Finite Group Databasebeta

← #492583

Subgroups — #492583

Order 512.

446 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 126 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#33947C211yesmaximal, characteristic, complemented
2#33849C211yesmaximal, characteristic
3#19980C212yesmaximal, complemented
4#12504C211yesmaximal, characteristic, complemented
5#12452C211yesmaximal, characteristic, complemented
6#12440C211yesmaximal, characteristic, complemented
7#12420C212yesmaximal, complemented
8#12375C212yesmaximal, complemented
9#12359C212yesmaximal, complemented
10C16×SD16C211yesmaximal, characteristic, complemented
11C8×D16C211yesmaximal, characteristic, complemented
12C2×#508—21no—
13#3380C2212yes—
14#3335—22no—
15#3283C2212yes—
16#2253—21no—
17#2252C2211yescharacteristic
18#2247C2211yescharacteristic
19#2241C2212yes—
20#2228—22no—
21#2226C2212yes—
22#2216—21no—
23#2208C2212yes—
24#2189—22no—
25Q8×C16C2211yescharacteristic
26C4×D16—21no—
27C4×D16C2211yescharacteristic
28D4×C16C2211yescharacteristic
29#1707C2212yes—
30#1675C2212yes—
31#1671—22no—
32#1670C2212yes—
33#1643—22no—
34#1635—22no—
35C8×SD16C2211yescharacteristic
36C8×D8C2212yes—
37#1371C2211yescharacteristic
38#1367C2211yescharacteristic
39#1365C2212yes—
40#1362C2212yes—
41Q8⋊C16C2211yescharacteristic
42#1331C2212yes—
43D4⋊C16C2211yescharacteristic
44C8×C16C2211yescharacteristic
45C2×#135—22no—
46#554—22no—
47#547—22no—
48#508—44no—
49C2×Q32—21no—
50C2×Q32—21no—
51C2×SD32—22no—
52C2×D16D411yescharacteristic
53C22×C16—21no—
54#500C2312yes—
55#499—22no—
56#498—22no—
57#494C2312yes—
58#487—22no—
59#483—42no—
60#481—42no—
61#478—42no—
62#476—22no—
63#476—22no—
64#475—22no—
65#467—22no—
66#465—42no—
67#461—22no—
68#452—42no—
69C8×Q8C2311yescharacteristic
70C4×SD16D411yescharacteristic
71C4×D8—22no—
72C4×D8C2312yes—
73C8×D4—22no—
74C8×D4C2311yescharacteristic
75C2×#106—42no—
76#367—22no—
77#366—21no—
78#366—21no—
79#366D411yescharacteristic
80C4⋊C16—21no—
81C4⋊C16C2311yescharacteristic
82C4⋊C16C2311yescharacteristic
83#358—22no—
84#358—42no—
85#357—22no—
86#357D412yes—
87C22⋊C16—21no—
88C4×C16—21no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335C2312yes—
92#334D411yescharacteristic
93#333—22no—
94#332—22no—
95Q8⋊C8D411yescharacteristic
96D4⋊C8—22no—
97D4⋊C8D411yescharacteristic
98C82C2311yescharacteristic
99C2×#41—42no—
100#135—44no—
101#135—82no—
102C2×Q16—22no—
103C2×Q16—22no—
104C2×SD16—42no—
105C2×SD16D811yescharacteristic
106C2×D8C2×D412yes—
107C22×C8—21no—
108C22×C8—42no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111#125—42no—
112#123—82no—
113C22⋊Q8—42no—
114C22⋊Q8—42no—
115C22⋊Q8—82no—
116#121—42no—
117C4×Q8C2×D411yescharacteristic
118C4×D4—22no—
119C4×D4—42no—
120C4×D4C2×D411yescharacteristic
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123Q32—42no—
124Q32—81no—
125SD32—44no—
126D16SD1614yescomplemented
127C2×C16—21no—
128C2×C16—21no—
129C2×C16—21no—
130C2×C16—42no—
131C2×C16C2×D411yescharacteristic
132C2×C16C2×D411yescharacteristic
133#107—22no—
134#107—22no—
135#107C2×D412yes—
136#106—42no—
137#106—42no—
138#106—42no—
139#106—42no—
140#106—42no—
141#106D811yescharacteristic
142C4⋊C8—21no—
143C4⋊C8—22no—
144C4⋊C8C2×D411yescharacteristic
145C4⋊C8C2×D411yescharacteristic
146#103—42no—
147#103—42no—
148#103—82no—
149#103D811yescharacteristic
150#102—22no—
151#102—42no—
152#102—42no—
153#102D811yescharacteristic
154C22⋊C8—21no—
155C22⋊C8—42no—
156C4×C8—21no—
157C4×C8C2×D411yescharacteristic
158C4×C8C2×D411yescharacteristic
159C4×C8C2411yescharacteristic
160#41—84no—
161#41—162no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8C2×D811yescharacteristic
165C2×D4—22no—
166C2×D4—82no—
167C2×D4C2×D811yescharacteristic
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—82no—
171C22×C4—82no—
172Q16—42no—
173Q16—42no—
174Q16—82no—
175SD16—22no—
176SD16—84no—
177D8—22no—
178D8C2×SD1614yes—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—84no—
186C2×C8C2×D811yescharacteristic
187C2×C8C2×D811yescharacteristic
188C2×C8C22×D411yescharacteristic
189C2×C8H(5,𝔽2)11yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C4⋊C4C2×D811yescharacteristic
200C4⋊C4C2×D811yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—42no—
203C22⋊C4—82no—
204C22⋊C4—82no—
205C42—21no—
206C42C22×D411yescharacteristic
207C16—21no—
208C16—41no—
209C16—41no—
210C16—41no—
211C16C2×SD1612yes—
212C23—21no—
213C23—42no—
214Q8—21no—
215Q8—21no—
216Q8—82no—
217Q8—82no—
218Q8—162no—
219D4—21no—
220D4—21no—
221D4—24no—
222D4—42no—
223D4—164no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—164no—
235C2×C4—322no—
236C2×C4D4211yescharacteristic
237C2×C4C22×D811yescharacteristic
238C2×C4#57611yescharacteristic
239C8—21no—
240C8—21no—
241C8—41no—
242C8—41no—
243C8—41no—
244C8—82no—
245C8#50812yescomplemented
246C8C22×SD1611yescharacteristic
247C8#57711yescharacteristic
248C22—41no—
249C22—44no—
250C22—41no—
251C22—41no—
252C22—82no—
253C22—82no—
254C22D4×D811yescharacteristic
255C4—21no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—82no—
260C4—162no—
261C4—162no—
262C4D4×SD1611yescharacteristic
263C4#329011yescharacteristic
264C4C2×#50811yescharacteristic
265C4#341711yescharacteristic
266C2—81no—
267C2—84no—
268C2D8×SD1611yescharacteristic
269C2#3384911yescharacteristic
270C2#3385911yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radical#33849C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33947C2semidirect
3. #19980C2semidirect
4. #12504C2semidirect
5. #12452C2semidirect
6. #12440C2semidirect
7. #12420C2semidirect
8. #12375C2semidirect
9. #12359C2semidirect
10. C16×SD16C2semidirect
11. C8×D16C2semidirect
126. D16SD16semidirect
245. C8#508semidirect

Maximal subgroups

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

subgroupindex
#339472
#338492
#199802
#125042
#124522
#124402
#124202
#123752
#123592
C16×SD162
C8×D162