Finite Group Databasebeta

← #173930

Subgroups — #173930

Order 512.

1113 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1113 conjugacy classes of subgroups — too many to draw. Only its 192 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#59817C211yesmaximal, characteristic
2#28689C212yesmaximal, complemented
3#28508C211yesmaximal, characteristic
4#7797C218yesmaximal, complemented
5#7570C212yesmaximal, complemented
6#7569C211yesmaximal, characteristic
7C2×D4×Q8—22no—
8C2×#500—22no—
9C2×#474—22no—
10#2868C2211yescharacteristic
11#2683C2214yes—
12#2551—21no—
13#2538—22no—
14#2452—24no—
15#2409—22no—
16#2366—21no—
17C4×C4⋊Q8C2212yes—
18C4×#127C2211yescharacteristic
19#2093C2214yes—
20#2083C2218yes—
21#1990—22no—
22#1957—21no—
23#1947—24no—
24#1845—21no—
25C4×C4⋊C8C2212yes—
26C4×C4⋊C8C2211yescharacteristic
27#1271C2218yes—
28#1270C2214yes—
29C23×Q8—22no—
30D4×Q8—48no—
31D4×Q8—48no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8D412yes—
36C2×#127D411yescharacteristic
37C2×C22⋊Q8—24no—
38C2×C22⋊Q8—28no—
39C2×C4×Q8—22no—
40C2×C4×D4—24no—
41C2×C4×D4—22no—
42C2×C4×D4D412yes—
43#500—48no—
44#474—48no—
45C2×#107—28no—
46C2×#107—24no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8—21no—
49C2×#102—24no—
50C2×C4×C8—21no—
51C2×C4×C8—22no—
52#396—28no—
53#395—24no—
54#393—24no—
55#391—22no—
56#390—24no—
57#390—22no—
58#390C2312yes—
59#389—24no—
60#389—22no—
61#389C2314yes—
62#386—24no—
63#385—24no—
64#384—21no—
65#384—22no—
66#384D411yescharacteristic
67#384D412yes—
68#384C2312yes—
69#380—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C43C2311yescharacteristic
73#340—28no—
74#336C2314yes—
75#336C2312yes—
76C22×Q8—24no—
77C22×Q8—22no—
78C22×Q8—416no—
79C22×Q8—41no—
80C22×Q8—84no—
81C22×D4—22no—
82C22×D4C2×D412yes—
83C23×C4—24no—
84C23×C4—22no—
85C22×C8—24no—
86C22×C8—22no—
87C4⋊Q8—24no—
88C4⋊Q8—44no—
89C4⋊Q8—416no—
90C4⋊Q8—44no—
91C4⋊Q8—44no—
92C4⋊Q8D818yes—
93#127—22no—
94#127Q1614yes—
95C22⋊Q8—416no—
96C22⋊Q8—416no—
97C22⋊Q8—416no—
98C4×Q8—44no—
99C4×Q8—44no—
100C4×D4—44no—
101C4×D4—416no—
102C4×D4—44no—
103C4×D4—44no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D414yes—
117C2×C22⋊C4—28no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—22no—
121C2×C42—24no—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D412yes—
124C2×C42C2×D412yes—
125C2×C42C2×D411yescharacteristic
126C2×C42C2411yescharacteristic
127#107—48no—
128#107—416no—
129C4⋊C8—28no—
130C4⋊C8—24no—
131#102—48no—
132C4×C8—44no—
133C4×C8—42no—
134#95—24no—
135#95—24no—
136#95—24no—
137C24—22no—
138C2×Q8—28no—
139C2×Q8—416no—
140C2×Q8—44no—
141C2×Q8—44no—
142C2×Q8—84no—
143C2×Q8—832no—
144C2×Q8—84no—
145C2×Q8—82no—
146C2×D4—22no—
147C2×D4—28no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—44no—
151C2×D4—44no—
152C2×D4C2×Q1614yes—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—416no—
157C22×C4—48no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—84no—
161C22×C4#12011yescharacteristic
162C22×C4#12011yescharacteristic
163C22×C4#12012yes—
164C22×C4C22×D412yes—
165C22×C4C22×D411yescharacteristic
166C2×C8—24no—
167C2×C8—28no—
168C2×C8—44no—
169C2×C8—42no—
170C2×C8—44no—
171C2×C8—42no—
172C4⋊C4—24no—
173C4⋊C4—28no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C4⋊C4—416no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—82no—
186C4⋊C4—84no—
187C4⋊C4C2×D818yes—
188C22⋊C4—416no—
189C22⋊C4—44no—
190C22⋊C4—44no—
191C22⋊C4—82no—
192C22⋊C4—84no—
193C42—22no—
194C42—24no—
195C42—41no—
196C42—48no—
197C42C2×D814yes—
198C42C2×Q1612yes—
199C42AGL(1,ℤ/8)14yes—
200C42#13712yes—
201C23—24no—
202C23—44no—
203C23—42no—
204C23—44no—
205C23—42no—
206C23C2×#12011yescharacteristic
207Q8—416no—
208Q8—816no—
209Q8—164no—
210D4—44no—
211D4—416no—
212D4—82no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—28no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—816no—
228C2×C4—88no—
229C2×C4—84no—
230C2×C4—82no—
231C2×C4#44714yes—
232C2×C4#44714yes—
233C2×C4#45114yes—
234C2×C4#45312yes—
235C2×C4#45614yes—
236C2×C4C22×D812yes—
237C2×C4C22×Q1611yescharacteristic
238C2×C4C2×AGL(1,ℤ/8)12yes—
239C2×C4C2×#13711yescharacteristic
240C8—44no—
241C8—48no—
242C22—48no—
243C22—44no—
244C22—82no—
245C22—82no—
246C22—84no—
247C22C2×#44712yes—
248C22C2×#45111yescharacteristic
249C22C2×#45311yescharacteristic
250C22C2×#45612yes—
251C22#302811yescharacteristic
252C4—21no—
253C4—21no—
254C4—22no—
255C4—22no—
256C4—48no—
257C4—41no—
258C4—82no—
259C4—84no—
260C4—162no—
261C4#161914yes—
262C4#164918yes—
263C2—84no—
264C2#1839711yescharacteristic
265C2#1840812yes—
266C2C2×#161911yescharacteristic
267C2C2×#164912yes—
268C2#1907811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28689C2semidirect
4. #7797C2semidirect
5. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#598172
#286892
#285082
#77972
#75702
#75692