Finite Group Databasebeta

← #492793

Subgroups — #492793

Order 512.

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

normal subgroups only

This group has 420 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#33953C212yesmaximal
2#20010C211yesmaximal, characteristic, complemented
3D8×Q16C211yesmaximal, characteristic
4#12521C211yesmaximal, characteristic, complemented
5#12460C212yesmaximal, complemented
6#12446C211yesmaximal, characteristic, complemented
7#12445C211yesmaximal, characteristic
8#12405C212yesmaximal, complemented
9#12404C212yesmaximal
10#12054C211yesmaximal, characteristic, complemented
11C8×SD32C211yesmaximal, characteristic
12#3395—21no—
13#3383C2212yes—
14Q8×D8C2212yes—
15D4×Q16—21no—
16#2255—22no—
17#2252C2212yes—
18#2252C2212yes—
19#2228—22no—
20#2227—22no—
21#2225C2212yes—
22#2224C2212yes—
23#2183C2212yes—
24#2175—22no—
25C4×SD32C2211yescharacteristic
26#1713C2211yescharacteristic
27#1713C2211yescharacteristic
28#1674C2212yes—
29#1674C2212yes—
30#1670—21no—
31#1669—21no—
32#1649—22no—
33#1648—22no—
34C8×Q16C2211yescharacteristic
35C8×Q16C2211yescharacteristic
36C8×D8C2211yescharacteristic
37#1372C2211yescharacteristic
38#1370C2211yescharacteristic
39#1364C2212yes—
40#1364C2212yes—
41Q8⋊C16C2212yes—
42#1332C2211yescharacteristic
43#1331C2211yescharacteristic
44C8×C16C2211yescharacteristic
45C22×Q16—41no—
46#557—22no—
47D4×Q8—22no—
48C2×SD32D411yescharacteristic
49#500—22no—
50#500—21no—
51#500—21no—
52#500C2312yes—
53#500C2312yes—
54#499—22no—
55#494—21no—
56#494—21no—
57#494—41no—
58#479—42no—
59#475—22no—
60#474—22no—
61#474—42no—
62#467—41no—
63#462—22no—
64#462—22no—
65#462—42no—
66#451—42no—
67C8×Q8—21no—
68C8×Q8C2312yes—
69C4×Q16—22no—
70C4×Q16—41no—
71C4×Q16D411yescharacteristic
72C4×Q16C2311yescharacteristic
73C4×D8—22no—
74C4×D8C2311yescharacteristic
75C8×D4—21no—
76#367—22no—
77#367D411yescharacteristic
78#366—22no—
79#366—22no—
80#366—22no—
81#366—42no—
82C4⋊C16—22no—
83C4⋊C16C2312yes—
84#358—22no—
85#358D411yescharacteristic
86#357—22no—
87#357D411yescharacteristic
88#346—22no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335D411yescharacteristic
92#335C2311yescharacteristic
93#335C2311yescharacteristic
94#332—22no—
95#332—22no—
96Q8⋊C8—21no—
97Q8⋊C8D412yes—
98D4⋊C8—21no—
99C82C2311yescharacteristic
100C22×Q8—42no—
101C2×Q16—42no—
102C2×Q16—42no—
103C2×Q16—41no—
104C2×Q16—41no—
105C2×Q16—41no—
106C2×Q16—41no—
107C2×Q16—84no—
108C2×Q16D811yescharacteristic
109C2×Q16C2×D411yescharacteristic
110C2×D8C2×D411yescharacteristic
111C22×C8—41no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—42no—
115C4⋊Q8—42no—
116#125—42no—
117#125—82no—
118C22⋊Q8—42no—
119C22⋊Q8—82no—
120C4×Q8—21no—
121C4×Q8—42no—
122C4×Q8—42no—
123C4×Q8C2×D412yes—
124C4×D4—21no—
125C4×D4—42no—
126SD32Q1614yescomplemented
127C2×C16—21no—
128C2×C16—22no—
129C2×C16C2×D411yescharacteristic
130C2×C16C2×D411yescharacteristic
131#107—22no—
132#107—22no—
133#107—22no—
134#107—22no—
135#107—42no—
136#107—81no—
137#107—81no—
138#107D811yescharacteristic
139#107C2×D411yescharacteristic
140#107C2×D411yescharacteristic
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×Q8—82no—
164C2×Q8—84no—
165C2×Q8—162no—
166C2×Q8C2×D812yes—
167C2×D4—21no—
168C22×C4—41no—
169C22×C4—82no—
170Q16—21no—
171Q16—22no—
172Q16—84no—
173Q16—82no—
174Q16—82no—
175Q16—161no—
176Q16—161no—
177Q16—162no—
178Q16C2×Q1612yescomplemented
179D8—21no—
180D8C2×Q1612yes—
181C2×C8—21no—
182C2×C8—22no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—82no—
186C2×C8C2×D811yescharacteristic
187C2×C8C2×D811yescharacteristic
188C2×C8C22×D411yescharacteristic
189C2×C8H(5,𝔽2)11yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C4⋊C4C2×D812yes—
203C22⋊C4—41no—
204C22⋊C4—82no—
205C42—22no—
206C42—41no—
207C42—82no—
208C42C22×D411yescharacteristic
209C16—22no—
210C16—21no—
211C16—42no—
212C16C2×Q1612yescomplemented
213C23—41no—
214Q8—22no—
215Q8—22no—
216Q8—22no—
217Q8—41no—
218Q8—82no—
219Q8—82no—
220Q8—82no—
221Q8—82no—
222Q8—164no—
223D4—22no—
224D4—41no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—164no—
235C2×C4D4211yescharacteristic
236C2×C4C22×D811yescharacteristic
237C2×C4#57611yescharacteristic
238C8—22no—
239C8—21no—
240C8—21no—
241C8—42no—
242C8—81no—
243C8—81no—
244C8#50912yes—
245C8C22×Q1611yescharacteristic
246C8#57811yescharacteristic
247C22—42no—
248C22—81no—
249C22—81no—
250C22D4×D811yescharacteristic
251C4—21no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—81no—
256C4—81no—
257C4—162no—
258C4—162no—
259C4D4×Q1611yescharacteristic
260C4#329311yescharacteristic
261C4C2×#50911yescharacteristic
262C4#341511yescharacteristic
263C2—82no—
264C2D8×Q1611yescharacteristic
265C2#3384011yescharacteristic
266C2#3386011yescharacteristic
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. #12521C2semidirect
5. #12460C2semidirect
6. #12446C2semidirect
8. #12405C2semidirect
10. #12054C2semidirect
126. SD32Q16semidirect
178. Q16C2×Q16semidirect
212. C16C2×Q16semidirect

Maximal subgroups

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

subgroupindex
#339532
#200102
D8×Q162
#125212
#124602
#124462
#124452
#124052
#124042
#120542
C8×SD322