Finite Group Databasebeta

← #237998

Subgroups — #237998

Order 512.

903 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 903 conjugacy classes of subgroups — too many to draw. Only its 182 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×#2402C211yesmaximal, characteristic, complemented
2#21417C211yesmaximal, characteristic
3#17546C212yesmaximal
4#10373C214yesmaximal, complemented
5#10372C214yesmaximal, complemented
6#8098C212yesmaximal, complemented
7#7855C211yesmaximal, characteristic, complemented
8#3444C2211yescharacteristic
9#2920C2212yes—
10#2920C2211yescharacteristic
11#2402—24no—
12#2402—22no—
13#2402C2214yes—
14C2×#400—22no—
15C2×#398—22no—
16C2×C4×C4⋊C4—22no—
17C2×C4×C4⋊C4C2212yes—
18C2×#376C2211yescharacteristic
19#1995C2212yes—
20#1963C2212yes—
21#1896—22no—
22#1759—22no—
23#1509—24no—
24#1504—22no—
25#1504—22no—
26#1503C2214yes—
27#1502C2214yes—
28#1303C2218yes—
29#1274C2214yes—
30#533Q812yes—
31#533C2311yescharacteristic
32#532Q812yes—
33#532C2311yescharacteristic
34#529—22no—
35#529—22no—
36#529—22no—
37#525—21no—
38#525—22no—
39C4×#41—21no—
40C4×#41—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C42—22no—
45C22×C42D412yes—
46C22×C42C2311yescharacteristic
47#432—22no—
48#432—22no—
49#432—22no—
50#431—22no—
51#431—22no—
52#431C2314yes—
53#423—22no—
54#423C2312yes—
55C2×#104—24no—
56#407—22no—
57#407—21no—
58C4×M16—22no—
59C4×M16—21no—
60#400—28no—
61#400—28no—
62#398—28no—
63#398—28no—
64C4×C4⋊C4—28no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—28no—
68C4×C4⋊C4D414yes—
69C4×C4⋊C4C2314yes—
70#376—22no—
71#376D412yes—
72#376C2312yes—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95D412yes—
77C2×#95D412yes—
78#339—24no—
79#328—24no—
80#328—24no—
81#327—24no—
82#327—24no—
83C2×#41—22no—
84C2×#41—21no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D412yes—
90C2×M16—22no—
91C2×M16—42no—
92C4⋊Q8C2×Q812yes—
93C4⋊Q8C2×Q811yescharacteristic
94#127C2×Q811yescharacteristic
95#126—42no—
96#125—22no—
97#125—21no—
98#124—22no—
99#124—21no—
100#123—41no—
101#123—42no—
102C22⋊Q8—22no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105#121—22no—
106#121—22no—
107#121—22no—
108#120—41no—
109C4×Q8—22no—
110C4×Q8—22no—
111C4×Q8—22no—
112C4×D4—22no—
113C4×D4—22no—
114C4×D4—22no—
115C4×D4—42no—
116C4×D4—42no—
117C4×D4—42no—
118#117—22no—
119#117—21no—
120#117—41no—
121#117—42no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—48no—
127C2×C4⋊C4—48no—
128C2×C4⋊C4—48no—
129C2×C42—22no—
130C2×C42—24no—
131C2×C42—24no—
132C2×C42—44no—
133C2×C42—48no—
134C2×C42—41no—
135C2×C42C2×D414yes—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D412yes—
138C2×C42C2×Q812yes—
139C2×C42#4112yes—
140C2×C42#4112yes—
141C2×C42#4112yes—
142C2×C42C2411yescharacteristic
143C4⋊C8—24no—
144C4⋊C8—24no—
145C4⋊C8—42no—
146#104—48no—
147C22⋊C8—24no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150#97—42no—
151#97—41no—
152C4×C8—24no—
153C4×C8—41no—
154#95—28no—
155#95—28no—
156#95—24no—
157#95—22no—
158#95—22no—
159#95—24no—
160#95C2×D414yes—
161#95C2×D414yes—
162C24#12011yescharacteristic
163#41—44no—
164#41—84no—
165C2×Q8—21no—
166C2×Q8—22no—
167C2×D4—21no—
168C2×D4—22no—
169C2×D4—41no—
170C2×D4—42no—
171C22×C4—24no—
172C22×C4—28no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4#12011yescharacteristic
188C22×C4#12012yes—
189C22×C4C22⋊Q812yes—
190C22×C4C22⋊Q812yes—
191C22×C4C22⋊Q812yes—
192C22×C4#12412yes—
193C22×C4#12412yes—
194C22×C4#12412yes—
195C22×C4C22×D412yes—
196C22×C4C22×D411yescharacteristic
197M16—44no—
198M16—84no—
199C2×C8—44no—
200C2×C8—42no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—48no—
205C4⋊C4—48no—
206C4⋊C4—416no—
207C4⋊C4—416no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C22⋊C4—41no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—24no—
217C42—24no—
218C42—48no—
219C42—44no—
220C42—41no—
221C42—48no—
222C42—42no—
223C42C22⋊Q812yes—
224C42C22⋊Q812yes—
225C42C22⋊Q812yes—
226C42#12412yes—
227C42#12412yes—
228C42#12411yescharacteristic
229C42#12411yescharacteristic
230C42C22×Q811yescharacteristic
231C42C2×#4111yescharacteristic
232C42C2×#4112yes—
233C23—22no—
234C23—24no—
235C23—41no—
236C23#39312yes—
237C23C2×#12011yescharacteristic
238Q8—42no—
239Q8—82no—
240D4—42no—
241D4—82no—
242D4—82no—
243D4—84no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—28no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—416no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—84no—
266C2×C4—88no—
267C2×C4—88no—
268C2×C4—82no—
269C2×C4#39312yes—
270C2×C4#39312yes—
271C2×C4#39312yes—
272C2×C4C2×C22⋊Q812yes—
273C2×C4C2×C22⋊Q811yescharacteristic
274C2×C4C2×#12412yes—
275C2×C4C2×#12411yescharacteristic
276C2×C4#57514yes—
277C8—84no—
278C8—82no—
279C22—28no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—81no—
284C22—82no—
285C22#203114yes—
286C22C2×#39311yescharacteristic
287C22#300512yes—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—22no—
293C4—48no—
294C4—84no—
295C4—84no—
296C4—81no—
297C4—82no—
298C4#308012yes—
299C4#314212yes—
300C2—22no—
301C2—24no—
302C2—161no—
303C2#2345311yescharacteristic
304C2C2×#203112yes—
subgroupG/H
centerC2×C4C2×#124
commutator, Frattini, cosocleC2×C42C24
socleC22C2×#393
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
1. C2×#2402C2semidirect
4. #10373C2semidirect
5. #10372C2semidirect
6. #8098C2semidirect
7. #7855C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24022
#214172
#175462
#103732
#103722
#80982
#78552