Finite Group Databasebeta

← #138136

Subgroups — #138136

Order 512.

406 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 112 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#33212C211yesmaximal, characteristic
2#17787C211yesmaximal, characteristic
3#12785C211yesmaximal, characteristic
4#11940C212yesmaximal, complemented
5#7278C212yesmaximal, complemented
6C2×#500C2211yescharacteristic
7C2×#431—21no—
8C2×C4×M16C2211yescharacteristic
9#2857—21no—
10#2150C2212yes—
11C2×#349—21no—
12#1990—21no—
13#1944—21no—
14#1944C412yes—
15#1844C2211yescharacteristic
16#1809—21no—
17#1809C412yes—
18#1565—24no—
19#1564—21no—
20#1564C2212yes—
21#1377—21no—
22#1364—24no—
23C22×M16C2×C411yescharacteristic
24C22×M16D411yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C22⋊Q8—41no—
27C22×C42D411yescharacteristic
28C2×M32—22no—
29#500—22no—
30#500—22no—
31#431—22no—
32#431—22no—
33#431—41no—
34#431—41no—
35C2×#107—21no—
36C2×#107C2×C412yes—
37C2×#106—21no—
38C2×#106C2×C412yes—
39C2×C4⋊C8—22no—
40C2×C22⋊C8—41no—
41C4×M16—22no—
42C4×M16—22no—
43C2×#97C2×C411yescharacteristic
44C2×#97D411yescharacteristic
45C2×C4×C8—41no—
46C2×C4×C8D411yescharacteristic
47C2×C4×C8C2311yescharacteristic
48#391—41no—
49#389—22no—
50#389—41no—
51#384—21no—
52#380—22no—
53#380—41no—
54C4⋊C16—22no—
55C4⋊C16—22no—
56#349—42no—
57#340—22no—
58#340—22no—
59#335—22no—
60#335D414yes—
61#334—22no—
62#334—22no—
63C22×Q8—41no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4D811yescharacteristic
66C23×C4SD1611yescharacteristic
67C2×M16—22no—
68C2×M16—22no—
69C2×M16—22no—
70C2×M16—22no—
71C2×M16—41no—
72C2×M16SD1612yes—
73C2×M16Q1612yes—
74C22×C8—42no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22×C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C4⋊Q8—41no—
80C4⋊Q8—41no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C22⋊Q8—84no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C22⋊C4—42no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42D811yescharacteristic
97C2×C42SD1611yescharacteristic
98C2×C42C2×D411yescharacteristic
99M32—42no—
100M32—42no—
101C2×C16—42no—
102#107—24no—
103#107—42no—
104#106—42no—
105#106—42no—
106C4⋊C8—24no—
107C4⋊C8—42no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110#97—22no—
111#97—22no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—42no—
115C4×C8—42no—
116C4×C8C2×D412yes—
117C4×C8#4112yes—
118#95—42no—
119#95—42no—
120C24#10211yescharacteristic
121C2×Q8—82no—
122C2×Q8—82no—
123C2×Q8—81no—
124C2×Q8—81no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4#10112yes—
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4#10211yescharacteristic
141C22×C4#10312yes—
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4C2×D811yescharacteristic
144C22×C4C2×SD1611yescharacteristic
145M16—24no—
146M16—24no—
147M16—44no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—44no—
156C2×C8—84no—
157C2×C8#10312yes—
158C2×C8C4×D412yes—
159C2×C8#12112yes—
160C2×C8C2×SD1611yescharacteristic
161C2×C8C2×Q1611yescharacteristic
162C2×C8#13512yes—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—84no—
170C4⋊C4—81no—
171C4⋊C4—81no—
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#33212yes—
189C23C2×#10211yescharacteristic
190Q8—162no—
191Q8—162no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4#33212yes—
215C2×C4#38611yescharacteristic
216C2×C4C2×#10111yescharacteristic
217C2×C4#41311yescharacteristic
218C2×C4C2×#10311yescharacteristic
219C2×C4#41611yescharacteristic
220C2×C4#41711yescharacteristic
221C2×C4#44711yescharacteristic
222C2×C4#44811yescharacteristic
223C2×C4#48011yescharacteristic
224C2×C4#48111yescharacteristic
225C8—22no—
226C8—22no—
227C8—41no—
228C8—84no—
229C8#46512yes—
230C8#46712yes—
231C22—21no—
232C22—21no—
233C22—22no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22C2×#33211yescharacteristic
239C22#154011yescharacteristic
240C22#187411yescharacteristic
241C4—22no—
242C4—21no—
243C4—21no—
244C4—41no—
245C4—41no—
246C4—162no—
247C4—162no—
248C4#155011yescharacteristic
249C4#155211yescharacteristic
250C4#189211yescharacteristic
251C4#190111yescharacteristic
252C4#219012yes—
253C4#223912yes—
254C2—21no—
255C2—21no—
256C2—42no—
257C2#913111yescharacteristic
258C2#1299911yescharacteristic
259C2#1300911yescharacteristic
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×C4×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11940C2semidirect
5. #7278C2semidirect

Maximal subgroups

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

subgroupindex
#332122
#177872
#127852
#119402
#72782