Finite Group Databasebeta

← #173971

Subgroups — #173971

Order 512.

1077 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1077 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#28509C211yesmaximal, characteristic
4#7826C218yesmaximal, complemented
5#7570C212yesmaximal, complemented
6#7569C211yesmaximal, characteristic
7C2×D4×Q8—22no—
8C2×#476—22no—
9#2868C2211yescharacteristic
10#2683C2214yes—
11#2551—21no—
12#2538—22no—
13#2452—24no—
14#2409—22no—
15#2366—21no—
16C4×C4⋊Q8C2212yes—
17C4×#127C2211yescharacteristic
18#2100C2218yes—
19#2094C2214yes—
20#1990—22no—
21#1987—24no—
22#1957—21no—
23#1844—21no—
24#1834—22no—
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#476—48no—
44C2×#106—24no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—22no—
47C2×#102—24no—
48C2×C4×C8—22no—
49C2×C4×C8—21no—
50#396—28no—
51#395—24no—
52#393—24no—
53#391—22no—
54#390—24no—
55#390—22no—
56#390C2312yes—
57#389—24no—
58#389—22no—
59#389C2314yes—
60#386—24no—
61#385—24no—
62#384—22no—
63#384—21no—
64#384D411yescharacteristic
65#384D412yes—
66#384C2312yes—
67#380—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4D414yes—
70C43C2311yescharacteristic
71#340—28no—
72#340—28no—
73#336C2312yes—
74#336C2314yes—
75C22×Q8—24no—
76C22×Q8—22no—
77C22×Q8—416no—
78C22×Q8—41no—
79C22×Q8—84no—
80C22×D4—22no—
81C22×D4C2×D412yes—
82C23×C4—24no—
83C23×C4—22no—
84C22×C8—22no—
85C22×C8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—48no—
88C4⋊Q8—44no—
89C4⋊Q8—416no—
90C4⋊Q8SD1618yes—
91#127—22no—
92#127Q1614yes—
93C22⋊Q8—416no—
94C22⋊Q8—416no—
95C22⋊Q8—416no—
96C4×Q8—44no—
97C4×Q8—44no—
98C4×D4—44no—
99C4×D4—416no—
100C4×D4—44no—
101C4×D4—44no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4C2×D414yes—
113C2×C4⋊C4C2×D412yes—
114C2×C4⋊C4C2×D414yes—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—22no—
119C2×C42—24no—
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D412yes—
122C2×C42C2×D412yes—
123C2×C42C2×D411yescharacteristic
124C2×C42C2411yescharacteristic
125#106—48no—
126C4⋊C8—24no—
127C4⋊C8—44no—
128#102—48no—
129C4×C8—44no—
130C4×C8—42no—
131#95—24no—
132#95—24no—
133#95—24no—
134C24—22no—
135C2×Q8—28no—
136C2×Q8—416no—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—82no—
140C2×Q8—84no—
141C2×Q8—84no—
142C2×Q8—832no—
143C2×D4—22no—
144C2×D4—28no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—44no—
148C2×D4—44no—
149C2×D4C2×Q1614yes—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—416no—
154C22×C4—48no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—84no—
158C22×C4#12011yescharacteristic
159C22×C4#12011yescharacteristic
160C22×C4#12012yes—
161C22×C4C22×D412yes—
162C22×C4C22×D411yescharacteristic
163C2×C8—24no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—44no—
167C2×C8—44no—
168C2×C8—44no—
169C4⋊C4—24no—
170C4⋊C4—28no—
171C4⋊C4—42no—
172C4⋊C4—44no—
173C4⋊C4—416no—
174C4⋊C4—44no—
175C4⋊C4—48no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—82no—
182C4⋊C4—84no—
183C4⋊C4C2×SD1618yes—
184C22⋊C4—416no—
185C22⋊C4—44no—
186C22⋊C4—44no—
187C22⋊C4—82no—
188C22⋊C4—84no—
189C42—22no—
190C42—24no—
191C42—41no—
192C42—48no—
193C42C2×SD1614yes—
194C42C2×Q1612yes—
195C42AGL(1,ℤ/8)14yes—
196C42#13712yes—
197C23—24no—
198C23—44no—
199C23—42no—
200C23—44no—
201C23—42no—
202C23C2×#12011yescharacteristic
203Q8—416no—
204Q8—816no—
205Q8—164no—
206D4—44no—
207D4—416no—
208D4—82no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—28no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—816no—
224C2×C4—88no—
225C2×C4—82no—
226C2×C4—84no—
227C2×C4#45014yes—
228C2×C4#45014yes—
229C2×C4#45114yes—
230C2×C4#45312yes—
231C2×C4#45414yes—
232C2×C4C22×SD1612yes—
233C2×C4C22×Q1611yescharacteristic
234C2×C4C2×AGL(1,ℤ/8)12yes—
235C2×C4C2×#13711yescharacteristic
236C8—44no—
237C8—84no—
238C22—48no—
239C22—44no—
240C22—82no—
241C22—82no—
242C22—84no—
243C22C2×#45012yes—
244C22C2×#45111yescharacteristic
245C22C2×#45311yescharacteristic
246C22C2×#45412yes—
247C22#302811yescharacteristic
248C4—21no—
249C4—21no—
250C4—22no—
251C4—22no—
252C4—48no—
253C4—41no—
254C4—82no—
255C4—84no—
256C4—162no—
257C4#162914yes—
258C4#163718yes—
259C2—84no—
260C2#1839711yescharacteristic
261C2#1840912yes—
262C2C2×#162911yescharacteristic
263C2C2×#163712yes—
264C2#1907711yescharacteristic
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. #29218C2semidirect
4. #7826C2semidirect
5. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#598172
#292182
#285092
#78262
#75702
#75692