Finite Group Databasebeta

← #244455

Subgroups — #244455

Order 512.

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

normal subgroups only

This group has 455 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#21716C211yesmaximal, characteristic, complemented
2#21663C211yesmaximal, characteristic
3#10689C212yesmaximal, complemented
4#10628C211yesmaximal, characteristic, complemented
5#10627C211yesmaximal, characteristic
6#10547C212yesmaximal
7#10371C211yesmaximal, characteristic, complemented
8#10370C211yesmaximal, characteristic
9#8254C212yesmaximal, complemented
10#8253C212yesmaximal
11#8139C211yesmaximal, characteristic, complemented
12#3135C2211yescharacteristic
13#2960C2211yescharacteristic
14C2×#446—21no—
15#2931C2211yescharacteristic
16#2399C2211yescharacteristic
17#1995C2211yescharacteristic
18#1985—21no—
19#1963C2211yescharacteristic
20#1940—21no—
21#1940C2212yes—
22#1937—21no—
23#1937—21no—
24#1843C2211yescharacteristic
25#1838C2211yescharacteristic
26#1770—21no—
27#1770C2212yes—
28#1765—21no—
29#1765—21no—
30#1726C2211yescharacteristic
31#1542C2212yes—
32#1540C2212yes—
33#1503C2212yes—
34#1502C2212yes—
35#1452C2212yes—
36#1303C2212yes—
37#1303C2212yes—
38#1301C2212yes—
39#1301C2212yes—
40#1275C2212yes—
41#1274C2211yescharacteristic
42#1274C2211yescharacteristic
43C2×#131—21no—
44#533C2311yescharacteristic
45#532C2311yescharacteristic
46C2×C4×Q8—21no—
47#491—21no—
48#490—21no—
49#489—22no—
50#446—22no—
51#446—44no—
52C8×Q8—22no—
53#436—22no—
54C8×D4—22no—
55#431C2312yes—
56#431C2312yes—
57#424—21no—
58#423C2311yescharacteristic
59#423C2311yescharacteristic
60C2×C4⋊C8—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2312yes—
64C2×#104—21no—
65C2×#104—21no—
66#416—21no—
67#416—21no—
68#408—21no—
69#405—21no—
70C2×#97—22no—
71C2×#97D411yescharacteristic
72C2×#97D411yescharacteristic
73C2×#97C2311yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#398—22no—
78#395—22no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4C2312yes—
81#376C2311yescharacteristic
82#341—21no—
83#341—21no—
84#339—21no—
85#339—21no—
86#336—22no—
87#336—22no—
88#333—22no—
89#332—22no—
90#332—22no—
91#331—22no—
92#328—22no—
93#328—22no—
94#327—22no—
95#327—22no—
96#322—22no—
97#322—22no—
98C2×#41—21no—
99C22×Q8—21no—
100#131—42no—
101#131—42no—
102C2×M16—21no—
103C2×M16—21no—
104C2×M16—41no—
105C22×C8—22no—
106C22×C8—41no—
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C4⋊Q8#4111yescharacteristic
112C4⋊Q8#4112yes—
113#127#4111yescharacteristic
114#125—21no—
115#124—21no—
116C22⋊Q8—22no—
117#121—22no—
118C4×Q8—22no—
119C4×Q8—42no—
120C4×Q8—42no—
121C4×D4—22no—
122#117—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C42—22no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2411yescharacteristic
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—42no—
138C4⋊C8—44no—
139C4⋊C8—42no—
140C4⋊C8—42no—
141#104—42no—
142#104—42no—
143#103—22no—
144#103—22no—
145#102—22no—
146#102—22no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—42no—
150#97—21no—
151#97—21no—
152#97—41no—
153#97—41no—
154#97—44no—
155#97—41no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—41no—
161C4×C8#4112yes—
162C4×C8#4112yes—
163#95—21no—
164#95—21no—
165#95—22no—
166#95#4112yes—
167#95#4112yes—
168#41—41no—
169#41—41no—
170#41—41no—
171#41—41no—
172C2×Q8—21no—
173C2×Q8—22no—
174C2×Q8—42no—
175C2×Q8—82no—
176C2×D4—21no—
177C2×D4—22no—
178C2×D4—41no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—41no—
184C22×C4C22×D411yescharacteristic
185C22×C4C2×#4112yes—
186M16—41no—
187M16—41no—
188M16—41no—
189M16—41no—
190M16—82no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—82no—
209C2×C8#12411yescharacteristic
210C2×C8#12411yescharacteristic
211C2×C8#12412yes—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—42no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C22⋊C4—41no—
221C22⋊C4—42no—
222C42—21no—
223C42—21no—
224C42—41no—
225C42—42no—
226C42—42no—
227C42AGL(1,ℤ/8)12yes—
228C42#13712yes—
229C42C2×#4111yescharacteristic
230C42C2×#4111yescharacteristic
231C42C2×#4112yes—
232C23—41no—
233C23#52511yescharacteristic
234Q8—41no—
235Q8—41no—
236Q8—82no—
237Q8—84no—
238D4—41no—
239D4—41no—
240D4—82no—
241D4—81no—
242D4—81no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4#52511yescharacteristic
259C2×C4C2×#12411yescharacteristic
260C2×C4#52912yes—
261C2×C4#52912yes—
262C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
263C2×C4C2×#13711yescharacteristic
264C2×C4#57511yescharacteristic
265C2×C4#57511yescharacteristic
266C8—21no—
267C8—21no—
268C8—22no—
269C8—41no—
270C8—41no—
271C8—82no—
272C8—82no—
273C8—82no—
274C8—81no—
275C8—81no—
276C22—22no—
277C22—81no—
278C22—82no—
279C22#239811yescharacteristic
280C22#311411yescharacteristic
281C22#311711yescharacteristic
282C4—22no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—82no—
287C4—82no—
288C4—81no—
289C4—82no—
290C4#311511yescharacteristic
291C4#311611yescharacteristic
292C4#314011yescharacteristic
293C4#314111yescharacteristic
294C2—21no—
295C2—21no—
296C2—161no—
297C2#2382111yescharacteristic
298C2#2383412yes—
subgroupG/H
centerC2×C4C2×#124
commutator, Frattini, cosocleC2×C42C24
socleC22#2398
prime radical#532C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21716C2semidirect
3. #10689C2semidirect
4. #10628C2semidirect
7. #10371C2semidirect
9. #8254C2semidirect
11. #8139C2semidirect

Maximal subgroups

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

subgroupindex
#217162
#216632
#106892
#106282
#106272
#105472
#103712
#103702
#82542
#82532
#81392