Finite Group Databasebeta

← #145281

Subgroups — #145281

Order 512.

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

normal subgroups only

This group has 266 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#19281C211yesmaximal, characteristic
2#12019C211yesmaximal, characteristic, complemented
3#11984C211yesmaximal, characteristic, complemented
4#11679C211yesmaximal, characteristic, complemented
5C2×#1330C211yesmaximal, characteristic
6#8624C211yesmaximal, characteristic
7#7243C211yesmaximal, characteristic, complemented
8#3170C2211yescharacteristic
9#3077—21no—
10C2×#436C2211yescharacteristic
11C2×C4⋊C16C2211yescharacteristic
12C2×#346C2211yescharacteristic
13#1703—22no—
14#1695—21no—
15#1689—21no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18#1566C2211yescharacteristic
19#1550—21no—
20#1466—21no—
21#1368—22no—
22#1364—21no—
23#1362—21no—
24#1334—21no—
25#1334—21no—
26#1330—21no—
27#1330—21no—
28#1290—21no—
29C22×M16—21no—
30#533—21no—
31C2×C4×D4D411yescharacteristic
32C22×C16—21no—
33C22×C16—21no—
34#500D411yescharacteristic
35#494D411yescharacteristic
36#487—21no—
37#478—42no—
38#476—22no—
39#474—22no—
40#467—21no—
41#467—21no—
42C4×Q16—21no—
43C4×Q16—21no—
44#436—21no—
45#436—21no—
46#436—21no—
47#436—21no—
48#432—21no—
49#432—21no—
50#428—41no—
51#423—21no—
52C2×C4⋊C8D411yescharacteristic
53#418—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#333—22no—
68#332—22no—
69#323—42no—
70C22×D4SD1611yescharacteristic
71C23×C4—21no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—81no—
78C2×M16—81no—
79C22×C8—21no—
80C22×C8#4111yescharacteristic
81C22×C8#4111yescharacteristic
82C4⋊Q8—21no—
83C4⋊Q8—21no—
84#125—41no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4—21no—
93#117—41no—
94C2×C4⋊C4SD1611yescharacteristic
95C2×C22⋊C4—21no—
96C2×C42C2×D411yescharacteristic
97C2×C16—21no—
98C2×C16—21no—
99C2×C16—42no—
100C2×C16—41no—
101C2×C16—81no—
102#107—21no—
103#107—21no—
104#107—42no—
105#106—42no—
106C4⋊C8—21no—
107C4⋊C8—22no—
108C4⋊C8—21no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112#103—41no—
113#103—41no—
114#102—42no—
115#102—42no—
116C22⋊C8—41no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119C22⋊C8—41no—
120#97—41no—
121#97—41no—
122C4×C8C2×D411yescharacteristic
123C4×C8C2×D411yescharacteristic
124C4×C8C2×D411yescharacteristic
125C4×C8#4111yescharacteristic
126C24—21no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—42no—
132C22×C4—21no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—81no—
136C22×C4—81no—
137C22×C4#12311yescharacteristic
138C22×C4C2×SD1611yescharacteristic
139C22×C4AGL(1,ℤ/8)11yescharacteristic
140Q16—41no—
141Q16—41no—
142Q16—81no—
143M16—82no—
144M16—82no—
145M16—81no—
146M16—81no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—42no—
152C2×C8—81no—
153C2×C8—81no—
154C2×C8—81no—
155C2×C8#12111yescharacteristic
156C2×C8#12111yescharacteristic
157C2×C8#12311yescharacteristic
158C2×C8#12411yescharacteristic
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C22⋊C4—81no—
170C42—81no—
171C42#12011yescharacteristic
172C42#12111yescharacteristic
173C42#13511yescharacteristic
174C42AGL(1,ℤ/8)11yescharacteristic
175C16—22no—
176C16—41no—
177C16—81no—
178C23—41no—
179C23—42no—
180C23—81no—
181C23—81no—
182C23#48111yescharacteristic
183Q8—81no—
184Q8—81no—
185Q8—161no—
186D4—42no—
187D4—81no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—82no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4#39411yescharacteristic
200C2×C4#44911yescharacteristic
201C2×C4#45011yescharacteristic
202C2×C4#46511yescharacteristic
203C2×C4#46911yescharacteristic
204C2×C4#48211yescharacteristic
205C8—21no—
206C8—21no—
207C8—21no—
208C8—82no—
209C8—81no—
210C8—161no—
211C8—161no—
212C8#47111yescharacteristic
213C8#47211yescharacteristic
214C22—21no—
215C22—21no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22—162no—
220C22#203411yescharacteristic
221C22#223912yes—
222C4—21no—
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—81no—
228C4—161no—
229C4—161no—
230C4#205011yescharacteristic
231C4#205111yescharacteristic
232C2—22no—
233C2—161no—
234C2#1045711yescharacteristic
235C2#1339911yescharacteristic
236C2#1340111yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
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. #11984C2semidirect
4. #11679C2semidirect
7. #7243C2semidirect

Maximal subgroups

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

subgroupindex
#192812
#120192
#119842
#116792
C2×#13302
#86242
#72432