Finite Group Databasebeta

← #145953

Subgroups — #145953

Order 512.

243 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 76 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#13262C211yesmaximal, characteristic
2#13207C211yesmaximal, characteristic
3#13187C211yesmaximal, characteristic
4#13162C211yesmaximal, characteristic
5C2×#1369C211yesmaximal, characteristic
6C2×#1358C211yesmaximal, characteristic
7#7271C211yesmaximal, characteristic
8C2×#494C2211yescharacteristic
9C2×#365—21no—
10C2×#364C2211yescharacteristic
11C2×#364C2211yescharacteristic
12C2×#364C2211yescharacteristic
13#2151—21no—
14C2×#362—21no—
15#2141—21no—
16C2×#358—21no—
17C2×#358—21no—
18C2×#346C2211yescharacteristic
19#1846C2211yescharacteristic
20#1772C2211yescharacteristic
21#1386—21no—
22#1379—21no—
23#1369—22no—
24#1358—22no—
25C22×Q16—21no—
26C22×Q16—21no—
27C2×C4⋊Q8—21no—
28C2×M32—21no—
29C2×M32—21no—
30C2×M32—21no—
31C22×C16D411yescharacteristic
32C22×C16D411yescharacteristic
33#494—21no—
34#494—21no—
35#494—21no—
36#494Q812yes—
37C2×#108D411yescharacteristic
38#427D411yescharacteristic
39C2×#107—21no—
40C2×#106—21no—
41#405—21no—
42C2×C4×C8C2311yescharacteristic
43#382—21no—
44#365—42no—
45#364—21no—
46#364—21no—
47#364—21no—
48#364—21no—
49#364—21no—
50#364D412yes—
51C4⋊C16—42no—
52#362—42no—
53#358—42no—
54#358—42no—
55#358—42no—
56#346—21no—
57#346—21no—
58#339—21no—
59C22×Q8—41no—
60C2×Q16—22no—
61C2×Q16—22no—
62C2×Q16—41no—
63C2×Q16—41no—
64C2×Q16—41no—
65C2×Q16—41no—
66C2×Q16—41no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×Q16—41no—
70C2×Q16—41no—
71C2×Q16—41no—
72C2×M16—21no—
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C4⋊Q8—22no—
76C4⋊Q8—41no—
77C4⋊Q8—81no—
78#117—21no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—81no—
81C2×C42#4111yescharacteristic
82M32—22no—
83M32—41no—
84M32—41no—
85M32—41no—
86M32—41no—
87M32—41no—
88C2×C16—21no—
89C2×C16—21no—
90C2×C16—21no—
91C2×C16—21no—
92C2×C16—21no—
93C2×C16—41no—
94C2×C16—41no—
95C2×C16—41no—
96C2×C16SD1612yes—
97#108—21no—
98#108—21no—
99#107—41no—
100#107—41no—
101#107SD1612yes—
102#106—21no—
103#106—41no—
104#106—41no—
105#97—41no—
106C4×C8—41no—
107C4×C8C2×D411yescharacteristic
108C4×C8C2×Q811yescharacteristic
109C4×C8#4111yescharacteristic
110C4×C8#4111yescharacteristic
111#95—81no—
112C2×Q8—42no—
113C2×Q8—81no—
114C2×Q8—81no—
115C2×Q8—81no—
116C2×Q8—81no—
117C2×Q8—81no—
118C22×C4—21no—
119C22×C4—81no—
120C22×C4—81no—
121C22×C4#12111yescharacteristic
122Q16—42no—
123Q16—42no—
124Q16—42no—
125Q16—42no—
126Q16—81no—
127Q16—81no—
128Q16—81no—
129Q16—81no—
130M16—41no—
131M16—41no—
132C2×C8—41no—
133C2×C8#12111yescharacteristic
134C2×C8#12111yescharacteristic
135C2×C8C22⋊Q811yescharacteristic
136C2×C8C22⋊Q811yescharacteristic
137C2×C8C2×SD1611yescharacteristic
138C2×C8C2×SD1611yescharacteristic
139C2×C8#13511yescharacteristic
140C2×C8#13511yescharacteristic
141C2×C8AGL(1,ℤ/8)11yescharacteristic
142C2×C8AGL(1,ℤ/8)11yescharacteristic
143C2×C8AGL(1,ℤ/8)11yescharacteristic
144C2×C8AGL(1,ℤ/8)11yescharacteristic
145C4⋊C4—22no—
146C4⋊C4—82no—
147C4⋊C4—82no—
148C4⋊C4—81no—
149C4⋊C4—81no—
150C22⋊C4—41no—
151C42—21no—
152C42—41no—
153C42C22⋊Q811yescharacteristic
154C42#12611yescharacteristic
155C16—22no—
156C16—42no—
157C16—41no—
158C16—81no—
159C16—81no—
160C23#47211yescharacteristic
161Q8—82no—
162Q8—82no—
163Q8—161no—
164Q8—161no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—82no—
170C2×C4—161no—
171C2×C4—161no—
172C2×C4—162no—
173C2×C4—161no—
174C2×C4#39611yescharacteristic
175C2×C4#46011yescharacteristic
176C2×C4#46311yescharacteristic
177C2×C4#46511yescharacteristic
178C2×C4#46911yescharacteristic
179C2×C4#47111yescharacteristic
180C8—81no—
181C8#46011yescharacteristic
182C8#46311yescharacteristic
183C8#46511yescharacteristic
184C8#46911yescharacteristic
185C8#47611yescharacteristic
186C8#47611yescharacteristic
187C8#47811yescharacteristic
188C8#47811yescharacteristic
189C22—22no—
190C22#167711yescharacteristic
191C22#168811yescharacteristic
192C22#206311yescharacteristic
193C4—41no—
194C4—41no—
195C4—82no—
196C4—162no—
197C4—321no—
198C4#166611yescharacteristic
199C4#170111yescharacteristic
200C4#206511yescharacteristic
201C4#206511yescharacteristic
202C2—21no—
203C2—21no—
204C2#1059511yescharacteristic
205C2#1246612yes—
subgroupG/H
center, socleC22#1677
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#472
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#132622
#132072
#131872
#131622
C2×#13692
C2×#13582
#72712