Finite Group Databasebeta

← #394844

Subgroups — #394844

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2#34249C211yesmaximal, characteristic
3#23552C212yesmaximal
4C2×#1857C212yesmaximal
5#11340C214yesmaximal
6#11226C212yesmaximal
7#10763C211yesmaximal, characteristic
8#10750C212yesmaximal
9C22×C4⋊Q8C2211yescharacteristic
10C2×#519C2211yescharacteristic
11C2×#519C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#428C2214yes—
14C22×#101C2212yes—
15#2567—28no—
16C2×#395—22no—
17#2351—24no—
18Q8×C4⋊C4—24no—
19#2317—22no—
20#2317C2212yes—
21#2296—22no—
22#2296C2212yes—
23C2×#391C2212yes—
24C2×#389—22no—
25C2×#384C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#2058—24no—
28#2057—24no—
29#1857—28no—
30C2×#343C2212yes—
31#1736C2214yes—
32C2×#339C2214yes—
33C2×#339C2214yes—
34#1728C2212yes—
35C23×Q8—21no—
36C23×Q8C2311yescharacteristic
37C22×M16C2314yes—
38C2×C4⋊Q8—24no—
39C2×C4⋊Q8—24no—
40C2×C4⋊Q8—42no—
41#519—22no—
42#519—24no—
43#519—24no—
44#519—42no—
45#519Q814yescomplemented
46C2×C4×Q8—24no—
47C2×C4×Q8—24no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—24no—
51C2×C4×Q8—42no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117C2312yes—
55C2×#117C2312yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C42D412yes—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65#428—28no—
66#428—44no—
67C2×#107—24no—
68C2×#106—24no—
69C2×#101—24no—
70C2×#101—24no—
71C2×#101Q818yescomplemented
72#395—216no—
73#391—24no—
74#391D418yes—
75#389—216no—
76#389—28no—
77#389—24no—
78#389—24no—
79#389—24no—
80#389D414yes—
81#389D414yes—
82#385—22no—
83#385—24no—
84#385—22no—
85#384—24no—
86#384—24no—
87#384—22no—
88#384—24no—
89#382—24no—
90#382—22no—
91C4×C4⋊C4—28no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C2×#95—21no—
95C2×#95—24no—
96#343—42no—
97#340—28no—
98#339—28no—
99#339—28no—
100#336—24no—
101C22×Q8—24no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—84no—
109C22×Q8C2×D414yes—
110C22×Q8C2×Q812yes—
111C22×Q8C2×Q814yes—
112C22×Q8#4112yes—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16C2×Q818yes—
126C22×C8—24no—
127C4⋊Q8—416no—
128C4×Q8—216no—
129C4×Q8—216no—
130C4×Q8—416no—
131C4×Q8—48no—
132C4×Q8—48no—
133#117—28no—
134#117—28no—
135#117—22no—
136#117—22no—
137#117—24no—
138#117—24no—
139#117—44no—
140#117—44no—
141#117C2×Q814yes—
142#117#4114yes—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—216no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—88no—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D414yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—42no—
183C2×C42C2×D414yes—
184C2×C42C2×D414yes—
185#107—48no—
186#106—48no—
187#101—216no—
188#95—24no—
189#95—28no—
190#95—216no—
191#95—24no—
192#95—24no—
193C24C22×D411yescharacteristic
194C2×Q8—24no—
195C2×Q8—24no—
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—28no—
199C2×Q8—28no—
200C2×Q8—28no—
201C2×Q8—28no—
202C2×Q8—48no—
203C2×Q8—88no—
204C2×Q8—88no—
205C2×Q8—88no—
206C2×Q8C22⋊Q818yes—
207C2×Q8C4⋊Q818yes—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—28no—
216C22×C4—24no—
217C22×C4—21no—
218C22×C4—24no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—48no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4#12012yes—
235C22×C4#12014yes—
236C22×C4#12012yes—
237C22×C4C22⋊Q814yes—
238C22×C4C4⋊Q814yes—
239C22×C4C22×D412yes—
240C22×C4C22×D412yes—
241C22×C4C22×Q811yescharacteristic
242C22×C4C22×Q812yes—
243C22×C4C2×#4111yescharacteristic
244C22×C4C2×#4112yes—
245C22×C4H(5,𝔽2)12yes—
246C22×C42−1+412yes—
247M16—28no—
248M16—28no—
249M16—44no—
250M16—44no—
251C2×C8—28no—
252C2×C8—44no—
253C2×C8—44no—
254C4⋊C4—216no—
255C4⋊C4—216no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—28no—
259C4⋊C4—24no—
260C4⋊C4—48no—
261C4⋊C4—48no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—48no—
267C4⋊C4—84no—
268C4⋊C4—88no—
269C4⋊C4—816no—
270C4⋊C4C22⋊Q818yes—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C42—216no—
276C42—24no—
277C42—24no—
278C42—28no—
279C42—28no—
280C42—48no—
281C42—48no—
282C42—44no—
283C42—44no—
284C23—22no—
285C23—22no—
286C23C2×#12011yescharacteristic
287C23C2×#12011yescharacteristic
288C23C2×C22⋊Q811yescharacteristic
289C23#52511yescharacteristic
290C23C2×C4⋊Q811yescharacteristic
291C23#53411yescharacteristic
292C23#53611yescharacteristic
293Q8—216no—
294Q8—216no—
295Q8—816no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—28no—
307C2×C4—28no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—48no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4—82no—
330C2×C4#39518yes—
331C2×C4#45614yes—
332C2×C4C2×C22⋊Q812yes—
333C2×C4C2×C22⋊Q812yes—
334C2×C4#52512yes—
335C2×C4C2×C4⋊Q812yes—
336C2×C4#53912yes—
337C2×C4#54412yes—
338C2×C4D4214yes—
339C2×C4#54714yes—
340C2×C4D4×Q814yes—
341C2×C4#55414yes—
342C8—48no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22C2×#39511yescharacteristic
352C22#240311yescharacteristic
353C22#240611yescharacteristic
354C22#243411yescharacteristic
355C22#258311yescharacteristic
356C22#262711yescharacteristic
357C22#284811yescharacteristic
358C22C2×#45612yes—
359C22#302112yes—
360C4—24no—
361C4—28no—
362C4—48no—
363C4—42no—
364C4—48no—
365C4—82no—
366C4—82no—
367C4—82no—
368C4—84no—
369C4#244014yes—
370C4#244914yes—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2#1472011yescharacteristic
376C2#2391111yescharacteristic
377C2#2391511yescharacteristic
378C2#2582512yes—
379C2#3297212yes—
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
45. #519Q8semidirect
71. C2×#101Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
#342492
#235522
C2×#18572
#113402
#112262
#107632
#107502