Finite Group Databasebeta

← #492779

Subgroups — #492779

Order 512.

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

normal subgroups only

This group has 386 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#33950C211yesmaximal, characteristic
2Q8×Q32C211yesmaximal, characteristic
3#20008C212yesmaximal
4#12522C211yesmaximal, characteristic
5#12458C211yesmaximal, characteristic
6#12457C211yesmaximal, characteristic
7#12450C212yesmaximal
8#12409C212yesmaximal
9#12406C212yesmaximal
10C16×Q16C211yesmaximal, characteristic
11C8×Q32C211yesmaximal, characteristic
12#3383C2212yes—
13Q8×Q16C2212yes—
14#3363—22no—
15#2254C2212yes—
16#2253—21no—
17#2252—21no—
18#2252C2211yescharacteristic
19#2252C2211yescharacteristic
20#2229—22no—
21#2227—22no—
22#2227C2212yes—
23#2225C2212yes—
24Q8×C16C2211yescharacteristic
25Q8×C16C2211yescharacteristic
26C4×Q32—21no—
27C4×Q32—21no—
28C4×Q32C2211yescharacteristic
29#1710C2212yes—
30#1684—22no—
31#1676C2212yes—
32#1674C2212yes—
33#1645—22no—
34#1636—22no—
35C8×Q16C2212yes—
36C8×Q16C2211yescharacteristic
37#1371C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1364C2212yes—
41Q8⋊C16C2211yescharacteristic
42Q8⋊C16C2211yescharacteristic
43#1332C2212yes—
44C8×C16C2211yescharacteristic
45Q82—22no—
46#557—22no—
47C2×Q32D411yescharacteristic
48#500—22no—
49#500C2312yes—
50#500C2312yes—
51#499—22no—
52#487—22no—
53#487—22no—
54#487—42no—
55#479—42no—
56#477—22no—
57#477—22no—
58#477—42no—
59#477—42no—
60#475—22no—
61#462—22no—
62#462—42no—
63C8×Q8—22no—
64C8×Q8C2311yescharacteristic
65C8×Q8C2311yescharacteristic
66C4×Q16—22no—
67C4×Q16—22no—
68C4×Q16—42no—
69C4×Q16D411yescharacteristic
70C4×Q16C2312yes—
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#358—22no—
84#358—22no—
85#358D412yes—
86C4×C16—21no—
87C4×C16—21no—
88C4×C16C2311yescharacteristic
89C4×C16C2311yescharacteristic
90#335D411yescharacteristic
91#335C2312yes—
92#333—22no—
93#332—22no—
94Q8⋊C8—22no—
95Q8⋊C8D411yescharacteristic
96Q8⋊C8D411yescharacteristic
97C82C2311yescharacteristic
98C2×Q16—42no—
99C2×Q16D811yescharacteristic
100C2×Q16C2×D412yes—
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C4⋊Q8—42no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—82no—
107#125—42no—
108#125—82no—
109C4×Q8—22no—
110C4×Q8—42no—
111C4×Q8—42no—
112C4×Q8—42no—
113C4×Q8—42no—
114C4×Q8C2×D411yescharacteristic
115C4×Q8C2×D411yescharacteristic
116Q32Q1614yescomplemented
117C2×C16—21no—
118C2×C16—21no—
119C2×C16—21no—
120C2×C16C2×D411yescharacteristic
121C2×C16C2×D411yescharacteristic
122#107—22no—
123#107—22no—
124#107—22no—
125#107—22no—
126#107—42no—
127#107—42no—
128#107D811yescharacteristic
129#107C2×D412yes—
130#106—42no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—22no—
134C4⋊C8—42no—
135C4⋊C8C2×D411yescharacteristic
136C4⋊C8C2×D411yescharacteristic
137#103—22no—
138#103—42no—
139#103—42no—
140#103—42no—
141#103—42no—
142#103—42no—
143#103—42no—
144#103—82no—
145#103D811yescharacteristic
146#103D811yescharacteristic
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—42no—
150C4×C8C2×D411yescharacteristic
151C4×C8C2×D411yescharacteristic
152C4×C8C2411yescharacteristic
153C2×Q8—22no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—82no—
157C2×Q8C2×D811yescharacteristic
158C2×Q8C2×D811yescharacteristic
159Q16—22no—
160Q16—22no—
161Q16—84no—
162Q16C2×Q1614yes—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—42no—
167C2×C8C2×D811yescharacteristic
168C2×C8C2×D811yescharacteristic
169C2×C8C22×D411yescharacteristic
170C2×C8H(5,𝔽2)11yescharacteristic
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C4⋊C4—82no—
184C4⋊C4C2×D811yescharacteristic
185C4⋊C4C2×D811yescharacteristic
186C42—21no—
187C42—21no—
188C42—42no—
189C42—82no—
190C42—82no—
191C42C22×D411yescharacteristic
192C16—22no—
193C16—21no—
194C16—41no—
195C16—41no—
196C16C2×Q1612yes—
197Q8—21no—
198Q8—21no—
199Q8—21no—
200Q8—24no—
201Q8—42no—
202Q8—82no—
203Q8—82no—
204Q8—162no—
205Q8—322no—
206Q8#50812yes—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4D4211yescharacteristic
216C2×C4C22×D811yescharacteristic
217C2×C4#57611yescharacteristic
218C8—21no—
219C8—21no—
220C8—21no—
221C8—41no—
222C8—41no—
223C8—41no—
224C8—82no—
225C8C22×Q1611yescharacteristic
226C8#57811yescharacteristic
227C22D4×D811yescharacteristic
228C4—22no—
229C4—21no—
230C4—44no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—82no—
235C4—162no—
236C4—162no—
237C4D4×Q1611yescharacteristic
238C4#329311yescharacteristic
239C4C2×#50811yescharacteristic
240C4#341711yescharacteristic
241C2D8×Q1611yescharacteristic
242C2#3385111yescharacteristic
243C2#3385911yescharacteristic
subgroupG/H
center, socle, prime radicalC22D4×D8
commutator, Frattini, cosocleC4×C8C24
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
116. Q32Q16semidirect

Maximal subgroups

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

subgroupindex
#339502
Q8×Q322
#200082
#125222
#124582
#124572
#124502
#124092
#124062
C16×Q162
C8×Q322