Finite Group Databasebeta

← #124409

Subgroups — #124409

Order 512.

286 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 80 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#21662C211yesmaximal, characteristic
2#8801C211yesmaximal, characteristic, complemented
3#8268C211yesmaximal, characteristic, complemented
4#7134C212yesmaximal, complemented
5#7106C212yesmaximal
6#3135C411yescharacteristic
7#2961C411yescharacteristic
8#2932C411yescharacteristic
9#1994C2211yescharacteristic
10#1962C2211yescharacteristic
11#1938—21no—
12#1937—21no—
13#1838C411yescharacteristic
14#1765—21no—
15#1764—21no—
16C4×#97C2211yescharacteristic
17#1564—21no—
18#1450—21no—
19#1281—21no—
20#1276C2212yes—
21#1276C2212yes—
22#1275—21no—
23C2×#131—21no—
24#533C2×C411yescharacteristic
25#532C2×C411yescharacteristic
26#491C812yescomplemented
27#490C812yescomplemented
28#489—22no—
29#446—42no—
30#435—42no—
31#431—22no—
32#431—22no—
33#424—21no—
34#423C2×C411yescharacteristic
35#423C2×C411yescharacteristic
36C2×#104—21no—
37C2×#104—21no—
38#416—21no—
39#416—21no—
40#408—21no—
41#405—21no—
42C2×#97C2×C411yescharacteristic
43C2×#97C2×C411yescharacteristic
44C2×#97D411yescharacteristic
45C2×#97D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C43C2311yescharacteristic
49#341—21no—
50#341—21no—
51#339—21no—
52#339—21no—
53#335—22no—
54#334—22no—
55#322—41no—
56C82—41no—
57C2×#41—21no—
58#131—41no—
59#131—41no—
60#131—41no—
61#131—41no—
62C2×M16—21no—
63C2×M16—21no—
64C2×M16—41no—
65C22×C8—41no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C4⋊Q8—21no—
71C4⋊Q8C2×C811yescharacteristic
72#127C2×C811yescharacteristic
73#125C2×C812yes—
74#124—21no—
75C22⋊Q8—41no—
76#121—41no—
77C4×Q8—41no—
78C4×D4—41no—
79#117—21no—
80C2×C42—41no—
81C2×C42C22×C411yescharacteristic
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C4⋊C8—24no—
85C4⋊C8—41no—
86C4⋊C8—41no—
87C4⋊C8—42no—
88C4⋊C8—42no—
89#104—42no—
90#104—41no—
91#104—41no—
92#103—22no—
93#103—22no—
94#102—22no—
95#102—22no—
96C22⋊C8—42no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99#97—22no—
100#97—22no—
101#97—41no—
102#97—41no—
103#97C2×C812yes—
104#97M1612yes—
105C4×C8—21no—
106C4×C8—21no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—41no—
110C4×C8—42no—
111C4×C8—42no—
112#41—41no—
113#41—41no—
114#41—41no—
115#41—41no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×D4—21no—
119C2×D4—41no—
120C2×D4—41no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4C2×C22⋊C411yescharacteristic
126M16—41no—
127M16—41no—
128M16—41no—
129M16—41no—
130M16—81no—
131M16—81no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—44no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—81no—
144C2×C8—81no—
145C2×C8—82no—
146C2×C8C22⋊C812yes—
147C2×C8C22⋊C812yes—
148C2×C8#10012yes—
149C2×C8#10411yescharacteristic
150C2×C8#10411yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—41no—
153C4⋊C4—42no—
154C4⋊C4—41no—
155C22⋊C4—42no—
156C22⋊C4—41no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—41no—
161C42—82no—
162C42C22×C811yescharacteristic
163C42C2×M1611yescharacteristic
164C42#13512yes—
165C42AGL(1,ℤ/8)11yescharacteristic
166C42#13711yescharacteristic
167C23—41no—
168C23#41011yescharacteristic
169Q8—41no—
170Q8—41no—
171Q8—161no—
172D4—41no—
173D4—41no—
174D4—81no—
175D4—81no—
176D4—161no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4C2×C22⋊C811yescharacteristic
190C2×C4C2×#10011yescharacteristic
191C2×C4#41611yescharacteristic
192C2×C4#41711yescharacteristic
193C2×C4C2×#10411yescharacteristic
194C2×C4#42111yescharacteristic
195C8—22no—
196C8—22no—
197C8—22no—
198C8—21no—
199C8—21no—
200C8—84no—
201C8—81no—
202C8—81no—
203C8—162no—
204C22—41no—
205C22—82no—
206C22—81no—
207C22#146011yescharacteristic
208C22#150911yescharacteristic
209C22#152111yescharacteristic
210C4—22no—
211C4—42no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—82no—
216C4—81no—
217C4#147611yescharacteristic
218C4#147711yescharacteristic
219C4#152711yescharacteristic
220C4#155111yescharacteristic
221C2—21no—
222C2—21no—
223C2—21no—
224C2—161no—
225C2#758411yescharacteristic
subgroupG/H
centerC4#1551
commutatorC42C22×C8
Frattini, cosocleC43C23
socleC2#7584
prime radical#532C2×C4
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8801C2semidirect
3. #8268C2semidirect
4. #7134C2semidirect
26. #491C8semidirect
27. #490C8semidirect

Maximal subgroups

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

subgroupindex
#216622
#88012
#82682
#71342
#71062