Finite Group Databasebeta

← #136426

Subgroups — #136426

Order 512.

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

normal subgroups only

This group has 359 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#22657C211yesmaximal, characteristic
2#13185C211yesmaximal, characteristic, complemented
3#12886C211yesmaximal, characteristic, complemented
4#11583C211yesmaximal, characteristic
5#8658C211yesmaximal, characteristic
6#7278C211yesmaximal, characteristic, complemented
7#7174C211yesmaximal, characteristic, complemented
8C2×C8×D4C2211yescharacteristic
9C2×#427—21no—
10#2354—21no—
11C2×#365—21no—
12#2150C2211yescharacteristic
13C4×M32C2211yescharacteristic
14#1947—21no—
15#1854—22no—
16#1848C2211yescharacteristic
17#1844C2211yescharacteristic
18#1807—21no—
19C4×#106C2211yescharacteristic
20#1564C2211yescharacteristic
21#1486—21no—
22#1383—22no—
23#1383—21no—
24D4⋊C16C414yescomplemented
25#1301—21no—
26C23×C8—21no—
27C2×C4×D4D411yescharacteristic
28C2×#117—22no—
29C2×M32—21no—
30C2×M32—21no—
31C8×D4—21no—
32C8×D4—21no—
33C8×D4C2×C412yes—
34C8×D4Q812yes—
35#431—21no—
36C2×#108—21no—
37C2×#108—21no—
38#427—42no—
39#427—42no—
40C2×#107—22no—
41C2×#106—21no—
42C2×#106D411yescharacteristic
43C2×#106D411yescharacteristic
44C2×#106D412yes—
45#423—21no—
46C2×C4⋊C8D411yescharacteristic
47C2×C22⋊C8—21no—
48C2×C4×C8C2311yescharacteristic
49#390—41no—
50#388—42no—
51#384—21no—
52#380—22no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4—41no—
55#365—42no—
56C4⋊C16—21no—
57C4⋊C16C2×C412yes—
58#346—21no—
59C4×C16C2×C412yes—
60#340—22no—
61#340—21no—
62#335—21no—
63#334—21no—
64D4⋊C8—42no—
65C22×D4SD1611yescharacteristic
66C23×C4—21no—
67C2×M16—41no—
68C22×C8—21no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8C2×D411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C4×D4—21no—
78C4×D4—21no—
79C4×D4C4⋊C412yes—
80C4×D4SD1612yes—
81#117—44no—
82#117—82no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4SD1611yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C42—42no—
93C2×C42—41no—
94C2×C42C2×D411yescharacteristic
95M32—41no—
96M32—41no—
97M32—41no—
98M32—41no—
99C2×C16—22no—
100C2×C16—22no—
101#108—42no—
102#108—41no—
103#108—41no—
104#107—44no—
105#106—21no—
106#106—21no—
107#106—21no—
108#106—41no—
109#106—41no—
110#106—42no—
111#106D812yes—
112C4⋊C8—21no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8C4⋊C412yes—
117C22⋊C8—22no—
118C22⋊C8—22no—
119C22⋊C8—41no—
120C4×C8—41no—
121C4×C8C22×C411yescharacteristic
122C4×C8C2×Q811yescharacteristic
123C4×C8#4111yescharacteristic
124C4×C8#4111yescharacteristic
125#95—42no—
126#95—41no—
127C24—21no—
128C2×D4—21no—
129C2×D4—42no—
130C2×D4#10612yes—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4#12011yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143C22×C4AGL(1,ℤ/8)11yescharacteristic
144M16—81no—
145M16—81no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—81no—
156C2×C8—81no—
157C2×C8C4×D411yescharacteristic
158C2×C8C4×D411yescharacteristic
159C2×C8C22⋊Q811yescharacteristic
160C2×C8C22⋊Q811yescharacteristic
161C2×C8C2×D811yescharacteristic
162C2×C8#13511yescharacteristic
163C2×C8AGL(1,ℤ/8)11yescharacteristic
164C2×C8#13711yescharacteristic
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—82no—
173C4⋊C4—161no—
174C4⋊C4#10612yes—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—84no—
178C22⋊C4—162no—
179C42—82no—
180C42—84no—
181C42C2×C4⋊C411yescharacteristic
182C42#12311yescharacteristic
183C42C2×SD1611yescharacteristic
184C42#13711yescharacteristic
185C16—42no—
186C16—42no—
187C23—22no—
188C23—42no—
189C23—42no—
190C23—41no—
191C23—41no—
192C23#45011yescharacteristic
193D4—24no—
194D4—42no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—81no—
212C2×C4—82no—
213C2×C4#38511yescharacteristic
214C2×C4C2×#10611yescharacteristic
215C2×C4#42811yescharacteristic
216C2×C4#44711yescharacteristic
217C2×C4#44811yescharacteristic
218C2×C4#45211yescharacteristic
219C2×C4#45411yescharacteristic
220C2×C4#45511yescharacteristic
221C2×C4#48111yescharacteristic
222C2×C4#48311yescharacteristic
223C8—21no—
224C8—21no—
225C8—42no—
226C8—161no—
227C8—161no—
228C8C4×D811yescharacteristic
229C8#44011yescharacteristic
230C8#47411yescharacteristic
231C8#47811yescharacteristic
232C22—21no—
233C22—21no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—82no—
238C22—82no—
239C22#163511yescharacteristic
240C22#163911yescharacteristic
241C22#185311yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—42no—
246C4—82no—
247C4—162no—
248C4—161no—
249C4#162711yescharacteristic
250C4#164311yescharacteristic
251C4#186011yescharacteristic
252C4#186211yescharacteristic
253C4#215512yes—
254C2—21no—
255C2—21no—
256C2—82no—
257C2#889411yescharacteristic
258C2#1293011yescharacteristic
259C2#1293411yescharacteristic
subgroupG/H
center, socleC22#1853
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13185C2semidirect
3. #12886C2semidirect
6. #7278C2semidirect
7. #7174C2semidirect
24. D4⋊C16C4semidirect

Maximal subgroups

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

subgroupindex
#226572
#131852
#128862
#115832
#86582
#72782
#71742