Finite Group Databasebeta

← #150883

Subgroups — #150883

Order 512.

251 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 68 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#33611C211yesmaximal, characteristic
2#13556C211yesmaximal, characteristic, complemented
3#12840C211yesmaximal, characteristic, complemented
4#7425C212yesmaximal, complemented
5#7424C212yesmaximal, complemented
6C2×#508C411yescharacteristic
7C2×#504C2211yescharacteristic
8#3061—21no—
9C2×M64—21no—
10#2219C411yescharacteristic
11#2219C2211yescharacteristic
12#2218C411yescharacteristic
13#2218C2211yescharacteristic
14#2217—21no—
15#2217—21no—
16C2×#368—21no—
17C2×#367C411yescharacteristic
18#2139—21no—
19#1813—21no—
20#1400—21no—
21#1400C2212yes—
22#1385C2212yes—
23C2×#135—21no—
24C2×#131D411yescharacteristic
25#537—41no—
26#508—41no—
27#508—41no—
28C2×Q32C2×C411yescharacteristic
29C2×SD32—21no—
30C2×D16C2×C411yescharacteristic
31#504—41no—
32#504—41no—
33C2×M32C2×C411yescharacteristic
34C2×M32D411yescharacteristic
35C2×M32C2311yescharacteristic
36C22×C16C2×C411yescharacteristic
37C22×C16D411yescharacteristic
38C22×C16D411yescharacteristic
39#468—41no—
40#468—41no—
41#467—21no—
42#467—21no—
43#466—21no—
44#466—21no—
45C2×#108—22no—
46C2×#107—21no—
47#417—41no—
48#413—41no—
49M64—22no—
50M64—41no—
51C2×C32—41no—
52#368—41no—
53#368—41no—
54#367—21no—
55#367C2×C411yescharacteristic
56#367C2×C411yescharacteristic
57#358—21no—
58#358—21no—
59#357—21no—
60#357—21no—
61#338—41no—
62C2×#41—41no—
63C2×#41D811yescharacteristic
64#135—42no—
65#135—42no—
66C2×Q16—21no—
67C2×SD16—41no—
68C2×D8—21no—
69#131—41no—
70#131—41no—
71C2×M16—42no—
72C2×M16D811yescharacteristic
73C2×M16SD1611yescharacteristic
74C2×M16SD1611yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22⋊Q8—41no—
79C22⋊Q8—41no—
80C22⋊Q8—81no—
81#121—41no—
82#121—41no—
83#121—81no—
84#121—81no—
85#121—81no—
86C2×C4⋊C4—41no—
87Q32—41no—
88SD32—41no—
89SD32—41no—
90D16—41no—
91M32—41no—
92M32—41no—
93M32—41no—
94C2×C16—21no—
95C2×C16—21no—
96C2×C16—21no—
97C2×C16—21no—
98C2×C16—21no—
99C2×C16—21no—
100C2×C16C22⋊C411yescharacteristic
101C2×C16C22⋊C411yescharacteristic
102C2×C16C22×C411yescharacteristic
103C2×C16C2×D411yescharacteristic
104#108—42no—
105#108—42no—
106#107—21no—
107#107—21no—
108#107—41no—
109#103—41no—
110#103—41no—
111#102—41no—
112#102—41no—
113#101—81no—
114#100—81no—
115C32—42no—
116#41—41no—
117#41—41no—
118#41—82no—
119#41—82no—
120C2×Q8—41no—
121C2×Q8D1611yescharacteristic
122C2×D4—41no—
123C2×D4—81no—
124C2×D4—81no—
125C2×D4—81no—
126C2×D4D1611yescharacteristic
127C2×D4SD3211yescharacteristic
128C2×D4SD3211yescharacteristic
129C22×C4—81no—
130C22×C4—81no—
131C22×C4#10211yescharacteristic
132C22×C4#10211yescharacteristic
133C22×C4C2×D811yescharacteristic
134Q16—41no—
135Q16—41no—
136SD16—82no—
137D8—41no—
138D8—41no—
139M16—41no—
140M16—41no—
141M16—41no—
142M16—82no—
143M16—82no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—82no—
151C2×C8#10211yescharacteristic
152C2×C8#10211yescharacteristic
153C2×C8C2×C22⋊C411yescharacteristic
154C2×C8C2×SD1611yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—81no—
158C4⋊C4—81no—
159C4⋊C4—81no—
160C22⋊C4—81no—
161C22⋊C4—81no—
162C22⋊C4—81no—
163C22⋊C4—81no—
164C16—21no—
165C16—21no—
166C16—21no—
167C16#9911yescharacteristic
168C16#9911yescharacteristic
169C23—81no—
170C23#35711yescharacteristic
171C23#35711yescharacteristic
172C23C2×SD3211yescharacteristic
173Q8—41no—
174Q8—81no—
175Q8—81no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—81no—
180D4—81no—
181D4—162no—
182D4—321no—
183D4—321no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—162no—
194C2×C4—321no—
195C2×C4#35711yescharacteristic
196C2×C4#35711yescharacteristic
197C2×C4C2×#10211yescharacteristic
198C2×C4C2×D1611yescharacteristic
199C8—21no—
200C8—21no—
201C8—21no—
202C8—162no—
203C8C2×#9911yescharacteristic
204C8#42111yescharacteristic
205C22—21no—
206C22—21no—
207C22—21no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—161no—
212C22—161no—
213C22—161no—
214C22C2×#35711yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—161no—
219C4—161no—
220C4—161no—
221C4#151911yescharacteristic
222C4#214511yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—321no—
228C2#1188611yescharacteristic
subgroupG/H
center, socleC2#11886
commutatorC2×C16C22×C4
Frattini, cosocleC2×M32C23
prime radical#33611C2
LCSG ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ C2×#504 ◁ G
LECSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13556C2semidirect
3. #12840C2semidirect
4. #7425C2semidirect
5. #7424C2semidirect

Maximal subgroups

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

subgroupindex
#336112
#135562
#128402
#74252
#74242