Finite Group Databasebeta

← #116369

Subgroups — #116369

Order 512.

732 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 130 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#21935C211yesmaximal, characteristic
2#20705C212yesmaximal
3#20688C211yesmaximal, characteristic
4#8100C211yesmaximal, characteristic
5#8049C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2212yes—
8#2280C2211yescharacteristic
9#2280C2212yes—
10#2280C2211yescharacteristic
11#1851—24no—
12#1826—24no—
13#1826—24no—
14#1820C414yescomplemented
15#1819—22no—
16#1798—22no—
17#1740—22no—
18#1733—24no—
19#1304—22no—
20#1297—24no—
21C24×C4C2311yescharacteristic
22C22×M16D414yes—
23C2×#117—21no—
24C2×#117—21no—
25C2×#117—22no—
26C2×#117—21no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117C2×C412yes—
32C2×#117D412yes—
33C2×#117D412yes—
34C22×C4⋊C4Q812yes—
35C22×C22⋊C4—21no—
36C22×C42—21no—
37C2×C4⋊C8—24no—
38#407—22no—
39#407—22no—
40#407—22no—
41#407—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44#381—22no—
45#381C2×C412yes—
46#381C2×C412yes—
47#380—22no—
48#380—22no—
49#380—22no—
50#380—22no—
51#379—22no—
52#379—21no—
53#379D412yes—
54C4×C22⋊C4—22no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4D414yes—
58#342—28no—
59#339—216no—
60#336—24no—
61C25C2×D411yescharacteristic
62C23×C4—22no—
63C23×C4—44no—
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D412yes—
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C23×C4C2×D412yes—
69C23×C4C2×Q811yescharacteristic
70C23×C4#4111yescharacteristic
71C23×C4#4111yescharacteristic
72C23×C4#4112yes—
73C23×C4#4112yes—
74C2×M16—24no—
75C2×M16—24no—
76C2×M16—48no—
77C22×C8—24no—
78#117—24no—
79#117—24no—
80#117—24no—
81#117—24no—
82#117—41no—
83#117—42no—
84#117—41no—
85#117—42no—
86#117—41no—
87#117—41no—
88#117—48no—
89#117—44no—
90#117—44no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4C4⋊C414yes—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4C22⋊C414yes—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—24no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—41no—
118C4⋊C8—84no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—42no—
127C24—42no—
128C24—82no—
129C24C2×C22⋊C411yescharacteristic
130C24#12112yes—
131C24#12111yescharacteristic
132C24#12112yes—
133C24C22⋊Q811yescharacteristic
134C22×C4—24no—
135C22×C4—28no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—42no—
139C22×C4—44no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—84no—
151C22×C4C2×C4⋊C411yescharacteristic
152C22×C4#11711yescharacteristic
153C22×C4C4×D412yes—
154C22×C4C4×D412yes—
155C22×C4#12012yes—
156C22×C4#12011yescharacteristic
157C22×C4#12112yes—
158C22×C4#12111yescharacteristic
159C22×C4C22⋊Q812yes—
160C22×C4C22⋊Q811yescharacteristic
161C22×C4C22⋊Q812yes—
162C22×C4#12311yescharacteristic
163C22×C4#12312yes—
164C22×C4#12312yes—
165C22×C4#12412yes—
166C22×C4#12411yescharacteristic
167C22×C4#12612yes—
168C22×C4#12711yescharacteristic
169M16—48no—
170M16—84no—
171C2×C8—44no—
172C2×C8—48no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—82no—
178C4⋊C4—84no—
179C4⋊C4—82no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—44no—
183C22⋊C4—44no—
184C22⋊C4—44no—
185C22⋊C4—84no—
186C22⋊C4—84no—
187C22⋊C4—84no—
188C22⋊C4—84no—
189C22⋊C4—84no—
190C42—48no—
191C42—44no—
192C42—42no—
193C42—42no—
194C42—44no—
195C42—82no—
196C23—22no—
197C23—24no—
198C23—41no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—41no—
206C23—42no—
207C23—42no—
208C23—88no—
209C23#38011yescharacteristic
210C23#38611yescharacteristic
211C23#39011yescharacteristic
212C23#39412yes—
213C23#39612yes—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—48no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—84no—
235C2×C4—84no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4—82no—
239C2×C4—84no—
240C2×C4—84no—
241C2×C4—84no—
242C2×C4#38512yes—
243C2×C4#38812yes—
244C2×C4#39211yescharacteristic
245C2×C4#39311yescharacteristic
246C2×C4#39711yescharacteristic
247C2×C4#39911yescharacteristic
248C2×C4#45814yes—
249C8—84no—
250C22—22no—
251C22—21no—
252C22—22no—
253C22—21no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—44no—
262C22—82no—
263C22—84no—
264C22—84no—
265C22—82no—
266C22#143911yescharacteristic
267C22#189012yes—
268C4—22no—
269C4—22no—
270C4—21no—
271C4—21no—
272C4—84no—
273C4—84no—
274C4—164no—
275C4—162no—
276C4#202212yes—
277C4#203012yes—
278C2—22no—
279C2—41no—
280C2—42no—
281C2—41no—
282C2—42no—
283C2#1107011yescharacteristic
284C2#1109412yes—
subgroupG/H
centerC2×C4#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1439
prime radicalC25C2×D4
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
14. #1820C4semidirect

Maximal subgroups

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

subgroupindex
#219352
#207052
#206882
#81002
#80492