Finite Group Databasebeta

← #326590

Subgroups — #326590

Order 512.

446 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 126 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#22304C211yesmaximal, characteristic, complemented
2#22282C211yesmaximal, characteristic
3#10667C211yesmaximal, characteristic, complemented
4#10666C211yesmaximal, characteristic, complemented
5#10372C211yesmaximal, characteristic, complemented
6#10372C211yesmaximal, characteristic, complemented
7#9753C212yesmaximal
8#9632C212yesmaximal, complemented
9#8880C212yesmaximal, complemented
10#8878C212yesmaximal, complemented
11#8783C211yesmaximal, characteristic, complemented
12#3147C2211yescharacteristic
13#2932C2211yescharacteristic
14#2932C2211yescharacteristic
15#2402C2211yescharacteristic
16#1963C2211yescharacteristic
17#1963C2211yescharacteristic
18#1940—22no—
19#1940C2212yes—
20#1917—22no—
21#1876—22no—
22#1876—22no—
23#1843C2211yescharacteristic
24#1837C2211yescharacteristic
25#1770C2212yes—
26#1746C2211yescharacteristic
27#1571C2212yes—
28#1570C2212yes—
29#1564C2212yes—
30#1502C2212yes—
31#1502C2212yes—
32#1480C2212yes—
33#1480C2212yes—
34#1303C2212yes—
35#1303C2212yes—
36#1281C2212yes—
37#1274C2211yescharacteristic
38#1274C2211yescharacteristic
39C2×#135—22no—
40C2×#131—21no—
41C2×#131—21no—
42#532C2311yescharacteristic
43#532C2311yescharacteristic
44#495—22no—
45#494Q811yescharacteristic
46#493Q811yescharacteristic
47#492Q811yescharacteristic
48#492Q811yescharacteristic
49#435—42no—
50#435—42no—
51#431C2312yes—
52#431C2312yes—
53#423C2311yescharacteristic
54#423C2311yescharacteristic
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×#104—22no—
59C2×#104—22no—
60#408—21no—
61#408—21no—
62C2×#97—22no—
63C2×#97C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#400—22no—
66#398—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4C2312yes—
69#376C2311yescharacteristic
70#341—22no—
71#336—22no—
72#336—22no—
73#336D412yes—
74#336D412yes—
75#335—22no—
76#335—42no—
77#334—22no—
78#334—42no—
79#327—22no—
80#327—22no—
81#327—22no—
82#327—22no—
83D4⋊C8—42no—
84D4⋊C8—42no—
85C2×#41—21no—
86C2×#41—21no—
87#135—82no—
88#135—82no—
89C2×Q16—22no—
90C2×SD16—22no—
91C2×SD16—22no—
92C2×D8—22no—
93#131—82no—
94#131—82no—
95C2×M16—21no—
96C2×M16—21no—
97C2×M16—41no—
98C2×M16—41no—
99C22×C8—22no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D412yes—
105C4⋊Q8C2×Q811yescharacteristic
106C4⋊Q8C2×Q811yescharacteristic
107#127C2×Q811yescharacteristic
108#127C2×Q811yescharacteristic
109#124—21no—
110#124—21no—
111#121—22no—
112#121—22no—
113C4×D4—22no—
114C4×D4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C42—22no—
119C2×C42C2×D412yes—
120C2×C42C2411yescharacteristic
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—82no—
128C4⋊C8—82no—
129#104—82no—
130#104—82no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135#97—22no—
136#97—82no—
137C4×C8—22no—
138C4×C8—22no—
139C4×C8C2×Q811yescharacteristic
140C4×C8C2×Q811yescharacteristic
141C4×C8#4112yes—
142#95—22no—
143#95—22no—
144#95—22no—
145#95#4112yes—
146#95#4112yes—
147#41—42no—
148#41—42no—
149#41—42no—
150#41—42no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—41no—
158C2×D4—41no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4C22×D411yescharacteristic
165C22×C4C2×#4112yes—
166Q16—82no—
167SD16—82no—
168SD16—82no—
169D8—82no—
170M16—42no—
171M16—42no—
172M16—82no—
173M16—82no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8C22⋊Q812yes—
190C2×C8C22⋊Q812yes—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C42—22no—
199C42—22no—
200C42—42no—
201C42—42no—
202C42C22×Q811yescharacteristic
203C42C2×#4111yescharacteristic
204C42H(5,𝔽2)12yes—
205C23—41no—
206C23—41no—
207C23#52511yescharacteristic
208Q8—42no—
209Q8—42no—
210D4—42no—
211D4—42no—
212D4—82no—
213D4—82no—
214D4—82no—
215D4—82no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4C2×C22⋊Q811yescharacteristic
233C2×C4#53511yescharacteristic
234C2×C4#55412yes—
235C2×C4#55912yes—
236C2×C4#57612yes—
237C2×C4#57712yes—
238C8—22no—
239C8—22no—
240C8—41no—
241C8—41no—
242C8—82no—
243C8—82no—
244C8—82no—
245C8—81no—
246C8—81no—
247C22—22no—
248C22—81no—
249C22—81no—
250C22—82no—
251C22—82no—
252C22#258011yescharacteristic
253C22#311912yes—
254C4—22no—
255C4—22no—
256C4—82no—
257C4—82no—
258C4—82no—
259C4—81no—
260C4—81no—
261C4#308112yes—
262C4#322011yescharacteristic
263C4#322211yescharacteristic
264C2—22no—
265C2—161no—
266C2—161no—
267C2#2926511yescharacteristic
268C2#2927112yes—
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2580
prime radical#3147C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22304C2semidirect
3. #10667C2semidirect
4. #10666C2semidirect
5. #10372C2semidirect
6. #10372C2semidirect
8. #9632C2semidirect
9. #8880C2semidirect
10. #8878C2semidirect
11. #8783C2semidirect

Maximal subgroups

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

subgroupindex
#223042
#222822
#106672
#106662
#103722
#97532
#96322
#88802
#88782
#87832