Finite Group Databasebeta

← #510024

Subgroups — #510024

Order 512.

561 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 140 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×#2209C212yesmaximal
2#26255C212yesmaximal
3#13477C212yesmaximal
4#13450C212yesmaximal
5#13330C212yesmaximal
6#13240C211yesmaximal, characteristic
7#13240C211yesmaximal, characteristic
8#13160C211yesmaximal, characteristic
9#12938C212yesmaximal
10C22×Q32C2211yescharacteristic
11C22×Q32C2212yes—
12C22×Q32C2211yescharacteristic
13C2×#494C2212yes—
14C2×C4×Q16C2212yes—
15#2446—22no—
16#2209—28no—
17C2×#367C2211yescharacteristic
18C2×#367C2211yescharacteristic
19C2×C4⋊C16C2212yes—
20C2×C4⋊C16C2211yescharacteristic
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24C2×#358C2212yes—
25C2×#346C2211yescharacteristic
26#2056—22no—
27#2043—22no—
28#1969C2212yes—
29#1969C2212yes—
30#1969C2212yes—
31#1942C2212yes—
32#1878—22no—
33C8×C4⋊C4C2211yescharacteristic
34#1380C2212yes—
35#1374C2212yes—
36C22×Q16—22no—
37C22×Q16C2312yes—
38C22×Q16C2312yes—
39C2×C4⋊Q8—22no—
40C2×C4×Q8—22no—
41C2×Q32—24no—
42C2×Q32—22no—
43C2×Q32—41no—
44C2×Q32—41no—
45C2×Q32—41no—
46C2×Q32—41no—
47C2×Q32—42no—
48C2×Q32—42no—
49C2×Q32D414yes—
50C22×C16C2312yes—
51C22×C16C2311yescharacteristic
52C22×C16C2311yescharacteristic
53#494—24no—
54#494—42no—
55C4×Q16—24no—
56C4×Q16—42no—
57C2×#107C2312yes—
58C2×#107C2312yes—
59C2×C4⋊C8D411yescharacteristic
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#103—22no—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2312yes—
66#400—42no—
67#398—42no—
68#391—22no—
69#391—22no—
70#391—22no—
71#389—22no—
72#387—42no—
73#387—42no—
74C4×C4⋊C4D411yescharacteristic
75#367—41no—
76#367—41no—
77C4⋊C16—24no—
78C4⋊C16—41no—
79#358—24no—
80#358—24no—
81#358—24no—
82#358—42no—
83#346—41no—
84#340—22no—
85#336D412yes—
86C22×Q8—22no—
87C22×Q8—22no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—22no—
93C2×Q16—22no—
94C2×Q16—22no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—22no—
99C2×Q16—44no—
100C2×Q16—44no—
101C2×Q16—82no—
102C2×Q16C2×D414yes—
103C22×C8C2×D412yes—
104C22×C8C2×D411yescharacteristic
105C22×C8C2411yescharacteristic
106C4⋊Q8—44no—
107C4⋊Q8—82no—
108C4×Q8—44no—
109C4×Q8—82no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4D812yes—
117C2×C42—42no—
118C2×C42C2×D412yes—
119C2×C42C2×D411yescharacteristic
120Q32—44no—
121Q32—44no—
122Q32—48no—
123C2×C16—21no—
124C2×C16—21no—
125C2×C16—21no—
126C2×C16—21no—
127C2×C16—22no—
128C2×C16—22no—
129C2×C16—22no—
130C2×C16C2×D412yes—
131C2×C16#4112yes—
132#107—22no—
133#107—22no—
134#107—22no—
135#107—22no—
136C4⋊C8—41no—
137#103—44no—
138#103—82no—
139#103—82no—
140#103—82no—
141C4×C8—22no—
142C4×C8—21no—
143C4×C8—22no—
144C4×C8—21no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—42no—
149#95—42no—
150#95—42no—
151#95—42no—
152#95D811yescharacteristic
153#95D811yescharacteristic
154C2×Q8—24no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4C2×D811yescharacteristic
171C22×C4C2×D811yescharacteristic
172C22×C4C2×D812yes—
173C22×C4C2×D812yes—
174C22×C4C22×D411yescharacteristic
175Q16—24no—
176Q16—24no—
177Q16—42no—
178Q16—42no—
179Q16—42no—
180Q16—42no—
181Q16—42no—
182Q16—42no—
183Q16—88no—
184C2×C8—21no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—22no—
190C2×C8C22×D411yescharacteristic
191C2×C8C2×#4111yescharacteristic
192C2×C8H(5,𝔽2)11yescharacteristic
193C2×C8H(5,𝔽2)11yescharacteristic
194C2×C8H(5,𝔽2)11yescharacteristic
195C2×C82−1+411yescharacteristic
196C4⋊C4—24no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—84no—
202C4⋊C4—84no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C42—21no—
206C42—22no—
207C42—21no—
208C42—22no—
209C42—84no—
210C16—22no—
211C16—22no—
212C16—42no—
213C23C22×D811yescharacteristic
214Q8—44no—
215Q8—44no—
216Q8—82no—
217Q8—82no—
218Q8—82no—
219Q8—82no—
220Q8—82no—
221Q8—82no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—44no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4#51014yes—
244C2×C4#53711yescharacteristic
245C2×C4D4211yescharacteristic
246C2×C4#55011yescharacteristic
247C2×C4#57511yescharacteristic
248C2×C4#57611yescharacteristic
249C2×C4#57611yescharacteristic
250C8—41no—
251C8—42no—
252C8#53411yescharacteristic
253C8#53811yescharacteristic
254C8#54711yescharacteristic
255C8#56211yescharacteristic
256C22#324511yescharacteristic
257C22D4×D811yescharacteristic
258C22#339111yescharacteristic
259C22C2×#51012yes—
260C22#341712yes—
261C4—24no—
262C4—41no—
263C4—41no—
264C4—42no—
265C4—84no—
266C4—162no—
267C4—162no—
268C4—162no—
269C4#268211yescharacteristic
270C4#325911yescharacteristic
271C4#327311yescharacteristic
272C4#340311yescharacteristic
273C2#3120811yescharacteristic
274C2#3382111yescharacteristic
275C2#3382711yescharacteristic
276C2#3383612yes—
277C2#3396512yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ 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
C2×#22092
#262552
#134772
#134502
#133302
#132402
#131602
#129382