Finite Group Databasebeta

← #416757

Subgroups — #416757

Order 512.

1076 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21172C211yesmaximal, characteristic
3#20765C211yesmaximal, characteristic
4#14348C211yesmaximal, characteristic
5#11392C214yesmaximal
6#11217C214yesmaximal
7#10857C211yesmaximal, characteristic
8#10840C211yesmaximal, characteristic
9#10701C211yesmaximal, characteristic
10C2×#544C2212yes—
11C2×#428C2212yes—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2868—21no—
15#2859—21no—
16#2660—22no—
17#2552—22no—
18#2551—22no—
19#2440—24no—
20#2317C2212yes—
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1949—22no—
27#1949—22no—
28#1836—21no—
29#1836—21no—
30#1835—22no—
31C2×#343C2214yes—
32#1736C2214yes—
33#1736C2212yes—
34C2×#339C2212yes—
35C2×#339C2212yes—
36C2×#339C2214yes—
37#1442—24no—
38C4⋊#95C2214yes—
39C22×M16C2312yes—
40C22×M16C2312yes—
41#544—28no—
42#544—44no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×#125—22no—
47C2×#125—22no—
48C2×C22⋊Q8—24no—
49C2×#117C2312yes—
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C22×C4⋊C4D414yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57#428—28no—
58C2×#107—22no—
59C2×#106—22no—
60#423—22no—
61#423Q814yescomplemented
62C2×C4⋊C8—22no—
63C4×M16—22no—
64C4×M16Q814yescomplemented
65C2×#97—21no—
66C2×C4×C8—21no—
67#398—28no—
68#395—28no—
69#389—24no—
70#389—24no—
71#389—24no—
72#385—24no—
73#384—24no—
74#384—22no—
75#384—21no—
76#384—21no—
77#384—22no—
78#384D411yescharacteristic
79#384D411yescharacteristic
80#382—22no—
81#382—21no—
82#382—21no—
83#382—22no—
84#382D412yes—
85#380—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4D412yes—
88C4×C4⋊C4D412yes—
89C4×C22⋊C4—22no—
90#376—21no—
91#376—22no—
92#376Q814yescomplemented
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—28no—
96C43—21no—
97#343—44no—
98#340—24no—
99#340—24no—
100#340—22no—
101#340—22no—
102#339—24no—
103#339—24no—
104#339—44no—
105C22×Q8—24no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4C2×D414yes—
109C23×C4C2×D412yes—
110C23×C4C2411yescharacteristic
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16C2×Q814yes—
120C2×M16C2×Q814yes—
121C22×C8—22no—
122C22×C8—22no—
123C4⋊Q8—48no—
124C4⋊Q8—44no—
125C4⋊Q8—44no—
126C4⋊Q8—88no—
127#125—48no—
128#125—44no—
129#125—44no—
130C22⋊Q8—416no—
131#117—22no—
132#117—22no—
133#117—24no—
134#117—24no—
135#117#4114yes—
136#117#4114yes—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—84no—
154C2×C4⋊C4C2×D412yes—
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D412yes—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D412yes—
175C2×C42C2×Q814yes—
176C2×C42C2×Q812yes—
177C2×C42C2×Q812yes—
178C2×C42C2×Q812yes—
179C2×C42#4112yes—
180#107—44no—
181#106—44no—
182C4⋊C8—28no—
183#97—24no—
184C4×C8—24no—
185#95—28no—
186#95—22no—
187#95—22no—
188#95—24no—
189#95—28no—
190#95—42no—
191#95—42no—
192#95—88no—
193C24C22×D411yescharacteristic
194C2×Q8—48no—
195C2×Q8—88no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—28no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—48no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4#12012yes—
216C22×C4#12012yes—
217C22×C4C22⋊Q814yes—
218C22×C4C4⋊Q814yes—
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4C22×Q811yescharacteristic
222C22×C4C22×Q811yescharacteristic
223C22×C4C22×Q811yescharacteristic
224C22×C4C2×#4111yescharacteristic
225C22×C4C2×#4112yes—
226C22×C4C2×#4112yes—
227C22×C4H(5,𝔽2)12yes—
228C22×C42−1+412yes—
229M16—24no—
230M16—24no—
231M16—28no—
232M16—44no—
233M16—42no—
234M16—42no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—22no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—44no—
247C4⋊C4—48no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4—48no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—84no—
259C4⋊C4—88no—
260C4⋊C4#12414yes—
261C4⋊C4#12412yes—
262C4⋊C4#12412yes—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—44no—
266C42—24no—
267C42—24no—
268C42—24no—
269C42—24no—
270C42—44no—
271C42—44no—
272C42—42no—
273C42—42no—
274C42—42no—
275C42—42no—
276C42C22⋊Q812yes—
277C42C22⋊Q812yes—
278C42C22⋊Q814yes—
279C42C4⋊Q814yes—
280C42C4⋊Q812yes—
281C42C4⋊Q812yes—
282C23—22no—
283C23—22no—
284C23C2×#12011yescharacteristic
285C23C2×C22⋊Q811yescharacteristic
286C23C2×C4⋊Q811yescharacteristic
287C23#54612yes—
288C23#54812yes—
289Q8—168no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—48no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—48no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—84no—
314C2×C4—88no—
315C2×C4—88no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4#39514yes—
320C2×C4#39514yes—
321C2×C4C2×C22⋊Q811yescharacteristic
322C2×C4C2×C22⋊Q811yescharacteristic
323C2×C4C2×#12411yescharacteristic
324C2×C4C2×#12411yescharacteristic
325C2×C4C2×C4⋊Q811yescharacteristic
326C2×C4C2×C4⋊Q811yescharacteristic
327C2×C4#53312yes—
328C2×C4#53312yes—
329C2×C4#53511yescharacteristic
330C2×C4#53611yescharacteristic
331C2×C4#54412yes—
332C2×C4#54412yes—
333C2×C4#54612yes—
334C2×C4#54812yes—
335C2×C4#55412yes—
336C2×C4#55512yes—
337C2×C4#55612yes—
338C2×C4#55712yes—
339C8—44no—
340C8—44no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—24no—
346C22C2×#39511yescharacteristic
347C22#243511yescharacteristic
348C22#243611yescharacteristic
349C22#243912yes—
350C22#255012yes—
351C22#301912yes—
352C22#302012yes—
353C4—24no—
354C4—24no—
355C4—42no—
356C4—42no—
357C4—44no—
358C4—84no—
359C4—84no—
360C4—82no—
361C4—82no—
362C4—162no—
363C4#243912yes—
364C4#255012yes—
365C4#256311yescharacteristic
366C4#256811yescharacteristic
367C4#266011yescharacteristic
368C4#286211yescharacteristic
369C2—22no—
370C2—22no—
371C2#1523011yescharacteristic
372C2#2356112yes—
373C2#2587811yescharacteristic
374C2#2587911yescharacteristic
375C2#2592612yes—
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
61. #423Q8semidirect
64. C4×M16Q8semidirect
92. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#211722
#207652
#143482
#113922
#112172
#108572
#108402
#107012