Finite Group Databasebeta

← #136367

Subgroups — #136367

Order 512.

403 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 102 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
1D4×#106C211yesmaximal, characteristic
2#13185C211yesmaximal, characteristic, complemented
3#12903C211yesmaximal, characteristic, complemented
4#11583C211yesmaximal, characteristic
5#8658C211yesmaximal, characteristic
6#7278C211yesmaximal, characteristic, complemented
7#7174C211yesmaximal, characteristic, complemented
8C2×C8×D4C2211yescharacteristic
9C22×#106—21no—
10D4×C4⋊C4—21no—
11C2×#365—22no—
12C2×#365—21no—
13#2150C2211yescharacteristic
14C4×M32C2211yescharacteristic
15#1948—21no—
16#1853—22no—
17#1848C2211yescharacteristic
18#1844C2211yescharacteristic
19#1807—21no—
20C4×#106C2211yescharacteristic
21#1564C2211yescharacteristic
22#1486—21no—
23#1383—21no—
24D4⋊C16C414yescomplemented
25#1301—21no—
26C23×C8—21no—
27C2×C4×D4—41no—
28C2×C4×D4D411yescharacteristic
29C22×C4⋊C4—22no—
30C2×M32—21no—
31C2×M32—21no—
32C8×D4—21no—
33C8×D4—21no—
34C8×D4C2×C412yes—
35C8×D4Q812yes—
36#431—21no—
37C2×#108—21no—
38C2×#108—21no—
39C2×#106—22no—
40C2×#106—21no—
41C2×#106—42no—
42C2×#106—42no—
43C2×#106D411yescharacteristic
44C2×#106D411yescharacteristic
45C2×#106D412yes—
46#423—21no—
47C2×C4⋊C8D411yescharacteristic
48C2×C22⋊C8—21no—
49C2×C4×C8C2311yescharacteristic
50#389—41no—
51#385—42no—
52#384—21no—
53#380—22no—
54C4×C4⋊C4—21no—
55#365—42no—
56#365—44no—
57C4⋊C16—21no—
58C4⋊C16C2×C412yescomplemented
59#346—21no—
60C4×C16C2×C412yescomplemented
61#340—22no—
62#340—21no—
63#335—21no—
64#334—21no—
65D4⋊C8—42no—
66C22×D4SD1611yescharacteristic
67C23×C4—21no—
68C23×C4—42no—
69C2×M16—41no—
70C22×C8—21no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8C2×D411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C4×D4—21no—
80C4×D4—21no—
81C4×D4—84no—
82C4×D4C4⋊C412yes—
83C4×D4SD1612yes—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—82no—
92C2×C4⋊C4SD1611yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—42no—
95C2×C42—41no—
96C2×C42C2×D411yescharacteristic
97M32—22no—
98M32—22no—
99M32—41no—
100M32—41no—
101C2×C16—22no—
102C2×C16—22no—
103#108—42no—
104#108—41no—
105#108—41no—
106#106—21no—
107#106—21no—
108#106—21no—
109#106—44no—
110#106—41no—
111#106—41no—
112#106—42no—
113#106SD1612yes—
114C4⋊C8—21no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8C4⋊C412yes—
119C22⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—41no—
122C4×C8—41no—
123C4×C8C22×C411yescharacteristic
124C4×C8C2×Q811yescharacteristic
125C4×C8#4111yescharacteristic
126C4×C8#4111yescharacteristic
127#95—42no—
128#95—41no—
129C24—21no—
130C2×D4—24no—
131C2×D4—21no—
132C2×D4#10612yes—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—84no—
144C22×C4—84no—
145C22×C4—162no—
146C22×C4#12011yescharacteristic
147C22×C4C2×SD1611yescharacteristic
148C22×C4AGL(1,ℤ/8)11yescharacteristic
149M16—81no—
150M16—81no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—22no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—81no—
161C2×C8—81no—
162C2×C8C4×D411yescharacteristic
163C2×C8C4×D411yescharacteristic
164C2×C8C22⋊Q811yescharacteristic
165C2×C8C22⋊Q811yescharacteristic
166C2×C8C2×SD1611yescharacteristic
167C2×C8#13511yescharacteristic
168C2×C8AGL(1,ℤ/8)11yescharacteristic
169C2×C8AGL(1,ℤ/8)11yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—82no—
178C4⋊C4—84no—
179C4⋊C4—161no—
180C4⋊C4#10612yes—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—84no—
184C42—82no—
185C42C2×C4⋊C411yescharacteristic
186C42#12311yescharacteristic
187C42C2×SD1611yescharacteristic
188C42#13711yescharacteristic
189C16—24no—
190C16—42no—
191C23—24no—
192C23—22no—
193C23—42no—
194C23—41no—
195C23—41no—
196C23#45011yescharacteristic
197D4—24no—
198D4—22no—
199D4#36514yescomplemented
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—81no—
218C2×C4—168no—
219C2×C4#38511yescharacteristic
220C2×C4C2×#10611yescharacteristic
221C2×C4#42811yescharacteristic
222C2×C4#44811yescharacteristic
223C2×C4#44911yescharacteristic
224C2×C4#45011yescharacteristic
225C2×C4#45311yescharacteristic
226C2×C4#45611yescharacteristic
227C2×C4#48111yescharacteristic
228C2×C4#48311yescharacteristic
229C8—21no—
230C8—21no—
231C8—42no—
232C8—161no—
233C8—161no—
234C8C4×SD1611yescharacteristic
235C8#44211yescharacteristic
236C8#47611yescharacteristic
237C8#47811yescharacteristic
238C22—21no—
239C22—21no—
240C22—24no—
241C22—44no—
242C22—44no—
243C22—42no—
244C22—42no—
245C22#162911yescharacteristic
246C22#165411yescharacteristic
247C22#185311yescharacteristic
248C4—21no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—82no—
253C4—161no—
254C4—162no—
255C4#162211yescharacteristic
256C4#165111yescharacteristic
257C4#185911yescharacteristic
258C4#186411yescharacteristic
259C4C2×#36512yes—
260C2—21no—
261C2—21no—
262C2—44no—
263C2#888611yescharacteristic
264C2#1292911yescharacteristic
265C2#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. #12903C2semidirect
6. #7278C2semidirect
7. #7174C2semidirect
24. D4⋊C16C4semidirect
58. C4⋊C16C2×C4semidirect
60. C4×C16C2×C4semidirect
199. D4#365semidirect

Maximal subgroups

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

subgroupindex
D4×#1062
#131852
#129032
#115832
#86582
#72782
#71742