Finite Group Databasebeta

← #305061

Subgroups — #305061

Order 512.

1135 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1135 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#29218C212yesmaximal, complemented
3#28438C211yesmaximal, characteristic
4#10349C214yesmaximal, complemented
5#9865C212yesmaximal, complemented
6#9064C214yesmaximal
7C4×#331C211yesmaximal, characteristic
8C2×D4×Q8—22no—
9C2×#492C2212yes—
10C2×#461—22no—
11#2868C2211yescharacteristic
12#2683—22no—
13#2551C2212yes—
14#2538—22no—
15#2452—24no—
16#2409—22no—
17#2366C2212yes—
18C4×C4⋊Q8C2212yes—
19C4×#127C2211yescharacteristic
20#2100—24no—
21#2013—22no—
22#1990—22no—
23#1987C2218yes—
24#1956C2212yes—
25#1834—22no—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#331C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C23×Q8—22no—
32C22×SD16—24no—
33D4×Q8—48no—
34D4×Q8—48no—
35C2×C4⋊Q8—24no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8C2312yes—
40C2×#127C2311yescharacteristic
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—28no—
43C2×C4×Q8—22no—
44C2×C4×D4—24no—
45C2×C4×D4—22no—
46C2×C4×D4D412yes—
47#492—28no—
48#461—48no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8C2314yes—
51C2×#103—24no—
52C2×C4×C8—22no—
53C2×C4×C8C2312yes—
54#396—28no—
55#395—24no—
56#393—24no—
57#391—22no—
58#390—24no—
59#390—22no—
60#390D412yes—
61#389—24no—
62#389—22no—
63#389D414yes—
64#386—24no—
65#385—24no—
66#384—22no—
67#384—22no—
68#384D412yes—
69#384C2311yescharacteristic
70#380—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4C2314yes—
73C43C2311yescharacteristic
74#340—28no—
75#340—28no—
76#336—22no—
77#336—21no—
78#331—24no—
79C22×Q8—24no—
80C22×Q8—22no—
81C22×Q8—22no—
82C22×Q8—416no—
83C22×Q8—84no—
84C22×D4—22no—
85C22×D4C2×D412yes—
86C23×C4—24no—
87C23×C4—22no—
88C2×SD16—48no—
89C2×SD16—48no—
90C2×SD16—84no—
91C22×C8—22no—
92C22×C8—24no—
93C4⋊Q8—24no—
94C4⋊Q8—24no—
95C4⋊Q8—24no—
96C4⋊Q8—44no—
97C4⋊Q8—48no—
98C4⋊Q8—48no—
99C4⋊Q8—82no—
100C4⋊Q8SD1618yescomplemented
101#127—22no—
102#127#4114yes—
103C22⋊Q8—432no—
104C22⋊Q8—416no—
105C4×Q8—44no—
106C4×Q8—44no—
107C4×D4—44no—
108C4×D4—48no—
109C4×D4—48no—
110C4×D4—44no—
111C4×D4—44no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4C2×D414yes—
123C2×C4⋊C4C2×D414yes—
124C2×C4⋊C4C2×D412yes—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—22no—
129C2×C42—24no—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D412yes—
134C2×C42C2411yescharacteristic
135C4⋊C8—24no—
136C4⋊C8—24no—
137C4⋊C8—24no—
138#103—48no—
139C4×C8—24no—
140C4×C8—44no—
141#95—24no—
142#95—24no—
143#95—24no—
144C24—22no—
145C2×Q8—28no—
146C2×Q8—416no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—84no—
154C2×Q8—832no—
155C2×D4—22no—
156C2×D4—28no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—44no—
162C2×D4#12414yes—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—416no—
167C22×C4—48no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—84no—
171C22×C4#12011yescharacteristic
172C22×C4#12012yes—
173C22×C4#12011yescharacteristic
174C22×C4C22×D411yescharacteristic
175C22×C4C22×D412yes—
176SD16—816no—
177C2×C8—24no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—44no—
183C4⋊C4—24no—
184C4⋊C4—28no—
185C4⋊C4—28no—
186C4⋊C4—42no—
187C4⋊C4—416no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—48no—
191C4⋊C4—44no—
192C4⋊C4—48no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—84no—
196C4⋊C4—82no—
197C4⋊C4C2×SD1618yes—
198C22⋊C4—416no—
199C22⋊C4—48no—
200C22⋊C4—82no—
201C22⋊C4—84no—
202C42—24no—
203C42—22no—
204C42—41no—
205C42—44no—
206C42—44no—
207C42C2×SD1614yes—
208C42AGL(1,ℤ/8)14yes—
209C42C2×#4112yes—
210C422−1+412yes—
211C23—24no—
212C23—44no—
213C23—44no—
214C23—42no—
215C23—42no—
216C23C2×#12011yescharacteristic
217Q8—416no—
218Q8—84no—
219Q8—84no—
220Q8—816no—
221D4—44no—
222D4—48no—
223D4—48no—
224D4—82no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—28no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—816no—
240C2×C4—88no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4#45014yes—
245C2×C4#45014yes—
246C2×C4#45312yes—
247C2×C4C2×#12411yescharacteristic
248C2×C4#53611yescharacteristic
249C2×C4#54814yes—
250C2×C4C22×SD1612yes—
251C2×C4C2×AGL(1,ℤ/8)12yes—
252C2×C4#57814yes—
253C8—44no—
254C8—84no—
255C22—44no—
256C22—48no—
257C22—82no—
258C22—82no—
259C22—84no—
260C22#243611yescharacteristic
261C22C2×#45012yes—
262C22#301112yes—
263C22C2×#45311yescharacteristic
264C22#302111yescharacteristic
265C4—21no—
266C4—21no—
267C4—24no—
268C4—41no—
269C4—48no—
270C4—82no—
271C4—84no—
272C4—162no—
273C4#162914yes—
274C4#335914yes—
275C4#336714yes—
276C2—84no—
277C2C2×#162911yescharacteristic
278C2#1871911yescharacteristic
279C2#1891312yes—
280C2#2591811yescharacteristic
281C2#2592912yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29218C2semidirect
4. #10349C2semidirect
5. #9865C2semidirect
100. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#598172
#292182
#284382
#103492
#98652
#90642
C4×#3312