Finite Group Databasebeta

← #302885

Subgroups — #302885

Order 512.

1109 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 1109 conjugacy classes of subgroups — too many to draw. Only its 208 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#29206C212yesmaximal, complemented
3#28509C211yesmaximal, characteristic
4#10574C214yesmaximal, complemented
5#9865C212yesmaximal, complemented
6#8924C214yesmaximal
7C4×#331C211yesmaximal, characteristic
8C2×D4×Q8—22no—
9C2×#476—22no—
10#2868C2211yescharacteristic
11#2683—22no—
12#2551C2212yes—
13#2538—22no—
14#2452—24no—
15#2409—22no—
16#2366C2212yes—
17C4×C4⋊Q8C2212yes—
18C4×#127C2211yescharacteristic
19#2094—22no—
20#2058—24no—
21#1990—22no—
22#1957C2212yes—
23#1948C2218yes—
24#1844—22no—
25#1844C2212yes—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#331C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C23×Q8—22no—
32D4×Q8—48no—
33D4×Q8—48no—
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8D412yes—
38C2×#127C2311yescharacteristic
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—28no—
41C2×C4×Q8—22no—
42C2×C4×D4—24no—
43C2×C4×D4—22no—
44C2×C4×D4D412yes—
45#476—48no—
46C2×#106—24no—
47C2×#106—28no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8C2314yes—
50C2×#102—24no—
51C2×C4×C8—22no—
52C2×C4×C8C2312yes—
53#396—28no—
54#395—24no—
55#393—24no—
56#391—22no—
57#390—24no—
58#390—22no—
59#390D412yes—
60#389—24no—
61#389—22no—
62#389D414yes—
63#386—24no—
64#385—24no—
65#384—21no—
66#384—22no—
67#384D412yes—
68#384C2312yes—
69#384C2311yescharacteristic
70#380—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4C2314yes—
73C43C2311yescharacteristic
74#340—28no—
75#336—22no—
76#336—21no—
77#331Q818yescomplemented
78C22×Q8—24no—
79C22×Q8—22no—
80C22×Q8—416no—
81C22×Q8—41no—
82C22×Q8—84no—
83C22×D4—22no—
84C22×D4C2×D412yes—
85C23×C4—24no—
86C23×C4—22no—
87C22×C8—24no—
88C22×C8—22no—
89C4⋊Q8—24no—
90C4⋊Q8—44no—
91C4⋊Q8—48no—
92C4⋊Q8—48no—
93C4⋊Q8—84no—
94C4⋊Q8SD1618yes—
95#127—22no—
96#127C2×Q814yes—
97C22⋊Q8—432no—
98C22⋊Q8—416no—
99C4×Q8—44no—
100C4×Q8—44no—
101C4×D4—44no—
102C4×D4—48no—
103C4×D4—48no—
104C4×D4—44no—
105C4×D4—44no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C2×D414yes—
117C2×C4⋊C4C2×D414yes—
118C2×C4⋊C4C2×D412yes—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—22no—
123C2×C42—24no—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2411yescharacteristic
129#106—48no—
130#106—88no—
131C4⋊C8—24no—
132C4⋊C8—24no—
133C4⋊C8C2×Q818yescomplemented
134#102—48no—
135C4×C8—24no—
136C4×C8—44no—
137#95—24no—
138#95—24no—
139#95—24no—
140C24—22no—
141C2×Q8—28no—
142C2×Q8—416no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—82no—
146C2×Q8—84no—
147C2×Q8—84no—
148C2×Q8—832no—
149C2×D4—22no—
150C2×D4—28no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—44no—
156C2×D4C4⋊Q814yes—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—416no—
161C22×C4—48no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—84no—
165C22×C4#12011yescharacteristic
166C22×C4#12012yes—
167C22×C4#12011yescharacteristic
168C22×C4C22×D411yescharacteristic
169C22×C4C22×D412yes—
170C2×C8—24no—
171C2×C8—28no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—44no—
175C2×C8—44no—
176C4⋊C4—24no—
177C4⋊C4—28no—
178C4⋊C4—28no—
179C4⋊C4—42no—
180C4⋊C4—416no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—48no—
185C4⋊C4—44no—
186C4⋊C4—48no—
187C4⋊C4—82no—
188C4⋊C4—84no—
189C4⋊C4—82no—
190C4⋊C4C2×SD1618yes—
191C22⋊C4—416no—
192C22⋊C4—48no—
193C22⋊C4—82no—
194C22⋊C4—84no—
195C42—24no—
196C42—22no—
197C42—41no—
198C42—44no—
199C42—44no—
200C42C2×SD1614yes—
201C42AGL(1,ℤ/8)14yes—
202C42C22×Q812yes—
203C422−1+412yes—
204C23—24no—
205C23—44no—
206C23—44no—
207C23—42no—
208C23—42no—
209C23C2×#12011yescharacteristic
210Q8—416no—
211Q8—816no—
212Q8—164no—
213D4—44no—
214D4—48no—
215D4—48no—
216D4—82no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—28no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—816no—
232C2×C4—88no—
233C2×C4—84no—
234C2×C4—82no—
235C2×C4#45014yes—
236C2×C4#45014yes—
237C2×C4#45312yes—
238C2×C4C2×C4⋊Q811yescharacteristic
239C2×C4#53611yescharacteristic
240C2×C4D4×Q814yes—
241C2×C4C22×SD1612yes—
242C2×C4C2×AGL(1,ℤ/8)12yes—
243C2×C4#57714yes—
244C8—44no—
245C8—48no—
246C22—44no—
247C22—48no—
248C22—82no—
249C22—82no—
250C22—84no—
251C22#243411yescharacteristic
252C22C2×#45012yes—
253C22#301012yes—
254C22C2×#45311yescharacteristic
255C22#302211yescharacteristic
256C4—21no—
257C4—21no—
258C4—24no—
259C4—41no—
260C4—48no—
261C4—82no—
262C4—84no—
263C4—162no—
264C4#162914yes—
265C4Q8×SD1614yes—
266C4#338414yes—
267C2—84no—
268C2C2×#162911yescharacteristic
269C2#1871811yescharacteristic
270C2#1892312yes—
271C2#2582411yescharacteristic
272C2#2583412yes—
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. #29206C2semidirect
4. #10574C2semidirect
5. #9865C2semidirect
77. #331Q8semidirect
133. C4⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#598172
#292062
#285092
#105742
#98652
#89242
C4×#3312