Finite Group Databasebeta

← #416677

Subgroups — #416677

Order 512.

1000 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#35286C211yesmaximal, characteristic
2#21172C211yesmaximal, characteristic
3#20801C211yesmaximal, characteristic
4#14348C211yesmaximal, characteristic
5#11342C214yesmaximal
6#11217C214yesmaximal
7#10857C211yesmaximal, characteristic
8#10840C211yesmaximal, characteristic
9#10701C211yesmaximal, characteristic
10C2×#428C2214yes—
11C2×#428C2212yes—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2859—22no—
15#2647—22no—
16#2571—22no—
17#2552—22no—
18#2442—24no—
19#2317C2212yes—
20#2316C2212yes—
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1949—22no—
27#1949—22no—
28#1847—22no—
29#1846—22no—
30C2×#343C2214yes—
31#1736C2212yes—
32C2×#339C2212yes—
33C2×#339C2212yes—
34C2×#339C2214yes—
35#1442—24no—
36C4⋊#95C2214yes—
37C22×M16C2312yes—
38C22×M16C2312yes—
39C2×#125—24no—
40C2×#117C2311yescharacteristic
41C2×#117C2312yes—
42C2×#117C2311yescharacteristic
43C22×C4⋊C4D414yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C42C2312yes—
47C22×C42C2311yescharacteristic
48#428—28no—
49#428—44no—
50#428—44no—
51C2×#107—24no—
52C2×#107—22no—
53C2×#106—24no—
54C2×#106—22no—
55#423—22no—
56#423Q814yes—
57C2×C4⋊C8—22no—
58C4×M16—22no—
59C4×M16Q814yes—
60C2×#97—21no—
61C2×C4×C8—21no—
62#400—28no—
63#398—28no—
64#389—24no—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—24no—
69#385—24no—
70#384—22no—
71#384—22no—
72#384—21no—
73#384—21no—
74#384D412yes—
75#382—24no—
76#382—22no—
77#382—21no—
78#382—21no—
79#382—22no—
80#382D412yes—
81#380—24no—
82#380—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D412yes—
85C4×C4⋊C4D412yes—
86C4×C22⋊C4—22no—
87#376—21no—
88#376—22no—
89#376Q814yes—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—28no—
93C43—21no—
94#343—28no—
95#340—22no—
96#340—22no—
97#339—24no—
98#339—24no—
99#339—44no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4C2×D414yes—
103C23×C4C2×D412yes—
104C23×C4C2411yescharacteristic
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—24no—
110C2×M16—24no—
111C2×M16C2×Q814yes—
112C2×M16C2×Q814yes—
113C2×M16#4114yes—
114C2×M16#4114yes—
115C22×C8—22no—
116C22×C8—22no—
117#125—88no—
118#117—22no—
119#117—22no—
120#117—24no—
121#117—24no—
122#117—22no—
123#117—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—84no—
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C4⋊C4C2×D411yescharacteristic
141C2×C4⋊C4C2×D412yes—
142C2×C4⋊C4C2×D414yes—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D412yes—
159C2×C42C2×Q814yes—
160C2×C42C2×Q812yes—
161C2×C42C2×Q812yes—
162C2×C42C2×Q812yes—
163C2×C42#4112yes—
164#107—48no—
165#107—44no—
166#106—48no—
167#106—44no—
168C4⋊C8—28no—
169#97—24no—
170C4×C8—24no—
171#95—28no—
172#95—28no—
173#95—22no—
174#95—22no—
175#95—24no—
176#95—28no—
177#95—42no—
178#95—42no—
179#95—88no—
180C24C22×D411yescharacteristic
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—24no—
187C22×C4—28no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—48no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4#12012yes—
200C22×C4#12012yes—
201C22×C4C22⋊Q814yes—
202C22×C4C4⋊Q814yes—
203C22×C4C22×D411yescharacteristic
204C22×C4C22×D411yescharacteristic
205C22×C4C22×Q811yescharacteristic
206C22×C4C22×Q811yescharacteristic
207C22×C4C22×Q811yescharacteristic
208C22×C4C2×#4111yescharacteristic
209C22×C4C2×#4112yes—
210C22×C4C2×#4112yes—
211C22×C4H(5,𝔽2)12yes—
212C22×C42−1+412yes—
213M16—24no—
214M16—24no—
215M16—24no—
216M16—28no—
217M16—44no—
218M16#12518yescomplemented
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—24no—
222C2×C8—24no—
223C2×C8—42no—
224C2×C8—42no—
225C4⋊C4—22no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—44no—
234C4⋊C4—48no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—84no—
243C4⋊C4—88no—
244C4⋊C4—88no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—44no—
248C42—24no—
249C42—24no—
250C42—44no—
251C42—44no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42—42no—
256C42—42no—
257C42—42no—
258C42C22⋊Q812yes—
259C42C22⋊Q812yes—
260C42C22⋊Q814yes—
261C42C4⋊Q814yes—
262C42C4⋊Q812yes—
263C42C4⋊Q812yes—
264C23—22no—
265C23—22no—
266C23C2×#12011yescharacteristic
267C23C2×C22⋊Q811yescharacteristic
268C23C2×C4⋊Q811yescharacteristic
269C23#54712yes—
270C23#55012yes—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—84no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—88no—
295C2×C4—88no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4#39514yes—
300C2×C4#39514yes—
301C2×C4C2×C22⋊Q811yescharacteristic
302C2×C4C2×C22⋊Q811yescharacteristic
303C2×C4C2×#12512yes—
304C2×C4C2×C4⋊Q811yescharacteristic
305C2×C4C2×C4⋊Q811yescharacteristic
306C2×C4#53211yescharacteristic
307C2×C4#53211yescharacteristic
308C2×C4#53312yes—
309C2×C4#53511yescharacteristic
310C2×C4#53611yescharacteristic
311C2×C4#54412yes—
312C2×C4#54412yes—
313C2×C4#54612yes—
314C2×C4#54812yes—
315C2×C4#55212yes—
316C2×C4#55312yes—
317C2×C4#55412yes—
318C2×C4#55712yes—
319C8—28no—
320C8—44no—
321C22—22no—
322C22—22no—
323C22—24no—
324C22—22no—
325C22—22no—
326C22C2×#39511yescharacteristic
327C22#243111yescharacteristic
328C22#243211yescharacteristic
329C22#244112yes—
330C22#255212yes—
331C22#301912yes—
332C22#302012yes—
333C4—24no—
334C4—24no—
335C4—84no—
336C4—82no—
337C4—84no—
338C4—81no—
339C4—84no—
340C4—81no—
341C4—162no—
342C4#244212yes—
343C4#255012yes—
344C4#256511yescharacteristic
345C4#256611yescharacteristic
346C4#266011yescharacteristic
347C4#285911yescharacteristic
348C2—22no—
349C2—22no—
350C2#1522811yescharacteristic
351C2#2356112yes—
352C2#2561812yes—
353C2#2569011yescharacteristic
354C2#2569111yescharacteristic
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

Complements

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

NH ≅ G/Ntype
218. M16#125semidirect

Maximal subgroups

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

subgroupindex
#352862
#211722
#208012
#143482
#113422
#112172
#108572
#108402
#107012