Finite Group Databasebeta

← #451778

Subgroups — #451778

Order 512.

1460 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 1460 conjugacy classes of subgroups — too many to draw. Only its 240 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#13872C212yesmaximal
2#11316C214yesmaximal, complemented
3#11200C214yesmaximal, complemented
4#10964C211yesmaximal, characteristic
5#10730C214yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7C23×C4⋊C4C2212yes—
8C22×C22⋊C8C2214yes—
9C2×#386—22no—
10C2×#380C2212yes—
11C2×#380C2214yes—
12C2×#340C2214yes—
13C2×#340C2214yes—
14C2×#340C2214yes—
15C2×#340C2214yes—
16C2×#336C2212yes—
17#1441—22no—
18#1439—22no—
19C4⋊#95C2214yes—
20C22⋊#95—22no—
21C23×D4D411yescharacteristic
22C24×C4—22no—
23C24×C4C2312yes—
24C23×C8C2314yes—
25C2×C4×D4—28no—
26C2×C4×D4—42no—
27C22×C4⋊C4—28no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—42no—
30C22×C4⋊C4—42no—
31C22×C4⋊C4—44no—
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4C2314yes—
34C22×C4⋊C4C2314yes—
35C22×C22⋊C4—22no—
36C22×C22⋊C4—24no—
37C22×C22⋊C4D412yes—
38C22×C42C2311yescharacteristic
39C2×C22⋊C8—28no—
40C2×C22⋊C8—44no—
41C2×C22⋊C8—44no—
42#386—48no—
43#380—28no—
44#380—28no—
45#380—44no—
46C2×#95—24no—
47C2×#95—24no—
48C2×#95—22no—
49C2×#95—24no—
50#340—28no—
51#340—44no—
52#340—44no—
53#340—44no—
54#336—42no—
55C25C2×D412yes—
56C22×D4—24no—
57C22×D4—24no—
58C22×D4—41no—
59C22×D4—41no—
60C22×D4Q1614yes—
61C23×C4—24no—
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—24no—
67C23×C4—24no—
68C23×C4—24no—
69C23×C4—24no—
70C23×C4—48no—
71C23×C4—44no—
72C23×C4—44no—
73C23×C4—44no—
74C23×C4—84no—
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D412yes—
77C23×C4C2×D411yescharacteristic
78C23×C4#4118yes—
79C23×C4C2411yescharacteristic
80C22×C8—24no—
81C22×C8—24no—
82C22×C8—24no—
83C22×C8—24no—
84C22×C8—24no—
85C22×C8—24no—
86C22×C8—24no—
87C4×D4—416no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—48no—
101C2×C4⋊C4—416no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—48no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—48no—
106C2×C4⋊C4#4118yes—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—48no—
111C2×C22⋊C4—44no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—48no—
115C2×C22⋊C4—84no—
116C2×C42—24no—
117C2×C42—22no—
118C22⋊C8—416no—
119#95—48no—
120#95—48no—
121#95—44no—
122#95—84no—
123C24—22no—
124C24—22no—
125C24—24no—
126C24—24no—
127C24—22no—
128C24—22no—
129C24—24no—
130C24—24no—
131C24#12314yes—
132C24C2×Q1614yes—
133C24AGL(1,ℤ/8)14yes—
134C24C22×D411yescharacteristic
135C2×D4—216no—
136C2×D4—44no—
137C2×D4—44no—
138C2×D4—48no—
139C2×D4—44no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—28no—
149C22×C4—28no—
150C22×C4—28no—
151C22×C4—28no—
152C22×C4—28no—
153C22×C4—28no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—28no—
157C22×C4—24no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—416no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—816no—
183C22×C4—816no—
184C22×C4—88no—
185C22×C4—88no—
186C22×C4#12314yes—
187C22×C4C2×Q1612yes—
188C22×C4C2×#4114yes—
189C22×C4C2×#4114yes—
190C22×C4H(5,𝔽2)14yes—
191C22×C4H(5,𝔽2)12yes—
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—44no—
197C2×C8—44no—
198C2×C8—44no—
199C4⋊C4—28no—
200C4⋊C4—28no—
201C4⋊C4—44no—
202C4⋊C4—416no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C22⋊C4—48no—
210C22⋊C4—816no—
211C22⋊C4—88no—
212C42—44no—
213C23—24no—
214C23—24no—
215C23—24no—
216C23—28no—
217C23—24no—
218C23—24no—
219C23—28no—
220C23—24no—
221C23—28no—
222C23—28no—
223C23—28no—
224C23—28no—
225C23—44no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—44no—
231C23—44no—
232C23—44no—
233C23—44no—
234C23—44no—
235C23#48418yes—
236C23C2×#12312yes—
237C23#53411yescharacteristic
238C23#54614yes—
239C23C22×Q1611yescharacteristic
240C23C2×AGL(1,ℤ/8)12yes—
241C23#57511yescharacteristic
242C23#57714yes—
243D4—416no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—28no—
247C2×C4—24no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—41no—
254C2×C4—48no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—48no—
261C2×C4—41no—
262C2×C4—48no—
263C2×C4—48no—
264C2×C4—832no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4—84no—
275C2×C4—84no—
276C2×C4#52914yes—
277C2×C4#53814yes—
278C2×C4#53814yes—
279C2×C4#55112yes—
280C2×C4#55114yes—
281C2×C4#55314yes—
282C2×C4#55314yes—
283C2×C4#56012yes—
284C8—84no—
285C22—216no—
286C22—28no—
287C22—48no—
288C22—48no—
289C22—48no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22—44no—
295C22—44no—
296C22#244414yes—
297C22#248311yescharacteristic
298C22#269712yes—
299C22C2×#48412yes—
300C22#309412yes—
301C22#309914yes—
302C22#318911yescharacteristic
303C22#319211yescharacteristic
304C22#320012yes—
305C22#329914yes—
306C22#330614yes—
307C22#331014yes—
308C22#332614yes—
309C4—44no—
310C4—42no—
311C4—88no—
312C4—81no—
313C4—164no—
314C4#257814yes—
315C4#277814yes—
316C2—48no—
317C2#1601411yescharacteristic
318C2#2620414yes—
319C2#2621914yes—
320C2#2760311yescharacteristic
321C2#2760911yescharacteristic
322C2#3166912yes—
323C2#3168112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11316C2semidirect
3. #11200C2semidirect
5. #10730C2semidirect

Maximal subgroups

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

subgroupindex
#138722
#113162
#112002
#109642
#107302