Finite Group Databasebeta

← #512167

Subgroups — #512167

Order 512.

628 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C2×#2227C211yesmaximal, characteristic
2C2×#2209C211yesmaximal, characteristic
3#33242C211yesmaximal, characteristic
4C42×Q16C211yesmaximal, characteristic
5#13276C211yesmaximal, characteristic
6#13253C211yesmaximal, characteristic
7#13133C214yesmaximal
8#13015C214yesmaximal
9#12879C211yesmaximal, characteristic
10C22×Q32C2212yes—
11C2×#500C2211yescharacteristic
12C2×#494C2211yescharacteristic
13C2×#487—21no—
14C2×C4×Q16C2212yes—
15C2×C4×Q16C2211yescharacteristic
16#2862—21no—
17C42×Q8—21no—
18#2227—24no—
19#2209—24no—
20C2×#366C2212yes—
21C2×C4⋊C16C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×#358C2212yes—
24C2×#358C2214yes—
25C2×#358C2212yes—
26#1969—22no—
27#1969C2214yes—
28C4×#107C2212yes—
29C4×#107C2211yescharacteristic
30C4×#103—21no—
31C4×#103—22no—
32C42×C8C2211yescharacteristic
33#1380C2214yes—
34#1374C2214yes—
35C22×Q16—21no—
36C22×Q16C2312yes—
37C22×Q16C2311yescharacteristic
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4×Q8—22no—
41C2×C4×Q8—21no—
42C2×Q32—24no—
43C2×Q32—28no—
44C22×C16C2312yes—
45C22×C16C2312yes—
46#500—22no—
47#500—22no—
48#494—22no—
49#494—22no—
50#487—44no—
51C4×Q16—24no—
52C4×Q16—24no—
53C4×Q16—22no—
54C4×Q16D414yes—
55C2×#107—21no—
56C2×#107C2312yes—
57C2×#107C2311yescharacteristic
58C2×#107C2312yes—
59C2×#103—22no—
60C2×#103—21no—
61C2×#103—24no—
62C2×C4×C8D411yescharacteristic
63C2×C4×C8D412yes—
64C2×C4×C8C2312yes—
65C2×C4×C8C2311yescharacteristic
66#391—24no—
67#391—42no—
68#382—41no—
69#382—41no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—41no—
73C4×C4⋊C4—42no—
74C43—41no—
75C43D411yescharacteristic
76#366—24no—
77C4⋊C16—22no—
78C4⋊C16—22no—
79#358—24no—
80#358—24no—
81#358—28no—
82#340—22no—
83#340—21no—
84C22×Q8—21no—
85C22×Q8—22no—
86C22×Q8—41no—
87C2×Q16—22no—
88C2×Q16—21no—
89C2×Q16—22no—
90C2×Q16—24no—
91C2×Q16—22no—
92C2×Q16—24no—
93C2×Q16—44no—
94C2×Q16—42no—
95C2×Q16C2×D412yes—
96C2×Q16#4114yes—
97C22×C8C2×D412yes—
98C22×C8C2×D411yescharacteristic
99C22×C8C2411yescharacteristic
100C4⋊Q8—42no—
101C4⋊Q8—42no—
102C4⋊Q8—42no—
103C4⋊Q8—42no—
104C4×Q8—28no—
105C4×Q8—24no—
106C4×Q8—44no—
107C4×Q8—42no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C42—42no—
117C2×C42—41no—
118C2×C42—42no—
119C2×C42—41no—
120C2×C42D811yescharacteristic
121C2×C42D812yes—
122C2×C42D811yescharacteristic
123C2×C42C2×D412yes—
124C2×C42C2×D411yescharacteristic
125Q32—48no—
126C2×C16—22no—
127C2×C16—22no—
128C2×C16—24no—
129C2×C16—24no—
130#107—21no—
131#107—22no—
132#107—22no—
133#107—22no—
134#107—42no—
135#107C2×D412yes—
136#107#4114yes—
137#103—28no—
138#103—24no—
139#103—48no—
140C4×C8—22no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—42no—
144C4×C8C2×D412yes—
145C4×C8#4112yes—
146#95—42no—
147#95—41no—
148#95—44no—
149#95—44no—
150C2×Q8—22no—
151C2×Q8—24no—
152C2×Q8—41no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—44no—
156C2×Q8—42no—
157C2×Q8—44no—
158C2×Q8—84no—
159C2×Q8—82no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4C2×D812yes—
166C22×C4C2×D811yescharacteristic
167C22×C4C2×D812yes—
168C22×C4C2×D811yescharacteristic
169C22×C4C22×D411yescharacteristic
170Q16—24no—
171Q16—22no—
172Q16—42no—
173Q16—44no—
174Q16—42no—
175Q16—84no—
176Q16#12414yes—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8#12112yes—
182C2×C8C22×D411yescharacteristic
183C2×C8C2×#4112yes—
184C2×C8C2×#4111yescharacteristic
185C2×C8C2×#4112yes—
186C4⋊C4—22no—
187C4⋊C4—24no—
188C4⋊C4—48no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C42—22no—
198C42—22no—
199C42—22no—
200C42—42no—
201C42—48no—
202C42—44no—
203C42—84no—
204C42—82no—
205C42Q3214yescomplemented
206C42#12112yes—
207C42AGL(1,ℤ/8)12yes—
208C16—42no—
209C16—42no—
210C23C22×D811yescharacteristic
211Q8—24no—
212Q8—44no—
213Q8—42no—
214Q8—82no—
215Q8—84no—
216Q8—82no—
217Q8—164no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—84no—
232C2×C4—84no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4#45912yes—
236C2×C4#45912yes—
237C2×C4C2×Q3214yes—
238C2×C4C2×Q3212yes—
239C2×C4#51012yes—
240C2×C4C2×#12111yescharacteristic
241C2×C4#52512yes—
242C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
243C2×C4#57512yes—
244C8—22no—
245C8—42no—
246C8C2×#12411yescharacteristic
247C8#52912yes—
248C8#53211yescharacteristic
249C22C2×#45911yescharacteristic
250C22#310012yes—
251C22C22×Q3211yescharacteristic
252C22C2×#51011yescharacteristic
253C22#341412yes—
254C4—22no—
255C4—24no—
256C4—44no—
257C4—42no—
258C4—82no—
259C4—84no—
260C4—162no—
261C4—162no—
262C4#220914yes—
263C4#285011yescharacteristic
264C4#304011yescharacteristic
265C4#311512yes—
266C2#3302411yescharacteristic
267C2C2×#220911yescharacteristic
268C2#3358411yescharacteristic
269C2#3366412yes—
270C2#3367612yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
205. C42Q32semidirect

Maximal subgroups

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

subgroupindex
C2×#22272
C2×#22092
#332422
C42×Q162
#132762
#132532
#131332
#130152
#128792