Finite Group Databasebeta

← #492473

Subgroups — #492473

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
2Q8×Q32C211yesmaximal, characteristic, complemented
3#19942C212yesmaximal, complemented
4#12503C211yesmaximal, characteristic
5#12486C212yesmaximal, complemented
6#12458C211yesmaximal, characteristic, complemented
7#12439C211yesmaximal, characteristic
8#12385C212yesmaximal, complemented
9#12354C212yesmaximal, complemented
10#12083C211yesmaximal, characteristic, complemented
11C16×Q16C211yesmaximal, characteristic, complemented
12#3396C2212yes—
13Q8×Q16C2212yes—
14#3375—22no—
15#2252—21no—
16#2252C2211yescharacteristic
17#2249—21no—
18#2246C2211yescharacteristic
19#2240C2212yes—
20#2227—22no—
21#2227C2212yes—
22#2210—22no—
23#2207C2212yes—
24Q8×C16C2211yescharacteristic
25Q8×C16C2211yescharacteristic
26C4×Q32—21no—
27C4×Q32C2211yescharacteristic
28C4×D16—21no—
29#1706C2212yes—
30#1698—22no—
31#1676C2212yes—
32#1668C2212yes—
33#1630—22no—
34#1623—22no—
35#1591C2212yes—
36C8×Q16C2211yescharacteristic
37#1371C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1361C2212yes—
41Q8⋊C16C2211yescharacteristic
42Q8⋊C16C2211yescharacteristic
43#1332C2212yes—
44C8×C16C2211yescharacteristic
45#559—22no—
46Q82—22no—
47C2×Q32D411yescharacteristic
48C2×SD32—22no—
49C2×D16—21no—
50C2×D16—21no—
51#500—22no—
52#500C2312yes—
53#495—22no—
54#493C2312yes—
55#492—22no—
56#489—42no—
57#486—22no—
58#477—22no—
59#477—22no—
60#477—42no—
61#474—42no—
62#464—42no—
63#463—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#366—21no—
75#366—21no—
76#366—22no—
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#333—22no—
93#331—22no—
94Q8⋊C8—22no—
95Q8⋊C8D411yescharacteristic
96Q8⋊C8D411yescharacteristic
97C82C2311yescharacteristic
98C2×Q16D811yescharacteristic
99C2×Q16C2×D412yes—
100C2×SD16—42no—
101C2×SD16—42no—
102C2×SD16—82no—
103C2×D8—22no—
104C2×D8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108C4⋊Q8—82no—
109#127—22no—
110#124—42no—
111#124—42no—
112#123—82no—
113C4×Q8—22no—
114C4×Q8—42no—
115C4×Q8—42no—
116C4×Q8C2×D411yescharacteristic
117C4×Q8C2×D411yescharacteristic
118C4×D4—42no—
119C4×D4—42no—
120Q32—22no—
121SD32—82no—
122D16—42no—
123D16—81no—
124C2×C16—21no—
125C2×C16—21no—
126C2×C16—21no—
127C2×C16C2×D411yescharacteristic
128C2×C16C2×D411yescharacteristic
129#107—22no—
130#107—22no—
131#107—42no—
132#107—42no—
133#107D811yescharacteristic
134#107C2×D412yes—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8—22no—
138C4⋊C8—42no—
139C4⋊C8C2×D411yescharacteristic
140C4⋊C8C2×D411yescharacteristic
141#103—22no—
142#103—42no—
143#103—42no—
144#103D811yescharacteristic
145#103D811yescharacteristic
146#102—42no—
147#102—42no—
148#102—42no—
149#102—42no—
150#97—42no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8C2×D411yescharacteristic
154C4×C8C2×D411yescharacteristic
155C4×C8C2411yescharacteristic
156C2×Q8—22no—
157C2×Q8—42no—
158C2×Q8—82no—
159C2×Q8C2×D811yescharacteristic
160C2×Q8C2×D811yescharacteristic
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C22×C4—82no—
165Q16—22no—
166Q16—22no—
167Q16—22no—
168SD16—84no—
169SD16—164no—
170SD16—162no—
171D8—42no—
172D8—42no—
173D8—82no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—42no—
178C2×C8C2×D811yescharacteristic
179C2×C8C2×D811yescharacteristic
180C2×C8C22×D411yescharacteristic
181C2×C8H(5,𝔽2)11yescharacteristic
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C4⋊C4C2×D811yescharacteristic
191C4⋊C4C2×D811yescharacteristic
192C22⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—82no—
195C42—21no—
196C42—21no—
197C42—42no—
198C42—82no—
199C42C22×D411yescharacteristic
200C16—21no—
201C16—41no—
202C16—41no—
203C16—41no—
204C16#13712yes—
205C23—82no—
206Q8—21no—
207Q8—21no—
208Q8—21no—
209Q8—42no—
210Q8—42no—
211Q8—162no—
212Q8—322no—
213Q8#50812yes—
214D4—82no—
215D4—82no—
216D4—82no—
217D4—82no—
218D4—162no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—82no—
226C2×C4—322no—
227C2×C4D4211yescharacteristic
228C2×C4C22×D811yescharacteristic
229C2×C4#57611yescharacteristic
230C8—22no—
231C8—21no—
232C8—21no—
233C8—21no—
234C8—41no—
235C8—41no—
236C8—82no—
237C8C2×#13711yescharacteristic
238C8#57711yescharacteristic
239C22—162no—
240C22—162no—
241C22—162no—
242C22D4×D811yescharacteristic
243C4—22no—
244C4—21no—
245C4—41no—
246C4—41no—
247C4—41no—
248C4—82no—
249C4—82no—
250C4—162no—
251C4#327811yescharacteristic
252C4#329011yescharacteristic
253C4C2×#50811yescharacteristic
254C4#341711yescharacteristic
255C2—322no—
256C2#1988911yescharacteristic
257C2#3385111yescharacteristic
258C2#3385811yescharacteristic
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. Q8×Q32C2semidirect
3. #19942C2semidirect
5. #12486C2semidirect
6. #12458C2semidirect
8. #12385C2semidirect
9. #12354C2semidirect
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
Q8×Q322
#199422
#125032
#124862
#124582
#124392
#123852
#123542
#120832
C16×Q162