Finite Group Databasebeta

← #508409

Subgroups — #508409

Order 512.

560 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 158 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×#2225C211yesmaximal, characteristic
2C2×#2209C211yesmaximal, characteristic
3#33242C211yesmaximal, characteristic
4#21276C211yesmaximal, characteristic
5#13477C214yesmaximal
6#13450C214yesmaximal
7#13275C211yesmaximal, characteristic
8#13253C211yesmaximal, characteristic
9#12898C211yesmaximal, characteristic
10C22×Q32C2212yes—
11C2×#500C2211yescharacteristic
12C2×#494C2211yescharacteristic
13C2×#487—21no—
14C2×C4×Q16C2211yescharacteristic
15#2862—21no—
16#2295—21no—
17#2225—24no—
18#2209—24no—
19C2×#367C2212yes—
20C2×C4⋊C16C2211yescharacteristic
21C2×C4⋊C16C2211yescharacteristic
22C2×C4⋊C16C2214yes—
23C2×#358C2212yes—
24C2×#358C2212yes—
25C2×#358C2214yes—
26#1969—22no—
27#1969C2212yes—
28#1969C2214yes—
29#1942C2212yes—
30#1933—22no—
31C4×#107C2211yescharacteristic
32C4×#103—21no—
33C42×C8C2211yescharacteristic
34#1380C2214yes—
35C22×Q16—21no—
36C22×Q16C2312yes—
37C22×Q16C2311yescharacteristic
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4×Q8—21no—
41C2×Q32—24no—
42C2×Q32—44no—
43C22×C16C2312yes—
44C22×C16C2312yes—
45#500—22no—
46#500—22no—
47#494—22no—
48#494—22no—
49#487—44no—
50C4×Q16—22no—
51C4×Q16D414yes—
52C2×#107—21no—
53C2×#107C2312yes—
54C2×#107C2312yes—
55C2×#107C2311yescharacteristic
56C2×#103—21no—
57C2×#103—24no—
58C2×#103—22no—
59C2×C4×C8D411yescharacteristic
60C2×C4×C8D412yes—
61C2×C4×C8C2312yes—
62C2×C4×C8C2311yescharacteristic
63#391—24no—
64#391—22no—
65#391—41no—
66#391—41no—
67#389—22no—
68#387—42no—
69#382—42no—
70C4×C4⋊C4—21no—
71C4×C4⋊C4—41no—
72#376—41no—
73C43D411yescharacteristic
74#367—24no—
75C4⋊C16—22no—
76C4⋊C16—22no—
77C4⋊C16—44no—
78#358—24no—
79#358—24no—
80#358—44no—
81#340—21no—
82#340—22no—
83C22×Q8—21no—
84C22×Q8—22no—
85C22×Q8—41no—
86C2×Q16—22no—
87C2×Q16—21no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—24no—
91C2×Q16—22no—
92C2×Q16—24no—
93C2×Q16—44no—
94C2×Q16—42no—
95C2×Q16C2×D412yes—
96C22×C8C2×D412yes—
97C22×C8C2×D411yescharacteristic
98C22×C8C2411yescharacteristic
99C4⋊Q8—42no—
100C4⋊Q8—42no—
101C4⋊Q8—42no—
102C4⋊Q8—42no—
103C4×Q8—24no—
104C4×Q8—42no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42D811yescharacteristic
116C2×C42D811yescharacteristic
117C2×C42D812yes—
118C2×C42C2×D412yes—
119C2×C42C2×D411yescharacteristic
120Q32—48no—
121C2×C16—22no—
122C2×C16—22no—
123C2×C16—24no—
124C2×C16—24no—
125#107—21no—
126#107—22no—
127#107—22no—
128#107—22no—
129#107—22no—
130#107—42no—
131#107C2×D412yes—
132#103—24no—
133#103—44no—
134#103—48no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—42no—
139C4×C8C2×D412yes—
140C4×C8#4112yes—
141#95—42no—
142#95—42no—
143#95—41no—
144#95—42no—
145#95—44no—
146#95—44no—
147C2×Q8—22no—
148C2×Q8—41no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—44no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—44no—
155C2×Q8—82no—
156C2×Q8—82no—
157C2×Q8—82no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4C2×D812yes—
164C22×C4C2×D811yescharacteristic
165C22×C4C2×D812yes—
166C22×C4C2×D811yescharacteristic
167C22×C4C22×D411yescharacteristic
168Q16—22no—
169Q16—22no—
170Q16—42no—
171Q16—42no—
172Q16—42no—
173Q16—44no—
174Q16—84no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8#12112yes—
180C2×C8C22×D411yescharacteristic
181C2×C8C2×#4111yescharacteristic
182C2×C8H(5,𝔽2)12yes—
183C2×C82−1+412yes—
184C4⋊C4—22no—
185C4⋊C4—41no—
186C4⋊C4—44no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C4⋊C4—84no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42—24no—
199C42—22no—
200C42—44no—
201C42—82no—
202C42#12112yes—
203C42AGL(1,ℤ/8)12yes—
204C16—42no—
205C16—42no—
206C23C22×D811yescharacteristic
207Q8—42no—
208Q8—42no—
209Q8—82no—
210Q8—82no—
211Q8—84no—
212Q8—82no—
213Q8—164no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—84no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4#45912yes—
233C2×C4#45912yes—
234C2×C4#50812yes—
235C2×C4#51014yes—
236C2×C4#51012yes—
237C2×C4C2×#12111yescharacteristic
238C2×C4#53712yes—
239C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
240C2×C4#57512yes—
241C8—22no—
242C8—42no—
243C8#53511yescharacteristic
244C8#53611yescharacteristic
245C8#54212yes—
246C22C2×#45911yescharacteristic
247C22#324512yes—
248C22C2×#50811yescharacteristic
249C22C2×#51012yes—
250C22C2×#51011yescharacteristic
251C4—24no—
252C4—22no—
253C4—42no—
254C4—82no—
255C4—82no—
256C4—162no—
257C4—162no—
258C4—162no—
259C4#221114yes—
260C4#262011yescharacteristic
261C4#304011yescharacteristic
262C4#325112yes—
263C2#2982411yescharacteristic
264C2C2×#221111yescharacteristic
265C2#3358311yescharacteristic
266C2#3381412yes—
267C2#3382112yes—
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
C2×#22252
C2×#22092
#332422
#212762
#134772
#134502
#132752
#132532
#128982