Finite Group Databasebeta

← #174039

Subgroups — #174039

Order 512.

779 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 779 conjugacy classes of subgroups — too many to draw. Only its 176 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#59869C211yesmaximal, characteristic
2#29206C212yesmaximal
3#28440C211yesmaximal, characteristic
4#7760C218yesmaximal
5#7571C211yesmaximal, characteristic
6#7570C212yesmaximal
7C2×#494—21no—
8C2×#462—22no—
9#2868C2211yescharacteristic
10#2809C2214yes—
11#2571—21no—
12#2568—22no—
13#2546—22no—
14#2508—24no—
15#2369—21no—
16#2355—22no—
17C4×C4⋊Q8C2212yes—
18C4×C4⋊Q8C2211yescharacteristic
19#2058C2218yes—
20#2016C2214yes—
21#1990—22no—
22#1956—21no—
23#1948—24no—
24#1844—22no—
25C4×C4⋊C8C2212yes—
26C4×C4⋊C8C2211yescharacteristic
27#1272C2214yes—
28#1271C2218yes—
29C22×Q16—24no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8D411yescharacteristic
34C2×C4⋊Q8D412yes—
35C2×C4⋊Q8C2312yes—
36C2×C4×Q8—22no—
37C2×C4×Q8D412yes—
38#494—44no—
39#462—48no—
40C2×#106—28no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—21no—
43C2×#103—24no—
44C2×C4×C8—22no—
45C2×C4×C8—21no—
46#400—24no—
47#400—24no—
48#398—28no—
49#391—24no—
50#391—24no—
51#391—22no—
52#391C2312yes—
53#389—24no—
54#389—24no—
55#389—22no—
56#389—22no—
57#389C2314yes—
58#387—28no—
59#384—22no—
60#384—22no—
61#384D411yescharacteristic
62#384D412yes—
63#382—24no—
64#382—24no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4D414yes—
70C43C2311yescharacteristic
71#340—28no—
72#336C2314yes—
73#336C2312yes—
74C22×Q8—22no—
75C22×Q8—22no—
76C22×Q8C2×D412yes—
77C2×Q16—48no—
78C2×Q16—48no—
79C2×Q16—84no—
80C22×C8—22no—
81C22×C8—24no—
82C4⋊Q8—24no—
83C4⋊Q8—22no—
84C4⋊Q8—24no—
85C4⋊Q8—44no—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8—82no—
89C4⋊Q8—84no—
90C4⋊Q8SD1614yes—
91C4×Q8—44no—
92C4×Q8—44no—
93C4×Q8—44no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4C2×D414yes—
108C2×C4⋊C4C2×D412yes—
109C2×C4⋊C4C2×D414yes—
110C2×C42—24no—
111C2×C42—22no—
112C2×C42—24no—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D412yes—
117C2×C42C2411yescharacteristic
118#106—88no—
119C4⋊C8—28no—
120C4⋊C8—24no—
121#103—48no—
122C4×C8—42no—
123C4×C8—44no—
124#95—24no—
125#95—28no—
126#95—28no—
127#95—24no—
128#95—24no—
129#95—24no—
130C2×Q8—22no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—44no—
134C2×Q8—44no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8—44no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8C2×SD1614yes—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—42no—
146C22×C4#12011yescharacteristic
147C22×C4#12012yes—
148C22×C4#12011yescharacteristic
149C22×C4C22×D411yescharacteristic
150C22×C4C22×D412yes—
151Q16—816no—
152C2×C8—24no—
153C2×C8—28no—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—44no—
158C4⋊C4—28no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—42no—
162C4⋊C4—44no—
163C4⋊C4—48no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—84no—
173C4⋊C4—84no—
174C4⋊C4—84no—
175C4⋊C4—82no—
176C42—22no—
177C42—24no—
178C42—41no—
179C42—48no—
180C42—44no—
181C42—84no—
182C42C2×SD1612yes—
183C42#13514yes—
184C42#13712yes—
185C42#13714yes—
186C23C2×#12011yescharacteristic
187Q8—44no—
188Q8—82no—
189Q8—84no—
190Q8—84no—
191Q8—88no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—24no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4#44814yes—
213C2×C4#45214yes—
214C2×C4#45214yes—
215C2×C4#45312yes—
216C2×C4#45614yes—
217C2×C4C22×SD1611yescharacteristic
218C2×C4C2×#13512yes—
219C2×C4C2×#13711yescharacteristic
220C2×C4C2×#13712yes—
221C8—48no—
222C8—44no—
223C22C2×#44811yescharacteristic
224C22C2×#45212yes—
225C22C2×#45311yescharacteristic
226C22C2×#45612yes—
227C22#302711yescharacteristic
228C4—21no—
229C4—21no—
230C4—22no—
231C4—22no—
232C4—41no—
233C4—44no—
234C4—84no—
235C4—82no—
236C4—84no—
237C4—162no—
238C4#163414yes—
239C4#165318yes—
240C2#1838911yescharacteristic
241C2#1839212yes—
242C2C2×#165312yes—
243C2C2×#163411yescharacteristic
244C2#1907811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#598692
#292062
#284402
#77602
#75712
#75702