Finite Group Databasebeta

← #142060

Subgroups — #142060

Order 512.

349 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 82 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#33213C211yesmaximal, characteristic
2#12606C211yesmaximal, characteristic
3#10804C211yesmaximal, characteristic
4#7263C212yesmaximal
5#7262C212yesmaximal
6C2×#498C411yescharacteristic, complemented
7C2×#405C2211yescharacteristic
8#2628—21no—
9#2112C2212yes—
10C2×#347—21no—
11#1993—21no—
12#1943—21no—
13#1941C412yescomplemented
14#1846C411yescharacteristic, complemented
15#1809—21no—
16#1807C2212yes—
17C2×#344—21no—
18C2×#341C2211yescharacteristic
19#1735C2211yescharacteristic
20C2×H(3,ℤ/4)—21no—
21#1375—21no—
22C22×M16—21no—
23C22×M16D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×C4⋊Q8—21no—
26C2×#117—21no—
27C2×#117D411yescharacteristic
28C2×M32—22no—
29#498—42no—
30C2×#107—21no—
31C2×#106—21no—
32C2×#106C2×C412yes—
33C2×#106C2×C412yes—
34#405—42no—
35C2×#97D411yescharacteristic
36C2×#97D411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39#400—41no—
40#399—41no—
41#395—41no—
42#393—41no—
43#389—22no—
44#382—21no—
45#380—22no—
46C22⋊C16—22no—
47C22⋊C16Q814yescomplemented
48#347—42no—
49#344—24no—
50#341—22no—
51#340—22no—
52#340—22no—
53#340—22no—
54H(3,ℤ/4)—42no—
55C22×Q8—41no—
56C23×C4C2×D411yescharacteristic
57C2×M16—21no—
58C2×M16—21no—
59C2×M16—22no—
60C2×M16—22no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2×Q812yes—
75C22×C8#4112yes—
76C4⋊Q8—42no—
77C4⋊Q8—42no—
78#117—22no—
79#117—22no—
80#117—41no—
81#117—41no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—42no—
93C2×C42—41no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C22⋊C411yescharacteristic
96M32—42no—
97M32—42no—
98C2×C16—24no—
99#107—42no—
100#106—42no—
101#106—42no—
102#106—42no—
103#97—22no—
104#97—41no—
105C4×C8—22no—
106C4×C8—41no—
107#95—41no—
108#95—41no—
109#95—42no—
110#95—42no—
111C24AGL(1,ℤ/8)11yescharacteristic
112C2×Q8—82no—
113C2×Q8—162no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—81no—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4#12411yescharacteristic
125C22×C4C4⋊Q811yescharacteristic
126C22×C4AGL(1,ℤ/8)11yescharacteristic
127C22×C4#13711yescharacteristic
128C22×C4#13711yescharacteristic
129M16—42no—
130M16—42no—
131M16—42no—
132M16—42no—
133M16—81no—
134M16—81no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8#10412yes—
150C2×C8C4×Q812yes—
151C2×C8C22⋊Q812yes—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—81no—
159C4⋊C4—82no—
160C4⋊C4—162no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—81no—
164C22⋊C4—162no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—42no—
169C42—81no—
170C42#9912yes—
171C16—44no—
172C23—21no—
173C23—21no—
174C23—42no—
175C23#41811yescharacteristic
176C23#48911yescharacteristic
177C23#50111yescharacteristic
178Q8—164no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—162no—
201C2×C4#39111yescharacteristic
202C2×C4C2×#9911yescharacteristic
203C2×C4#41011yescharacteristic
204C2×C4#41911yescharacteristic
205C2×C4C2×#10411yescharacteristic
206C2×C4#42111yescharacteristic
207C2×C4#48911yescharacteristic
208C2×C4#50111yescharacteristic
209C8—22no—
210C8—22no—
211C8—22no—
212C8—41no—
213C8—82no—
214C22—22no—
215C22—21no—
216C22—21no—
217C22—22no—
218C22—22no—
219C22—81no—
220C22—81no—
221C22#150511yescharacteristic
222C22#151011yescharacteristic
223C22#199311yescharacteristic
224C4—21no—
225C4—21no—
226C4—42no—
227C4—81no—
228C4—162no—
229C4—162no—
230C4—162no—
231C4#150811yescharacteristic
232C4#151511yescharacteristic
233C4#198611yescharacteristic
234C4#199611yescharacteristic
235C2—21no—
236C2—21no—
237C2—41no—
238C2—41no—
239C2#988911yescharacteristic
240C2#1329112yes—
subgroupG/H
center, socleC22#1993
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#498C4semidirect
13. #1941C4semidirect
14. #1846C4semidirect
47. C22⋊C16Q8semidirect

Maximal subgroups

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

subgroupindex
#332132
#126062
#108042
#72632
#72622