Finite Group Databasebeta

← #141118

Subgroups — #141118

Order 512.

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

normal subgroups only

This group has 566 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#33172C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3C2×#1348C214yesmaximal
4C4×#340C211yesmaximal, characteristic
5C2×#494C411yescharacteristic, complemented
6C2×#467—21no—
7C2×#467C2212yes—
8C22×C4×C8C2211yescharacteristic
9#2857—21no—
10C2×C4×C4⋊C4—21no—
11C2×C4⋊C16—21no—
12C2×C22⋊C16C2212yes—
13#1969—21no—
14#1969C412yescomplemented
15#1956—21no—
16#1845C411yescharacteristic, complemented
17C2×#340—21no—
18C2×#340C2212yes—
19#1374—21no—
20#1348—28no—
21C22×Q16—21no—
22C22×Q16C2×C412yes—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C22⋊Q8—22no—
27C22×C4⋊C4—22no—
28C22×C42—41no—
29C22×C42D411yescharacteristic
30C22×C16—22no—
31#494—42no—
32#467—24no—
33#467—24no—
34#467—44no—
35C2×#107—21no—
36C2×#107C2×C412yes—
37C2×#103—22no—
38C2×#103—22no—
39C2×C4×C8—42no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#391—22no—
45#389—41no—
46#389—41no—
47#384—21no—
48#380—41no—
49#380—41no—
50C4×C4⋊C4—24no—
51C4×C4⋊C4—42no—
52C2×#95—41no—
53C4⋊C16—42no—
54C22⋊C16—24no—
55#340—24no—
56#340—22no—
57#340D414yes—
58C22×Q8—22no—
59C23×C4—42no—
60C23×C4D811yescharacteristic
61C23×C4D811yescharacteristic
62C23×C4C2×D411yescharacteristic
63C2×Q16—24no—
64C2×Q16—42no—
65C2×Q16—42no—
66C2×Q16—42no—
67C2×Q16—42no—
68C2×Q16—42no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22×C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22×C8C2×D412yes—
79C22×C8#4112yes—
80C4⋊Q8—42no—
81C4⋊Q8—42no—
82C22⋊Q8—44no—
83C22⋊Q8—44no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—44no—
100C2×C42—82no—
101C2×C42—82no—
102C2×C42C22⋊C411yescharacteristic
103C2×C42C22⋊C411yescharacteristic
104C2×C42SD1611yescharacteristic
105C2×C42SD1611yescharacteristic
106C2×C16—42no—
107C2×C16—42no—
108C2×C16—42no—
109#107—24no—
110#107—42no—
111#103—44no—
112#103—42no—
113#103—42no—
114C4×C8—22no—
115C4×C8—22no—
116C4×C8—41no—
117C4×C8—41no—
118#95—44no—
119#95—42no—
120#95—42no—
121C24C2×D811yescharacteristic
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—42no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—41no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—82no—
143C22×C4—82no—
144C22×C4—82no—
145C22×C4—82no—
146C22×C4#10211yescharacteristic
147C22×C4#10211yescharacteristic
148C22×C4#10211yescharacteristic
149C22×C4#10211yescharacteristic
150C22×C4SD3212yes—
151C22×C4Q3212yes—
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4#12112yes—
154C22×C4C2×SD1611yescharacteristic
155C22×C4AGL(1,ℤ/8)12yes—
156Q16—84no—
157Q16—84no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—44no—
170C2×C8#10412yes—
171C2×C8C4×D412yes—
172C2×C8#12411yescharacteristic
173C2×C8#12711yescharacteristic
174C4⋊C4—28no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C4⋊C4—82no—
184C4⋊C4—84no—
185C22⋊C4—84no—
186C42—22no—
187C42—22no—
188C42—21no—
189C42—21no—
190C42—88no—
191C42#9912yes—
192C42#10412yes—
193C16—82no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23C2×#10211yescharacteristic
199C23#45912yes—
200C23C2×SD3211yescharacteristic
201C23C2×Q3211yescharacteristic
202C23#51012yes—
203Q8—82no—
204Q8—82no—
205Q8—82no—
206Q8—82no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—44no—
223C2×C4—48no—
224C2×C4—84no—
225C2×C4—84no—
226C2×C4—84no—
227C2×C4—84no—
228C2×C4—84no—
229C2×C4—84no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4#32712yes—
237C2×C4#32712yes—
238C2×C4#35812yes—
239C2×C4#35812yes—
240C2×C4#36012yes—
241C2×C4#39011yescharacteristic
242C2×C4C2×#9911yescharacteristic
243C2×C4#41011yescharacteristic
244C2×C4#41811yescharacteristic
245C2×C4C2×#10411yescharacteristic
246C2×C4C2×#10411yescharacteristic
247C2×C4#46012yes—
248C8—21no—
249C8—22no—
250C8—21no—
251C8—22no—
252C8—41no—
253C22—21no—
254C22—21no—
255C22—22no—
256C22—22no—
257C22—21no—
258C22—21no—
259C22—22no—
260C22—22no—
261C22C2×#32711yescharacteristic
262C22#150211yescharacteristic
263C22#198211yescharacteristic
264C22C2×#35811yescharacteristic
265C22#214111yescharacteristic
266C22C2×#36011yescharacteristic
267C22#214511yescharacteristic
268C22#220812yes—
269C22#220912yes—
270C4—22no—
271C4—22no—
272C4—22no—
273C4—41no—
274C4—88no—
275C4—162no—
276C4—162no—
277C4#134814yescomplemented
278C4#150511yescharacteristic
279C4#150711yescharacteristic
280C4#196211yescharacteristic
281C4#196411yescharacteristic
282C2—22no—
283C2—22no—
284C2#971211yescharacteristic
285C2C2×#134811yescharacteristic
286C2#1170611yescharacteristic
287C2#1174311yescharacteristic
288C2#1174511yescharacteristic
289C2#1325311yescharacteristic
290C2#1325511yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#494C4semidirect
14. #1969C4semidirect
16. #1845C4semidirect
277. C4#1348semidirect

Maximal subgroups

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

subgroupindex
#331722
#127832
C2×#13482
C4×#3402