Finite Group Databasebeta

← #418117

Subgroups — #418117

Order 512.

1714 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1714 conjugacy classes of subgroups — too many to draw. Only its 226 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×#2868C211yesmaximal, characteristic
2#35674C211yesmaximal, characteristic
3#23507C212yesmaximal
4C2×#1886C212yesmaximal
5#10983C211yesmaximal, characteristic
6#10978C212yesmaximal
7#10978C214yesmaximal
8#10878C212yesmaximal
9C22×C4⋊Q8C2212yes—
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2212yes—
12C22×C4⋊Q8C2211yescharacteristic
13C2×#519C2211yescharacteristic
14C2×#419C2214yes—
15C2×#419C2214yes—
16C22×#101C2212yes—
17C22×#101C2212yes—
18#2868—28no—
19#2484—24no—
20#2446—28no—
21#2310C2211yescharacteristic
22#2310C2212yes—
23#2296C2212yes—
24C2×#384—24no—
25C2×C43C2211yescharacteristic
26#2043—28no—
27#1886—28no—
28C2×#339C2214yes—
29C2×#339C2214yes—
30#1728C2212yes—
31C23×Q8C2312yes—
32C23×Q8C2311yescharacteristic
33C23×Q8C2311yescharacteristic
34C22×M16C2314yes—
35C2×C4⋊Q8—24no—
36C2×C4⋊Q8—24no—
37C2×C4⋊Q8—24no—
38C2×C4⋊Q8—24no—
39C2×C4⋊Q8—24no—
40C2×C4⋊Q8—24no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—28no—
43C2×C4⋊Q8—28no—
44C2×C4⋊Q8—44no—
45C2×C4⋊Q8—44no—
46C2×C4⋊Q8—42no—
47C2×C4⋊Q8—42no—
48C2×C4⋊Q8D418yes—
49#519—24no—
50#519—42no—
51C2×C4×Q8—24no—
52C2×#117—22no—
53C2×#117C2312yes—
54C2×#117C2312yes—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4D414yes—
59C22×C42—22no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62#419—28no—
63#419—44no—
64#419—44no—
65#419—44no—
66C2×#103—28no—
67C2×#103—28no—
68C2×#101—24no—
69C2×#101—44no—
70C2×#101—44no—
71C2×#101—42no—
72#400—28no—
73#398—28no—
74#391—24no—
75#391—24no—
76#391—24no—
77#389D418yes—
78#387—216no—
79#385—24no—
80#384—232no—
81#381—24no—
82#381—22no—
83C4×C4⋊C4—24no—
84C43—24no—
85#339—28no—
86#339—28no—
87#336—24no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—24no—
93C22×Q8—24no—
94C22×Q8—22no—
95C22×Q8—24no—
96C22×Q8—22no—
97C22×Q8—24no—
98C22×Q8—24no—
99C22×Q8—24no—
100C22×Q8—24no—
101C22×Q8—24no—
102C22×Q8—24no—
103C22×Q8—24no—
104C22×Q8C2×D414yes—
105C22×Q8C2×D414yes—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2411yescharacteristic
112C2×M16—24no—
113C2×M16—24no—
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C22×C8—24no—
119C4⋊Q8—216no—
120C4⋊Q8—48no—
121C4⋊Q8—416no—
122C4⋊Q8—416no—
123C4⋊Q8—416no—
124C4⋊Q8—416no—
125C4⋊Q8—48no—
126C4⋊Q8—48no—
127C4×Q8—416no—
128#117—24no—
129#117—24no—
130#117—24no—
131#117—24no—
132#117—44no—
133#117—44no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—216no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—216no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—416no—
143C2×C4⋊C4—416no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4C2×D414yes—
147C2×C4⋊C4C2×D414yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C42—28no—
151C2×C42—216no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—24no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42C2×D414yes—
160C2×C42C2×D414yes—
161C2×C42#4118yes—
162#103—416no—
163#103—416no—
164#101—48no—
165#101—48no—
166#95—28no—
167#95—216no—
168#95—28no—
169C24C22×D411yescharacteristic
170C2×Q8—24no—
171C2×Q8—24no—
172C2×Q8—28no—
173C2×Q8—28no—
174C2×Q8—24no—
175C2×Q8—24no—
176C2×Q8—28no—
177C2×Q8—28no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—44no—
191C2×Q8—44no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—44no—
206C22×C4—216no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—28no—
211C22×C4—28no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—28no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—28no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4#12014yes—
227C22×C4#12014yes—
228C22×C4C22×D412yes—
229C22×C4C22×D412yes—
230C22×C4C2×#4114yes—
231C22×C4H(5,𝔽2)14yes—
232C22×C4H(5,𝔽2)12yes—
233M16—44no—
234M16—44no—
235M16—44no—
236M16—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—44no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—28no—
243C4⋊C4—416no—
244C4⋊C4—432no—
245C4⋊C4—416no—
246C4⋊C4—432no—
247C4⋊C4—48no—
248C4⋊C4—416no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—44no—
252C42—232no—
253C42—24no—
254C42—28no—
255C42—28no—
256C42—48no—
257C42—44no—
258C42—44no—
259C42#12318yes—
260C23—24no—
261C23C2×#12011yescharacteristic
262C23C2×#12011yescharacteristic
263C23#53411yescharacteristic
264C23#54614yes—
265Q8—48no—
266Q8—48no—
267Q8—48no—
268Q8—48no—
269Q8—84no—
270Q8—84no—
271Q8—84no—
272Q8—84no—
273C2×C4—216no—
274C2×C4—28no—
275C2×C4—232no—
276C2×C4—28no—
277C2×C4—24no—
278C2×C4—28no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—28no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4#45618yes—
298C2×C4C2×#12312yes—
299C2×C4#53212yes—
300C2×C4#53412yes—
301C2×C4#53512yes—
302C2×C4#54014yes—
303C2×C4D4214yes—
304C2×C4#54614yes—
305C2×C4#55014yes—
306C2×C4#55912yes—
307C2×C4#56012yes—
308C8—84no—
309C22—22no—
310C22—22no—
311C22—28no—
312C22#240311yescharacteristic
313C22#240512yes—
314C22#243112yes—
315C22#270912yes—
316C22C2×#45614yes—
317C4—216no—
318C4—28no—
319C4—44no—
320C4—84no—
321C4—82no—
322C4—84no—
323C4—84no—
324C4#244514yes—
325C4#267314yes—
326C2—22no—
327C2—22no—
328C2#1526111yescharacteristic
329C2#2391112yes—
330C2#2558914yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#28682
#356742
#235072
C2×#18862
#109832
#109782
#108782