Finite Group Databasebeta

← #508950

Subgroups — #508950

Order 512.

582 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 186 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×#2254C211yesmaximal, characteristic
2C2×#2252C211yesmaximal, characteristic
3C2×#2241C211yesmaximal, characteristic
4C2×#2241C211yesmaximal, characteristic
5#33313C211yesmaximal, characteristic
6#33242C211yesmaximal, characteristic
7#13529C218yesmaximal
8#12590C211yesmaximal, characteristic
9C2×#500C2211yescharacteristic
10C2×#500C2211yescharacteristic
11C2×#500C2211yescharacteristic
12C2×#498—21no—
13C2×#494C2211yescharacteristic
14C2×#487—21no—
15#2862—21no—
16#2862—21no—
17#2254—24no—
18#2252—24no—
19#2241—24no—
20#2241—24no—
21C2×#367C2214yes—
22C2×#366C2214yes—
23C2×#358C2214yes—
24C2×#358C2214yes—
25C2×C4×C16C2211yescharacteristic
26C2×C4×C16C2211yescharacteristic
27#1969—22no—
28#1969C2214yes—
29#1846—22no—
30#1846C2214yes—
31C42×C8C2211yescharacteristic
32#1374C2214yes—
33C22×Q16—21no—
34C22×Q16C2312yes—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C22×C16C2312yes—
40C22×C16C2312yes—
41#500—22no—
42#500—22no—
43#500—22no—
44#500—22no—
45#500—22no—
46#500—22no—
47#498—44no—
48#494—22no—
49#494—22no—
50#487—44no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107C2312yes—
55C2×#107C2312yes—
56C2×#107C2312yes—
57C2×#106—24no—
58C2×#103—24no—
59C2×C4×C8D412yes—
60C2×C4×C8D411yescharacteristic
61C2×C4×C8C2312yes—
62C2×C4×C8C2311yescharacteristic
63#391—24no—
64#391—42no—
65#391—42no—
66#391—42no—
67#382—24no—
68#382—42no—
69C43D411yescharacteristic
70#367—28no—
71#366—28no—
72#358—28no—
73#358—28no—
74C4×C16D414yes—
75C4×C16D414yes—
76C22×Q8—22no—
77C22×Q8—41no—
78C22×Q8—41no—
79C22×Q8—41no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—28no—
83C2×Q16—42no—
84C2×Q16—44no—
85C22×C8C2×D412yes—
86C22×C8C2×D411yescharacteristic
87C22×C8C2411yescharacteristic
88C4⋊Q8—42no—
89C4⋊Q8—42no—
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C4⋊Q8—42no—
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C42D812yes—
105C2×C42D812yes—
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108C2×C16—24no—
109C2×C16—24no—
110C2×C16C2×D414yes—
111C2×C16C2×D414yes—
112#107—22no—
113#107—22no—
114#107—22no—
115#107—22no—
116#107—22no—
117#107—22no—
118#107—42no—
119#107—42no—
120#107—42no—
121#106—48no—
122#103—48no—
123C4×C8—24no—
124C4×C8—24no—
125C4×C8D814yes—
126C4×C8C2×D412yes—
127C4×C8C2×D412yes—
128#95—48no—
129#95—48no—
130C2×Q8—42no—
131C2×Q8—44no—
132C2×Q8—42no—
133C2×Q8—44no—
134C2×Q8—82no—
135C2×Q8—82no—
136C2×Q8—82no—
137C2×Q8—84no—
138C2×Q8—84no—
139C2×Q8—84no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4C2×D814yes—
145C22×C4C2×D811yescharacteristic
146C22×C4C2×D811yescharacteristic
147C22×C4C22×D411yescharacteristic
148Q16—44no—
149Q16—44no—
150Q16—84no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8#12712yes—
154C2×C8C2×D814yes—
155C2×C8#13714yes—
156C2×C8C22×D411yescharacteristic
157C2×C8C22×D411yescharacteristic
158C2×C82−1+414yes—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C42—22no—
171C42—22no—
172C42—24no—
173C42—24no—
174C42#12712yes—
175C42C2×D812yes—
176C16—24no—
177C16—24no—
178C23C22×D811yescharacteristic
179Q8—82no—
180Q8—82no—
181Q8—84no—
182Q8—164no—
183Q8—164no—
184Q8—164no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—84no—
193C2×C4—84no—
194C2×C4—84no—
195C2×C4—84no—
196C2×C4#49312yes—
197C2×C4#49312yes—
198C2×C4#50814yes—
199C2×C4#51014yes—
200C2×C4C2×#12711yescharacteristic
201C2×C4#53612yes—
202C2×C4C22×D811yescharacteristic
203C2×C4C2×#13712yes—
204C8—24no—
205C8#49714yes—
206C8#53612yes—
207C8#53612yes—
208C22C2×#49311yescharacteristic
209C22#322612yes—
210C22C2×#50812yes—
211C22C2×#51012yes—
212C4—44no—
213C4—162no—
214C4—162no—
215C4—162no—
216C4—162no—
217C4#224914yes—
218C4#225114yes—
219C4#265811yescharacteristic
220C4C2×#49711yescharacteristic
221C4#322612yes—
222C2#3027611yescharacteristic
223C2C2×#224911yescharacteristic
224C2C2×#225111yescharacteristic
225C2#3379714yes—
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×#22542
C2×#22522
C2×#22412
#333132
#332422
#135292
#125902