Finite Group Databasebeta

← #145408

Subgroups — #145408

Order 512.

291 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×#1690C211yesmaximal, characteristic
2#13268C211yesmaximal, characteristic
3#13212C211yesmaximal, characteristic
4#12000C211yesmaximal, characteristic
5#11932C211yesmaximal, characteristic
6#8663C211yesmaximal, characteristic
7#7280C211yesmaximal, characteristic
8C2×#494C2211yescharacteristic
9C2×#475—21no—
10C2×#446C2211yescharacteristic
11#2150C2211yescharacteristic
12C2×#362—21no—
13C2×#362—21no—
14C4×M32C2211yescharacteristic
15#1848C2211yescharacteristic
16#1690—22no—
17#1690—22no—
18#1564C2211yescharacteristic
19C2×#334C2211yescharacteristic
20C2×#332—21no—
21#1487—21no—
22#1377—21no—
23#1377—21no—
24#1368—22no—
25#1362—22no—
26#1301—21no—
27C22×Q16—21no—
28C22×Q16—21no—
29C2×C4⋊Q8—21no—
30C2×C4×Q8D411yescharacteristic
31C2×M32—21no—
32C2×M32D411yescharacteristic
33C2×M32D411yescharacteristic
34#494—21no—
35#494—21no—
36#494—21no—
37#494D412yes—
38#475—24no—
39#475—42no—
40#446—21no—
41#446—21no—
42#446—21no—
43#446—21no—
44#431—21no—
45C2×#108—21no—
46C2×#108—21no—
47C2×#106—41no—
48#423—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8D411yescharacteristic
52C2×#103—41no—
53C2×#97—21no—
54C2×C4×C8C2311yescharacteristic
55C4⋊C16—21no—
56C4⋊C16—21no—
57#362—42no—
58#362—42no—
59#346—21no—
60C4×C16—21no—
61#335—21no—
62#334—21no—
63#334—21no—
64#334D412yes—
65#332—24no—
66Q8⋊C8—42no—
67C22×Q8—41no—
68C22×Q8SD1611yescharacteristic
69C2×Q16—22no—
70C2×Q16—22no—
71C2×Q16—41no—
72C2×Q16—41no—
73C2×Q16—41no—
74C2×Q16—41no—
75C2×Q16—41no—
76C2×Q16—41no—
77C2×Q16—81no—
78C2×Q16—81no—
79C2×M16—41no—
80C22×C8—21no—
81C22×C8—41no—
82C22×C8C2×D411yescharacteristic
83C22×C8#4111yescharacteristic
84C4⋊Q8—22no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4SD1611yescharacteristic
96C2×C42—21no—
97C2×C42C2×D411yescharacteristic
98M32—21no—
99M32—21no—
100M32—21no—
101M32—41no—
102M32—41no—
103M32SD1612yes—
104C2×C16—21no—
105C2×C16—21no—
106C2×C16—41no—
107#108—41no—
108#108—41no—
109#108—81no—
110#106—44no—
111C4⋊C8—21no—
112C4⋊C8—22no—
113C4⋊C8—21no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120#103—44no—
121C22⋊C8—41no—
122#97—41no—
123#97—41no—
124C4×C8—41no—
125C4×C8C2×D411yescharacteristic
126C4×C8C2×D411yescharacteristic
127C4×C8#4111yescharacteristic
128C4×C8#4111yescharacteristic
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—81no—
134C2×Q8—81no—
135C2×Q8—81no—
136C2×Q8—81no—
137C2×Q8—81no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—81no—
141C22×C4#12111yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143C22×C4#13511yescharacteristic
144Q16—42no—
145Q16—42no—
146Q16—82no—
147Q16—82no—
148Q16—81no—
149Q16—81no—
150M16—81no—
151M16—81no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—41no—
155C2×C8—42no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—82no—
159C2×C8—81no—
160C2×C8—81no—
161C2×C8#12111yescharacteristic
162C2×C8#12111yescharacteristic
163C2×C8#12311yescharacteristic
164C2×C8#12411yescharacteristic
165C2×C8C2×SD1611yescharacteristic
166C2×C8#13511yescharacteristic
167C2×C8AGL(1,ℤ/8)11yescharacteristic
168C2×C8#13711yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C42—41no—
178C42—41no—
179C42#12011yescharacteristic
180C42#12311yescharacteristic
181C42AGL(1,ℤ/8)11yescharacteristic
182C42AGL(1,ℤ/8)11yescharacteristic
183C16—22no—
184C16—41no—
185C16—81no—
186C23#46511yescharacteristic
187Q8—42no—
188Q8—42no—
189Q8—82no—
190Q8—82no—
191Q8—161no—
192Q8—161no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—161no—
201C2×C4—161no—
202C2×C4#39411yescharacteristic
203C2×C4#44911yescharacteristic
204C2×C4#45011yescharacteristic
205C2×C4#46011yescharacteristic
206C2×C4#46411yescharacteristic
207C2×C4#46911yescharacteristic
208C2×C4#47111yescharacteristic
209C2×C4#47211yescharacteristic
210C2×C4#48111yescharacteristic
211C2×C4#48211yescharacteristic
212C8—21no—
213C8—21no—
214C8—82no—
215C8—81no—
216C8—161no—
217C8—161no—
218C8#46111yescharacteristic
219C8#46311yescharacteristic
220C8#46511yescharacteristic
221C8#46811yescharacteristic
222C22—21no—
223C22—21no—
224C22#166611yescharacteristic
225C22#167811yescharacteristic
226C22#203411yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—81no—
231C4—81no—
232C4—162no—
233C4#168811yescharacteristic
234C4#170211yescharacteristic
235C4#205111yescharacteristic
236C4#205511yescharacteristic
237C4#223912yes—
238C2—21no—
239C2—21no—
240C2#1047511yescharacteristic
241C2#1339911yescharacteristic
242C2#1340311yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#465
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
C2×#16902
#132682
#132122
#120002
#119322
#86632
#72802