Finite Group Databasebeta

← #140886

Subgroups — #140886

Order 512.

483 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 106 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#33176C211yesmaximal, characteristic
2C4×#349C211yesmaximal, characteristic
3C2×#1340C214yesmaximal
4#10837C211yesmaximal, characteristic
5C2×#494C411yescharacteristic, complemented
6C2×#470—21no—
7C2×#470C2212yes—
8C2×C4×M16C2211yescharacteristic
9#2857—21no—
10#2285—21no—
11C2×#349C2212yes—
12C4×M32—21no—
13#1971—21no—
14#1971C412yescomplemented
15#1956—21no—
16#1844C411yescharacteristic, complemented
17C2×H(3,ℤ/4)—21no—
18C2×H(3,ℤ/4)C2212yes—
19#1377—21no—
20#1340—28no—
21C22×Q16—21no—
22C22×Q16C2×C412yes—
23C22×M16D411yescharacteristic
24C22×M16C2311yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C22⋊Q8—22no—
27C2×#117—22no—
28C22×C42D411yescharacteristic
29C2×M32—22no—
30#494—41no—
31#494—41no—
32#470—24no—
33#470—24no—
34#470—42no—
35#470—42no—
36C2×#106—21no—
37C2×#106C2×C412yes—
38C2×#103—22no—
39C2×#103—22no—
40C4×M16—41no—
41C4×M16—41no—
42C2×#97C2×C411yescharacteristic
43C2×#97D411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#391—22no—
47#389—42no—
48#384—21no—
49#384—41no—
50#382—41no—
51#380—42no—
52#380—42no—
53C4×C4⋊C4—24no—
54#349—22no—
55#349D414yes—
56#346—41no—
57C4×C16—41no—
58H(3,ℤ/4)—24no—
59H(3,ℤ/4)—42no—
60C22×Q8—22no—
61C23×C4D812yes—
62C23×C4C2×D411yescharacteristic
63C2×Q16—24no—
64C2×Q16—42no—
65C2×Q16—42no—
66C2×Q16—42no—
67C2×Q16—42no—
68C2×Q16—42no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16D812yes—
74C2×M16SD1612yes—
75C2×M16C2×D412yes—
76C2×M16#4112yes—
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22×C411yescharacteristic
80C22×C8C2×D411yescharacteristic
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—42no—
84C22⋊Q8—42no—
85C22⋊Q8—42no—
86C22⋊Q8—42no—
87C22⋊Q8—42no—
88#117—42no—
89#117—42no—
90#117—42no—
91#117—42no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—22no—
103C2×C42—24no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C22⋊C411yescharacteristic
106C2×C42SD1612yes—
107M32—24no—
108M32—42no—
109C2×C16—42no—
110#106—24no—
111#106—42no—
112#103—44no—
113#103—44no—
114#97—41no—
115#97—41no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—21no—
120#95—44no—
121#95—44no—
122C24C2×D811yescharacteristic
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—42no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4#10212yes—
140C22×C4#10212yes—
141C22×C4C2×C22⋊C411yescharacteristic
142C22×C4#12411yescharacteristic
143C22×C4#12711yescharacteristic
144C22×C4C2×D811yescharacteristic
145C22×C4C2×SD1611yescharacteristic
146C22×C4C2×SD1611yescharacteristic
147Q16—84no—
148Q16—84no—
149M16—42no—
150M16—42no—
151M16—42no—
152M16—42no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8#10212yes—
162C2×C8#10212yes—
163C2×C8C4×D412yes—
164C2×C8#12112yes—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—84no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C4⋊C4—88no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C22⋊C4—82no—
176C22⋊C4—82no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C42—22no—
180C42—22no—
181C42—21no—
182C42—21no—
183C42—44no—
184C42—44no—
185C42#9911yescharacteristic
186C42#9911yescharacteristic
187C42#10412yes—
188C16—44no—
189C23—21no—
190C23—21no—
191C23—42no—
192C23C2×#10211yescharacteristic
193C23#48611yescharacteristic
194C23#49311yescharacteristic
195Q8—82no—
196Q8—82no—
197Q8—82no—
198Q8—82no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—82no—
214C2×C4—84no—
215C2×C4—84no—
216C2×C4—88no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4#32712yes—
222C2×C4#32712yes—
223C2×C4#39011yescharacteristic
224C2×C4C2×#9911yescharacteristic
225C2×C4#41011yescharacteristic
226C2×C4C2×#10211yescharacteristic
227C2×C4C2×#10411yescharacteristic
228C2×C4#42111yescharacteristic
229C2×C4#48611yescharacteristic
230C2×C4#49211yescharacteristic
231C8—22no—
232C8—22no—
233C8—22no—
234C8—22no—
235C22—22no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—42no—
241C22—81no—
242C22C2×#32711yescharacteristic
243C22#151711yescharacteristic
244C22#195711yescharacteristic
245C4—22no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—162no—
250C4—164no—
251C4—162no—
252C4#134014yescomplemented
253C4#150211yescharacteristic
254C4#151911yescharacteristic
255C4#196411yescharacteristic
256C4#198011yescharacteristic
257C2—21no—
258C2—21no—
259C2—41no—
260C2—41no—
261C2#968011yescharacteristic
262C2C2×#134012yes—
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#494C4semidirect
14. #1971C4semidirect
16. #1844C4semidirect
252. C4#1340semidirect

Maximal subgroups

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

subgroupindex
#331762
C4×#3492
C2×#13402
#108372