Finite Group Databasebeta

← #145301

Subgroups — #145301

Order 512.

247 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 62 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#19287C211yesmaximal, characteristic
2#12019C211yesmaximal, characteristic
3#11984C211yesmaximal, characteristic
4#11679C211yesmaximal, characteristic
5C2×#1338C211yesmaximal, characteristic
6#8625C211yesmaximal, characteristic
7#7243C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#3078—21no—
10C2×#446C2211yescharacteristic
11C2×C4⋊C16C2211yescharacteristic
12C2×#346C2211yescharacteristic
13#1704—22no—
14#1696—21no—
15#1690—21no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18#1566C2211yescharacteristic
19#1550—21no—
20#1467—21no—
21#1368—22no—
22#1364—21no—
23#1362—21no—
24#1338—21no—
25#1338—21no—
26#1334—21no—
27#1334—21no—
28#1290—21no—
29#533—21no—
30C2×C4×Q8D411yescharacteristic
31C22×C16—21no—
32C22×C16—21no—
33#500D411yescharacteristic
34#494D411yescharacteristic
35#487—21no—
36#479—42no—
37#477—22no—
38#475—22no—
39#467—21no—
40#467—21no—
41#446—21no—
42#446—21no—
43#446—21no—
44#446—21no—
45C4×Q16—21no—
46C4×Q16—21no—
47#432—21no—
48#432—21no—
49#428—41no—
50#423—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8D411yescharacteristic
53#419—41no—
54C2×#97—21no—
55C2×C4×C8C2311yescharacteristic
56C4⋊C16—21no—
57C4⋊C16—21no—
58#346—21no—
59#346D411yescharacteristic
60#346D411yescharacteristic
61#335—21no—
62#335—21no—
63#334—21no—
64#334—21no—
65#334—21no—
66#334D412yes—
67#333—22no—
68#332—22no—
69#324—42no—
70C22×Q8SD1611yescharacteristic
71C2×Q16—21no—
72C2×Q16—21no—
73C2×M16—41no—
74C22×C8—21no—
75C22×C8#4111yescharacteristic
76C22×C8#4111yescharacteristic
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79#125—41no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8—21no—
85C4×Q8—21no—
86C4×Q8—41no—
87C4×Q8—41no—
88#117—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4SD1611yescharacteristic
91C2×C42—21no—
92C2×C42C2×D411yescharacteristic
93C2×C16—21no—
94C2×C16—21no—
95C2×C16—42no—
96C2×C16—41no—
97C2×C16—81no—
98#107—21no—
99#107—21no—
100#107—42no—
101#106—42no—
102C4⋊C8—21no—
103C4⋊C8—22no—
104C4⋊C8—21no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110#103—42no—
111#103—42no—
112#103—41no—
113#103—41no—
114C22⋊C8—41no—
115C22⋊C8—41no—
116#97—41no—
117#97—41no—
118#97—41no—
119#97—41no—
120C4×C8C2×D411yescharacteristic
121C4×C8C2×D411yescharacteristic
122C4×C8C2×D411yescharacteristic
123C4×C8#4111yescharacteristic
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—42no—
127C2×Q8—41no—
128C2×Q8—41no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4#12311yescharacteristic
132C22×C4C2×SD1611yescharacteristic
133C22×C4AGL(1,ℤ/8)11yescharacteristic
134Q16—41no—
135Q16—41no—
136Q16—81no—
137M16—82no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—42no—
143C2×C8—81no—
144C2×C8—81no—
145C2×C8—81no—
146C2×C8#12111yescharacteristic
147C2×C8#12111yescharacteristic
148C2×C8#12311yescharacteristic
149C2×C8#12411yescharacteristic
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—81no—
159C4⋊C4—81no—
160C22⋊C4—81no—
161C42—41no—
162C42—41no—
163C42—81no—
164C42#12011yescharacteristic
165C42#12111yescharacteristic
166C42#13511yescharacteristic
167C42AGL(1,ℤ/8)11yescharacteristic
168C16—22no—
169C16—41no—
170C16—81no—
171C23#48111yescharacteristic
172Q8—42no—
173Q8—81no—
174Q8—81no—
175Q8—81no—
176Q8—161no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4#39411yescharacteristic
188C2×C4#44911yescharacteristic
189C2×C4#45011yescharacteristic
190C2×C4#46511yescharacteristic
191C2×C4#46911yescharacteristic
192C2×C4#48211yescharacteristic
193C8—21no—
194C8—21no—
195C8—21no—
196C8—82no—
197C8—81no—
198C8—161no—
199C8—161no—
200C8#47111yescharacteristic
201C8#47311yescharacteristic
202C22—21no—
203C22—21no—
204C22#203411yescharacteristic
205C22#223912yes—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—81no—
212C4—81no—
213C4—161no—
214C4—161no—
215C4#205011yescharacteristic
216C4#205211yescharacteristic
217C2—22no—
218C2#1045911yescharacteristic
219C2#1339911yescharacteristic
220C2#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×#446 ◁ 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
#192872
#120192
#119842
#116792
C2×#13382
#86252
#72432