Finite Group Databasebeta

← #415363

Subgroups — #415363

Order 512.

1444 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2#13924C211yesmaximal, characteristic
3#13924C212yesmaximal
4#11390C214yesmaximal
5#11287C214yesmaximal
6#10854C212yesmaximal
7C4×#342C211yesmaximal, characteristic
8C2×#543C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10#2857—22no—
11#2659—22no—
12#2549—24no—
13#2440—24no—
14#2315C2212yes—
15C2×#380C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×#342C2212yes—
20C2×#342C2218yes—
21C2×#342C2212yes—
22#1445—24no—
23#1445—22no—
24C4⋊#95—22no—
25C4⋊#95—24no—
26C22⋊#95C2212yes—
27C22⋊#95C2218yes—
28C22⋊#95C2212yes—
29C24×C4C2312yes—
30C24×C4C2311yescharacteristic
31#543—28no—
32#543—44no—
33C2×C4⋊Q8—22no—
34C2×C22⋊Q8—24no—
35C2×C22⋊Q8—28no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D414yes—
44C22×C4⋊C4Q814yescomplemented
45C22×C22⋊C4C2312yes—
46C22×C22⋊C4C2314yes—
47C22×C22⋊C4C2314yes—
48C22×C22⋊C4C2311yescharacteristic
49C22×C22⋊C4C2311yescharacteristic
50C22×C42—22no—
51C22×C42D412yes—
52#398—28no—
53#395—28no—
54#391—24no—
55#389—24no—
56#389—24no—
57#385—28no—
58#384—22no—
59#380—24no—
60#380—22no—
61#380—22no—
62#380—22no—
63#380—22no—
64#380—24no—
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—24no—
68C4×C22⋊C4—24no—
69C2×#95—22no—
70C2×#95—21no—
71C2×#95—216no—
72C2×#95—24no—
73C2×#95—24no—
74C2×#95D412yes—
75C2×#95D414yes—
76#342—24no—
77#342—48no—
78#342Q818yescomplemented
79C25C2411yescharacteristic
80C22×Q8—24no—
81C23×C4—24no—
82C23×C4—22no—
83C23×C4—24no—
84C23×C4—24no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—44no—
88C23×C4—84no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2×D418yes—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×Q814yes—
97C23×C4C2×Q812yes—
98C23×C4#4112yes—
99C23×C4#4114yes—
100C4⋊Q8—84no—
101C22⋊Q8—432no—
102C22⋊Q8—48no—
103C22⋊Q8—48no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—48no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4C2×Q818yes—
141C2×C22⋊C4C2×Q814yes—
142C2×C22⋊C4#4114yes—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—44no—
146C2×C42—48no—
147C2×C42—44no—
148#95—28no—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—44no—
153#95—42no—
154#95—84no—
155#95—84no—
156#95—816no—
157C24—42no—
158C24—42no—
159C24C22×D411yescharacteristic
160C24C22×D412yes—
161C24C22×Q811yescharacteristic
162C24C22×Q812yes—
163C24C2×#4112yes—
164C24C2×#4111yescharacteristic
165C24C2×#4112yes—
166C24H(5,𝔽2)12yes—
167C242−1+412yes—
168C2×Q8—48no—
169C2×Q8—84no—
170C2×Q8—84no—
171C22×C4—24no—
172C22×C4—28no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—28no—
178C22×C4—24no—
179C22×C4—28no—
180C22×C4—28no—
181C22×C4—216no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—48no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—48no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—416no—
195C22×C4—48no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—48no—
202C22×C4—48no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—84no—
209C22×C4—88no—
210C22×C4—84no—
211C22×C4#12011yescharacteristic
212C22×C4#12012yes—
213C22×C4#12011yescharacteristic
214C22×C4C22⋊Q812yes—
215C22×C4C22⋊Q814yes—
216C22×C4C22⋊Q812yes—
217C22×C4C22⋊Q814yes—
218C22×C4#12414yes—
219C22×C4#12412yes—
220C22×C4#12414yes—
221C22×C4#12412yes—
222C22×C4#12414yes—
223C22×C4C4⋊Q814yes—
224C4⋊C4—48no—
225C4⋊C4—416no—
226C4⋊C4—44no—
227C4⋊C4—48no—
228C4⋊C4—44no—
229C4⋊C4—48no—
230C4⋊C4—84no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C22⋊C4—28no—
235C22⋊C4—28no—
236C22⋊C4—28no—
237C22⋊C4—28no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—48no—
242C22⋊C4—44no—
243C22⋊C4—48no—
244C42—44no—
245C42—82no—
246C42—88no—
247C23—22no—
248C23—24no—
249C23—22no—
250C23—24no—
251C23—41no—
252C23—42no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—41no—
257C23—44no—
258C23—44no—
259C23—44no—
260C23—44no—
261C23C2×#12011yescharacteristic
262C23C2×C22⋊Q811yescharacteristic
263C23C2×C22⋊Q812yes—
264C23C2×#12411yescharacteristic
265C23C2×#12411yescharacteristic
266C23C2×#12412yes—
267C23C2×C4⋊Q811yescharacteristic
268C23#53314yes—
269C23#53511yescharacteristic
270C23#53611yescharacteristic
271C23#54214yes—
272C23#54614yes—
273C23D4×Q814yes—
274C23#55412yes—
275C23#55612yes—
276C23#55712yes—
277C23Q8212yes—
278Q8—168no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—28no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—416no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—416no—
301C2×C4—44no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—84no—
309C2×C4—88no—
310C2×C4—84no—
311C2×C4—88no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—816no—
315C2×C4—82no—
316C2×C4—84no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4—82no—
320C2×C4#39312yes—
321C2×C4#39314yes—
322C2×C4#39312yes—
323C2×C4#45814yes—
324C22—22no—
325C22—22no—
326C22—28no—
327C22—22no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—22no—
333C22—22no—
334C22—21no—
335C22—22no—
336C22—28no—
337C22—22no—
338C22—44no—
339C22—44no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22C2×#39311yescharacteristic
345C22#243411yescharacteristic
346C22#243511yescharacteristic
347C22#245114yes—
348C22#249214yes—
349C22#256711yescharacteristic
350C22#256811yescharacteristic
351C22#264011yescharacteristic
352C22#286211yescharacteristic
353C22C2×#45812yes—
354C22#302712yes—
355C4—22no—
356C4—24no—
357C4—44no—
358C4—44no—
359C4—88no—
360C4—88no—
361C4—84no—
362C4—162no—
363C4#203014yes—
364C2—24no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—24no—
370C2#1519111yescharacteristic
371C2C2×#203011yescharacteristic
372C2#2346911yescharacteristic
373C2#2585312yes—
374C2#2590012yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
44. C22×C4⋊C4Q8semidirect
78. #342Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#139242
#113902
#112872
#108542
C4×#3422