Finite Group Databasebeta

← #124681

Subgroups — #124681

Order 512.

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

normal subgroups only

This group has 321 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#21286C211yesmaximal, characteristic
2#12595C211yesmaximal, characteristic, complemented
3#8800C211yesmaximal, characteristic, complemented
4#7127C211yesmaximal, characteristic, complemented
5#7126C211yesmaximal, characteristic, complemented
6#7110C211yesmaximal, characteristic
7#7110C211yesmaximal, characteristic
8C2×#442C411yescharacteristic
9#2293—21no—
10C2×#347—21no—
11#1972C2211yescharacteristic
12#1970C411yescharacteristic
13#1944C2211yescharacteristic
14#1941C411yescharacteristic
15#1936—21no—
16#1934—21no—
17#1776C411yescharacteristic
18#1762—21no—
19C4×#97C2211yescharacteristic
20#1377C2211yescharacteristic
21#1377C2211yescharacteristic
22#1376—21no—
23#1281—21no—
24#1276—21no—
25#1275C2211yescharacteristic
26#1275C2211yescharacteristic
27C22×D8C2×C411yescharacteristic
28C2×C4×D4—21no—
29C2×#117—41no—
30C2×M32—21no—
31C2×M32—21no—
32#442—22no—
33#442—22no—
34#431—21no—
35#431—21no—
36C2×#107C2×C411yescharacteristic
37C2×#106C2×C411yescharacteristic
38C2×#106C2×C411yescharacteristic
39C2×#102—21no—
40C2×#102—21no—
41C2×#102—21no—
42C2×#97C2×C411yescharacteristic
43C2×#97D411yescharacteristic
44C2×#97D411yescharacteristic
45C2×#97C2311yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#390—21no—
49#390—21no—
50#389—21no—
51#389—21no—
52#388—41no—
53#388—41no—
54C4×C4⋊C4—21no—
55C4×C22⋊C4—41no—
56C43D411yescharacteristic
57#347—24no—
58#340—21no—
59#340—21no—
60#340—21no—
61C22×D4—21no—
62C23×C4—41no—
63C2×D8—21no—
64C2×D8—41no—
65C2×D8—41no—
66C2×D8M1612yes—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22×C411yescharacteristic
70C22×C8C2×D411yescharacteristic
71C4×D4—24no—
72C4×D4—42no—
73#117—84no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—41no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—41no—
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22⋊C411yescharacteristic
89C2×C42C2×D411yescharacteristic
90M32—22no—
91M32—22no—
92M32—41no—
93M32—41no—
94C2×C16—41no—
95C2×C16—41no—
96#107—41no—
97#106—21no—
98#106—41no—
99#106M1612yes—
100C4⋊C8—41no—
101C4⋊C8—41no—
102#102—24no—
103#102—42no—
104#102—41no—
105#102—41no—
106C22⋊C8—41no—
107C22⋊C8—41no—
108#97—21no—
109#97—21no—
110#97—22no—
111#97—41no—
112#97—41no—
113C4×C8—21no—
114C4×C8—21no—
115C4×C8—41no—
116C4×C8—41no—
117C4×C8—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121C24—41no—
122C2×D4—22no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—44no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—82no—
135C22×C4—82no—
136C22×C4C2×C22⋊C411yescharacteristic
137C22×C4AGL(1,ℤ/8)11yescharacteristic
138C22×C4AGL(1,ℤ/8)11yescharacteristic
139D8—42no—
140D8—42no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8C2×M1611yescharacteristic
154C2×C8C2×M1611yescharacteristic
155C2×C8#13511yescharacteristic
156C2×C8#13511yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—82no—
164C4⋊C4—82no—
165C22⋊C4—44no—
166C22⋊C4—82no—
167C22⋊C4—82no—
168C22⋊C4—82no—
169C22⋊C4—82no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—42no—
176C42—81no—
177C42—82no—
178C42#10412yes—
179C16—42no—
180C16—42no—
181C23—42no—
182C23—81no—
183C23—81no—
184C23—81no—
185C23—81no—
186C23—81no—
187C23—81no—
188C23#41811yescharacteristic
189D4—42no—
190D4—42no—
191D4—81no—
192D4—81no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—88no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4#40711yescharacteristic
214C2×C4#41011yescharacteristic
215C2×C4#41311yescharacteristic
216C2×C4#41611yescharacteristic
217C2×C4C2×#10411yescharacteristic
218C2×C4#42111yescharacteristic
219C8—21no—
220C8—21no—
221C8—21no—
222C8—21no—
223C8—41no—
224C8—41no—
225C8—161no—
226C8—161no—
227C22—21no—
228C22—21no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22—161no—
233C22—161no—
234C22—161no—
235C22—161no—
236C22#149011yescharacteristic
237C22#151411yescharacteristic
238C22#155311yescharacteristic
239C4—42no—
240C4—42no—
241C4—41no—
242C4—41no—
243C4—82no—
244C4—161no—
245C4—161no—
246C4#146411yescharacteristic
247C4#148811yescharacteristic
248C4#150011yescharacteristic
249C4#152911yescharacteristic
250C2—21no—
251C2—21no—
252C2—162no—
253C2#760811yescharacteristic
254C2#1183611yescharacteristic
255C2#1183711yescharacteristic
subgroupG/H
center, socleC22#1514
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12595C2semidirect
3. #8800C2semidirect
4. #7127C2semidirect
5. #7126C2semidirect

Maximal subgroups

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

subgroupindex
#212862
#125952
#88002
#71272
#71262
#71102