Finite Group Databasebeta

← #401698

Subgroups — #401698

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34249C211yesmaximal, characteristic
2#23746C212yesmaximal
3#13942C211yesmaximal, characteristic
4#11276C214yesmaximal
5#11226C212yesmaximal
6#10763C211yesmaximal, characteristic
7#10750C212yesmaximal
8#10750C212yesmaximal
9C2×#519C2211yescharacteristic
10C2×#519C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×#101C2212yes—
13#2351—24no—
14Q8×C4⋊C4—24no—
15#2317—22no—
16#2317C2212yes—
17#2296—22no—
18#2296C2212yes—
19C2×#391C2212yes—
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#2101—24no—
23#2101—24no—
24C2×#343C2212yes—
25#1736C2214yes—
26#1736C2214yes—
27C2×#339C2214yes—
28C2×#339C2214yes—
29#1728C2212yes—
30#1441—22no—
31C4⋊#95C2212yes—
32#1434—22no—
33C23×Q8C2311yescharacteristic
34C22×M16C2314yes—
35#519—22no—
36#519—24no—
37#519—24no—
38#519—42no—
39#519Q814yes—
40C2×C4×Q8—24no—
41C2×C4×Q8—24no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—24no—
45C2×C4×Q8—42no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117C2312yes—
49C2×#117C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C42D412yes—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57C2×#101—24no—
58C2×#101—24no—
59C2×#101—24no—
60#391—28no—
61#389—28no—
62#389—24no—
63#389—24no—
64#389—24no—
65#389D414yes—
66#389D414yes—
67#385—22no—
68#385—24no—
69#385—22no—
70#384—24no—
71#384—22no—
72#382—24no—
73#382—22no—
74#382—24no—
75C4×C4⋊C4—28no—
76C4×C4⋊C4—24no—
77C4×C4⋊C4—42no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—24no—
81C2×#95—24no—
82C2×#95—21no—
83#343—42no—
84#340—24no—
85#340—24no—
86#340—28no—
87#339—28no—
88#339—44no—
89#336—24no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8C2×Q812yes—
95C22×Q8#4112yes—
96C22×Q8#4114yes—
97C22×Q8#4114yes—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2411yescharacteristic
105C2×M16—24no—
106C2×M16—24no—
107C2×M16—24no—
108C2×M16—24no—
109C2×M16—24no—
110C2×M16—24no—
111C22×C8—24no—
112C4×Q8—216no—
113C4×Q8—216no—
114C4×Q8—416no—
115C4×Q8—48no—
116C4×Q8—48no—
117#117—28no—
118#117—28no—
119#117—22no—
120#117—22no—
121#117—24no—
122#117—24no—
123#117—44no—
124#117—44no—
125#117C2×Q814yes—
126#117#4114yes—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C4⋊C4C2×D414yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C42—24no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—22no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—42no—
161C2×C42C2×D414yes—
162#101—48no—
163#95—28no—
164#95—24no—
165#95—24no—
166#95—48no—
167#95—42no—
168#95—42no—
169#95—42no—
170#95—48no—
171C24C22×D411yescharacteristic
172C2×Q8—24no—
173C2×Q8—24no—
174C2×Q8—24no—
175C2×Q8—24no—
176C2×Q8—28no—
177C2×Q8—28no—
178C2×Q8—28no—
179C2×Q8—28no—
180C2×Q8#12518yes—
181C2×Q8#12618yes—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4#12012yes—
212C22×C4#12012yes—
213C22×C4C22⋊Q814yes—
214C22×C4#12314yes—
215C22×C4#12414yes—
216C22×C4C22×D412yes—
217C22×C4C22×Q811yescharacteristic
218C22×C4C2×#4112yes—
219C22×C4C2×#4111yescharacteristic
220C22×C4C2×#4112yes—
221C22×C4C2×#4112yes—
222C22×C4H(5,𝔽2)12yes—
223C22×C42−1+412yes—
224M16—44no—
225M16—44no—
226M16—44no—
227M16—44no—
228C2×C8—44no—
229C2×C8—44no—
230C2×C8—44no—
231C4⋊C4—216no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—48no—
236C4⋊C4—48no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—48no—
240C4⋊C4—44no—
241C4⋊C4—48no—
242C4⋊C4—84no—
243C4⋊C4—88no—
244C4⋊C4C22⋊Q818yes—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C42—216no—
250C42—24no—
251C42—24no—
252C42—28no—
253C42—48no—
254C42—48no—
255C42—44no—
256C42—48no—
257C23—22no—
258C23—22no—
259C23C2×#12011yescharacteristic
260C23C2×C22⋊Q811yescharacteristic
261C23C2×#12311yescharacteristic
262C23#52511yescharacteristic
263C23C2×#12411yescharacteristic
264C23#53411yescharacteristic
265C23#53611yescharacteristic
266Q8—216no—
267Q8—216no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—28no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4#40018yes—
307C2×C4C2×C22⋊Q812yes—
308C2×C4#52512yes—
309C2×C4C2×#12512yes—
310C2×C4C2×#12612yes—
311C2×C4#53912yes—
312C2×C4#54412yes—
313C2×C4#54614yes—
314C2×C4#54814yes—
315C2×C4#55214yes—
316C2×C4#55314yes—
317C8—84no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22C2×#40011yescharacteristic
327C22#240511yescharacteristic
328C22#243611yescharacteristic
329C22#248411yescharacteristic
330C22#259111yescharacteristic
331C22#262711yescharacteristic
332C22#284811yescharacteristic
333C22#301812yes—
334C22#302012yes—
335C4—28no—
336C4—48no—
337C4—42no—
338C4—42no—
339C4—84no—
340C4—82no—
341C4—82no—
342C4—82no—
343C4—162no—
344C4#244214yes—
345C4#248114yes—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—21no—
350C2#1482211yescharacteristic
351C2#2402011yescharacteristic
352C2#2402311yescharacteristic
353C2#2592112yes—
354C2#3297412yes—
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
#342492
#237462
#139422
#112762
#112262
#107632
#107502