Finite Group Databasebeta

← #506347

Subgroups — #506347

Order 512.

545 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 152 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×#2229C212yesmaximal
2#21723C211yesmaximal, characteristic
3#21698C211yesmaximal, characteristic
4#13507C212yesmaximal
5#13409C212yesmaximal
6#13133C212yesmaximal
7#13132C212yesmaximal
8#12880C211yesmaximal, characteristic
9#12812C212yesmaximal
10C2×#499C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#441—21no—
13C2×C4×Q16C2212yes—
14#2330—21no—
15#2327—21no—
16#2229—28no—
17C2×#367C2212yes—
18C2×#366C2212yes—
19C2×C4⋊C16C2212yes—
20C2×#358C2212yes—
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24#1992—21no—
25#1971—21no—
26#1969C2212yes—
27#1960—21no—
28#1944—21no—
29#1942C2212yes—
30#1846C2211yescharacteristic
31#1845C2211yescharacteristic
32#1776—21no—
33C4×#107C2212yes—
34#1726C2211yescharacteristic
35#1380C2212yes—
36#1380C2212yes—
37#1374C2211yescharacteristic
38#1374C2211yescharacteristic
39#1374C2212yes—
40C22×Q16—21no—
41C22×Q16C2312yes—
42C2×C4⋊Q8—21no—
43C2×#125—21no—
44C2×C4×Q8—22no—
45C22×C16C2312yes—
46C22×C16C2312yes—
47#499—22no—
48#499D414yes—
49#494—24no—
50#441—44no—
51C4×Q16—24no—
52C4×Q16—24no—
53C2×#107—21no—
54C2×#107C2312yes—
55C2×#107C2312yes—
56C2×#107C2312yes—
57C2×#106—21no—
58C2×#106—21no—
59C2×#103—22no—
60C2×#103—22no—
61C2×#97D411yescharacteristic
62C2×#97D411yescharacteristic
63C2×#97D411yescharacteristic
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#391—22no—
68#389—22no—
69#389—42no—
70#387—42no—
71#387—42no—
72#387—42no—
73#384—21no—
74#382—21no—
75C4×C4⋊C4—22no—
76#376D411yescharacteristic
77#367—24no—
78#366—24no—
79C4⋊C16—24no—
80#358—24no—
81#358—24no—
82#358—24no—
83#358—24no—
84#340—22no—
85#340—22no—
86C22×Q8—22no—
87C2×Q16—24no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—22no—
93C2×Q16—42no—
94C2×Q16—42no—
95C2×Q16#4114yes—
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C22×C8C2411yescharacteristic
100C4⋊Q8—24no—
101C4⋊Q8—42no—
102#125—24no—
103#125—42no—
104C4×Q8—44no—
105C4×Q8—44no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C16—22no—
121C2×C16—22no—
122C2×C16—22no—
123C2×C16—22no—
124C2×C16—22no—
125C2×C16—22no—
126#107—22no—
127#107—22no—
128#107—22no—
129#107—22no—
130#107—22no—
131#107—42no—
132#107C2×D414yes—
133#106—24no—
134#106—42no—
135#103—44no—
136#103—44no—
137#97—22no—
138#97—22no—
139#97—41no—
140C4×C8—22no—
141C4×C8—22no—
142C4×C8C2×D412yes—
143C4×C8#4112yes—
144#95—42no—
145#95—41no—
146#95—41no—
147#95—42no—
148#95—44no—
149#95D812yes—
150#95D812yes—
151C2×Q8—24no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4C2×D812yes—
162C22×C4C2×D811yescharacteristic
163C22×C4C2×D811yescharacteristic
164C22×C4C2×D812yes—
165C22×C4C22×D411yescharacteristic
166Q16—24no—
167Q16—24no—
168Q16—48no—
169Q16—42no—
170Q16—42no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8#12112yes—
179C2×C8#13712yes—
180C2×C8C22×D411yescharacteristic
181C2×C8C2×#4112yes—
182C2×C8C2×#4111yescharacteristic
183C2×C82−1+412yes—
184C4⋊C4—24no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—42no—
188C4⋊C4—48no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—84no—
194C4⋊C4—162no—
195C4⋊C4—162no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—84no—
201C42—162no—
202C42#12112yes—
203C42AGL(1,ℤ/8)12yes—
204C16—42no—
205C16—42no—
206C23C22×D811yescharacteristic
207Q8—44no—
208Q8—44no—
209Q8—82no—
210Q8—82no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4#45914yes—
233C2×C4C2×#12111yescharacteristic
234C2×C4#52511yescharacteristic
235C2×C4#53611yescharacteristic
236C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
237C2×C4C2×#13711yescharacteristic
238C2×C4#57511yescharacteristic
239C8—22no—
240C8—22no—
241C8—41no—
242C8#54812yes—
243C8#55212yes—
244C22C2×#45911yescharacteristic
245C22#310011yescharacteristic
246C22#322611yescharacteristic
247C22#341612yes—
248C22#341712yes—
249C4—24no—
250C4—42no—
251C4—41no—
252C4—41no—
253C4—84no—
254C4—84no—
255C4—162no—
256C4—162no—
257C4#253611yescharacteristic
258C4#303711yescharacteristic
259C4#311011yescharacteristic
260C4#323511yescharacteristic
261C2#2820111yescharacteristic
262C2#3359012yes—
263C2#3368611yescharacteristic
264C2#3368711yescharacteristic
265C2#3380812yes—
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
#217232
#216982
#135072
#134092
#131332
#131322
#128802
#128122