Finite Group Databasebeta

← #512218

Subgroups — #512218

Order 512.

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

normal subgroups only

This group has 592 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
1#33258C211yesmaximal, characteristic, complemented
2C42×D8C211yesmaximal, characteristic
3#13176C212yesmaximal, complemented
4#13109C214yesmaximal, complemented
5#12935C214yesmaximal
6#12879C211yesmaximal, characteristic, complemented
7#12811C212yesmaximal
8C2×C4×D8C2212yes—
9C2×C4×D8C2211yescharacteristic
10#2859—21no—
11C42×D4—21no—
12C2×#366C2214yes—
13C2×C4⋊C16C2212yes—
14C2×#357C2214yes—
15#1942—22no—
16#1942C2214yes—
17#1845C2212yes—
18#1834—21no—
19C4×#107C2212yes—
20C4×#107C2211yescharacteristic
21C4×#102—21no—
22C4×#102—22no—
23C42×C8C2211yescharacteristic
24#1380C2214yes—
25#1380C2214yes—
26#1374C2214yes—
27C22×D8C2311yescharacteristic
28C2×#125—41no—
29C2×C4×D4—22no—
30C2×C4×D4—21no—
31C22×C42—41no—
32C22×C16C2314yes—
33C4×D8—24no—
34C4×D8—24no—
35C4×D8—22no—
36C4×D8Q814yes—
37C2×#107—22no—
38C2×#107C2314yes—
39C2×#107C2311yescharacteristic
40C2×#107C2312yes—
41C2×#102—22no—
42C2×#102—21no—
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D412yes—
45C2×C4×C8C2312yes—
46C2×C4×C8C2311yescharacteristic
47#389—24no—
48#389—42no—
49#384—22no—
50#382—41no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4—21no—
53C4×C22⋊C4—41no—
54C4×C22⋊C4—42no—
55C43D411yescharacteristic
56#366—28no—
57C4⋊C16—24no—
58#357—28no—
59#340—24no—
60#340—22no—
61#340—21no—
62C22×D4—21no—
63C23×C4—42no—
64C23×C4—41no—
65C2×D8—21no—
66C2×D8—22no—
67C2×D8C2×Q812yes—
68C2×D8#4114yes—
69C22×C8C2×D412yes—
70C22×C8C2×D411yescharacteristic
71C22×C8C2411yescharacteristic
72#125—84no—
73C4×D4—28no—
74C4×D4—24no—
75C4×D4—44no—
76C4×D4—42no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—41no—
85C2×C42—42no—
86C2×C42—41no—
87C2×C42—84no—
88C2×C42D811yescharacteristic
89C2×C42D812yes—
90C2×C42D811yescharacteristic
91C2×C42C2×D412yes—
92C2×C42C2×D411yescharacteristic
93C2×C16—24no—
94C2×C16—24no—
95C2×C16—24no—
96#107—21no—
97#107—22no—
98#107—24no—
99#107—24no—
100#107—44no—
101#107C2×Q812yes—
102#107#4114yes—
103#102—28no—
104#102—24no—
105C4×C8—22no—
106C4×C8—22no—
107C4×C8—22no—
108C4×C8—42no—
109C4×C8C2×Q812yes—
110C4×C8#4112yes—
111#95—42no—
112#95—41no—
113#95—44no—
114C24—41no—
115C2×D4—22no—
116C2×D4—24no—
117C2×D4—41no—
118C2×D4—42no—
119C22×C4—48no—
120C22×C4—42no—
121C22×C4—41no—
122C22×C4—44no—
123C22×C4—44no—
124C22×C4—84no—
125C22×C4—84no—
126C22×C4—82no—
127C22×C4—82no—
128C22×C4C2×D812yes—
129C22×C4C2×D811yescharacteristic
130C22×C4C2×D812yes—
131C22×C4C2×D811yescharacteristic
132C22×C4C22×D411yescharacteristic
133D8—24no—
134D8—22no—
135D8#12514yescomplemented
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8C22⋊Q812yes—
141C2×C8C22×Q811yescharacteristic
142C2×C8C2×#4112yes—
143C2×C8C2×#4111yescharacteristic
144C2×C8C2×#4112yes—
145C4⋊C4—22no—
146C4⋊C4—24no—
147C4⋊C4—41no—
148C4⋊C4—42no—
149C4⋊C4—44no—
150C4⋊C4—44no—
151C4⋊C4—84no—
152C4⋊C4—88no—
153C4⋊C4—84no—
154C22⋊C4—48no—
155C22⋊C4—44no—
156C42—22no—
157C42—22no—
158C42—22no—
159C42—42no—
160C42—84no—
161C42—82no—
162C42D1614yes—
163C42C22⋊Q812yes—
164C42#13712yes—
165C16—44no—
166C23—44no—
167C23—42no—
168C23—81no—
169C23—81no—
170C23—82no—
171C23C22×D811yescharacteristic
172D4—24no—
173D4—42no—
174D4—44no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—816no—
184C2×C4—81no—
185C2×C4—82no—
186C2×C4—81no—
187C2×C4—82no—
188C2×C4—88no—
189C2×C4—84no—
190C2×C4—84no—
191C2×C4—84no—
192C2×C4#47412yes—
193C2×C4#47412yes—
194C2×C4C2×D1614yes—
195C2×C4C2×D1612yes—
196C2×C4#51012yes—
197C2×C4C2×C22⋊Q811yescharacteristic
198C2×C4#52512yes—
199C2×C4C2×#13711yescharacteristic
200C2×C4#57512yes—
201C8—22no—
202C8—42no—
203C8C2×#12511yescharacteristic
204C8#52912yes—
205C8#53311yescharacteristic
206C22—44no—
207C22—82no—
208C22—82no—
209C22—84no—
210C22—84no—
211C22C2×#47411yescharacteristic
212C22#310012yes—
213C22C22×D1611yescharacteristic
214C22C2×#51011yescharacteristic
215C22#341412yes—
216C4—22no—
217C4—24no—
218C4—42no—
219C4—82no—
220C4—84no—
221C4—164no—
222C4#222414yes—
223C4#285111yescharacteristic
224C4#308011yescharacteristic
225C4#311612yes—
226C2—84no—
227C2#3306611yescharacteristic
228C2C2×#222411yescharacteristic
229C2#3363011yescharacteristic
230C2#3366012yes—
231C2#3367612yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #33258C2semidirect
3. #13176C2semidirect
4. #13109C2semidirect
6. #12879C2semidirect
135. D8#125semidirect

Maximal subgroups

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

subgroupindex
#332582
C42×D82
#131762
#131092
#129352
#128792
#128112