Finite Group Databasebeta

← #492776

Subgroups — #492776

Order 512.

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

normal subgroups only

This group has 412 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
2Q8×D16C211yesmaximal, characteristic
3#19993C212yesmaximal, complemented
4#12522C211yesmaximal, characteristic, complemented
5#12458C211yesmaximal, characteristic, complemented
6#12457C211yesmaximal, characteristic, complemented
7#12449C212yesmaximal, complemented
8#12408C212yesmaximal, complemented
9#12407C212yesmaximal, complemented
10C16×Q16C211yesmaximal, characteristic, complemented
11C8×D16C211yesmaximal, characteristic
12#3380C2212yes—
13Q8×D8C2212yes—
14#3299—22no—
15#2254C2212yes—
16#2253—21no—
17#2252—21no—
18#2252C2211yescharacteristic
19#2252C2211yescharacteristic
20#2228—22no—
21#2226C2212yes—
22#2224—22no—
23#2224C2212yes—
24Q8×C16C2211yescharacteristic
25Q8×C16C2211yescharacteristic
26C4×D16—21no—
27C4×D16—21no—
28C4×D16C2211yescharacteristic
29#1710C2212yes—
30#1683—22no—
31#1676C2212yes—
32#1674C2212yes—
33#1644—22no—
34#1637—22no—
35C8×Q16C2211yescharacteristic
36C8×D8C2212yes—
37#1371C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1364C2212yes—
41Q8⋊C16C2211yescharacteristic
42Q8⋊C16C2211yescharacteristic
43#1331C2212yes—
44C8×C16C2211yescharacteristic
45#554—22no—
46D4×Q8—22no—
47C2×D16D411yescharacteristic
48#500—22no—
49#500C2312yes—
50#500C2312yes—
51#499—22no—
52#487—22no—
53#487—22no—
54#484—42no—
55#478—42no—
56#477—22no—
57#476—22no—
58#474—22no—
59#474—42no—
60#467—42no—
61#462—22no—
62#451—42no—
63C8×Q8C2311yescharacteristic
64C8×Q8C2311yescharacteristic
65C4×Q16D411yescharacteristic
66C4×D8—22no—
67C4×D8—22no—
68C4×D8C2312yes—
69C8×D4—22no—
70C2×#103—42no—
71#367—22no—
72#367—42no—
73#366—21no—
74#366—21no—
75#366—21no—
76#366—21no—
77#366—22no—
78#366D411yescharacteristic
79C4⋊C16—21no—
80C4⋊C16—21no—
81C4⋊C16C2311yescharacteristic
82C4⋊C16C2311yescharacteristic
83#357—22no—
84#357—22no—
85#357D412yes—
86C4×C16—21no—
87C4×C16—21no—
88C4×C16C2311yescharacteristic
89C4×C16C2311yescharacteristic
90#335D411yescharacteristic
91#335C2312yes—
92#333—22no—
93#332—22no—
94Q8⋊C8D411yescharacteristic
95Q8⋊C8D411yescharacteristic
96D4⋊C8—22no—
97C82C2311yescharacteristic
98C22×Q8—42no—
99C2×Q16—42no—
100C2×Q16D811yescharacteristic
101C2×D8C2×D412yes—
102C22×C8—42no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—42no—
106#125—42no—
107C22⋊Q8—42no—
108C22⋊Q8—42no—
109C22⋊Q8—82no—
110C22⋊Q8—82no—
111C4×Q8C2×D411yescharacteristic
112C4×Q8C2×D411yescharacteristic
113C4×D4—22no—
114C4×D4—42no—
115C4×D4—42no—
116C2×C4⋊C4—42no—
117D16Q1614yescomplemented
118C2×C16—21no—
119C2×C16—21no—
120C2×C16—21no—
121C2×C16C2×D411yescharacteristic
122C2×C16C2×D411yescharacteristic
123#107—22no—
124#107—22no—
125#107—22no—
126#107—22no—
127#107—42no—
128#107—42no—
129#107D811yescharacteristic
130#107C2×D412yes—
131#106—42no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—22no—
135C4⋊C8C2×D411yescharacteristic
136C4⋊C8C2×D411yescharacteristic
137#103—42no—
138#103—42no—
139#103—42no—
140#103—42no—
141#103—82no—
142#103D811yescharacteristic
143#103D811yescharacteristic
144#102—22no—
145#102—42no—
146#102—42no—
147C22⋊C8—42no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8C2×D411yescharacteristic
151C4×C8C2×D411yescharacteristic
152C4×C8C2411yescharacteristic
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—82no—
156C2×Q8—84no—
157C2×Q8—162no—
158C2×Q8C2×D811yescharacteristic
159C2×Q8C2×D811yescharacteristic
160C2×D4—22no—
161C22×C4—42no—
162C22×C4—82no—
163C22×C4—82no—
164Q16—22no—
165Q16—162no—
166D8—22no—
167D8C2×Q1614yes—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—42no—
172C2×C8—162no—
173C2×C8C2×D811yescharacteristic
174C2×C8C2×D811yescharacteristic
175C2×C8C22×D411yescharacteristic
176C2×C8H(5,𝔽2)11yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—82no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C4⋊C4C2×D811yescharacteristic
188C4⋊C4C2×D811yescharacteristic
189C22⋊C4—42no—
190C22⋊C4—82no—
191C22⋊C4—82no—
192C42—21no—
193C42—21no—
194C42C22×D411yescharacteristic
195C16—22no—
196C16—21no—
197C16—41no—
198C16—41no—
199C16C2×Q1612yes—
200C23—42no—
201Q8—21no—
202Q8—21no—
203Q8—21no—
204Q8—82no—
205Q8—82no—
206Q8—164no—
207Q8—162no—
208Q8#50812yes—
209D4—24no—
210D4—42no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—164no—
221C2×C4—322no—
222C2×C4D4211yescharacteristic
223C2×C4C22×D811yescharacteristic
224C2×C4#57611yescharacteristic
225C8—21no—
226C8—21no—
227C8—21no—
228C8—41no—
229C8—41no—
230C8—41no—
231C8—82no—
232C8C22×Q1611yescharacteristic
233C8#57811yescharacteristic
234C22—44no—
235C22—82no—
236C22—82no—
237C22D4×D811yescharacteristic
238C4—22no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—82no—
244C4—162no—
245C4—162no—
246C4D4×Q1611yescharacteristic
247C4#329311yescharacteristic
248C4C2×#50811yescharacteristic
249C4#341511yescharacteristic
250C2—84no—
251C2D8×Q1611yescharacteristic
252C2#3384611yescharacteristic
253C2#3385911yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radicalC2×D16D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #19993C2semidirect
4. #12522C2semidirect
5. #12458C2semidirect
6. #12457C2semidirect
7. #12449C2semidirect
8. #12408C2semidirect
9. #12407C2semidirect
10. C16×Q16C2semidirect
117. D16Q16semidirect

Maximal subgroups

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

subgroupindex
#339472
Q8×D162
#199932
#125222
#124582
#124572
#124492
#124082
#124072
C16×Q162
C8×D162