Finite Group Databasebeta

← #492780

Subgroups — #492780

Order 512.

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

normal subgroups only

This group has 397 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#33948C212yesmaximal
2#20010C211yesmaximal, characteristic, complemented
3#20006C211yesmaximal, characteristic
4#12523C211yesmaximal, characteristic, complemented
5#12459C212yesmaximal, complemented
6#12448C211yesmaximal, characteristic, complemented
7#12445C211yesmaximal, characteristic
8#12409C212yesmaximal, complemented
9#12407C212yesmaximal
10C16×Q16C211yesmaximal, characteristic, complemented
11C8×SD32C211yesmaximal, characteristic
12#3395—21no—
13#3383C2212yes—
14#3380C2212yes—
15#3338—21no—
16#2254C2212yes—
17#2254C2212yes—
18#2253—22no—
19#2229—22no—
20#2228—22no—
21#2227C2212yes—
22#2224C2212yes—
23Q8×C16C2212yes—
24C4×SD32—22no—
25C4×SD32C2211yescharacteristic
26#1713C2211yescharacteristic
27#1713C2211yescharacteristic
28#1676C2212yes—
29#1675—21no—
30#1674C2212yes—
31#1670—21no—
32#1648—22no—
33#1632—22no—
34C8×Q16C2211yescharacteristic
35C8×Q16C2211yescharacteristic
36C8×D8C2211yescharacteristic
37#1372C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1364C2212yes—
41Q8⋊C16C2212yes—
42#1332C2211yescharacteristic
43#1331C2211yescharacteristic
44C8×C16C2211yescharacteristic
45#557—22no—
46#554—22no—
47C2×SD32D411yescharacteristic
48#500—21no—
49#500—21no—
50#500C2312yes—
51#500C2312yes—
52#499—22no—
53#499—22no—
54#494—21no—
55#494—21no—
56#494—41no—
57#484—42no—
58#479—42no—
59#478—42no—
60#477—22no—
61#476—22no—
62#475—22no—
63#467—41no—
64#462—22no—
65#462—42no—
66C8×Q8—21no—
67C8×Q8C2312yes—
68C4×Q16—22no—
69C4×Q16—41no—
70C4×Q16D411yescharacteristic
71C4×Q16C2311yescharacteristic
72C4×D8—22no—
73C4×D8C2311yescharacteristic
74C8×D4—21no—
75C2×#107—41no—
76#367—22no—
77#367—22no—
78#367—42no—
79#367D411yescharacteristic
80#366—22no—
81#366—22no—
82C4⋊C16—22no—
83C4⋊C16C2312yes—
84#358—22no—
85#358D411yescharacteristic
86#357—22no—
87#357D411yescharacteristic
88C4×C16—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—42no—
101C2×Q16—41no—
102C2×Q16—41no—
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—
123SD32Q1614yescomplemented
124C2×C16—21no—
125C2×C16—22no—
126C2×C16C2×D411yescharacteristic
127C2×C16C2×D411yescharacteristic
128#107—22no—
129#107—22no—
130#107—22no—
131#107—22no—
132#107—41no—
133#107—41no—
134#107—42no—
135#107—81no—
136#107D811yescharacteristic
137#107C2×D411yescharacteristic
138#107C2×D411yescharacteristic
139#106—42no—
140#106—42no—
141C4⋊C8—22no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—41no—
145C4⋊C8C2×D412yes—
146#103—21no—
147#103—42no—
148#103—42no—
149#103—42no—
150#103D812yes—
151#102—21no—
152#102—42no—
153C22⋊C8—41no—
154C4×C8—22no—
155C4×C8—41no—
156C4×C8C2×D411yescharacteristic
157C4×C8C2×D411yescharacteristic
158C4×C8C2411yescharacteristic
159C2×Q8—21no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8C2×D812yes—
163C2×D4—21no—
164C22×C4—41no—
165C22×C4—82no—
166Q16—21no—
167Q16—22no—
168Q16—82no—
169Q16—161no—
170Q16—162no—
171Q16—162no—
172Q16C2×Q1612yes—
173D8—21no—
174D8C2×Q1612yes—
175C2×C8—21no—
176C2×C8—22no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—82no—
180C2×C8C2×D811yescharacteristic
181C2×C8C2×D811yescharacteristic
182C2×C8C22×D411yescharacteristic
183C2×C8H(5,𝔽2)11yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4C2×D812yes—
198C22⋊C4—41no—
199C22⋊C4—82no—
200C42—22no—
201C42—41no—
202C42—82no—
203C42C22×D411yescharacteristic
204C16—22no—
205C16—21no—
206C16—42no—
207C16C2×Q1612yes—
208C23—41no—
209Q8—22no—
210Q8—22no—
211Q8—22no—
212Q8—41no—
213Q8—82no—
214Q8—82no—
215Q8—162no—
216D4—22no—
217D4—41no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—322no—
228C2×C4D4211yescharacteristic
229C2×C4C22×D811yescharacteristic
230C2×C4#57611yescharacteristic
231C8—22no—
232C8—21no—
233C8—21no—
234C8—42no—
235C8—81no—
236C8—81no—
237C8#50812yes—
238C8C22×Q1611yescharacteristic
239C8#57811yescharacteristic
240C22—42no—
241C22—81no—
242C22—81no—
243C22D4×D811yescharacteristic
244C4—21no—
245C4—42no—
246C4—42no—
247C4—42no—
248C4—81no—
249C4—81no—
250C4—162no—
251C4—162no—
252C4D4×Q1611yescharacteristic
253C4#329311yescharacteristic
254C4C2×#50811yescharacteristic
255C4#341611yescharacteristic
256C2—82no—
257C2D8×Q1611yescharacteristic
258C2#3385011yescharacteristic
259C2#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. #12523C2semidirect
5. #12459C2semidirect
6. #12448C2semidirect
8. #12409C2semidirect
10. C16×Q16C2semidirect
123. SD32Q16semidirect

Maximal subgroups

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

subgroupindex
#339482
#200102
#200062
#125232
#124592
#124482
#124452
#124092
#124072
C16×Q162
C8×SD322