Finite Group Databasebeta

← #215148

Subgroups — #215148

Order 512.

1015 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1015 conjugacy classes of subgroups — too many to draw. Only its 202 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×#2663C211yesmaximal, characteristic
2#21906C212yesmaximal, complemented
3#9654C214yesmaximal, complemented
4#9652C214yesmaximal, complemented
5#8308C212yesmaximal, complemented
6#8304C212yesmaximal
7C22×#123C2211yescharacteristic
8C22×C22⋊Q8C2211yescharacteristic
9#2663—28no—
10C2×#400C411yescharacteristic
11C2×#399C411yescharacteristic
12C2×#397C412yes—
13C2×#395C411yescharacteristic
14C2×#394C412yes—
15C2×#393C411yescharacteristic
16C2×#389C2212yes—
17C2×#388C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19#1924—28no—
20#1797C2214yes—
21#1779C2214yes—
22#1749C2214yes—
23C2×#324C2214yes—
24C2×#324C2218yes—
25C2×#323C2214yes—
26C23×Q8C2×C411yescharacteristic
27C23×D4C2×C411yescharacteristic
28C24×C4C2311yescharacteristic
29C2×#123—24no—
30C2×#123—44no—
31C2×C22⋊Q8—24no—
32C2×C22⋊Q8—44no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2312yes—
38C22×C4⋊C4C2311yescharacteristic
39C22×C22⋊C4C2×C411yescharacteristic
40C22×C22⋊C4C2×C412yes—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2×C411yescharacteristic
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C412yes—
45C22×C22⋊C4C2311yescharacteristic
46C22×C42C2312yes—
47C2×C4⋊C8—24no—
48C2×C22⋊C8—24no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8C2318yes—
51C2×#97—24no—
52#400—24no—
53#399—24no—
54#397—28no—
55#395—24no—
56#394—28no—
57#393—24no—
58#389—24no—
59#389—24no—
60#388—24no—
61#388—24no—
62C4×C22⋊C4D418yes—
63C2×#95C2×C412yes—
64C2×#95C2×C412yes—
65C2×#95C2×C412yes—
66C2×#95C2×C412yes—
67C2×#95C2×C411yescharacteristic
68C2×#95C2×C411yescharacteristic
69#336—24no—
70#336—24no—
71#324—48no—
72#324—44no—
73#323—44no—
74C25C22×C411yescharacteristic
75C25C22×C411yescharacteristic
76C22×Q8—22no—
77C22×Q8—42no—
78C22×Q8—82no—
79C22×D4—22no—
80C22×D4—42no—
81C22×D4—82no—
82C23×C4—24no—
83C23×C4—82no—
84C23×C4C22×C411yescharacteristic
85C23×C4C22×C412yes—
86C23×C4C22×C412yes—
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C22×C412yes—
90C23×C4C22×C412yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D414yes—
93C23×C4C2411yescharacteristic
94C22×C8—28no—
95#123—416no—
96C22⋊Q8—416no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—82no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4—44no—
119C2×C22⋊C4—44no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—82no—
123C2×C22⋊C4C22⋊C418yes—
124C2×C42—82no—
125C2×C42C2×D414yes—
126C2×C42C2×D414yes—
127C4⋊C8—84no—
128C22⋊C8—28no—
129C22⋊C8—28no—
130C22⋊C8—84no—
131C22⋊C8—84no—
132#97—84no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—24no—
137#95C22⋊C414yes—
138#95C22⋊C414yes—
139C24—22no—
140C24—24no—
141C24—21no—
142C24—22no—
143C24—24no—
144C24—42no—
145C24—41no—
146C24—41no—
147C24—42no—
148C24—41no—
149C24—41no—
150C24C2×C22⋊C412yes—
151C24C23×C411yescharacteristic
152C2×Q8—44no—
153C2×Q8—84no—
154C2×Q8—88no—
155C2×D4—44no—
156C2×D4—84no—
157C2×D4—88no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—28no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—88no—
189C22×C4C2×C22⋊C414yes—
190C22×C4C2×C22⋊C414yes—
191C22×C4C2×C22⋊C412yes—
192C22×C4C22×D412yes—
193C22×C4H(5,𝔽2)14yes—
194C2×C8—48no—
195C2×C8—48no—
196C2×C8—48no—
197C4⋊C4—48no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—84no—
204C22⋊C4—48no—
205C22⋊C4—48no—
206C22⋊C4—44no—
207C22⋊C4—48no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—84no—
211C42—44no—
212C42—44no—
213C23—22no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—24no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—44no—
222C23—44no—
223C23—44no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23—44no—
230C23—84no—
231C23—84no—
232C23—82no—
233C23—82no—
234C23—84no—
235C23—84no—
236C23C22×C22⋊C411yescharacteristic
237C23#51812yes—
238C23#56814yes—
239Q8—88no—
240D4—88no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—44no—
252C2×C4—48no—
253C2×C4—44no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—44no—
268C2×C4—48no—
269C2×C4—88no—
270C2×C4—84no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—84no—
274C2×C4#53514yes—
275C8—88no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—24no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—44no—
284C22—44no—
285C22—88no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—82no—
290C22—82no—
291C22—164no—
292C22#230911yescharacteristic
293C22#287912yes—
294C22#288312yes—
295C22#288911yescharacteristic
296C22#288911yescharacteristic
297C22#298014yes—
298C4—24no—
299C4—42no—
300C4—44no—
301C4—44no—
302C4—81no—
303C4—82no—
304C4—81no—
305C4—82no—
306C4—81no—
307C2—24no—
308C2—82no—
309C2—161no—
310C2#1740612yes—
311C2#1740712yes—
312C2#2179111yescharacteristic
313C2#2179511yescharacteristic
314C2#2180211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21906C2semidirect
3. #9654C2semidirect
4. #9652C2semidirect
5. #8308C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26632
#219062
#96542
#96522
#83082
#83042