Finite Group Databasebeta

← #492403

Subgroups — #492403

Order 512.

411 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 118 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
1Q8×D16C211yesmaximal, characteristic, complemented
2#33908C211yesmaximal, characteristic
3#19905C212yesmaximal, complemented
4#12522C211yesmaximal, characteristic, complemented
5#12501C212yesmaximal, complemented
6#12458C211yesmaximal, characteristic, complemented
7#12451C211yesmaximal, characteristic, complemented
8#12408C212yesmaximal, complemented
9#12395C212yesmaximal, complemented
10#12081C211yesmaximal, characteristic, complemented
11C16×SD16C211yesmaximal, characteristic, complemented
12Q8×D8C2212yes—
13#3341C2212yes—
14#3329—22no—
15#2254C2212yes—
16#2252—21no—
17#2252C2211yescharacteristic
18#2252C2211yescharacteristic
19#2234—22no—
20#2226C2212yes—
21#2224—22no—
22#2224C2212yes—
23#2215—21no—
24Q8×C16C2211yescharacteristic
25C4×D16—21no—
26C4×D16C2211yescharacteristic
27D4×C16C2211yescharacteristic
28#2158—21no—
29#1710C2212yes—
30#1703—22no—
31#1676C2212yes—
32#1673C2212yes—
33#1646—22no—
34#1637—22no—
35#1589C2212yes—
36C8×SD16C2211yescharacteristic
37#1371C2211yescharacteristic
38#1367C2211yescharacteristic
39#1365C2212yes—
40#1364C2212yes—
41Q8⋊C16C2211yescharacteristic
42#1331C2212yes—
43D4⋊C16C2211yescharacteristic
44C8×C16C2211yescharacteristic
45#555—22no—
46D4×Q8—22no—
47C2×D16D411yescharacteristic
48C22×C16—21no—
49#500—22no—
50#500C2312yes—
51#500C2312yes—
52#487—22no—
53#487—22no—
54#485—42no—
55#482—42no—
56#477—22no—
57#476—22no—
58#474—22no—
59#474—42no—
60#468—42no—
61#466—22no—
62#461—22no—
63#448—42no—
64C8×Q8C2311yescharacteristic
65C4×SD16D411yescharacteristic
66C4×D8—22no—
67C4×D8C2312yes—
68#436—22no—
69C8×D4C2311yescharacteristic
70#427—22no—
71#419—42no—
72#367—22no—
73#367—42no—
74#366—21no—
75#366—21no—
76#366—21no—
77#366—21no—
78#366—22no—
79#366D411yescharacteristic
80C4⋊C16—21no—
81C4⋊C16C2311yescharacteristic
82C4⋊C16C2311yescharacteristic
83#357—22no—
84#357—22no—
85#357D412yes—
86C22⋊C16—21no—
87C4×C16—21no—
88C4×C16C2311yescharacteristic
89C4×C16C2311yescharacteristic
90#335C2312yes—
91#334D411yescharacteristic
92#333—22no—
93#332—22no—
94Q8⋊C8D411yescharacteristic
95D4⋊C8—22no—
96D4⋊C8D411yescharacteristic
97C82C2311yescharacteristic
98C22×Q8—42no—
99C2×SD16—42no—
100C2×SD16D811yescharacteristic
101C2×D8C2×D412yes—
102C2×M16—42no—
103C22×C8—21no—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—42no—
107#124—82no—
108C22⋊Q8—42no—
109C22⋊Q8—82no—
110#121—42no—
111#121—42no—
112C4×Q8C2×D411yescharacteristic
113C4×D4—22no—
114C4×D4—42no—
115C4×D4C2×D411yescharacteristic
116#117—42no—
117#117—42no—
118D16—22no—
119C2×C16—21no—
120C2×C16—21no—
121C2×C16—21no—
122C2×C16—42no—
123C2×C16C2×D411yescharacteristic
124C2×C16C2×D411yescharacteristic
125#107—22no—
126#107—22no—
127#107—22no—
128#107—22no—
129#107—42no—
130#107—42no—
131#107C2×D412yes—
132#106—42no—
133#106D811yescharacteristic
134C4⋊C8—21no—
135C4⋊C8—22no—
136C4⋊C8C2×D411yescharacteristic
137C4⋊C8C2×D411yescharacteristic
138#103—42no—
139#103—42no—
140#103—42no—
141#103—42no—
142#103—82no—
143#103D811yescharacteristic
144#102—22no—
145#102—42no—
146#102—42no—
147#102D811yescharacteristic
148C22⋊C8—21no—
149C22⋊C8—42no—
150C4×C8—21no—
151C4×C8C2×D411yescharacteristic
152C4×C8C2×D411yescharacteristic
153C4×C8C2411yescharacteristic
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—82no—
157C2×Q8—84no—
158C2×Q8—162no—
159C2×Q8C2×D811yescharacteristic
160C2×D4—22no—
161C2×D4—82no—
162C2×D4C2×D811yescharacteristic
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—82no—
166SD16—22no—
167SD16—162no—
168D8—22no—
169D8—22no—
170M16—162no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—42no—
177C2×C8C2×D811yescharacteristic
178C2×C8C2×D811yescharacteristic
179C2×C8C22×D411yescharacteristic
180C2×C8H(5,𝔽2)11yescharacteristic
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—82no—
188C4⋊C4C2×D811yescharacteristic
189C4⋊C4C2×D811yescharacteristic
190C22⋊C4—21no—
191C22⋊C4—42no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—82no—
195C42—21no—
196C42—82no—
197C42C22×D411yescharacteristic
198C16—22no—
199C16—21no—
200C16—41no—
201C16—41no—
202C16#13712yes—
203C23—21no—
204C23—42no—
205Q8—21no—
206Q8—82no—
207Q8—82no—
208Q8—164no—
209Q8—162no—
210Q8#50812yes—
211D4—21no—
212D4—21no—
213D4—42no—
214D4—42no—
215D4—322no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—164no—
227C2×C4D4211yescharacteristic
228C2×C4C22×D811yescharacteristic
229C2×C4#57611yescharacteristic
230C8—21no—
231C8—21no—
232C8—21no—
233C8—41no—
234C8—41no—
235C8—41no—
236C8—82no—
237C8C2×#13711yescharacteristic
238C8#57811yescharacteristic
239C22—41no—
240C22—41no—
241C22—41no—
242C22—82no—
243C22—82no—
244C22—82no—
245C22D4×D811yescharacteristic
246C4—22no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—82no—
251C4—162no—
252C4—162no—
253C4#327611yescharacteristic
254C4#329311yescharacteristic
255C4C2×#50811yescharacteristic
256C4#341511yescharacteristic
257C2—81no—
258C2—162no—
259C2#1988711yescharacteristic
260C2#3384611yescharacteristic
261C2#3385911yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radical#33908C2
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. Q8×D16C2semidirect
3. #19905C2semidirect
4. #12522C2semidirect
5. #12501C2semidirect
6. #12458C2semidirect
7. #12451C2semidirect
8. #12408C2semidirect
9. #12395C2semidirect
10. #12081C2semidirect
11. C16×SD16C2semidirect

Maximal subgroups

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

subgroupindex
Q8×D162
#339082
#199052
#125222
#125012
#124582
#124512
#124082
#123952
#120812
C16×SD162