Finite Group Databasebeta

← #124678

Subgroups — #124678

Order 512.

287 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 68 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#21297C211yesmaximal, characteristic
2#12896C211yesmaximal, characteristic, complemented
3#8134C211yesmaximal, characteristic, complemented
4#7119C212yesmaximal, complemented
5#7114C212yesmaximal
6#2943C411yescharacteristic
7#2923—21no—
8#1975C411yescharacteristic
9#1974C2211yescharacteristic
10#1946C411yescharacteristic
11#1945C2211yescharacteristic
12#1938—21no—
13#1937—21no—
14#1778C411yescharacteristic
15C4×#104—21no—
16#1744—21no—
17C4×#97C2211yescharacteristic
18#1377—21no—
19#1376C2212yes—
20#1375—21no—
21#1273—21no—
22#1273C2212yes—
23C2×#135C2×C411yescharacteristic
24C4×#41—21no—
25C2×M32—22no—
26#442—21no—
27#441—21no—
28#440—22no—
29#433—42no—
30#432—42no—
31C2×#108C2×C411yescharacteristic
32C2×#108C2×C411yescharacteristic
33#427C2×C411yescharacteristic
34#424—21no—
35#424—21no—
36C2×#104—21no—
37C2×#104—21no—
38#416—21no—
39#408—21no—
40#408—21no—
41C4×M16—21no—
42C2×#97C2×C411yescharacteristic
43C2×#97C2×C411yescharacteristic
44C2×#97D411yescharacteristic
45C2×#97D411yescharacteristic
46C2×C4×C8—22no—
47C2×C4×C8D411yescharacteristic
48C2×C4×C8C2311yescharacteristic
49C43D411yescharacteristic
50#341—21no—
51#339—21no—
52#339—21no—
53#336—24no—
54C2×#41—21no—
55#135—41no—
56#135—41no—
57C2×Q16M1611yescharacteristic
58C2×SD16—21no—
59C2×D8M1611yescharacteristic
60C2×M16—21no—
61C2×M16—21no—
62C22×C8—24no—
63C22×C8C22⋊C411yescharacteristic
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22×C411yescharacteristic
66C22×C8C2×D411yescharacteristic
67C4×Q8—22no—
68C4×D4—22no—
69C4×D4—42no—
70#117—21no—
71#117—41no—
72C2×C42—22no—
73C2×C42—41no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22⋊C411yescharacteristic
76C2×C42C2×D411yescharacteristic
77M32—42no—
78M32—42no—
79C2×C16—42no—
80#108—41no—
81#108—41no—
82#107M1612yes—
83#106—21no—
84C4⋊C8—42no—
85C4⋊C8—42no—
86#104—42no—
87#104—42no—
88#103—22no—
89#102—22no—
90C22⋊C8—42no—
91C22⋊C8—42no—
92#97—21no—
93#97—21no—
94#97—22no—
95#97—41no—
96#97—41no—
97#97—41no—
98C4×C8—22no—
99C4×C8—41no—
100C4×C8—42no—
101C4×C8—41no—
102C4×C8—42no—
103#41—41no—
104#41—41no—
105#41—41no—
106#41—41no—
107C2×Q8—21no—
108C2×D4—21no—
109C2×D4—41no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—42no—
113C22×C4—41no—
114C22×C4#10412yes—
115C22×C4C2×C22⋊C411yescharacteristic
116Q16—41no—
117SD16—42no—
118D8—41no—
119M16—41no—
120M16—41no—
121M16—41no—
122M16—41no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—44no—
134C2×C8—48no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8C2×M1611yescharacteristic
138C2×C8C2×M1611yescharacteristic
139C2×C8AGL(1,ℤ/8)12yes—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C22⋊C4—42no—
143C22⋊C4—41no—
144C42—21no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42—44no—
150C42—42no—
151C42—41no—
152C42#13512yes—
153C16—82no—
154C23—41no—
155C23C2×#10411yescharacteristic
156Q8—41no—
157Q8—41no—
158D4—41no—
159D4—41no—
160D4—81no—
161D4—81no—
162C2×C4—24no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—162no—
173C2×C4#40711yescharacteristic
174C2×C4#41011yescharacteristic
175C2×C4#41311yescharacteristic
176C2×C4#41611yescharacteristic
177C2×C4#41811yescharacteristic
178C2×C4#42111yescharacteristic
179C8—21no—
180C8—21no—
181C8—22no—
182C8—41no—
183C8—41no—
184C8—84no—
185C8—81no—
186C8—81no—
187C22—22no—
188C22—82no—
189C22—81no—
190C22#149011yescharacteristic
191C22#150011yescharacteristic
192C22#152911yescharacteristic
193C4—24no—
194C4—42no—
195C4—81no—
196C4—82no—
197C4—81no—
198C4#146411yescharacteristic
199C4#148811yescharacteristic
200C4#151411yescharacteristic
201C4#155311yescharacteristic
202C2—21no—
203C2—21no—
204C2—161no—
205C2#760811yescharacteristic
206C2#1167012yes—
subgroupG/H
centerC2×C4#418
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1490
prime radicalC2×#135C2×C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C4×#97 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12896C2semidirect
3. #8134C2semidirect
4. #7119C2semidirect

Maximal subgroups

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

subgroupindex
#212972
#128962
#81342
#71192
#71142