Finite Group Databasebeta

← #4131006

Subgroups — #4131006

Order 512.

2083 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17229C211yesmaximal, characteristic, complemented
2#16218C214yesmaximal, complemented
3#15619C214yesmaximal
4#15616C211yesmaximal, characteristic, complemented
5#15384C212yesmaximal
6#15353C214yesmaximal, complemented
7#14253C212yesmaximal, complemented
8#14204C214yesmaximal, complemented
9#14167C211yesmaximal, characteristic
10#14077C214yesmaximal, complemented
11#14005C214yesmaximal, complemented
12C22×#123C2211yescharacteristic
13C2×#399C2212yes—
14C2×#397C2212yes—
15C2×#396C2214yes—
16C2×#394C2214yes—
17C2×#389C2214yes—
18C2×#389C2214yes—
19C2×#389C2214yes—
20C2×#389C2214yes—
21C2×#388C2214yes—
22C2×#388C2214yes—
23C2×#385C2214yes—
24C2×#384C2212yes—
25C2×C4×C4⋊C4C2214yes—
26C2×C4×C22⋊C4C2212yes—
27#1445C2212yes—
28#1444C2214yes—
29#1444C2214yes—
30#1443C2214yes—
31#1443C2214yes—
32#1443C2214yes—
33#1442C2214yes—
34#1441C2214yes—
35#1441C2214yes—
36#1441C2214yes—
37#1440C2211yescharacteristic
38#1440C2211yescharacteristic
39#1439C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1438C2212yes—
44#1437C2212yes—
45#1437C2214yes—
46#1437C2214yes—
47#1437C2212yes—
48#1437C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54C4⋊#95C2214yes—
55#1434C2214yes—
56C4×#95C2214yes—
57C23×D4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C2×#123—24no—
60C2×#123—44no—
61C22×C4⋊C4C2314yes—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2312yes—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2311yescharacteristic
79C22×C42C2312yes—
80C22×C42C2314yes—
81C22×C42C2314yes—
82C22×C42C2314yes—
83#399—44no—
84#397—28no—
85#396—216no—
86#394—48no—
87#389—216no—
88#389—48no—
89#389—48no—
90#389—48no—
91#388—216no—
92#388—48no—
93#385—48no—
94#384—44no—
95C4×C4⋊C4—48no—
96C4×C22⋊C4—44no—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2312yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2312yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C25C2411yescharacteristic
119C25C2411yescharacteristic
120C22×D4—22no—
121C22×D4—42no—
122C22×D4—44no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—44no—
126C23×C4C2414yes—
127C23×C4C2414yes—
128C23×C4C2414yes—
129C23×C4C2411yescharacteristic
130C23×C4C2414yes—
131C23×C4C2412yes—
132C23×C4C2412yes—
133C23×C4C2414yes—
134C23×C4C2414yes—
135#123—416no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4#41116yes—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4#4114yes—
187C2×C42—28no—
188C2×C42—28no—
189C2×C42—28no—
190C2×C42—28no—
191C2×C42—44no—
192C2×C42—42no—
193C2×C42—44no—
194C2×C42—44no—
195C2×C42—44no—
196C2×C42—44no—
197C2×C42—44no—
198#95—28no—
199#95—28no—
200#95—28no—
201#95—28no—
202#95—28no—
203#95—24no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—28no—
209#95—28no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—42no—
219C24—24no—
220C24—24no—
221C24—22no—
222C24—24no—
223C24—24no—
224C24—21no—
225C24—24no—
226C24—22no—
227C24—24no—
228C24C2×#4112yes—
229C24C2511yescharacteristic
230C2×D4—44no—
231C2×D4—84no—
232C2×D4—88no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—88no—
283C22×C4—84no—
284C22×C4C2×#4118yes—
285C22×C4C2×#4114yes—
286C22×C4C2×#4118yes—
287C22×C4C2×#4118yes—
288C22×C4H(5,𝔽2)14yes—
289C22×C4H(5,𝔽2)18yes—
290C22×C42−1+418yes—
291C4⋊C4—232no—
292C4⋊C4—416no—
293C4⋊C4—416no—
294C4⋊C4—48no—
295C4⋊C4—416no—
296C4⋊C4—88no—
297C4⋊C4—84no—
298C4⋊C4—88no—
299C4⋊C4—88no—
300C22⋊C4—44no—
301C22⋊C4—48no—
302C22⋊C4—416no—
303C22⋊C4—48no—
304C22⋊C4—48no—
305C22⋊C4—416no—
306C22⋊C4—416no—
307C22⋊C4—88no—
308C22⋊C4—88no—
309C42—48no—
310C42—416no—
311C42—88no—
312C42—88no—
313C23—28no—
314C23—22no—
315C23—24no—
316C23—44no—
317C23—42no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—42no—
323C23—44no—
324C23—44no—
325C23—42no—
326C23—44no—
327C23—41no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23—42no—
334C23—44no—
335C23C22×#4114yes—
336C23C22×#4111yescharacteristic
337C23C2×H(5,𝔽2)14yes—
338C23C2×H(5,𝔽2)12yes—
339C23C2×2−1+414yes—
340D4—88no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—28no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—88no—
396C2×C4#553116yes—
397C22—48no—
398C22—42no—
399C22—44no—
400C22—84no—
401C22—84no—
402C22—84no—
403C22—84no—
404C22—81no—
405C22—82no—
406C22—84no—
407C22C2×#55314yes—
408C22#348112yes—
409C22#348214yes—
410C22#349011yescharacteristic
411C22#350812yes—
412C22#351114yes—
413C22#352014yes—
414C22#352214yes—
415C22#353714yes—
416C22#353812yes—
417C22#355214yes—
418C4—216no—
419C4—48no—
420C4—44no—
421C4—44no—
422C4—48no—
423C4—84no—
424C4—84no—
425C4—84no—
426C4—81no—
427C2—82no—
428C2—161no—
429C2#4645411yescharacteristic
430C2#4658414yes—
431C2#5967614yes—
432C2#5969714yes—
433C2#6003512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17229C2semidirect
2. #16218C2semidirect
4. #15616C2semidirect
6. #15353C2semidirect
7. #14253C2semidirect
8. #14204C2semidirect
10. #14077C2semidirect
11. #14005C2semidirect

Maximal subgroups

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

subgroupindex
#172292
#162182
#156192
#156162
#153842
#153532
#142532
#142042
#141672
#140772
#140052