Finite Group Databasebeta

← #138508

Subgroups — #138508

Order 512.

446 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 84 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#29806C211yesmaximal, characteristic
2#17818C211yesmaximal, characteristic, complemented
3#11964C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic, complemented
5#7222C212yesmaximal, complemented
6C2×#493C2211yescharacteristic
7C2×#432—21no—
8C2×#431C2211yescharacteristic
9#2856—21no—
10#2163C2212yes—
11#2054—21no—
12#2054C412yes—
13#1999—21no—
14#1999C412yes—
15#1957—21no—
16#1834C2211yescharacteristic
17#1573—24no—
18#1568—21no—
19#1568C2212yes—
20#1361—24no—
21#1322—21no—
22#1314—21no—
23C22×D8—21no—
24C22×D8C2×C412yes—
25C2×#127—21no—
26C2×#121—22no—
27C2×#121—41no—
28C2×#121—41no—
29C22×C42D411yescharacteristic
30#493—22no—
31#493—22no—
32#432—22no—
33#432—22no—
34#432—41no—
35#432—41no—
36#431—22no—
37#431—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41C2×#102—22no—
42C2×#102—22no—
43C2×C22⋊C8—41no—
44C2×C22⋊C8C2×C411yescharacteristic
45C2×C22⋊C8D411yescharacteristic
46C2×#97—41no—
47C2×C4×C8D411yescharacteristic
48C2×C4×C8C2311yescharacteristic
49#390—22no—
50#390—41no—
51#390—41no—
52#384—21no—
53C4⋊C16—22no—
54C22⋊C16—22no—
55#346—22no—
56#340—21no—
57#340C2×C412yes—
58#335—22no—
59#335D414yescomplemented
60#334—22no—
61#334—22no—
62C22×D4—22no—
63C22×D4—42no—
64C22×D4—41no—
65C22×D4—41no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4D811yescharacteristic
68C23×C4SD1611yescharacteristic
69C2×D8—24no—
70C2×D8—42no—
71C2×D8—82no—
72C2×D8—82no—
73C22×C8—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22×C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78#127—41no—
79#127—41no—
80#127—41no—
81#127—41no—
82#121—82no—
83#121—82no—
84#121—84no—
85#121—84no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C22⋊C411yescharacteristic
97C2×C42D811yescharacteristic
98C2×C42SD1611yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C16—42no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—24no—
104C4⋊C8—42no—
105#102—82no—
106#102—82no—
107C22⋊C8—22no—
108C22⋊C8—22no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111#97—42no—
112#97—42no—
113C4×C8—21no—
114C4×C8—21no—
115C4×C8C2×D412yes—
116C4×C8#4112yes—
117C24—42no—
118C24#10211yescharacteristic
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4—82no—
128C2×D4—82no—
129C2×D4—84no—
130C2×D4—84no—
131C2×D4—84no—
132C2×D4—81no—
133C2×D4—81no—
134C2×D4—81no—
135C2×D4—81no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4#10211yescharacteristic
149C22×C4#10211yescharacteristic
150C22×C4#10211yescharacteristic
151C22×C4C2×C22⋊C411yescharacteristic
152C22×C4C2×D811yescharacteristic
153C22×C4C2×SD1611yescharacteristic
154D8—84no—
155D8—84no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—44no—
163C2×C8—84no—
164C2×C8C4×D412yes—
165C2×C8#12112yes—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C22⋊C4—84no—
171C22⋊C4—84no—
172C22⋊C4—84no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42#12011yescharacteristic
180C42#12311yescharacteristic
181C42AGL(1,ℤ/8)11yescharacteristic
182C42#13711yescharacteristic
183C16—82no—
184C23—21no—
185C23—21no—
186C23—41no—
187C23—41no—
188C23—82no—
189C23—82no—
190C23—82no—
191C23—82no—
192C23—82no—
193C23—82no—
194C23—82no—
195C23#36112yes—
196C23C2×#10211yescharacteristic
197D4—82no—
198D4—82no—
199D4—82no—
200D4—82no—
201D4—162no—
202D4—162no—
203D4—162no—
204D4—168no—
205D4—162no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4#38611yescharacteristic
229C2×C4#41711yescharacteristic
230C2×C4#44711yescharacteristic
231C2×C4#44811yescharacteristic
232C2×C4#48011yescharacteristic
233C2×C4#48111yescharacteristic
234C8—22no—
235C8—22no—
236C8—41no—
237C8—41no—
238C8—84no—
239C22—21no—
240C22—21no—
241C22—22no—
242C22—42no—
243C22—41no—
244C22—41no—
245C22—81no—
246C22—162no—
247C22—162no—
248C22—162no—
249C22—162no—
250C22—162no—
251C22—162no—
252C22—162no—
253C22#187411yescharacteristic
254C22C2×#36111yescharacteristic
255C22#214811yescharacteristic
256C4—21no—
257C4—21no—
258C4—21no—
259C4—21no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—162no—
264C4#191011yescharacteristic
265C4#191411yescharacteristic
266C4#219012yes—
267C2—21no—
268C2—21no—
269C2—42no—
270C2—322no—
271C2#919711yescharacteristic
272C2#1299911yescharacteristic
273C2#1300411yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#29806C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17818C2semidirect
3. #11964C2semidirect
4. #11460C2semidirect
5. #7222C2semidirect
59. #335D4semidirect

Maximal subgroups

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

subgroupindex
#298062
#178182
#119642
#114602
#72222