Finite Group Databasebeta

← #508953

Subgroups — #508953

Order 512.

568 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C2×#2255C212yesmaximal
2C2×#2245C212yesmaximal
3#33311C211yesmaximal, characteristic
4#33242C211yesmaximal, characteristic
5#13531C212yesmaximal
6#13530C212yesmaximal
7#13529C214yesmaximal
8#12591C211yesmaximal, characteristic
9C2×#500C2211yescharacteristic
10C2×#499—21no—
11C2×#499C2212yes—
12C2×#494C2211yescharacteristic
13C2×#487—21no—
14#2862—21no—
15#2859—21no—
16#2255—28no—
17#2245—28no—
18C2×#367C2214yes—
19C2×#366C2214yes—
20C2×#358C2214yes—
21C2×#358C2214yes—
22C2×#346C2212yes—
23#1969—21no—
24#1969—21no—
25#1969C2212yes—
26#1969C2212yes—
27#1846—21no—
28#1846C2212yes—
29#1845C2211yescharacteristic
30#1845C2211yescharacteristic
31#1844—21no—
32C42×C8C2211yescharacteristic
33#1374C2212yes—
34#1374C2212yes—
35C22×Q16—21no—
36C22×Q16C2312yes—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×#125—22no—
40C22×C16C2314yes—
41#500—24no—
42#499—24no—
43#499—24no—
44#499—44no—
45#494—24no—
46#487—44no—
47C2×#107—21no—
48C2×#107—21no—
49C2×#107—21no—
50C2×#107C2312yes—
51C2×#107C2312yes—
52C2×#107C2312yes—
53C2×#106—22no—
54C2×#106—22no—
55C2×#103—22no—
56C2×#103—22no—
57C2×C4×C8D411yescharacteristic
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63#391—22no—
64#391—22no—
65#391—42no—
66#389—42no—
67#389—42no—
68#384—21no—
69#384—21no—
70#382—22no—
71#382—42no—
72C43D411yescharacteristic
73#367—28no—
74#366—28no—
75#358—28no—
76#358—28no—
77#346D418yes—
78C22×Q8—22no—
79C22×Q8—41no—
80C2×Q16—24no—
81C2×Q16—24no—
82C2×Q16—24no—
83C2×Q16—42no—
84C2×Q16—42no—
85C2×Q16—42no—
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22×C8C2411yescharacteristic
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C4⋊Q8—42no—
94#125—44no—
95#125—44no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C42D812yes—
107C2×C42D812yes—
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C16—24no—
112C2×C16—24no—
113C2×C16C2×D418yes—
114#107—22no—
115#107—22no—
116#107—22no—
117#107—24no—
118#107—22no—
119#107—42no—
120#107—42no—
121#107—42no—
122#106—44no—
123#106—44no—
124#103—44no—
125#103—44no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—22no—
130C4×C8D814yes—
131C4×C8C2×D414yes—
132#95—44no—
133#95—44no—
134#95—44no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—84no—
142C2×Q8—82no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4C2×D812yes—
148C22×C4C2×D811yescharacteristic
149C22×C4C2×D811yescharacteristic
150C22×C4C2×D812yes—
151C22×C4C22×D411yescharacteristic
152Q16—48no—
153Q16—84no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8#12712yes—
159C2×C8C2×D812yes—
160C2×C8C2×D812yes—
161C2×C8#13712yes—
162C2×C8#13712yes—
163C2×C8C22×D412yes—
164C2×C82−1+412yes—
165C2×C82−1+412yes—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—84no—
173C4⋊C4—84no—
174C4⋊C4—84no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C4⋊C4—82no—
178C4⋊C4—84no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—24no—
184C42—42no—
185C42#12712yes—
186C42C2×D812yes—
187C16—28no—
188C23C22×D811yescharacteristic
189Q8—82no—
190Q8—82no—
191Q8—82no—
192Q8—82no—
193Q8—164no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—84no—
206C2×C4—84no—
207C2×C4—84no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4#49312yes—
211C2×C4#49312yes—
212C2×C4#51014yes—
213C2×C4C2×#12711yescharacteristic
214C2×C4#53611yescharacteristic
215C2×C4#53611yescharacteristic
216C2×C4C22×D811yescharacteristic
217C2×C4C2×#13711yescharacteristic
218C2×C4C2×#13711yescharacteristic
219C8—22no—
220C8—22no—
221C8#49714yes—
222C8#53614yes—
223C22C2×#49311yescharacteristic
224C22#322611yescharacteristic
225C22#322611yescharacteristic
226C22C2×#51012yes—
227C22#341412yes—
228C4—24no—
229C4—22no—
230C4—22no—
231C4—42no—
232C4—162no—
233C4—162no—
234C4—162no—
235C4—162no—
236C4#265811yescharacteristic
237C4C2×#49711yescharacteristic
238C4#322611yescharacteristic
239C4#322611yescharacteristic
240C2#3027611yescharacteristic
241C2#3371811yescharacteristic
242C2#3371911yescharacteristic
243C2#3380212yes—
244C2#3380312yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#22552
C2×#22452
#333112
#332422
#135312
#135302
#135292
#125912