12512yes—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4C2×C22⋊C411yescharacteristic
154C22×C4C2×C4⋊C411yescharacteristic
155C22×C4C4×D412yes—
156C22×C4C4×Q812yes—
157C22×C4#12011yescharacteristic
158C22×C4#12112yes—
159C22×C4#12112yes—
160C22×C4#12111yescharacteristic
161C22×C4C22⋊Q812yes—
162C22×C4C22⋊Q811yescharacteristic
163C22×C4C22⋊Q812yes—
164C22×C4C22⋊Q812yes—
165C22×C4#12311yescharacteristic
166C22×C4#12411yescharacteristic
167C22×C4#12512yes—
168C22×C4#12612yes—
169C22×C4C4⋊Q812yes—
170C22×C4C4⋊Q811yescharacteristic
171M16—28no—
172M16—44no—
173M16—44no—
174M16—44no—
175M16—84no—
176M16—82no—
177M16—82no—
178C2×C8—24no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—42no—
182C2×C8—48no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—48no—
189C2×C8—44no—
190C2×C8—81no—
191C2×C8—81no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C22⋊C4—48no—
198C22⋊C4—82no—
199C22⋊C4—84no—
200C22⋊C4—84no—
201C42—42no—
202C42—42no—
203C42—84no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—42no—
214C23—41no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—88no—
220C23#38011yescharacteristic
221C23#38712yes—
222C23#39411yescharacteristic
223C23#39812yes—
224C23#39811yescharacteristic
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—84no—
250C2×C4—84no—
251C2×C4—84no—
252C2×C4#38611yescharacteristic
253C2×C4#38912yes—
254C2×C4#39111yescharacteristic
255C2×C4#39511yescharacteristic
256C2×C4#39612yes—
257C2×C4#40011yescharacteristic
258C2×C4B(4,𝔽2)12yes—
259C2×C4#45812yes—
260C8—44no—
261C8—42no—
262C8—42no—
263C8—84no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—22no—
269C22—22no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—21no—
274C22—22no—
275C22—21no—
276C22—22no—
277C22—44no—
278C22—44no—
279C22—82no—
280C22—84no—
281C22—84no—
282C22#144111yescharacteristic
283C22#181412yes—
284C22#204412yes—
285C4—22no—
286C4—22no—
287C4—21no—
288C4—21no—
289C4—84no—
290C4—164no—
291C4#188911yescharacteristic
292C4#189011yescharacteristic
293C4#206212yes—
294C2—22no—
295C2—22no—
296C2—41no—
297C2—42no—
298C2—42no—
299C2#1122211yescharacteristic
300C2#1124111yescharacteristic
301C2#1124511yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1851C4semidirect

Maximal subgroups

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

subgroupindex
#219392
#208362
#107372
#80082
#79462