Finite Group Databasebeta

← #102574

Subgroups — #102574

Order 512.

472 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 90 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#29272C212yesmaximal
2#23801C211yesmaximal, characteristic
3#9641C211yesmaximal, characteristic
4#7652C212yesmaximal
5#7018C211yesmaximal, characteristic
6C2×C4×Q16—22no—
7C4×C4⋊Q8C2211yescharacteristic
8C4×C4⋊Q8C2212yes—
9#2099—22no—
10#2095—22no—
11#2057—22no—
12#2016—22no—
13#1988—22no—
14#1947—22no—
15#1918—24no—
16#1831C2211yescharacteristic
17#1831C2212yes—
18#1744C2211yescharacteristic
19C2×Q8⋊C8—22no—
20#1271—24no—
21C22×Q16—22no—
22C2×C4⋊Q8D411yescharacteristic
23C2×C4⋊Q8D412yes—
24C2×C4×Q8—22no—
25C2×C4×Q8—21no—
26C4×Q16—44no—
27C4×Q16—44no—
28C2×#107—24no—
29C2×#107—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—21no—
32C2×#103—24no—
33C2×C4×C8D412yes—
34#391—22no—
35#391—21no—
36#389—22no—
37#389—22no—
38#389—22no—
39#384D412yes—
40#384D411yescharacteristic
41C4×C4⋊C4—22no—
42C4×C4⋊C4—22no—
43C4×C4⋊C4—22no—
44C43C2311yescharacteristic
45#340—24no—
46#340—24no—
47#340—24no—
48#336—22no—
49#336—22no—
50#336—22no—
51#336D414yes—
52Q8⋊C8—44no—
53C22×Q8—22no—
54C22×Q8—21no—
55C2×Q16—44no—
56C2×Q16—48no—
57C22×C8—22no—
58C22×C8—24no—
59C22×C8—41no—
60C4⋊Q8—21no—
61C4⋊Q8—22no—
62C4⋊Q8—44no—
63C4⋊Q8Q1612yescomplemented
64C4×Q8—42no—
65C4×Q8—44no—
66C4×Q8—42no—
67C4×Q8—44no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—42no—
78C2×C42—42no—
79C2×C42—41no—
80C2×C42C2×D412yes—
81C2×C42C2×D412yes—
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C42#4111yescharacteristic
85#107—48no—
86#107—44no—
87C4⋊C8—42no—
88C4⋊C8—42no—
89C4⋊C8—42no—
90#103—48no—
91C4×C8—22no—
92C4×C8D814yes—
93#95—42no—
94#95—42no—
95#95—42no—
96C2×Q8—24no—
97C2×Q8—22no—
98C2×Q8—42no—
99C2×Q8—44no—
100C2×Q8—41no—
101C2×Q8—44no—
102C2×Q8—42no—
103C2×Q8—42no—
104C22×C4—41no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—41no—
108C22×C4#12011yescharacteristic
109C22×C4#12012yes—
110C22×C4#12112yes—
111C22×C4#12111yescharacteristic
112C22×C4#12711yescharacteristic
113Q16—416no—
114C2×C8—24no—
115C2×C8—44no—
116C2×C8—44no—
117C2×C8—44no—
118C2×C8—44no—
119C2×C8—82no—
120C2×C8—81no—
121C4⋊C4—24no—
122C4⋊C4—24no—
123C4⋊C4—24no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—44no—
127C4⋊C4—44no—
128C4⋊C4—42no—
129C4⋊C4—44no—
130C4⋊C4—42no—
131C4⋊C4—82no—
132C4⋊C4—82no—
133C4⋊C4—82no—
134C4⋊C4—82no—
135C42—22no—
136C42—42no—
137C42—42no—
138C42—84no—
139C42—81no—
140C42—81no—
141C42C2×D812yes—
142C42C2×Q1611yescharacteristic
143C42#13511yescharacteristic
144C42AGL(1,ℤ/8)12yes—
145C42#13712yes—
146C23#39211yescharacteristic
147Q8—48no—
148Q8—44no—
149Q8—84no—
150Q8—82no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—44no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—81no—
168C2×C4—82no—
169C2×C4#44712yes—
170C2×C4#45112yes—
171C2×C4#45212yes—
172C2×C4#45912yes—
173C2×C4#46711yescharacteristic
174C2×C4#47011yescharacteristic
175C2×C4#47312yes—
176C2×C4#49311yescharacteristic
177C2×C4#49611yescharacteristic
178C8—44no—
179C8—84no—
180C8—161no—
181C22#200111yescharacteristic
182C22#200912yes—
183C22#201012yes—
184C22#201111yescharacteristic
185C22#202011yescharacteristic
186C4—22no—
187C4—22no—
188C4—22no—
189C4—21no—
190C4—21no—
191C4—42no—
192C4—84no—
193C4—82no—
194C4—161no—
195C4—162no—
196C4#166912yes—
197C4#168912yes—
198C2#1000312yes—
199C2#1001211yescharacteristic
200C2#1004512yes—
201C2#1022511yescharacteristic
202C2#1027411yescharacteristic
subgroupG/H
center, socle, prime radicalC23#392
commutator, Frattini, cosocleC43C23
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
63. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#292722
#238012
#96412
#76522
#70182