Finite Group Databasebeta

← #508721

Subgroups — #508721

Order 512.

520 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 146 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#33258C211yesmaximal, characteristic
2C4×#494C211yesmaximal, characteristic
3#13530C212yesmaximal
4#13505C212yesmaximal
5#13481C212yesmaximal
6#13409C212yesmaximal
7#12936C214yesmaximal
8#12894C211yesmaximal, characteristic
9C2×#494C2211yescharacteristic
10C2×C4×Q16—21no—
11C2×C4×Q16C2212yes—
12#2859—21no—
13C4×C4⋊Q8—21no—
14C2×#366C2214yes—
15C2×#358C2214yes—
16C2×#358C2214yes—
17#1969—21no—
18#1969C2212yes—
19#1956—21no—
20#1942—21no—
21#1942—21no—
22#1942C2212yes—
23#1942C2212yes—
24#1845C2211yescharacteristic
25#1845C2211yescharacteristic
26#1845C2211yescharacteristic
27#1834—21no—
28C42×C8C2211yescharacteristic
29#1380C2214yes—
30#1374C2212yes—
31#1374C2212yes—
32#1374C2212yes—
33C22×Q16—21no—
34C22×Q16C2312yes—
35C2×C4⋊Q8—21no—
36C2×#125—41no—
37C2×C4×Q8—22no—
38C22×C16C2314yes—
39#494—24no—
40C4×Q16—44no—
41C4×Q16—44no—
42C2×#107—21no—
43C2×#107—21no—
44C2×#107—21no—
45C2×#107C2312yes—
46C2×#107C2312yes—
47C2×#107C2312yes—
48C2×#103—22no—
49C2×#103—22no—
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#391—22no—
57#389—22no—
58#389—22no—
59#389—41no—
60#389—41no—
61#389—41no—
62#389—41no—
63#384—21no—
64#384—21no—
65#384—21no—
66#382—41no—
67C4×C4⋊C4—41no—
68C4×C4⋊C4—41no—
69C43D411yescharacteristic
70#366—44no—
71#358—28no—
72#358—44no—
73#340—22no—
74#340—22no—
75C22×Q8—22no—
76C2×Q16—24no—
77C2×Q16—22no—
78C2×Q16—24no—
79C2×Q16—24no—
80C2×Q16—42no—
81C2×Q16—42no—
82C2×Q16#4114yes—
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22×C8C2411yescharacteristic
87C4⋊Q8—24no—
88C4⋊Q8—42no—
89#125—84no—
90C4×Q8—44no—
91C4×Q8—44no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C42—42no—
107C2×C42D811yescharacteristic
108C2×C42D811yescharacteristic
109C2×C42D811yescharacteristic
110C2×C42D811yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C16—24no—
115C2×C16—24no—
116C2×C16—24no—
117#107—24no—
118#107—22no—
119#107—22no—
120#107—24no—
121#107—42no—
122#107—42no—
123#107—42no—
124#103—44no—
125#103—44no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—22no—
130C4×C8—22no—
131C4×C8—41no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136C2×Q8—24no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4C2×D811yescharacteristic
147C22×C4C2×D811yescharacteristic
148C22×C4C2×D811yescharacteristic
149C22×C4C2×D811yescharacteristic
150C22×C4C2×D811yescharacteristic
151C22×C4C2×D811yescharacteristic
152C22×C4C22×D411yescharacteristic
153Q16—28no—
154Q16—48no—
155Q16—44no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8AGL(1,ℤ/8)12yes—
164C2×C8#13712yes—
165C2×C8C2×#4112yes—
166C2×C8H(5,𝔽2)12yes—
167C2×C82−1+412yes—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—82no—
177C4⋊C4—82no—
178C4⋊C4—84no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—84no—
182C4⋊C4—82no—
183C4⋊C4—84no—
184C4⋊C4—82no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—22no—
192C42—22no—
193C42—41no—
194C42—84no—
195C42Q3214yes—
196C16—44no—
197C23C22×D811yescharacteristic
198Q8—44no—
199Q8—44no—
200Q8—82no—
201Q8—82no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—44no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4C2×Q3212yes—
228C2×C4C2×Q3212yes—
229C2×C4C2×Q3212yes—
230C2×C4#50912yes—
231C2×C4#51012yes—
232C2×C4#52511yescharacteristic
233C2×C4#53511yescharacteristic
234C2×C4#53611yescharacteristic
235C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
236C2×C4C2×#13711yescharacteristic
237C2×C4#57511yescharacteristic
238C8—22no—
239C8—22no—
240C8—41no—
241C8#53914yes—
242C22#310011yescharacteristic
243C22#320511yescharacteristic
244C22#322611yescharacteristic
245C22C22×Q3211yescharacteristic
246C22C2×#50911yescharacteristic
247C22C2×#51011yescharacteristic
248C22#341411yescharacteristic
249C4—22no—
250C4—22no—
251C4—22no—
252C4—22no—
253C4—22no—
254C4—41no—
255C4—84no—
256C4—162no—
257C4—162no—
258C4—162no—
259C4#263311yescharacteristic
260C4#311011yescharacteristic
261C4#321611yescharacteristic
262C4#323411yescharacteristic
263C2#3007511yescharacteristic
264C2#3366411yescharacteristic
265C2#3367211yescharacteristic
266C2#3377511yescharacteristic
267C2#3378011yescharacteristic
268C2#3379611yescharacteristic
269C2#3380211yescharacteristic
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

Maximal subgroups

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

subgroupindex
#332582
C4×#4942
#135302
#135052
#134812
#134092
#129362
#128942