Finite Group Databasebeta

← #416756

Subgroups — #416756

Order 512.

1056 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35330C211yesmaximal, characteristic
2#21171C211yesmaximal, characteristic
3#20764C211yesmaximal, characteristic
4#14348C211yesmaximal, characteristic
5#11392C214yesmaximal
6#11217C214yesmaximal
7#10858C211yesmaximal, characteristic
8#10840C211yesmaximal, characteristic
9#10701C211yesmaximal, characteristic
10C2×#544C2212yes—
11C2×#428C2212yes—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2862—21no—
15#2859—21no—
16#2660—22no—
17#2550—22no—
18#2550—22no—
19#2549—24no—
20#2317C2212yes—
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1948—22no—
27#1947—22no—
28#1836—21no—
29#1836—21no—
30#1834—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—22no—
44C2×C4⋊Q8—22no—
45C2×#125—22no—
46C2×#125—22no—
47C2×C22⋊Q8—24no—
48C2×#117C2312yes—
49C2×#117C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42C2312yes—
55C22×C42C2311yescharacteristic
56#428—28no—
57C2×#107—22no—
58C2×#106—22no—
59#423—22no—
60#423Q814yescomplemented
61C2×C4⋊C8—22no—
62C4×M16—22no—
63C4×M16Q814yescomplemented
64C2×#97—21no—
65C2×C4×C8—21no—
66#398—24no—
67#398—24no—
68#395—24no—
69#395—24no—
70#391—24no—
71#391—24no—
72#389—24no—
73#389—24no—
74#389—22no—
75#389—22no—
76#385—24no—
77#384—22no—
78#384—21no—
79#384—21no—
80#384—22no—
81#384D412yes—
82#382—22no—
83#382—21no—
84#382—21no—
85#382—22no—
86#382D411yescharacteristic
87#382D411yescharacteristic
88#380—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4D412yes—
91C4×C4⋊C4D412yes—
92C4×C22⋊C4—22no—
93#376—21no—
94#376—22no—
95#376Q814yescomplemented
96C2×#95—22no—
97C2×#95—28no—
98C2×#95D414yes—
99C43—21no—
100#343—44no—
101#340—24no—
102#340—24no—
103#340—24no—
104#339—24no—
105#339—24no—
106#339—44no—
107C22×Q8—24no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4C2×D414yes—
111C23×C4C2×D412yes—
112C23×C4C2411yescharacteristic
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16C2×Q814yes—
122C2×M16C2×Q814yes—
123C22×C8—22no—
124C22×C8—22no—
125C4⋊Q8—48no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128#125—48no—
129#125—44no—
130#125—44no—
131C22⋊Q8—416no—
132#117—22no—
133#117—22no—
134#117—24no—
135#117—24no—
136#117#4114yes—
137#117#4114yes—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—84no—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D412yes—
177C2×C42C2×Q814yes—
178C2×C42C2×Q812yes—
179C2×C42C2×Q812yes—
180C2×C42C2×Q812yes—
181C2×C42#4112yes—
182#107—44no—
183#106—44no—
184C4⋊C8—28no—
185#97—24no—
186C4×C8—24no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—28no—
196#95—88no—
197C24C22×D411yescharacteristic
198C2×Q8—48no—
199C2×Q8—84no—
200C2×Q8—84no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—28no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—48no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4#12012yes—
221C22×C4#12012yes—
222C22×C4C22⋊Q814yes—
223C22×C4C4⋊Q814yes—
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4C22×Q811yescharacteristic
227C22×C4C22×Q811yescharacteristic
228C22×C4C22×Q811yescharacteristic
229C22×C4C2×#4111yescharacteristic
230C22×C4C2×#4112yes—
231C22×C4C2×#4112yes—
232C22×C4H(5,𝔽2)12yes—
233C22×C42−1+412yes—
234M16—24no—
235M16—24no—
236M16—28no—
237M16—44no—
238M16—42no—
239M16—42no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C4⋊C4—22no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—48no—
252C4⋊C4—48no—
253C4⋊C4—48no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—84no—
263C4⋊C4—84no—
264C4⋊C4—84no—
265C4⋊C4—84no—
266C4⋊C4#12412yes—
267C4⋊C4#12412yes—
268C4⋊C4#12414yes—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—44no—
272C42—24no—
273C42—24no—
274C42—24no—
275C42—24no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42—82no—
281C42—82no—
282C42C22⋊Q812yes—
283C42C22⋊Q812yes—
284C42C22⋊Q814yes—
285C42C4⋊Q814yes—
286C42C4⋊Q812yes—
287C42C4⋊Q812yes—
288C23—22no—
289C23—22no—
290C23C2×#12011yescharacteristic
291C23C2×C22⋊Q811yescharacteristic
292C23C2×C4⋊Q811yescharacteristic
293C23#54612yes—
294C23#54812yes—
295Q8—168no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—48no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—88no—
324C2×C4—88no—
325C2×C4—82no—
326C2×C4—84no—
327C2×C4#39514yes—
328C2×C4#39514yes—
329C2×C4C2×C22⋊Q811yescharacteristic
330C2×C4C2×C22⋊Q811yescharacteristic
331C2×C4C2×#12411yescharacteristic
332C2×C4C2×#12411yescharacteristic
333C2×C4C2×C4⋊Q811yescharacteristic
334C2×C4C2×C4⋊Q811yescharacteristic
335C2×C4#53312yes—
336C2×C4#53312yes—
337C2×C4#53511yescharacteristic
338C2×C4#53611yescharacteristic
339C2×C4#54412yes—
340C2×C4#54412yes—
341C2×C4#54612yes—
342C2×C4#54812yes—
343C2×C4#55412yes—
344C2×C4#55512yes—
345C2×C4#55612yes—
346C2×C4#55712yes—
347C8—44no—
348C8—44no—
349C22—22no—
350C22—22no—
351C22—24no—
352C22—22no—
353C22—22no—
354C22C2×#39511yescharacteristic
355C22#243511yescharacteristic
356C22#243611yescharacteristic
357C22#243912yes—
358C22#255012yes—
359C22#302112yes—
360C22#302212yes—
361C4—24no—
362C4—24no—
363C4—44no—
364C4—42no—
365C4—42no—
366C4—84no—
367C4—82no—
368C4—84no—
369C4—82no—
370C4—162no—
371C4#243912yes—
372C4#255012yes—
373C4#256311yescharacteristic
374C4#256811yescharacteristic
375C4#266011yescharacteristic
376C4#286211yescharacteristic
377C2—22no—
378C2—22no—
379C2#1523011yescharacteristic
380C2#2356212yes—
381C2#2588012yes—
382C2#2592711yescharacteristic
383C2#2592811yescharacteristic
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
60. #423Q8semidirect
63. C4×M16Q8semidirect
95. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#353302
#211712
#207642
#143482
#113922
#112172
#108582
#108402
#107012