Finite Group Databasebeta

← #138491

Subgroups — #138491

Order 512.

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

normal subgroups only

This group has 352 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#29879C211yesmaximal, characteristic
2#17813C211yesmaximal, characteristic
3#11969C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic
5#7221C212yesmaximal, complemented
6C2×#499C2211yescharacteristic
7C2×#431C2211yescharacteristic
8C2×#423—21no—
9#2851—21no—
10#2163C2212yes—
11#2088—21no—
12#2087C411yescharacteristic
13#2086C411yescharacteristic
14#2077—21no—
15#2076C411yescharacteristic
16#2075C411yescharacteristic
17#1991—21no—
18#1835C2211yescharacteristic
19#1570—24no—
20#1566—21no—
21#1566C2212yes—
22#1363—24no—
23#1322—21no—
24#1314—21no—
25C22×M16—41no—
26C2×#125—21no—
27C22×C42D411yescharacteristic
28#499—22no—
29#499—22no—
30#431—22no—
31#431—22no—
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×#106—21no—
35#423—22no—
36#423—22no—
37#423—41no—
38#423—41no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—41no—
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8D411yescharacteristic
43C2×C22⋊C8C2×C411yescharacteristic
44C2×C22⋊C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8C2311yescharacteristic
47#389—21no—
48#389—21no—
49#387—41no—
50#387—41no—
51#385—41no—
52#385—41no—
53#382—21no—
54#380—21no—
55#380—21no—
56C4⋊C16—22no—
57C22⋊C16—22no—
58#346—22no—
59#340—21no—
60#340—21no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340C2×C411yescharacteristic
65#340C2×C411yescharacteristic
66#334—24no—
67#334—22no—
68#334D414yes—
69C23×C4C22⋊C411yescharacteristic
70C23×C4D811yescharacteristic
71C23×C4SD1611yescharacteristic
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—82no—
75C2×M16—82no—
76C22×C8—42no—
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22×C411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#125—41no—
82#125—41no—
83#125—41no—
84#125—41no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42C22⋊C411yescharacteristic
100C2×C42D811yescharacteristic
101C2×C42SD1611yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C16—42no—
104#107—22no—
105#107—22no—
106#106—42no—
107C4⋊C8—22no—
108C4⋊C8—22no—
109C4⋊C8—24no—
110C4⋊C8—42no—
111C4⋊C8—42no—
112C4⋊C8—42no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8C2×D412yes—
118C4×C8#4112yes—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125C24#10211yescharacteristic
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—81no—
141C22×C4#10211yescharacteristic
142C22×C4#10211yescharacteristic
143C22×C4#10211yescharacteristic
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4C2×D811yescharacteristic
146C22×C4C2×SD1611yescharacteristic
147M16—84no—
148M16—82no—
149M16—82no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—44no—
157C2×C8—84no—
158C2×C8C4×D412yes—
159C2×C8#12112yes—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C22⋊C4—82no—
175C22⋊C4—82no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42#12011yescharacteristic
183C42#12311yescharacteristic
184C42AGL(1,ℤ/8)11yescharacteristic
185C42#13711yescharacteristic
186C16—82no—
187C23—21no—
188C23—21no—
189C23—41no—
190C23—41no—
191C23#36012yes—
192C23C2×#10211yescharacteristic
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4#36012yes—
225C2×C4#38611yescharacteristic
226C2×C4#41711yescharacteristic
227C2×C4#44711yescharacteristic
228C2×C4#44811yescharacteristic
229C2×C4#48011yescharacteristic
230C2×C4#48111yescharacteristic
231C8—22no—
232C8—22no—
233C8—41no—
234C8—41no—
235C8—84no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—42no—
240C22—41no—
241C22—41no—
242C22—81no—
243C22#187411yescharacteristic
244C22C2×#36011yescharacteristic
245C22C2×#36011yescharacteristic
246C4—21no—
247C4—21no—
248C4—21no—
249C4—22no—
250C4—21no—
251C4—21no—
252C4—41no—
253C4—41no—
254C4—161no—
255C4—161no—
256C4—161no—
257C4—161no—
258C4#190911yescharacteristic
259C4#191511yescharacteristic
260C2—21no—
261C2—21no—
262C2—42no—
263C2#919511yescharacteristic
264C2#1299811yescharacteristic
265C2#1300111yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
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
3. #11969C2semidirect
5. #7221C2semidirect

Maximal subgroups

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

subgroupindex
#298792
#178132
#119692
#114602
#72212