Finite Group Databasebeta

← #128806

Subgroups — #128806

Order 512.

390 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 98 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#21550C211yesmaximal, characteristic
2#12608C211yesmaximal, characteristic, complemented
3#11685C212yesmaximal
4#10814C211yesmaximal, characteristic, complemented
5#7256C212yesmaximal, complemented
6#3054C2211yescharacteristic
7C2×#444C411yescharacteristic, complemented
8#2929—21no—
9C2×#405C2211yescharacteristic
10C2×#349C2212yes—
11C2×#347—21no—
12#1974C411yescharacteristic, complemented
13#1945C411yescharacteristic, complemented
14#1937—21no—
15#1897—21no—
16#1811C2211yescharacteristic
17#1773C411yescharacteristic, complemented
18#1765—21no—
19#1740—21no—
20#1735—21no—
21C2×H(3,ℤ/4)C2212yes—
22#1376—21no—
23#1340C414yescomplemented
24#1339C414yescomplemented
25C2×#135C2×C411yescharacteristic
26C2×#131—21no—
27C22×M16C2311yescharacteristic
28C23×C8D411yescharacteristic
29#525—21no—
30C2×#117—22no—
31C2×#117D411yescharacteristic
32C2×M32—22no—
33#470C2×C412yes—
34#469C2×C412yes—
35#468—22no—
36#444—42no—
37#435—42no—
38C8×D4—42no—
39C2×#108C2×C411yescharacteristic
40C2×#108C2×C411yescharacteristic
41#427C2×C411yescharacteristic
42#424—21no—
43C2×#104—21no—
44C2×#104—21no—
45#416—21no—
46#408—21no—
47#407—21no—
48#405—21no—
49#405—42no—
50C2×#97C2×C411yescharacteristic
51C2×#97D411yescharacteristic
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#349—22no—
55#349C2×C414yes—
56#347—42no—
57#342—42no—
58#341—21no—
59#340—42no—
60#339—21no—
61#339—21no—
62H(3,ℤ/4)—22no—
63H(3,ℤ/4)C2×C414yes—
64C2×#41—21no—
65C23×C4C2×D411yescharacteristic
66#135—41no—
67#135—41no—
68C2×Q16C4211yescharacteristic
69C2×SD16—21no—
70C2×D8C4211yescharacteristic
71#131—42no—
72#131—42no—
73C2×M16—21no—
74C2×M16—21no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16C22×C412yes—
79C2×M16#4112yes—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22×C411yescharacteristic
87C22×C8C2×D411yescharacteristic
88#123—41no—
89C22⋊Q8—22no—
90#121—22no—
91#120—41no—
92C4×D4—42no—
93#117—21no—
94#117—24no—
95#117—41no—
96#117—41no—
97#117—42no—
98#117—42no—
99#117—42no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—42no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—42no—
104C2×C42—42no—
105C2×C42C22⋊C411yescharacteristic
106C2×C42C22⋊C411yescharacteristic
107M32—24no—
108M32—42no—
109C2×C16—42no—
110#108—41no—
111#108—41no—
112#107—21no—
113#106C4212yes—
114C4⋊C8—42no—
115#104—42no—
116#104—42no—
117#103—22no—
118#102—22no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121#97—22no—
122#97—22no—
123#97—41no—
124C4×C8—41no—
125C4×C8—41no—
126C4×C8—41no—
127C24AGL(1,ℤ/8)11yescharacteristic
128#41—41no—
129#41—41no—
130#41—41no—
131#41—41no—
132C2×Q8—21no—
133C2×D4—21no—
134C2×D4—41no—
135C2×D4—42no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—82no—
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4C4×D412yes—
146C22×C4AGL(1,ℤ/8)12yes—
147C22×C4#13711yescharacteristic
148Q16—41no—
149SD16—42no—
150D8—41no—
151M16—42no—
152M16—42no—
153M16—41no—
154M16—41no—
155M16—41no—
156M16—41no—
157M16—82no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—44no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—82no—
171C2×C8#9912yes—
172C2×C8C2×C4211yescharacteristic
173C2×C8#11711yescharacteristic
174C2×C8C4×D412yes—
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—82no—
180C22⋊C4—42no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—44no—
185C22⋊C4—82no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—44no—
191C42—41no—
192C42—82no—
193C16—44no—
194C23—21no—
195C23—21no—
196C23—41no—
197C23—81no—
198C23#41711yescharacteristic
199C23#44212yes—
200Q8—41no—
201Q8—41no—
202D4—41no—
203D4—41no—
204D4—84no—
205D4—81no—
206D4—81no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—84no—
222C2×C4—164no—
223C2×C4C4×C22⋊C411yescharacteristic
224C2×C4C2×#9911yescharacteristic
225C2×C4#41011yescharacteristic
226C2×C4#41811yescharacteristic
227C2×C4#42111yescharacteristic
228C2×C4#42111yescharacteristic
229C2×C4#44012yes—
230C8—22no—
231C8—21no—
232C8—21no—
233C8—22no—
234C8—41no—
235C8—81no—
236C8—81no—
237C22—22no—
238C22—41no—
239C22—42no—
240C22—42no—
241C22—81no—
242C22—82no—
243C22—81no—
244C22#151211yescharacteristic
245C22#151311yescharacteristic
246C22#176211yescharacteristic
247C4—22no—
248C4—81no—
249C4—81no—
250C4—82no—
251C4—81no—
252C4—164no—
253C4#151011yescharacteristic
254C4#151411yescharacteristic
255C4C4×#9911yescharacteristic
256C4#175911yescharacteristic
257C2—21no—
258C2—21no—
259C2—21no—
260C2—42no—
261C2—161no—
262C2#821611yescharacteristic
subgroupG/H
centerC4#1514
commutatorC2×C8C2×C42
Frattini, cosocleC22×M16C23
socleC2#8216
prime radical#3054C22
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12608C2semidirect
4. #10814C2semidirect
5. #7256C2semidirect
7. C2×#444C4semidirect
12. #1974C4semidirect
13. #1945C4semidirect
17. #1773C4semidirect
23. #1340C4semidirect
24. #1339C4semidirect

Maximal subgroups

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

subgroupindex
#215502
#126082
#116852
#108142
#72562