Finite Group Databasebeta

← #415351

Subgroups — #415351

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21325C211yesmaximal, characteristic
3#21314C211yesmaximal, characteristic
4#14283C211yesmaximal, characteristic
5#11392C214yesmaximal
6#11276C214yesmaximal
7#10857C211yesmaximal, characteristic
8#10840C211yesmaximal, characteristic
9#10701C211yesmaximal, characteristic
10C2×#544C2212yes—
11C22×#125C2211yescharacteristic
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
23C4×#117C2211yescharacteristic
24C2×#376C2211yescharacteristic
25#1992—21no—
26#1992—21no—
27#1991—22no—
28#1989—22no—
29#1989—22no—
30C2×#343C2214yes—
31#1736C2212yes—
32#1736C2214yes—
33#1736C2212yes—
34C2×#339C2212yes—
35C2×#339C2214yes—
36C2×#339C2212yes—
37#1441—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×#125—22no—
49C2×#125—21no—
50C2×#125—21no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125Q814yescomplemented
55C2×C22⋊Q8—24no—
56C2×#117C2312yes—
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C22×C4⋊C4—22no—
60C22×C4⋊C4D414yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65#423—22no—
66#423—22no—
67C2×C4⋊C8—22no—
68C4×M16—22no—
69C4×M16—22no—
70C2×#97—21no—
71C2×C4×C8—21no—
72#398—28no—
73#395—28no—
74#389—24no—
75#389—24no—
76#389—24no—
77#385—24no—
78#384—24no—
79#384—22no—
80#384—21no—
81#384—21no—
82#384—22no—
83#384D411yescharacteristic
84#384D411yescharacteristic
85#382—22no—
86#382D412yes—
87#380—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D412yes—
91C4×C22⋊C4—22no—
92#376—21no—
93#376—22no—
94#376—22no—
95C2×#95—22no—
96C2×#95—28no—
97C43—21no—
98#343—44no—
99#340—22no—
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#339—24no—
105#339—44no—
106#339Q818yescomplemented
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—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16C2×Q814yes—
123C22×C8—22no—
124C22×C8—22no—
125C4⋊Q8—48no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C4⋊Q8—88no—
129#125—28no—
130#125—24no—
131#125—24no—
132#125—48no—
133#125—44no—
134#125—44no—
135#125—42no—
136#125—42no—
137#125—44no—
138C22⋊Q8—416no—
139#117—22no—
140#117—22no—
141#117—24no—
142#117—24no—
143#117C2×Q814yes—
144#117#4114yes—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×Q814yes—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D412yes—
189C2×C42C2×Q814yes—
190C2×C42C2×Q812yes—
191C2×C42#4112yes—
192C2×C42#4112yes—
193C2×C42#4112yes—
194C4⋊C8—44no—
195#97—42no—
196C4×C8—42no—
197#95—28no—
198#95—22no—
199#95—22no—
200#95—24no—
201#95—44no—
202#95—88no—
203C24C22×D411yescharacteristic
204C2×Q8—48no—
205C2×Q8—88no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—28no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—48no—
226C22×C4#12012yes—
227C22×C4#12012yes—
228C22×C4C22⋊Q814yes—
229C22×C4#12414yes—
230C22×C4C22×D411yescharacteristic
231C22×C4C22×D411yescharacteristic
232C22×C4C22×Q811yescharacteristic
233C22×C4C22×Q812yes—
234C22×C4C2×#4111yescharacteristic
235C22×C4C2×#4111yescharacteristic
236C22×C4C2×#4111yescharacteristic
237C22×C4C2×#4112yes—
238C22×C4H(5,𝔽2)12yes—
239C22×C42−1+412yes—
240M16—24no—
241M16—24no—
242M16—42no—
243M16—42no—
244M16—44no—
245M16—44no—
246C2×C8—24no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—44no—
260C4⋊C4—48no—
261C4⋊C4—48no—
262C4⋊C4—48no—
263C4⋊C4—48no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—84no—
273C4⋊C4—88no—
274C4⋊C4#12412yes—
275C4⋊C4#12412yes—
276C4⋊C4C4⋊Q814yes—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—44no—
280C42—24no—
281C42—24no—
282C42—24no—
283C42—24no—
284C42—44no—
285C42—44no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42C22⋊Q814yes—
291C42C22⋊Q812yes—
292C42C22⋊Q812yes—
293C42#12412yes—
294C42#12412yes—
295C42#12414yes—
296C23—22no—
297C23—22no—
298C23C2×#12011yescharacteristic
299C23C2×C22⋊Q811yescharacteristic
300C23C2×#12411yescharacteristic
301C23#54612yes—
302C23D4×Q812yes—
303Q8—168no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—88no—
334C2×C4#39314yes—
335C2×C4#39314yes—
336C2×C4C2×C22⋊Q811yescharacteristic
337C2×C4C2×C22⋊Q811yescharacteristic
338C2×C4C2×#12411yescharacteristic
339C2×C4C2×#12411yescharacteristic
340C2×C4C2×#12411yescharacteristic
341C2×C4C2×C4⋊Q811yescharacteristic
342C2×C4#53312yes—
343C2×C4#53312yes—
344C2×C4#53511yescharacteristic
345C2×C4#53611yescharacteristic
346C2×C4#54212yes—
347C2×C4#54212yes—
348C2×C4#54612yes—
349C2×C4D4×Q812yes—
350C2×C4#55412yes—
351C2×C4#55612yes—
352C2×C4#55712yes—
353C2×C4Q8212yes—
354C8—44no—
355C8—82no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—24no—
361C22C2×#39311yescharacteristic
362C22#243411yescharacteristic
363C22#243511yescharacteristic
364C22#245112yes—
365C22#249212yes—
366C22#301912yes—
367C22#302012yes—
368C4—24no—
369C4—24no—
370C4—42no—
371C4—42no—
372C4—44no—
373C4—84no—
374C4—84no—
375C4—82no—
376C4—82no—
377C4—84no—
378C4#245112yes—
379C4#249212yes—
380C4#256711yescharacteristic
381C4#256811yescharacteristic
382C4#264011yescharacteristic
383C4#286211yescharacteristic
384C2—22no—
385C2—22no—
386C2#1519111yescharacteristic
387C2#2346212yes—
388C2#2583212yes—
389C2#2587811yescharacteristic
390C2#2587911yescharacteristic
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
54. C2×#125Q8semidirect
106. #339Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#213252
#213142
#142832
#113922
#112762
#108572
#108402
#107012