Finite Group Databasebeta

← #129929

Subgroups — #129929

Order 512.

660 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C2×#3170C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3C2×#1350C214yesmaximal
4#10822C211yesmaximal, characteristic
5C2×#533—21no—
6#3170—28no—
7C2×#500C411yescharacteristic
8C2×#494C411yescharacteristic
9C2×#487—21no—
10C2×#467—21no—
11C2×#467C2212yes—
12C2×C4×Q16—21no—
13C2×C4×Q16C412yes—
14C22×C4×C8C2211yescharacteristic
15C2×C22⋊C16C2212yes—
16C2×#341—21no—
17C2×#341C2212yes—
18#1728—21no—
19#1374—21no—
20#1374—21no—
21#1350—28no—
22C22×Q16—21no—
23C22×Q16C2×C412yes—
24C22×M16—22no—
25C23×C8D411yescharacteristic
26C23×C8C2311yescharacteristic
27#533—28no—
28#533—44no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8—21no—
31C2×#125—41no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—41no—
34C2×C4×Q8—22no—
35C2×C4×Q8—41no—
36C2×#117—41no—
37C22×C42D411yescharacteristic
38C22×C16—22no—
39#500—24no—
40#494—24no—
41#487—28no—
42#467—28no—
43#467—24no—
44#467D418yes—
45C4×Q16—28no—
46C4×Q16—24no—
47C4×Q16—24no—
48C2×#107—21no—
49C2×#107C2×C412yes—
50C2×#103—22no—
51C2×#103—22no—
52C2×C4×C8—24no—
53C2×C4×C8C2×C411yescharacteristic
54C2×C4×C8C2×C411yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8D411yescharacteristic
57C22⋊C16—24no—
58#341—24no—
59#341—24no—
60#336—41no—
61#336—41no—
62C22×Q8—22no—
63C22×Q8—41no—
64C23×C4SD1611yescharacteristic
65C23×C4SD1611yescharacteristic
66C23×C4C2×D411yescharacteristic
67C2×Q16—24no—
68C2×Q16—22no—
69C2×Q16—42no—
70C2×Q16—42no—
71C2×Q16—42no—
72C2×Q16—42no—
73C2×Q16C22⋊C414yes—
74C2×M16—24no—
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—42no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—42no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22×C411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D414yes—
91C4⋊Q8—24no—
92C4⋊Q8—24no—
93C4⋊Q8—42no—
94C4⋊Q8—42no—
95#125—48no—
96C22⋊Q8—28no—
97C22⋊Q8—48no—
98C22⋊Q8—44no—
99C4×Q8—28no—
100C4×Q8—44no—
101C4×Q8—48no—
102#117—48no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—42no—
113C2×C42C22⋊C411yescharacteristic
114C2×C42C22⋊C411yescharacteristic
115C2×C42D811yescharacteristic
116C2×C42D811yescharacteristic
117C2×C16—42no—
118C2×C16—42no—
119C2×C16—42no—
120#107—24no—
121#107—22no—
122#107C22⋊C414yes—
123#103—28no—
124#103—28no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—22no—
129C24C2×SD1611yescharacteristic
130C2×Q8—24no—
131C2×Q8—44no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—82no—
138C2×Q8—82no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—41no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4#10211yescharacteristic
152C22×C4#10211yescharacteristic
153C22×C4#10211yescharacteristic
154C22×C4#10211yescharacteristic
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4#12012yes—
157C22×C4C2×D811yescharacteristic
158C22×C4AGL(1,ℤ/8)12yes—
159Q16—44no—
160Q16—44no—
161Q16—48no—
162M16—44no—
163M16—44no—
164M16—82no—
165M16—82no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—44no—
176C2×C8—44no—
177C2×C8—44no—
178C2×C8—82no—
179C2×C8—82no—
180C2×C8#10412yes—
181C2×C8C2×C22⋊C412yes—
182C2×C8#12012yes—
183C4⋊C4—24no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—48no—
187C4⋊C4—44no—
188C4⋊C4—42no—
189C4⋊C4—48no—
190C22⋊C4—48no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—22no—
196C42—22no—
197C42—48no—
198C16—82no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23C2×#10211yescharacteristic
206C23#45012yes—
207Q8—44no—
208Q8—44no—
209Q8—88no—
210Q8—82no—
211Q8—82no—
212C2×C4—28no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4#35912yes—
236C2×C4#36212yes—
237C2×C4#36212yes—
238C2×C4#37911yescharacteristic
239C2×C4#41011yescharacteristic
240C2×C4#41011yescharacteristic
241C2×C4#41811yescharacteristic
242C2×C4C2×#10411yescharacteristic
243C2×C4#42111yescharacteristic
244C2×C4#44712yes—
245C8—22no—
246C8—22no—
247C8—22no—
248C8—41no—
249C8—84no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22#150211yescharacteristic
258C22#151711yescharacteristic
259C22#178611yescharacteristic
260C22C2×#35911yescharacteristic
261C22#214311yescharacteristic
262C22C2×#36211yescharacteristic
263C22C2×#36211yescharacteristic
264C4—22no—
265C4—22no—
266C4—22no—
267C4—22no—
268C4—41no—
269C4—84no—
270C4—84no—
271C4#150511yescharacteristic
272C4#151411yescharacteristic
273C4#178211yescharacteristic
274C4#178911yescharacteristic
275C2—42no—
276C2#833211yescharacteristic
277C2#1174811yescharacteristic
278C2#1175211yescharacteristic
279C2#1186311yescharacteristic
280C2#1186511yescharacteristic
281C2#1273411yescharacteristic
282C2#1273711yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#31702
#128442
C2×#13502
#108222