Finite Group Databasebeta

← #145405

Subgroups — #145405

Order 512.

317 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 76 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#19203C211yesmaximal, characteristic
2#13267C211yesmaximal, characteristic, complemented
3#13206C211yesmaximal, characteristic
4#11999C211yesmaximal, characteristic, complemented
5#11931C211yesmaximal, characteristic
6#8677C211yesmaximal, characteristic, complemented
7#7279C211yesmaximal, characteristic, complemented
8#3208—21no—
9C2×#493C2211yescharacteristic
10#2960C2211yescharacteristic
11#2150C2211yescharacteristic
12C2×#357—21no—
13C2×#357—21no—
14C2×#346C2211yescharacteristic
15#1846C2211yescharacteristic
16#1688—22no—
17#1668—22no—
18#1564C2211yescharacteristic
19#1564C2211yescharacteristic
20#1545—21no—
21#1518—21no—
22#1379—21no—
23#1379—21no—
24#1368—22no—
25#1361—22no—
26#1286—21no—
27C22×D8—21no—
28C22×D8—21no—
29#533D411yescharacteristic
30C2×#127—21no—
31C2×M32—21no—
32C22×C16D411yescharacteristic
33C22×C16D411yescharacteristic
34#493—21no—
35#493—21no—
36#493—21no—
37#493D412yes—
38#492—41no—
39#486—41no—
40#460—24no—
41#450—42no—
42#446—21no—
43#446—21no—
44C8×Q8—21no—
45C8×Q8—21no—
46#431—21no—
47#431—21no—
48C2×#107—21no—
49C2×#106—21no—
50#424—21no—
51#423D411yescharacteristic
52#405—21no—
53C2×C4×C8C2311yescharacteristic
54#382—21no—
55C4⋊C16—21no—
56C4⋊C16—21no—
57#357—42no—
58#357—42no—
59#346—21no—
60#346—21no—
61#335—21no—
62#335—21no—
63#334—21no—
64#334D412yes—
65#331—24no—
66#328—42no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4—41no—
70C2×SD16—44no—
71C2×D8—22no—
72C2×D8—22no—
73C2×D8—41no—
74C2×D8—41no—
75C2×D8—41no—
76C2×D8—41no—
77C2×D8—41no—
78C2×D8—41no—
79C2×D8—81no—
80C2×D8—81no—
81C2×M16—21no—
82C22×C8C2×D411yescharacteristic
83C22×C8#4111yescharacteristic
84C4⋊Q8SD1611yescharacteristic
85C4⋊Q8SD1611yescharacteristic
86#127—22no—
87#127—41no—
88#127—41no—
89#127—41no—
90#125—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95#117—21no—
96C2×C4⋊C4—41no—
97C2×C42C2×D411yescharacteristic
98M32—41no—
99M32—41no—
100C2×C16—21no—
101C2×C16—21no—
102C2×C16—21no—
103C2×C16—21no—
104C2×C16—21no—
105C2×C16—41no—
106C2×C16SD1612yescomplemented
107#107—41no—
108#107—41no—
109#106—81no—
110C4⋊C8—21no—
111C4⋊C8—22no—
112C4⋊C8—21no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116#102—44no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119#97—41no—
120C4×C8—41no—
121C4×C8—41no—
122C4×C8—41no—
123C4×C8C2×D411yescharacteristic
124C4×C8C2×D411yescharacteristic
125C4×C8#4111yescharacteristic
126C4×C8#4111yescharacteristic
127#95—41no—
128#95—41no—
129C24—81no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4—81no—
136C2×D4—81no—
137C2×D4—81no—
138C2×D4—81no—
139C2×D4—81no—
140C2×D4—81no—
141C2×D4—81no—
142C2×D4—162no—
143C2×D4—162no—
144C22×C4—81no—
145C22×C4#12111yescharacteristic
146C22×C4AGL(1,ℤ/8)11yescharacteristic
147C22×C4AGL(1,ℤ/8)11yescharacteristic
148SD16—164no—
149D8—42no—
150D8—42no—
151D8—82no—
152D8—82no—
153D8—81no—
154D8—81no—
155M16—41no—
156M16—41no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—41no—
160C2×C8—42no—
161C2×C8—81no—
162C2×C8#12111yescharacteristic
163C2×C8#12111yescharacteristic
164C2×C8#12311yescharacteristic
165C2×C8#12411yescharacteristic
166C2×C8C2×SD1611yescharacteristic
167C2×C8#13511yescharacteristic
168C2×C8AGL(1,ℤ/8)11yescharacteristic
169C2×C8#13711yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C22⋊C4—41no—
179C42—41no—
180C42#12011yescharacteristic
181C42#12311yescharacteristic
182C42C2×SD1611yescharacteristic
183C42#13511yescharacteristic
184C16—22no—
185C16—41no—
186C16—81no—
187C23—82no—
188C23—162no—
189C23—162no—
190C23—161no—
191C23—161no—
192C23#46911yescharacteristic
193Q8—42no—
194Q8—81no—
195D4—84no—
196D4—82no—
197D4—82no—
198D4—162no—
199D4—162no—
200D4—162no—
201D4—161no—
202D4—161no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—161no—
208C2×C4—161no—
209C2×C4—161no—
210C2×C4#39411yescharacteristic
211C2×C4#44911yescharacteristic
212C2×C4#45011yescharacteristic
213C2×C4#46011yescharacteristic
214C2×C4#46411yescharacteristic
215C2×C4#46511yescharacteristic
216C2×C4#47111yescharacteristic
217C2×C4#47211yescharacteristic
218C2×C4#48111yescharacteristic
219C2×C4#48211yescharacteristic
220C8—21no—
221C8—21no—
222C8—82no—
223C8—81no—
224C8—161no—
225C8#46111yescharacteristic
226C8#46311yescharacteristic
227C8#46511yescharacteristic
228C8#46811yescharacteristic
229C22—21no—
230C22—21no—
231C22—162no—
232C22—162no—
233C22—162no—
234C22—322no—
235C22—322no—
236C22#168811yescharacteristic
237C22#170211yescharacteristic
238C22#203411yescharacteristic
239C4—21no—
240C4—21no—
241C4—21no—
242C4—21no—
243C4—81no—
244C4—81no—
245C4—321no—
246C4#166611yescharacteristic
247C4#167811yescharacteristic
248C4#205111yescharacteristic
249C4#205511yescharacteristic
250C2—21no—
251C2—21no—
252C2—322no—
253C2#1047511yescharacteristic
254C2#1339611yescharacteristic
255C2#1340411yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC2×#493C22
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13267C2semidirect
4. #11999C2semidirect
6. #8677C2semidirect
7. #7279C2semidirect
106. C2×C16SD16semidirect

Maximal subgroups

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

subgroupindex
#192032
#132672
#132062
#119992
#119312
#86772
#72792