Finite Group Databasebeta

← #145280

Subgroups — #145280

Order 512.

293 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 62 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#19280C211yesmaximal, characteristic
2#12019C211yesmaximal, characteristic, complemented
3#11982C211yesmaximal, characteristic, complemented
4#11678C211yesmaximal, characteristic, complemented
5C2×#1330C211yesmaximal, characteristic, complemented
6#8624C211yesmaximal, characteristic, complemented
7#7243C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9#3036—21no—
10C2×#436C2211yescharacteristic
11C2×C4⋊C16C2211yescharacteristic
12C2×#346C2211yescharacteristic
13#1701—22no—
14#1693—21no—
15#1687—21no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18#1566C2211yescharacteristic
19#1547—21no—
20#1466—21no—
21#1368—22no—
22#1365—21no—
23#1361—21no—
24#1333—21no—
25#1333—21no—
26#1330—21no—
27#1330—21no—
28#1290—21no—
29C2×AGL(1,ℤ/8)—41no—
30C22×M16—21no—
31#532—21no—
32C2×C4×D4D411yescharacteristic
33C22×C16—21no—
34C22×C16—21no—
35#500D411yescharacteristic
36#493D411yescharacteristic
37#486—21no—
38#466—21no—
39#466—21no—
40#463—42no—
41#461—22no—
42#459—22no—
43C4×D8—21no—
44C4×D8—21no—
45#436—21no—
46#436—21no—
47#436—21no—
48#436—21no—
49#432—21no—
50#432—21no—
51#423—21no—
52C2×C4⋊C8D411yescharacteristic
53#417—41no—
54C2×C22⋊C8—21no—
55C2×C4×C8C2311yescharacteristic
56C4⋊C16—21no—
57C4⋊C16—21no—
58#346—21no—
59#346D411yescharacteristic
60#346D411yescharacteristic
61#335—21no—
62#335—21no—
63#334—21no—
64#334—21no—
65#334—21no—
66#334D412yes—
67#332—22no—
68#331—22no—
69#323—42no—
70C2×#41—41no—
71C22×D4SD1611yescharacteristic
72C23×C4—21no—
73AGL(1,ℤ/8)—84no—
74C2×SD16—42no—
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—42no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—81no—
82C2×M16—81no—
83C22×C8—21no—
84C22×C8#4111yescharacteristic
85C22×C8#4111yescharacteristic
86C4⋊Q8—21no—
87#127—21no—
88#124—41no—
89#121—41no—
90#121—41no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—41no—
96C4×D4—41no—
97C2×C4⋊C4SD1611yescharacteristic
98C2×C22⋊C4—21no—
99C2×C42C2×D411yescharacteristic
100C2×C16—21no—
101C2×C16—21no—
102C2×C16—42no—
103C2×C16—41no—
104C2×C16—81no—
105#107—21no—
106#107—21no—
107C4⋊C8—21no—
108C4⋊C8—22no—
109C4⋊C8—21no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113#103—42no—
114#102—42no—
115#102—41no—
116#102—41no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121#97—41no—
122#97—41no—
123C4×C8C2×D411yescharacteristic
124C4×C8C2×D411yescharacteristic
125C4×C8C2×D411yescharacteristic
126C4×C8#4111yescharacteristic
127C24—21no—
128#41—82no—
129#41—82no—
130C2×Q8—41no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—42no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—81no—
138C22×C4—21no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—81no—
142C22×C4—81no—
143C22×C4—81no—
144C22×C4#12311yescharacteristic
145C22×C4C2×SD1611yescharacteristic
146C22×C4AGL(1,ℤ/8)11yescharacteristic
147SD16—84no—
148D8—41no—
149D8—41no—
150D8—81no—
151D8—84no—
152M16—82no—
153M16—82no—
154M16—81no—
155M16—81no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—42no—
161C2×C8—81no—
162C2×C8—81no—
163C2×C8—81no—
164C2×C8#12111yescharacteristic
165C2×C8#12111yescharacteristic
166C2×C8#12311yescharacteristic
167C2×C8#12411yescharacteristic
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C22⋊C4—41no—
173C22⋊C4—41no—
174C22⋊C4—81no—
175C22⋊C4—81no—
176C42#12011yescharacteristic
177C42#12111yescharacteristic
178C42#13511yescharacteristic
179C42AGL(1,ℤ/8)11yescharacteristic
180C16—22no—
181C16—41no—
182C16—81no—
183C23—41no—
184C23—42no—
185C23—81no—
186C23—81no—
187C23—81no—
188C23#48111yescharacteristic
189Q8—81no—
190Q8—81no—
191D4—42no—
192D4—81no—
193D4—81no—
194D4—81no—
195D4—81no—
196D4—81no—
197D4—161no—
198D4—162no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—82no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—162no—
210C2×C4#39411yescharacteristic
211C2×C4#44911yescharacteristic
212C2×C4#45011yescharacteristic
213C2×C4#46511yescharacteristic
214C2×C4#46911yescharacteristic
215C2×C4#48211yescharacteristic
216C8—21no—
217C8—21no—
218C8—21no—
219C8—82no—
220C8—81no—
221C8—161no—
222C8—161no—
223C8#47111yescharacteristic
224C8#47211yescharacteristic
225C22—21no—
226C22—21no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22—162no—
231C22—161no—
232C22—161no—
233C22—161no—
234C22#203411yescharacteristic
235C22#223812yes—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—21no—
241C4—81no—
242C4—161no—
243C4#205011yescharacteristic
244C4#205111yescharacteristic
245C2—22no—
246C2—161no—
247C2—321no—
248C2#1045711yescharacteristic
249C2#1339811yescharacteristic
250C2#1340111yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#19280C2
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#436 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12019C2semidirect
3. #11982C2semidirect
4. #11678C2semidirect
5. C2×#1330C2semidirect
6. #8624C2semidirect
7. #7243C2semidirect

Maximal subgroups

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

subgroupindex
#192802
#120192
#119822
#116782
C2×#13302
#86242
#72432