Finite Group Databasebeta

← #242929

Subgroups — #242929

Order 512.

663 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 663 conjugacy classes of subgroups — too many to draw. Only its 170 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#21417C211yesmaximal, characteristic
2#20792C211yesmaximal, characteristic, complemented
3#17737C212yesmaximal
4#10628C214yesmaximal, complemented
5#10627C214yesmaximal, complemented
6#8078C212yesmaximal, complemented
7#7855C211yesmaximal, characteristic, complemented
8#3444C2211yescharacteristic
9#2920C2212yes—
10#2920C2211yescharacteristic
11C2×#423—22no—
12C2×#423C2212yes—
13C2×#376C2211yescharacteristic
14#1995C2212yes—
15#1963C2212yes—
16#1896—22no—
17#1843—21no—
18#1843—21no—
19#1843C2214yes—
20#1759—21no—
21#1759—21no—
22#1561—22no—
23#1560—22no—
24#1559—22no—
25#1557—22no—
26#1542C2214yes—
27#1540C2214yes—
28#1301C2218yes—
29#1274C2214yes—
30C22×M16—22no—
31#533D412yes—
32#533C2311yescharacteristic
33#532D412yes—
34#532C2311yescharacteristic
35#529—22no—
36#529—22no—
37#529—22no—
38#525—21no—
39#525—22no—
40C4×#41—21no—
41C4×#41—22no—
42C22×C42D412yes—
43C22×C42C2311yescharacteristic
44#432—22no—
45#432—22no—
46#432—22no—
47#431—22no—
48#431—22no—
49#431C2314yes—
50#423—22no—
51#423—22no—
52#423—22no—
53#423—22no—
54#423—22no—
55#423—48no—
56#423C2312yes—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—24no—
59C2×C4⋊C8—24no—
60C2×C4⋊C8C2314yes—
61C2×#104—22no—
62C2×#104—22no—
63#407—22no—
64#407—21no—
65C4×M16—22no—
66C4×M16—21no—
67#376—22no—
68#376D411yescharacteristic
69#376D411yescharacteristic
70#376C2312yes—
71C2×#95—22no—
72#339—22no—
73#339—22no—
74#333—24no—
75#332—24no—
76#332—24no—
77#331—24no—
78C2×#41—22no—
79C2×#41—21no—
80C23×C4—22no—
81C23×C4C2×D411yescharacteristic
82C23×C4#4112yes—
83C2×M16—22no—
84C2×M16—48no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16—42no—
88C22×C8—24no—
89C4⋊Q8C2×D412yes—
90C4⋊Q8C2×D411yescharacteristic
91#127C2×D411yescharacteristic
92#126—42no—
93#125—22no—
94#125—21no—
95#124—22no—
96#124—21no—
97#123—41no—
98#123—42no—
99C22⋊Q8—22no—
100C22⋊Q8—22no—
101C22⋊Q8—22no—
102#121—22no—
103#121—22no—
104#121—22no—
105#120—41no—
106C4×Q8—22no—
107C4×Q8—22no—
108C4×Q8—22no—
109C4×D4—22no—
110C4×D4—22no—
111C4×D4—22no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115#117—22no—
116#117—21no—
117#117—41no—
118#117—42no—
119C2×C42—22no—
120C2×C42—24no—
121C2×C42—24no—
122C2×C42—41no—
123C2×C42C2×D412yes—
124C2×C42C2×D412yes—
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D412yes—
127C2×C42#4112yes—
128C2×C42#4112yes—
129C2×C42#4112yes—
130C2×C42#4112yes—
131C2×C42C2411yescharacteristic
132C4⋊C8—24no—
133C4⋊C8—24no—
134C4⋊C8—24no—
135C4⋊C8—24no—
136C4⋊C8—48no—
137C4⋊C8—42no—
138C4⋊C8—42no—
139C4⋊C8—42no—
140C4⋊C8—48no—
141#104—44no—
142#104—44no—
143C22⋊C8—24no—
144C22⋊C8—42no—
145C22⋊C8—42no—
146#97—42no—
147#97—41no—
148C4×C8—24no—
149C4×C8—41no—
150#95—22no—
151#95—22no—
152#95—24no—
153C24#12311yescharacteristic
154#41—42no—
155#41—42no—
156#41—84no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—41no—
162C2×D4—42no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—28no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—82no—
171C22×C4—84no—
172C22×C4#12112yes—
173C22×C4#12311yescharacteristic
174C22×C4#12312yes—
175C22×C4#12312yes—
176C22×C4#12312yes—
177C22×C4#12412yes—
178C22×C4#12612yes—
179C22×C4#12612yes—
180C22×C4C22×D411yescharacteristic
181C22×C4C2×#4112yes—
182M16—42no—
183M16—42no—
184M16—84no—
185M16—84no—
186M16—84no—
187C2×C8—44no—
188C2×C8—44no—
189C2×C8—42no—
190C2×C8—48no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C22⋊C4—41no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C42—28no—
203C42—22no—
204C42—22no—
205C42—41no—
206C42—42no—
207C42#12112yes—
208C42#12312yes—
209C42#12312yes—
210C42#12411yescharacteristic
211C42#12411yescharacteristic
212C42#12612yes—
213C42#12612yes—
214C42AGL(1,ℤ/8)14yes—
215C42#13714yes—
216C42C22×D411yescharacteristic
217C42C2×#4111yescharacteristic
218C42C2×#4112yes—
219C23—22no—
220C23—42no—
221C23—41no—
222C23#39712yes—
223C23C2×#12311yescharacteristic
224Q8—41no—
225Q8—41no—
226Q8—82no—
227D4—41no—
228D4—41no—
229D4—82no—
230D4—84no—
231D4—82no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—84no—
249C2×C4—88no—
250C2×C4—82no—
251C2×C4#39712yes—
252C2×C4#39712yes—
253C2×C4#39712yes—
254C2×C4C2×#12111yescharacteristic
255C2×C4C2×#12312yes—
256C2×C4C2×#12411yescharacteristic
257C2×C4C2×#12612yes—
258C2×C4C2×AGL(1,ℤ/8)12yes—
259C2×C4C2×#13712yes—
260C8—84no—
261C8—82no—
262C8—84no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—44no—
268C22—81no—
269C22—82no—
270C22C2×#39711yescharacteristic
271C22#309212yes—
272C4—22no—
273C4—22no—
274C4—22no—
275C4—22no—
276C4—22no—
277C4—44no—
278C4—44no—
279C4—81no—
280C4—82no—
281C4#303811yescharacteristic
282C4#303911yescharacteristic
283C4#314112yes—
284C2—22no—
285C2—42no—
286C2—161no—
287C2#2363011yescharacteristic
288C2#2364912yes—
subgroupG/H
centerC2×C4C2×#124
commutator, Frattini, cosocleC2×C42C24
socleC22C2×#397
prime radical#3444C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20792C2semidirect
4. #10628C2semidirect
5. #10627C2semidirect
6. #8078C2semidirect
7. #7855C2semidirect

Maximal subgroups

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

subgroupindex
#214172
#207922
#177372
#106282
#106272
#80782
#78552