Finite Group Databasebeta

← #391737

Subgroups — #391737

Order 512.

1215 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1215 conjugacy classes of subgroups — too many to draw. Only its 234 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#35540C211yesmaximal, characteristic
2#20638C212yesmaximal, complemented
3#11206C214yesmaximal, complemented
4#11167C212yesmaximal, complemented
5#11071C214yesmaximal, complemented
6#10967C212yesmaximal
7C22×#121C2211yescharacteristic
8C2×#518C2211yescharacteristic
9#2905C2214yes—
10C22×#100C2214yes—
11C2×#407C2214yes—
12#2483C414yes—
13#2422—24no—
14#2420—24no—
15#2405C414yes—
16#2317C2212yes—
17#2309C2211yescharacteristic
18#2280C2212yes—
19#1800—24no—
20#1800—24no—
21C2×#344C2218yes—
22#1742C2214yes—
23C2×#343C2214yes—
24C23×D4C2×C412yes—
25C23×D4C2311yescharacteristic
26C24×C4C2311yescharacteristic
27C22×M16C2318yes—
28C2×#123—24no—
29C2×#121—24no—
30C2×#121—44no—
31#518—44no—
32C2×C4×D4D414yes—
33C2×C4×D4D414yes—
34C2×#117C2312yes—
35C2×#117C2312yes—
36C22×C4⋊C4C2311yescharacteristic
37C22×C22⋊C4C2×C412yes—
38C22×C22⋊C4C2×C414yes—
39#424—48no—
40#423—28no—
41C2×#100—28no—
42C2×#100—48no—
43#407—48no—
44C2×C22⋊C8—44no—
45#397—28no—
46#394—28no—
47#390—24no—
48#389—24no—
49#386—24no—
50#385—24no—
51#385C2×C418yes—
52#384—22no—
53#382—22no—
54#381—24no—
55#381C2×C414yes—
56#380—22no—
57#380—22no—
58#379C2×C414yes—
59#379C2×C414yes—
60C4×C22⋊C4—24no—
61#344—48no—
62#343—24no—
63#343—24no—
64#343—44no—
65C25C22×C411yescharacteristic
66C25C22×C412yes—
67C22×D4—24no—
68C22×D4—24no—
69C22×D4—24no—
70C22×D4—24no—
71C22×D4—44no—
72C22×D4—84no—
73C22×D4C2×D414yes—
74C23×C4—22no—
75C23×C4—24no—
76C23×C4—24no—
77C23×C4—24no—
78C23×C4C22⋊C414yes—
79C23×C4C22⋊C414yes—
80C23×C4C22×C411yescharacteristic
81C23×C4C22×C412yes—
82C23×C4C2411yescharacteristic
83C2×M16—28no—
84C2×M16—28no—
85C2×M16—28no—
86C2×M16—28no—
87C2×M16—28no—
88C2×M16—28no—
89C22×C8—28no—
90#123—88no—
91#121—416no—
92C4×D4—48no—
93C4×D4—48no—
94#117—28no—
95#117—24no—
96#117—24no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—28no—
101C2×C4⋊C4C2×D412yes—
102C2×C4⋊C4C2×D412yes—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—48no—
109C2×C22⋊C4—44no—
110C2×C22⋊C4C22⋊C418yes—
111C2×C22⋊C4C22×C414yes—
112C2×C22⋊C4C22×C414yes—
113C2×C42—22no—
114C2×C42C2×D414yes—
115C4⋊C8—416no—
116#100—416no—
117C22⋊C8—416no—
118#95—24no—
119#95—24no—
120#95—28no—
121C24—21no—
122C24—22no—
123C24—24no—
124C24—28no—
125C24—24no—
126C24—42no—
127C24—81no—
128C24—82no—
129C24—82no—
130C24C2×C22⋊C414yes—
131C24C23×C411yescharacteristic
132C2×D4—28no—
133C2×D4—24no—
134C2×D4—24no—
135C2×D4—28no—
136C2×D4—48no—
137C2×D4—48no—
138C2×D4—44no—
139C2×D4—48no—
140C2×D4—44no—
141C2×D4—84no—
142C2×D4—88no—
143C2×D4—88no—
144C22×C4—21no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—48no—
158C22×C4—44no—
159C22×C4—82no—
160C22×C4—84no—
161C22×C4—84no—
162C22×C4C2×C22⋊C414yes—
163C22×C4C2×C22⋊C414yes—
164C22×C4C22×D412yes—
165C22×C4H(5,𝔽2)14yes—
166C22×C4H(5,𝔽2)14yes—
167C22×C42−1+414yes—
168M16—48no—
169M16—48no—
170M16—48no—
171M16—48no—
172C2×C8—48no—
173C2×C8—48no—
174C2×C8—48no—
175C4⋊C4—24no—
176C4⋊C4—24no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—88no—
180C22⋊C4—28no—
181C22⋊C4—28no—
182C22⋊C4—48no—
183C22⋊C4—48no—
184C22⋊C4—48no—
185C22⋊C4—84no—
186C22⋊C4—84no—
187C22⋊C4—84no—
188C42—28no—
189C42—42no—
190C42—42no—
191C23—22no—
192C23—24no—
193C23—24no—
194C23—28no—
195C23—41no—
196C23—41no—
197C23—42no—
198C23—41no—
199C23—42no—
200C23—44no—
201C23—44no—
202C23—44no—
203C23—41no—
204C23—42no—
205C23—44no—
206C23—48no—
207C23—84no—
208C23—84no—
209C23—88no—
210C23—168no—
211C23C22×C22⋊C411yescharacteristic
212C23#51812yes—
213C23#52014yes—
214D4—48no—
215D4—48no—
216D4—816no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—84no—
230C2×C4—84no—
231C2×C4—84no—
232C2×C4—164no—
233C2×C4—164no—
234C2×C4#53414yes—
235C2×C4#53512yes—
236C2×C4#53612yes—
237C2×C4#53714yes—
238C2×C4#56014yes—
239C2×C4#56218yes—
240C2×C4#56314yes—
241C8—88no—
242C22—21no—
243C22—22no—
244C22—21no—
245C22—22no—
246C22—44no—
247C22—44no—
248C22—48no—
249C22—44no—
250C22—42no—
251C22—44no—
252C22—44no—
253C22—88no—
254C22—82no—
255C22—84no—
256C22—81no—
257C22—81no—
258C22—168no—
259C22#230811yescharacteristic
260C22#231112yes—
261C22#237614yes—
262C4—44no—
263C4—44no—
264C4—84no—
265C4—82no—
266C4—81no—
267C4#281314yes—
268C4#281414yes—
269C2—22no—
270C2—24no—
271C2—82no—
272C2—84no—
273C2#1465811yescharacteristic
274C2#2177811yescharacteristic
275C2#2177911yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20638C2semidirect
3. #11206C2semidirect
4. #11167C2semidirect
5. #11071C2semidirect

Maximal subgroups

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

subgroupindex
#355402
#206382
#112062
#111672
#110712
#109672