Finite Group Databasebeta

← #492755

Subgroups — #492755

Order 512.

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

normal subgroups only

This group has 397 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#33939C212yesmaximal
2#20010C211yesmaximal, characteristic, complemented
3#20006C211yesmaximal, characteristic
4#12510C211yesmaximal, characteristic, complemented
5#12481C212yesmaximal, complemented
6#12448C211yesmaximal, characteristic, complemented
7#12426C211yesmaximal, characteristic
8#12386C212yesmaximal, complemented
9#12359C212yesmaximal
10#12054C211yesmaximal, characteristic, complemented
11C8×SD32C211yesmaximal, characteristic
12#3395C2212yes—
13#3383—21no—
14#3380C2212yes—
15#3338—21no—
16#2245—22no—
17#2241C2212yes—
18#2241C2212yes—
19#2229—22no—
20#2227C2212yes—
21#2208—22no—
22#2208C2212yes—
23#2183C2212yes—
24#2175—22no—
25C4×SD32C2211yescharacteristic
26#1713C2211yescharacteristic
27#1713C2211yescharacteristic
28#1676C2212yes—
29#1675—21no—
30#1674—21no—
31#1670C2212yes—
32#1648—22no—
33#1632—22no—
34C8×Q16C2211yescharacteristic
35C8×Q16C2211yescharacteristic
36C8×D8C2211yescharacteristic
37#1372C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1362C2212yes—
41Q8⋊C16C2212yes—
42#1332C2211yescharacteristic
43#1331C2211yescharacteristic
44C8×C16C2211yescharacteristic
45#557—22no—
46#554—22no—
47C2×SD32—22no—
48C2×SD32D411yescharacteristic
49#500—21no—
50#500—21no—
51#500—21no—
52#500C2312yes—
53#499—22no—
54#499—41no—
55#494—22no—
56#494—21no—
57#494C2312yes—
58#484—42no—
59#479—42no—
60#478—42no—
61#477—22no—
62#476—22no—
63#475—22no—
64#467—41no—
65#462—22no—
66#462—42no—
67C8×Q8—21no—
68C8×Q8C2312yes—
69C4×Q16—22no—
70C4×Q16—41no—
71C4×Q16D411yescharacteristic
72C4×Q16C2311yescharacteristic
73C4×D8—22no—
74C4×D8C2311yescharacteristic
75C8×D4—21no—
76C2×#107—41no—
77#367D411yescharacteristic
78#366—22no—
79C4⋊C16—22no—
80C4⋊C16C2312yes—
81#358—22no—
82#358—22no—
83#358—22no—
84#358—22no—
85#358—42no—
86#358D411yescharacteristic
87#357D411yescharacteristic
88#346—22no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335D411yescharacteristic
92#335C2311yescharacteristic
93#335C2311yescharacteristic
94#333—22no—
95#332—22no—
96Q8⋊C8—21no—
97Q8⋊C8D412yes—
98D4⋊C8—21no—
99C82C2311yescharacteristic
100C2×Q16—22no—
101C2×Q16—22no—
102C2×Q16—42no—
103C2×Q16—81no—
104C2×Q16D811yescharacteristic
105C2×Q16C2×D411yescharacteristic
106C2×D8C2×D411yescharacteristic
107C22×C8—41no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—42no—
111#125—42no—
112#125—42no—
113#125—82no—
114C22⋊Q8—42no—
115C22⋊Q8—82no—
116C4×Q8—21no—
117C4×Q8—42no—
118C4×Q8—42no—
119C4×Q8C2×D412yes—
120C4×D4—21no—
121C4×D4—42no—
122C2×C4⋊C4—42no—
123SD32—22no—
124SD32—44no—
125C2×C16—22no—
126C2×C16—21no—
127C2×C16C2×D411yescharacteristic
128C2×C16C2×D411yescharacteristic
129#107—22no—
130#107—22no—
131#107—41no—
132#107—41no—
133#107—41no—
134#107—41no—
135#107—42no—
136#107—81no—
137#107D811yescharacteristic
138#107C2×D411yescharacteristic
139#107C2×D411yescharacteristic
140#106—42no—
141#106—42no—
142C4⋊C8—22no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—41no—
146C4⋊C8C2×D412yes—
147#103—21no—
148#103—42no—
149#103—42no—
150#103—42no—
151#103D812yes—
152#102—21no—
153#102—42no—
154C22⋊C8—41no—
155C4×C8—22no—
156C4×C8—41no—
157C4×C8C2×D411yescharacteristic
158C4×C8C2×D411yescharacteristic
159C4×C8C2411yescharacteristic
160C2×Q8—21no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8C2×D812yes—
164C2×D4—21no—
165C22×C4—41no—
166C22×C4—82no—
167Q16—21no—
168Q16—21no—
169Q16—22no—
170Q16—42no—
171Q16—42no—
172Q16—82no—
173Q16—162no—
174Q16—162no—
175D8—21no—
176D8#13512yes—
177C2×C8—21no—
178C2×C8—22no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—82no—
182C2×C8C2×D811yescharacteristic
183C2×C8C2×D811yescharacteristic
184C2×C8C22×D411yescharacteristic
185C2×C8H(5,𝔽2)11yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C4⋊C4C2×D812yes—
200C22⋊C4—41no—
201C22⋊C4—82no—
202C42—22no—
203C42—41no—
204C42—82no—
205C42C22×D411yescharacteristic
206C16—21no—
207C16—21no—
208C16—41no—
209C16—42no—
210C23—41no—
211Q8—22no—
212Q8—22no—
213Q8—41no—
214Q8—41no—
215Q8—82no—
216Q8—82no—
217Q8—162no—
218D4—22no—
219D4—41no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—322no—
230C2×C4D4211yescharacteristic
231C2×C4C22×D811yescharacteristic
232C2×C4#57611yescharacteristic
233C8—22no—
234C8—21no—
235C8—21no—
236C8—42no—
237C8—81no—
238C8—81no—
239C8#51012yes—
240C8C2×#13511yescharacteristic
241C8#57811yescharacteristic
242C22—42no—
243C22—81no—
244C22—81no—
245C22D4×D811yescharacteristic
246C4—21no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—81no—
251C4—81no—
252C4—162no—
253C4—162no—
254C4#328511yescharacteristic
255C4#329011yescharacteristic
256C4C2×#51011yescharacteristic
257C4#341711yescharacteristic
258C2—82no—
259C2#1997311yescharacteristic
260C2#3384211yescharacteristic
261C2#3385611yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radicalC2×D8C2×D4
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. #20010C2semidirect
4. #12510C2semidirect
5. #12481C2semidirect
6. #12448C2semidirect
8. #12386C2semidirect
10. #12054C2semidirect

Maximal subgroups

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

subgroupindex
#339392
#200102
#200062
#125102
#124812
#124482
#124262
#123862
#123592
#120542
C8×SD322