Finite Group Databasebeta

← #123294

Subgroups — #123294

Order 512.

228 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 50 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#20020C211yesmaximal, characteristic
2#12510C211yesmaximal, characteristic, complemented
3#12448C211yesmaximal, characteristic, complemented
4#12447C211yesmaximal, characteristic
5#7097C211yesmaximal, characteristic, complemented
6#7069C211yesmaximal, characteristic, complemented
7#7068C211yesmaximal, characteristic
8#3393—21no—
9#3342—21no—
10#2241—22no—
11#2227—21no—
12#2226—21no—
13#1713C2211yescharacteristic
14#1713C2211yescharacteristic
15#1673—21no—
16#1670—21no—
17#1650—22no—
18C8×Q16C2211yescharacteristic
19C8×D8C2211yescharacteristic
20#1370C2211yescharacteristic
21#1370C2211yescharacteristic
22#1365—21no—
23#1365—21no—
24C8×C16C2211yescharacteristic
25#556—22no—
26#500—21no—
27#500—21no—
28#500—21no—
29#500D412yes—
30#495—41no—
31#494—21no—
32#494—21no—
33#485—42no—
34#475—22no—
35#467—41no—
36#464—42no—
37#461—22no—
38C8×Q8—21no—
39C4×Q16—41no—
40C4×Q16D411yescharacteristic
41C4×D8D411yescharacteristic
42C8×D4—21no—
43#427—41no—
44#367—41no—
45#366—41no—
46C4⋊C16—21no—
47C4⋊C16—21no—
48#358—44no—
49#358—41no—
50#357—41no—
51C4×C16—21no—
52#335D411yescharacteristic
53#335D411yescharacteristic
54#332—22no—
55Q8⋊C8—21no—
56D4⋊C8—21no—
57C82C2311yescharacteristic
58C2×Q16—42no—
59C2×Q16—41no—
60C2×Q16—41no—
61C2×Q16D811yescharacteristic
62C2×SD16—42no—
63C2×D8D811yescharacteristic
64C22×C8—41no—
65C4⋊Q8—22no—
66C4⋊Q8—41no—
67#126—82no—
68#124—42no—
69C22⋊Q8—42no—
70C4×Q8—21no—
71C4×Q8—42no—
72C4×D4—21no—
73#117—42no—
74C2×C16—22no—
75C2×C16—41no—
76C2×C16—41no—
77#107—41no—
78#107—41no—
79#107—42no—
80#107—41no—
81#107—41no—
82#107—41no—
83#107—41no—
84#107D811yescharacteristic
85#107D811yescharacteristic
86#106—42no—
87C4⋊C8—21no—
88C4⋊C8—21no—
89C4⋊C8—41no—
90#103—21no—
91#103—42no—
92#103—42no—
93#102—21no—
94C22⋊C8—41no—
95C4×C8—41no—
96C4×C8C2×D411yescharacteristic
97C4×C8C2×D411yescharacteristic
98C4×C8C2×D411yescharacteristic
99C2×Q8—21no—
100C2×Q8—42no—
101C2×Q8—81no—
102C2×D4—21no—
103C22×C4—41no—
104Q16—21no—
105Q16—21no—
106Q16—84no—
107Q16—81no—
108Q16—81no—
109Q16—161no—
110SD16—162no—
111D8—21no—
112D8—21no—
113C2×C8—21no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—81no—
117C2×C8—81no—
118C2×C8C2×D811yescharacteristic
119C2×C8C2×D811yescharacteristic
120C2×C8AGL(1,ℤ/8)11yescharacteristic
121C2×C8AGL(1,ℤ/8)11yescharacteristic
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—41no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—82no—
128C4⋊C4—81no—
129C4⋊C4—81no—
130C22⋊C4—41no—
131C22⋊C4—82no—
132C22⋊C4—82no—
133C42—41no—
134C42—82no—
135C42#12011yescharacteristic
136C16—42no—
137C16—81no—
138C16—81no—
139C23—41no—
140Q8—41no—
141Q8—41no—
142Q8—82no—
143Q8—82no—
144Q8—162no—
145D4—41no—
146D4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—82no—
153C2×C4—161no—
154C2×C4#44711yescharacteristic
155C2×C4#44711yescharacteristic
156C2×C4#45311yescharacteristic
157C8—22no—
158C8—21no—
159C8—21no—
160C8—81no—
161C8—81no—
162C8#50811yescharacteristic
163C8#50811yescharacteristic
164C8#51011yescharacteristic
165C8#51011yescharacteristic
166C22—81no—
167C22—81no—
168C22—81no—
169C22#161911yescharacteristic
170C4—21no—
171C4—81no—
172C4—81no—
173C4—81no—
174C4—162no—
175C4—321no—
176C4#218911yescharacteristic
177C4#218911yescharacteristic
178C4#219411yescharacteristic
179C4#219511yescharacteristic
180C2—161no—
181C2#1231111yescharacteristic
182C2#1232111yescharacteristic
183C2#1232311yescharacteristic
subgroupG/H
center, socleC22#1619
commutator, Frattini, cosocleC82C23
prime radicalC2×D8D8
LCSG ▷ C82 ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ G
LECSG ▷ C82 ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12510C2semidirect
3. #12448C2semidirect
5. #7097C2semidirect
6. #7069C2semidirect

Maximal subgroups

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

subgroupindex
#200202
#125102
#124482
#124472
#70972
#70692
#70682