Finite Group Databasebeta

← #508459

Subgroups — #508459

Order 512.

489 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 150 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×#2229C211yesmaximal, characteristic
2C2×#2227C211yesmaximal, characteristic
3#33319C211yesmaximal, characteristic
4#21281C211yesmaximal, characteristic
5#13507C212yesmaximal
6#13505C212yesmaximal
7#13483C212yesmaximal
8#13481C212yesmaximal
9#13277C211yesmaximal, characteristic
10#13275C211yesmaximal, characteristic
11#12880C211yesmaximal, characteristic
12C2×#501—21no—
13C2×#500C2211yescharacteristic
14C2×#499C2211yescharacteristic
15C2×C4×Q16C2211yescharacteristic
16#2660—21no—
17#2320—21no—
18#2229—24no—
19#2227—24no—
20C2×#367C2211yescharacteristic
21C2×#366C2212yes—
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#358C2212yes—
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28#1969C2212yes—
29#1942C2212yes—
30#1935—22no—
31#1847—22no—
32#1846C2212yes—
33#1845C2212yes—
34C4×#107C2211yescharacteristic
35#1763—21no—
36#1726C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2212yes—
42#1374C2212yes—
43C22×Q16C2311yescharacteristic
44C2×C4⋊Q8—21no—
45C2×#125—21no—
46C2×C4×Q8—21no—
47C22×C16C2312yes—
48C22×C16C2311yescharacteristic
49C22×C16C2311yescharacteristic
50#501—44no—
51#500—22no—
52#500—22no—
53#499—22no—
54#499—22no—
55C4×Q16—22no—
56C4×Q16D414yes—
57C2×#107—22no—
58C2×#107—21no—
59C2×#107C2311yescharacteristic
60C2×#107C2312yes—
61C2×#107C2311yescharacteristic
62C2×#107C2312yes—
63C2×#107C2311yescharacteristic
64C2×#106—22no—
65C2×#106—21no—
66C2×#103—21no—
67C2×#103—22no—
68C2×#97D412yes—
69C2×#97D411yescharacteristic
70C2×C4×C8C2312yes—
71C2×C4×C8C2311yescharacteristic
72#398—41no—
73#398—41no—
74#395—42no—
75#391—22no—
76#389—22no—
77#387—42no—
78#384—22no—
79#383—41no—
80#382—22no—
81#382—41no—
82C4×C4⋊C4—21no—
83#376D411yescharacteristic
84#367—22no—
85#366—24no—
86#366—22no—
87C4⋊C16—22no—
88C4⋊C16—22no—
89#358—24no—
90#358—22no—
91#358—22no—
92#340—21no—
93#340—22no—
94C22×Q8—21no—
95C22×Q8—41no—
96C2×Q16—21no—
97C2×Q16—22no—
98C2×Q16—22no—
99C2×Q16C2×D412yes—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D412yes—
102C22×C8C2411yescharacteristic
103C4⋊Q8—42no—
104C4⋊Q8—42no—
105#125—42no—
106#125—42no—
107C4×Q8—24no—
108C4×Q8—42no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42C2×D412yes—
124C2×C42C2×D411yescharacteristic
125C2×C16—22no—
126C2×C16—21no—
127C2×C16—21no—
128C2×C16—24no—
129C2×C16—22no—
130C2×C16—22no—
131#107—21no—
132#107—21no—
133#107—22no—
134#107—21no—
135#107—21no—
136#107—21no—
137#107—22no—
138#107—22no—
139#107—22no—
140#107—44no—
141#107—42no—
142#107C2×D412yes—
143#106—44no—
144#106—42no—
145#103—24no—
146#103—44no—
147#97—24no—
148#97—22no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8C2×D412yes—
152C4×C8#4112yes—
153#95—41no—
154#95—42no—
155#95—42no—
156#95—42no—
157#95—42no—
158#95—42no—
159#95D811yescharacteristic
160#95D811yescharacteristic
161#95D812yes—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—82no—
167C2×Q8—161no—
168C2×Q8—161no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4C2×D812yes—
176C22×C4C2×D812yes—
177C22×C4C2×D811yescharacteristic
178C22×C4C2×D811yescharacteristic
179C22×C4C22×D411yescharacteristic
180Q16—22no—
181Q16—22no—
182Q16—42no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8#12112yes—
187C2×C8#13714yes—
188C2×C8C22×D411yescharacteristic
189C2×C8C2×#4111yescharacteristic
190C2×C82−1+412yes—
191C2×C82−1+412yes—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—44no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C4⋊C4—162no—
207C4⋊C4—162no—
208C4⋊C4—162no—
209C42—22no—
210C42—22no—
211C42—44no—
212C42—161no—
213C42#12112yes—
214C42AGL(1,ℤ/8)12yes—
215C16—42no—
216C16—41no—
217C16—41no—
218C23C22×D811yescharacteristic
219Q8—42no—
220Q8—42no—
221Q8—82no—
222Q8—164no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—81no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4#45912yes—
249C2×C4#45912yes—
250C2×C4C2×#12111yescharacteristic
251C2×C4#53612yes—
252C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
253C2×C4C2×#13712yes—
254C8—24no—
255C8—22no—
256C8#53611yescharacteristic
257C8#53611yescharacteristic
258C8#54112yes—
259C22C2×#45911yescharacteristic
260C22#322612yes—
261C22#341511yescharacteristic
262C22#341611yescharacteristic
263C22#341712yes—
264C4—22no—
265C4—22no—
266C4—42no—
267C4—42no—
268C4—82no—
269C4—82no—
270C4—162no—
271C4—161no—
272C4—161no—
273C4—162no—
274C4#262111yescharacteristic
275C4#303911yescharacteristic
276C4#323912yes—
277C2#2986411yescharacteristic
278C2#3358811yescharacteristic
279C2#3358911yescharacteristic
280C2#3380812yes—
281C2#3380912yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ 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×#22292
C2×#22272
#333192
#212812
#135072
#135052
#134832
#134812
#132772
#132752
#128802