Finite Group Databasebeta

← #145258

Subgroups — #145258

Order 512.

244 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 60 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#19118C211yesmaximal, characteristic
2#12031C211yesmaximal, characteristic
3#11973C211yesmaximal, characteristic
4#11970C211yesmaximal, characteristic
5#11639C211yesmaximal, characteristic
6#8621C211yesmaximal, characteristic
7#7289C211yesmaximal, characteristic
8#3160C2211yescharacteristic
9#3104—21no—
10C2×C8×Q8C2211yescharacteristic
11#2163C2211yescharacteristic
12#2162C2211yescharacteristic
13#1676—21no—
14#1674—21no—
15#1672—22no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18#1564C2211yescharacteristic
19#1542—21no—
20#1467—21no—
21#1368—22no—
22#1365—21no—
23#1364—21no—
24#1363—22no—
25Q8⋊C16—22no—
26#1290—21no—
27#533—21no—
28C2×C4×Q8D411yescharacteristic
29#500D411yescharacteristic
30#500D411yescharacteristic
31#499—21no—
32#499—21no—
33#498—21no—
34#484—41no—
35#477—22no—
36#462—22no—
37#451—41no—
38C8×Q8—21no—
39C8×Q8—21no—
40C8×Q8—21no—
41C8×Q8—21no—
42C4×Q16—42no—
43#432—21no—
44#432—21no—
45#431—21no—
46#427—21no—
47#427—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D411yescharacteristic
50C2×C4×C8—21no—
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—21no—
53C4⋊C16D412yes—
54C22⋊C16—21no—
55C22⋊C16—21no—
56#346—21no—
57C4×C16—21no—
58#335—21no—
59#335—21no—
60#335—21no—
61#334—21no—
62#334—21no—
63#334D412yes—
64#333—22no—
65#332—22no—
66#324—42no—
67C22×Q8SD1611yescharacteristic
68C2×Q16—42no—
69C22×C8—21no—
70C22×C8#4111yescharacteristic
71C22×C8#4111yescharacteristic
72C4⋊Q8—21no—
73C4⋊Q8—21no—
74#125—41no—
75C22⋊Q8—41no—
76C22⋊Q8—41no—
77C4×Q8—21no—
78C4×Q8—21no—
79C4×Q8—21no—
80C4×Q8—21no—
81C4×Q8—41no—
82C4×Q8—41no—
83#117—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4SD1611yescharacteristic
86C2×C42—21no—
87C2×C42C2×D411yescharacteristic
88C2×C16—22no—
89C2×C16—41no—
90#107—21no—
91#107—21no—
92#107—21no—
93#107—21no—
94#107—42no—
95#106—41no—
96#106—41no—
97C4⋊C8—21no—
98C4⋊C8—22no—
99C4⋊C8—21no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104#103—42no—
105#103—42no—
106C22⋊C8—41no—
107C22⋊C8—41no—
108C22⋊C8—41no—
109#97—41no—
110#97—41no—
111C4×C8—41no—
112C4×C8—41no—
113C4×C8—41no—
114C4×C8C2×D411yescharacteristic
115C4×C8C2×D411yescharacteristic
116C4×C8C2×D411yescharacteristic
117C4×C8#4111yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—42no—
121C2×Q8—41no—
122C2×Q8—41no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4#12311yescharacteristic
126C22×C4C2×SD1611yescharacteristic
127C22×C4AGL(1,ℤ/8)11yescharacteristic
128Q16—84no—
129C2×C8—22no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—42no—
133C2×C8—81no—
134C2×C8—81no—
135C2×C8—81no—
136C2×C8#12111yescharacteristic
137C2×C8#12111yescharacteristic
138C2×C8#12311yescharacteristic
139C2×C8#12411yescharacteristic
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—81no—
149C4⋊C4—81no—
150C22⋊C4—81no—
151C42—41no—
152C42—41no—
153C42—81no—
154C42#12011yescharacteristic
155C42#12111yescharacteristic
156C42#13511yescharacteristic
157C42AGL(1,ℤ/8)11yescharacteristic
158C16—42no—
159C16—81no—
160C23#48111yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165Q8—82no—
166Q8—161no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4#39411yescharacteristic
179C2×C4#44911yescharacteristic
180C2×C4#45011yescharacteristic
181C2×C4#46511yescharacteristic
182C2×C4#46911yescharacteristic
183C2×C4#48211yescharacteristic
184C8—21no—
185C8—21no—
186C8—21no—
187C8—82no—
188C8—81no—
189C8—161no—
190C8—161no—
191C8#47312yes—
192C22—21no—
193C22—21no—
194C22—21no—
195C22#203411yescharacteristic
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—42no—
202C4—81no—
203C4—161no—
204C4—161no—
205C4#204711yescharacteristic
206C4#204911yescharacteristic
207C2—41no—
208C2#1044711yescharacteristic
209C2#1339511yescharacteristic
210C2#1340111yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#481
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ 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
#191182
#120312
#119732
#119702
#116392
#86212
#72892