Finite Group Databasebeta

← #492479

Subgroups — #492479

Order 512.

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

normal subgroups only

This group has 409 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
1#33963C211yesmaximal, characteristic
2#33950C211yesmaximal, characteristic, complemented
3#19966C212yesmaximal, complemented
4#12503C211yesmaximal, characteristic
5#12484C212yesmaximal, complemented
6#12457C211yesmaximal, characteristic, complemented
7#12439C211yesmaximal, characteristic
8#12381C212yesmaximal, complemented
9#12353C212yesmaximal, complemented
10#12083C211yesmaximal, characteristic, complemented
11C16×Q16C211yesmaximal, characteristic, complemented
12#3400—22no—
13#3396C2212yes—
14#3383C2212yes—
15#2253—21no—
16#2252C2211yescharacteristic
17#2249—21no—
18#2246C2211yescharacteristic
19#2240C2212yes—
20#2229—22no—
21#2225C2212yes—
22#2210—22no—
23#2207C2212yes—
24Q8×C16C2211yescharacteristic
25Q8×C16C2211yescharacteristic
26C4×Q32—21no—
27C4×Q32C2211yescharacteristic
28C4×D16—21no—
29#1706C2212yes—
30#1686—22no—
31#1674C2212yes—
32#1668C2212yes—
33#1640—22no—
34#1626—22no—
35#1591C2212yes—
36C8×Q16C2211yescharacteristic
37#1371C2211yescharacteristic
38#1370C2211yescharacteristic
39#1364C2212yes—
40#1361C2212yes—
41Q8⋊C16C2211yescharacteristic
42Q8⋊C16C2211yescharacteristic
43#1332C2212yes—
44C8×C16C2211yescharacteristic
45#559—22no—
46#557—22no—
47C2×Q32D411yescharacteristic
48C2×SD32—22no—
49C2×D16—21no—
50C2×D16—21no—
51#500C2312yes—
52#499—22no—
53#497—42no—
54#495—22no—
55#493C2312yes—
56#492—22no—
57#486—22no—
58#479—42no—
59#475—22no—
60#464—42no—
61#463—42no—
62#462—22no—
63#461—42no—
64#460—22no—
65#460—22no—
66#446—22no—
67C8×Q8C2311yescharacteristic
68C8×Q8C2311yescharacteristic
69#440—42no—
70C4×Q16—22no—
71C4×Q16D411yescharacteristic
72C4×Q16C2312yes—
73C4×D8—22no—
74#367—22no—
75#366—21no—
76#366—21no—
77#366D411yescharacteristic
78C4⋊C16—21no—
79C4⋊C16—21no—
80C4⋊C16C2311yescharacteristic
81C4⋊C16C2311yescharacteristic
82#358—22no—
83#358D412yes—
84#357—22no—
85#357—42no—
86C4×C16—21no—
87C4×C16—21no—
88C4×C16C2311yescharacteristic
89C4×C16C2311yescharacteristic
90#335D411yescharacteristic
91#335C2312yes—
92#332—22no—
93#331—22no—
94Q8⋊C8—22no—
95Q8⋊C8D411yescharacteristic
96Q8⋊C8D411yescharacteristic
97C82C2311yescharacteristic
98C2×Q16—42no—
99C2×Q16D811yescharacteristic
100C2×Q16C2×D412yes—
101C2×SD16—42no—
102C2×SD16—42no—
103C2×SD16—42no—
104C2×SD16—42no—
105C2×D8—22no—
106C2×D8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—42no—
109#127—22no—
110#125—42no—
111#125—82no—
112#124—42no—
113#124—42no—
114#123—82no—
115C4×Q8—22no—
116C4×Q8—42no—
117C4×Q8—42no—
118C4×Q8C2×D411yescharacteristic
119C4×Q8C2×D411yescharacteristic
120C4×D4—42no—
121C4×D4—42no—
122Q32—22no—
123SD32—82no—
124D16—42no—
125D16—81no—
126C2×C16—21no—
127C2×C16—21no—
128C2×C16—21no—
129C2×C16C2×D411yescharacteristic
130C2×C16C2×D411yescharacteristic
131#107—22no—
132#107—22no—
133#107D811yescharacteristic
134#107C2×D412yes—
135#106—42no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—22no—
139C4⋊C8—42no—
140C4⋊C8C2×D411yescharacteristic
141C4⋊C8C2×D411yescharacteristic
142#103—22no—
143#103—42no—
144#103—42no—
145#103D811yescharacteristic
146#103D811yescharacteristic
147#102—42no—
148#102—42no—
149#102—82no—
150#97—42no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8C2×D411yescharacteristic
154C4×C8C2×D411yescharacteristic
155C4×C8C2411yescharacteristic
156C2×Q8—22no—
157C2×Q8—42no—
158C2×Q8C2×D811yescharacteristic
159C2×Q8C2×D811yescharacteristic
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C22×C4—82no—
164Q16—22no—
165Q16—22no—
166Q16—22no—
167Q16—162no—
168SD16—84no—
169SD16—162no—
170SD16—162no—
171SD16—162no—
172D8—42no—
173D8—42no—
174D8—82no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—42no—
179C2×C8C2×D811yescharacteristic
180C2×C8C2×D811yescharacteristic
181C2×C8C22×D411yescharacteristic
182C2×C8H(5,𝔽2)11yescharacteristic
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—82no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C4⋊C4C2×D811yescharacteristic
193C4⋊C4C2×D811yescharacteristic
194C22⋊C4—82no—
195C22⋊C4—82no—
196C22⋊C4—82no—
197C42—21no—
198C42—21no—
199C42—42no—
200C42—82no—
201C42C22×D411yescharacteristic
202C16—21no—
203C16—41no—
204C16—41no—
205C16—41no—
206C16#13712yes—
207C23—82no—
208Q8—21no—
209Q8—21no—
210Q8—21no—
211Q8—21no—
212Q8—42no—
213Q8—42no—
214Q8—82no—
215Q8—82no—
216D4—82no—
217D4—82no—
218D4—82no—
219D4—82no—
220D4—162no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—322no—
229C2×C4D4211yescharacteristic
230C2×C4C22×D811yescharacteristic
231C2×C4#57611yescharacteristic
232C8—21no—
233C8—21no—
234C8—41no—
235C8—41no—
236C8—41no—
237C8—82no—
238C8#50812yes—
239C8C2×#13711yescharacteristic
240C8#57711yescharacteristic
241C22—162no—
242C22—162no—
243C22—162no—
244C22D4×D811yescharacteristic
245C4—21no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—82no—
251C4—82no—
252C4—162no—
253C4#327811yescharacteristic
254C4#329011yescharacteristic
255C4C2×#50811yescharacteristic
256C4#341511yescharacteristic
257C2—322no—
258C2#1988911yescharacteristic
259C2#3385311yescharacteristic
260C2#3385911yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radical#2246C22
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
2. #33950C2semidirect
3. #19966C2semidirect
5. #12484C2semidirect
6. #12457C2semidirect
8. #12381C2semidirect
9. #12353C2semidirect
10. #12083C2semidirect
11. C16×Q16C2semidirect

Maximal subgroups

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

subgroupindex
#339632
#339502
#199662
#125032
#124842
#124572
#124392
#123812
#123532
#120832
C16×Q162