Finite Group Databasebeta

← #416661

Subgroups — #416661

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#21171C211yesmaximal, characteristic
3#20764C211yesmaximal, characteristic
4#14687C211yesmaximal, characteristic
5#11339C214yesmaximal
6#11211C214yesmaximal
7#10857C211yesmaximal, characteristic
8#10839C211yesmaximal, characteristic
9C4×#339C211yesmaximal, characteristic
10C2×#428C2212yes—
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2859—22no—
14#2647—22no—
15#2571—22no—
16#2552—22no—
17#2442—24no—
18#2317C2212yes—
19#2316C2212yes—
20#2285C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2214yes—
24C4×#117C2211yescharacteristic
25C2×C43C2211yescharacteristic
26#1948—22no—
27#1947—22no—
28#1836—22no—
29#1834—22no—
30C2×#343C2214yes—
31#1736C2214yes—
32#1736C2212yes—
33C2×#339C2212yes—
34C2×#339C2214yes—
35C2×#339C2212yes—
36C4⋊#95—24no—
37C22×M16C2312yes—
38C22×M16C2312yes—
39C2×#125—24no—
40C2×#117C2311yescharacteristic
41C2×#117C2312yes—
42C2×#117C2311yescharacteristic
43C22×C4⋊C4—22no—
44C22×C4⋊C4D414yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C42—22no—
48C22×C42C2312yes—
49C22×C42C2311yescharacteristic
50#428—44no—
51C2×#107—22no—
52C2×#106—22no—
53#423—22no—
54#423Q814yes—
55C2×C4⋊C8—22no—
56C4×M16—22no—
57C4×M16Q814yes—
58C2×#97—21no—
59C2×C4×C8—21no—
60#400—28no—
61#398—28no—
62#389—24no—
63#389—24no—
64#389—24no—
65#389—24no—
66#389—24no—
67#385—24no—
68#384—22no—
69#384—22no—
70#384—21no—
71#384—21no—
72#384—21no—
73#384—21no—
74#384—22no—
75#384D412yes—
76#382—24no—
77#382—22no—
78#382—28no—
79#382—44no—
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—
88C2×#95—28no—
89C43—21no—
90C43—22no—
91C43Q814yes—
92#343—44no—
93#340—28no—
94#340—24no—
95#339—28no—
96#339—24no—
97#339—24no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4C2×D412yes—
101C23×C4C2×D414yes—
102C23×C4C2411yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—24no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—24no—
110C2×M16—22no—
111C2×M16C2×Q814yes—
112C2×M16C2×Q814yes—
113C22×C8—22no—
114C22×C8—22no—
115#125—88no—
116#117—24no—
117#117—22no—
118#117—22no—
119#117—22no—
120#117—24no—
121#117—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—48no—
137C2×C4⋊C4—48no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C2×D412yes—
141C2×C4⋊C4C2×D414yes—
142C2×C4⋊C4C2×D411yescharacteristic
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—28no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—22no—
156C2×C42—44no—
157C2×C42—44no—
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D412yes—
161C2×C42C2×Q814yes—
162C2×C42C2×Q812yes—
163C2×C42C2×Q812yes—
164C2×C42C2×Q812yes—
165C2×C42#4114yes—
166C2×C42#4114yes—
167C2×C42#4112yes—
168#107—44no—
169#106—44no—
170C4⋊C8—28no—
171#97—24no—
172C4×C8—24no—
173#95—28no—
174#95—28no—
175#95—22no—
176#95—22no—
177#95—24no—
178#95—416no—
179C24C22×D411yescharacteristic
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—21no—
190C22×C4—28no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—48no—
194C22×C4—48no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4#12012yes—
198C22×C4#12012yes—
199C22×C4C22⋊Q814yes—
200C22×C4C4⋊Q814yes—
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D411yescharacteristic
203C22×C4C22×Q811yescharacteristic
204C22×C4C22×Q811yescharacteristic
205C22×C4C22×Q811yescharacteristic
206C22×C4C2×#4111yescharacteristic
207C22×C4C2×#4112yes—
208C22×C4C2×#4112yes—
209C22×C4H(5,𝔽2)12yes—
210C22×C42−1+412yes—
211M16—24no—
212M16—24no—
213M16—28no—
214M16—42no—
215M16—44no—
216M16—42no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—22no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—22no—
229C4⋊C4—28no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—48no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—84no—
240C4⋊C4—84no—
241C4⋊C4—88no—
242C4⋊C4—88no—
243C4⋊C4—84no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—44no—
247C42—24no—
248C42—24no—
249C42—24no—
250C42—416no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42—44no—
256C42—44no—
257C42C22⋊Q812yes—
258C42C22⋊Q812yes—
259C42C22⋊Q814yes—
260C42#12518yescomplemented
261C42C4⋊Q814yes—
262C42C4⋊Q812yes—
263C42C4⋊Q812yes—
264C23—22no—
265C23—22no—
266C23C2×#12011yescharacteristic
267C23C2×C22⋊Q811yescharacteristic
268C23C2×C4⋊Q811yescharacteristic
269C23#54612yes—
270C23#54812yes—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—28no—
274C2×C4—28no—
275C2×C4—28no—
276C2×C4—48no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—88no—
293C2×C4—88no—
294C2×C4—82no—
295C2×C4—84no—
296C2×C4#39514yes—
297C2×C4#39514yes—
298C2×C4#45414yes—
299C2×C4#45514yes—
300C2×C4C2×C22⋊Q811yescharacteristic
301C2×C4C2×C22⋊Q811yescharacteristic
302C2×C4C2×#12512yes—
303C2×C4C2×C4⋊Q811yescharacteristic
304C2×C4C2×C4⋊Q811yescharacteristic
305C2×C4#53211yescharacteristic
306C2×C4#53211yescharacteristic
307C2×C4#53312yes—
308C2×C4#53511yescharacteristic
309C2×C4#53611yescharacteristic
310C2×C4#54412yes—
311C2×C4#54412yes—
312C2×C4#54712yes—
313C2×C4#55012yes—
314C2×C4#55212yes—
315C2×C4#55312yes—
316C2×C4#55412yes—
317C2×C4#55712yes—
318C8—44no—
319C8—44no—
320C22—22no—
321C22—22no—
322C22—24no—
323C22—22no—
324C22—22no—
325C22C2×#39511yescharacteristic
326C22#243111yescharacteristic
327C22#243211yescharacteristic
328C22#244212yes—
329C22#255012yes—
330C22C2×#45512yes—
331C22C2×#45412yes—
332C4—24no—
333C4—48no—
334C4—82no—
335C4—84no—
336C4—81no—
337C4—84no—
338C4—81no—
339C4—162no—
340C4#206118yes—
341C4#244112yes—
342C4#255212yes—
343C4#256511yescharacteristic
344C4#256611yescharacteristic
345C4#266011yescharacteristic
346C4#285911yescharacteristic
347C2—22no—
348C2—22no—
349C2#1522811yescharacteristic
350C2C2×#206112yes—
351C2#2559011yescharacteristic
352C2#2559111yescharacteristic
353C2#2566712yes—
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
260. C42#125semidirect

Maximal subgroups

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

subgroupindex
#352862
#211712
#207642
#146872
#113392
#112112
#108572
#108392
C4×#3392