Finite Group Databasebeta

← #6487680

Subgroups — #6487680

Order 512.

2284 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C212yesmaximal
2#35173C211yesmaximal, characteristic, complemented
3C2×#2349C211yesmaximal, characteristic
4#32905C214yesmaximal
5#29757C214yesmaximal, complemented
6#24833C218yesmaximal, complemented
7#24748C218yesmaximal, complemented
8#21955C211yesmaximal, characteristic, complemented
9#21866C212yesmaximal, complemented
10C2×#533C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×C4×D4C2212yes—
13#3110—216no—
14C2×#484C2214yes—
15C2×#474C2214yes—
16C2×#461C2214yes—
17C2×#461C2218yes—
18C2×#450C2214yes—
19C2×#440C2218yes—
20C2×#432C2212yes—
21#2857C2214yes—
22#2659—21no—
23#2659—21no—
24#2584—24no—
25#2567C2214yescomplemented
26#2546C2214yescomplemented
27#2349—24no—
28#2349C2218yes—
29#2316—22no—
30C2×#389C2212yes—
31C2×#385—22no—
32C2×#382C2211yescharacteristic
33C2×#380—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35C22×#95—21no—
36#2083C2218yescomplemented
37#2068C22116yescomplemented
38#2043C2218yescomplemented
39#1993C2214yescomplemented
40#1960C2214yescomplemented
41#1940C2218yescomplemented
42#1877C22116yes—
43#1872C2218yes—
44#1805C2218yescomplemented
45#1802C2218yes—
46#1797C2214yes—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4—22no—
50C22×SD16C2318yes—
51#533—28no—
52#533—28no—
53C2×C4⋊Q8C2312yes—
54C2×C4⋊Q8C2314yes—
55C2×C4⋊Q8C2312yes—
56C2×#125—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8C2314yes—
59C2×C4×Q8—22no—
60C2×C4×Q8C2314yes—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—48no—
65C2×C4×D4D418yes—
66C2×C4×D4C2318yes—
67C2×C4×D4C2314yes—
68C2×#117—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#484—216no—
78#474—216no—
79#461—216no—
80#461—416no—
81#450—216no—
82#440—232no—
83#432—28no—
84C2×#107C2318yes—
85C2×C4⋊C8C2314yes—
86C2×C4⋊C8C2318yes—
87C2×#103C2318yes—
88C2×#103C23116yes—
89C2×#102C2318yes—
90C2×C22⋊C8C2314yes—
91C2×#97C2314yes—
92#398—24no—
93#398—24no—
94#395—24no—
95#395—24no—
96#391C2318yes—
97#389—28no—
98#389—28no—
99#389—24no—
100#389—24no—
101#389—24no—
102#389C2318yes—
103#385—216no—
104#384C2314yes—
105#382—22no—
106#382C2314yes—
107#380—24no—
108#380—24no—
109#380—28no—
110C4×C4⋊C4D414yes—
111C4×C4⋊C4C2314yes—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—28no—
116C2×#95D411yescharacteristic
117C2×#95D411yescharacteristic
118#340C23116yes—
119#336C2318yes—
120C25—21no—
121C22×Q8—22no—
122C22×Q8C2414yes—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4C2×D414yes—
128C22×D4C2412yes—
129C23×C4—21no—
130C23×C4—28no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—28no—
134C23×C4—48no—
135C23×C4—48no—
136C23×C4—84no—
137C23×C4—84no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2411yescharacteristic
143C2×SD16—216no—
144C2×SD16—216no—
145C2×SD16—216no—
146C22×C8C2418yes—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—48no—
152#125—48no—
153C22⋊Q8—28no—
154C22⋊Q8—28no—
155C22⋊Q8—48no—
156C4×Q8—28no—
157C4×Q8—28no—
158C4×Q8—48no—
159C4×D4—216no—
160C4×D4—216no—
161C4×D4—28no—
162C4×D4—48no—
163C4×D4—416no—
164C4×D4C2×D4116yes—
165#117—48no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—84no—
182C2×C4⋊C4C2×D418yes—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2414yes—
185C2×C4⋊C4C2414yes—
186C2×C4⋊C4C2412yes—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—48no—
192C2×C42—24no—
193C2×C42—44no—
194C2×C42C2×D414yes—
195C2×C42C2×D412yes—
196C2×C42C2×D414yes—
197C2×C42#4114yes—
198C2×C42C2414yes—
199C2×C42C2412yes—
200#107—216no—
201C4⋊C8—28no—
202C4⋊C8—48no—
203#103—216no—
204#103—416no—
205#102—216no—
206C22⋊C8—28no—
207#97—28no—
208#95—24no—
209#95—28no—
210#95—24no—
211#95—28no—
212#95—28no—
213#95C2×D414yes—
214#95C2×D414yes—
215C24—24no—
216C24—24no—
217C24—24no—
218C24—22no—
219C24—44no—
220C24—44no—
221C24C22×D411yescharacteristic
222C2×Q8—24no—
223C2×Q8—24no—
224C2×Q8—28no—
225C2×Q8—28no—
226C2×Q8—44no—
227C2×Q8—84no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—216no—
231C2×D4—216no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4C22×D414yes—
237C2×D4H(5,𝔽2)14yes—
238C2×D42−1+414yes—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—216no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—416no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—416no—
260C22×C4—88no—
261C22×C4—88no—
262C22×C4—816no—
263C22×C4—832no—
264C22×C4#12114yes—
265C22×C4#12114yes—
266C22×C4#13718yes—
267C22×C4C22×D414yes—
268C22×C4C22×D412yes—
269C22×C4C22×D412yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D412yes—
272C22×C4C22×D414yes—
273C22×C4C2×#4112yes—
274C22×C4C2511yescharacteristic
275SD16—432no—
276C2×C8—28no—
277C2×C8—28no—
278C2×C8—28no—
279C4⋊C4—24no—
280C4⋊C4—216no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—28no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—44no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C4⋊C4—88no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4C22×D414yes—
297C22⋊C4—216no—
298C22⋊C4—416no—
299C22⋊C4—48no—
300C42—28no—
301C42—44no—
302C42—48no—
303C42#12118yes—
304C42C22×D414yes—
305C42C2×#4114yes—
306C23—22no—
307C23—21no—
308C23—21no—
309C23—216no—
310C23—24no—
311C23—24no—
312C23—24no—
313C23—48no—
314C23—416no—
315C23—416no—
316C23—44no—
317C23—48no—
318C23—44no—
319C23C2×#12112yes—
320C23C2×#13714yes—
321C23C23×D411yescharacteristic
322Q8—48no—
323Q8—48no—
324Q8—88no—
325D4—216no—
326D4—24no—
327D4—24no—
328D4—24no—
329D4#53718yes—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—48no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—816no—
356C2×C4—832no—
357C2×C4—84no—
358C2×C4—88no—
359C2×C4C2×#12114yes—
360C2×C4C2×#12112yes—
361C2×C4C2×#12114yes—
362C2×C4C2×#13718yes—
363C2×C4C23×D411yescharacteristic
364C2×C4C22×#4111yescharacteristic
365C2×C4C2×H(5,𝔽2)12yes—
366C2×C4C2×2−1+412yes—
367C8—48no—
368C22—22no—
369C22—24no—
370C22—24no—
371C22—22no—
372C22—28no—
373C22—432no—
374C22—44no—
375C22—48no—
376C22—48no—
377C22—44no—
378C22#303914yes—
379C22C22×#12111yescharacteristic
380C22#348411yescharacteristic
381C22#348611yescharacteristic
382C22C22×#13712yes—
383C22#358512yes—
384C4—24no—
385C4—28no—
386C4—22no—
387C4—24no—
388C4—44no—
389C4—44no—
390C4—84no—
391C4—84no—
392C4—84no—
393C4C2×#53712yes—
394C4#349212yes—
395C2—24no—
396C2—48no—
397C2#3522311yescharacteristic
398C2C2×#303911yescharacteristic
399C2#6042311yescharacteristic
400C2#6157012yes—
401C2#6182612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35173C2semidirect
5. #29757C2semidirect
6. #24833C2semidirect
7. #24748C2semidirect
8. #21955C2semidirect
9. #21866C2semidirect
25. #2567C22semidirect
26. #2546C22semidirect
36. #2083C22semidirect
37. #2068C22semidirect
38. #2043C22semidirect
39. #1993C22semidirect
40. #1960C22semidirect
41. #1940C22semidirect
44. #1805C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
#351732
C2×#23492
#329052
#297572
#248332
#247482
#219552
#218662