Finite Group Databasebeta

← #4465465

Subgroups — #4465465

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17243C213yesmaximal, complemented
2#17234C213yesmaximal, complemented
3#17204C212yesmaximal, complemented
4#17135C211yesmaximal, characteristic, complemented
5#17125C216yesmaximal, complemented
6#17114C211yesmaximal, characteristic, complemented
7#15074C216yesmaximal, complemented
8#15017C213yesmaximal, complemented
9#14996C213yesmaximal
10#14331C212yesmaximal, complemented
11C4×#399C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C2×#399C2211yescharacteristic
14C2×#398C2216yes—
15C2×#396C2212yes—
16C2×#396C2216yes—
17C2×#394C2216yes—
18C2×#388C2216yes—
19C2×#382C2213yes—
20C2×C4×C4⋊C4C2213yes—
21C2×C4×C22⋊C4C2213yes—
22#1445C2213yes—
23#1445C2216yes—
24#1444C2211yescharacteristic
25#1444C2213yes—
26#1444C2213yes—
27#1443C2212yes—
28#1443C2216yes—
29#1442C2213yes—
30#1442C2213yes—
31#1442C2216yes—
32#1441C2216yes—
33#1441C2216yes—
34#1440C2216yes—
35#1440C2216yes—
36#1439C2213yes—
37#1439C2216yes—
38#1439C2216yes—
39#1439C2216yes—
40#1439C2213yes—
41#1439C2213yes—
42#1438C2216yes—
43#1438C2213yes—
44#1438C2213yes—
45#1437C2216yes—
46#1436C2216yes—
47#1434C2213yes—
48C22⋊#95C2213yes—
49C4×#95C2211yescharacteristic
50C23×D4C2312yes—
51C23×D4C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C2×#121—46no—
54C22×C4⋊C4C2313yes—
55C22×C4⋊C4C2316yes—
56C22×C4⋊C4C2316yes—
57C22×C4⋊C4C2313yes—
58C22×C4⋊C4C2313yes—
59C22×C4⋊C4C2313yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4C2312yes—
62C22×C22⋊C4C2313yes—
63C22×C22⋊C4C2316yes—
64C22×C22⋊C4C2316yes—
65C22×C22⋊C4C2313yes—
66C22×C22⋊C4C2316yes—
67C22×C22⋊C4C2316yes—
68C22×C22⋊C4C2316yes—
69C22×C42C2313yes—
70C22×C42C2313yes—
71#399—24no—
72#398—412no—
73#396—28no—
74#396—412no—
75#394—412no—
76#388—412no—
77#382—46no—
78C4×C4⋊C4—46no—
79C4×C22⋊C4—46no—
80C2×#95C2313yes—
81C2×#95C2313yes—
82C2×#95C2316yes—
83C2×#95C2313yes—
84C2×#95C2316yes—
85C2×#95C2316yes—
86C2×#95C2316yes—
87C2×#95C2313yes—
88C2×#95C2313yes—
89C2×#95C2316yes—
90C2×#95C2316yes—
91C2×#95C2316yes—
92C2×#95C2316yes—
93C2×#95C2313yes—
94C2×#95C2313yes—
95C2×#95C2316yes—
96C2×#95C2316yes—
97C2×#95C2311yescharacteristic
98C25C2412yes—
99C25C2411yescharacteristic
100C22×D4—24no—
101C22×D4—22no—
102C22×D4—43no—
103C22×D4—46no—
104C22×D4—43no—
105C22×D4—46no—
106C23×C4—22no—
107C23×C4—26no—
108C23×C4—43no—
109C23×C4C2416yes—
110C23×C4C2416yes—
111C23×C4C2413yes—
112C23×C4C2413yes—
113C23×C4C2412yes—
114C23×C4C2413yes—
115C23×C4C2413yes—
116C23×C4C2411yescharacteristic
117C23×C4C2411yescharacteristic
118#121—416no—
119C2×C4⋊C4—26no—
120C2×C4⋊C4—212no—
121C2×C4⋊C4—212no—
122C2×C4⋊C4—212no—
123C2×C4⋊C4—43no—
124C2×C4⋊C4—43no—
125C2×C4⋊C4—43no—
126C2×C4⋊C4—43no—
127C2×C4⋊C4—46no—
128C2×C4⋊C4—43no—
129C2×C4⋊C4—43no—
130C2×C4⋊C4—46no—
131C2×C4⋊C4—46no—
132C2×C4⋊C4—46no—
133C2×C4⋊C4—46no—
134C2×C4⋊C4—46no—
135C2×C22⋊C4—26no—
136C2×C22⋊C4—26no—
137C2×C22⋊C4—212no—
138C2×C22⋊C4—26no—
139C2×C22⋊C4—212no—
140C2×C22⋊C4—212no—
141C2×C22⋊C4—43no—
142C2×C22⋊C4—46no—
143C2×C22⋊C4—46no—
144C2×C22⋊C4—43no—
145C2×C22⋊C4—43no—
146C2×C22⋊C4—46no—
147C2×C22⋊C4—46no—
148C2×C22⋊C4—46no—
149C2×C22⋊C4—46no—
150C2×C22⋊C4—46no—
151C2×C22⋊C4—46no—
152C2×C22⋊C4—46no—
153C2×C22⋊C4—46no—
154C2×C22⋊C4—46no—
155C2×C22⋊C4—46no—
156C2×C22⋊C4—46no—
157C2×C42—26no—
158C2×C42—26no—
159C2×C42—212no—
160C2×C42—46no—
161#95—212no—
162#95—26no—
163#95—26no—
164#95—26no—
165#95—43no—
166#95—43no—
167#95—46no—
168#95—46no—
169#95—46no—
170#95—43no—
171#95—43no—
172#95—46no—
173#95—46no—
174#95—46no—
175#95—46no—
176#95—46no—
177#95—46no—
178#95#4114yes—
179C24—21no—
180C24—23no—
181C24—23no—
182C24—21no—
183C24—23no—
184C24—23no—
185C24—22no—
186C24—26no—
187C24—26no—
188C24—22no—
189C24—22no—
190C24—26no—
191C24—26no—
192C242−1+412yes—
193C24C2511yescharacteristic
194C2×D4—44no—
195C2×D4—48no—
196C2×D4—86no—
197C2×D4—812no—
198C2×D4—86no—
199C2×D4—812no—
200C22×C4—21no—
201C22×C4—23no—
202C22×C4—26no—
203C22×C4—23no—
204C22×C4—23no—
205C22×C4—23no—
206C22×C4—26no—
207C22×C4—26no—
208C22×C4—23no—
209C22×C4—23no—
210C22×C4—23no—
211C22×C4—26no—
212C22×C4—23no—
213C22×C4—26no—
214C22×C4—26no—
215C22×C4—26no—
216C22×C4—26no—
217C22×C4—26no—
218C22×C4—23no—
219C22×C4—23no—
220C22×C4—26no—
221C22×C4—23no—
222C22×C4—26no—
223C22×C4—26no—
224C22×C4—26no—
225C22×C4—26no—
226C22×C4—26no—
227C22×C4—23no—
228C22×C4—22no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4—26no—
235C22×C4—26no—
236C22×C4—23no—
237C22×C4—26no—
238C22×C4—26no—
239C22×C4—412no—
240C22×C4—44no—
241C22×C4—86no—
242C22×C4C2×#4116yes—
243C22×C4H(5,𝔽2)16yes—
244C22×C4H(5,𝔽2)16yes—
245C22×C42−1+412yes—
246C4⋊C4—412no—
247C4⋊C4—412no—
248C4⋊C4—44no—
249C4⋊C4—812no—
250C4⋊C4—812no—
251C4⋊C4—86no—
252C4⋊C4—86no—
253C22⋊C4—412no—
254C22⋊C4—412no—
255C22⋊C4—424no—
256C22⋊C4—48no—
257C22⋊C4—812no—
258C22⋊C4—812no—
259C22⋊C4—812no—
260C22⋊C4—812no—
261C42—412no—
262C42—412no—
263C23—22no—
264C23—26no—
265C23—26no—
266C23—43no—
267C23—43no—
268C23—41no—
269C23—43no—
270C23—43no—
271C23—43no—
272C23—43no—
273C23—43no—
274C23—46no—
275C23—46no—
276C23—46no—
277C23—42no—
278C23—42no—
279C23—46no—
280C23—46no—
281C23—46no—
282C23—46no—
283C23—46no—
284C23—46no—
285C23—46no—
286C23—46no—
287C23—46no—
288C23C22×#4111yescharacteristic
289C23C2×H(5,𝔽2)13yes—
290C23C2×H(5,𝔽2)13yes—
291C23C2×2−1+411yescharacteristic
292C23C2×2−1+411yescharacteristic
293C23#58513yes—
294C23#58513yes—
295D4—88no—
296D4—816no—
297C2×C4—26no—
298C2×C4—26no—
299C2×C4—26no—
300C2×C4—212no—
301C2×C4—26no—
302C2×C4—26no—
303C2×C4—212no—
304C2×C4—43no—
305C2×C4—41no—
306C2×C4—43no—
307C2×C4—43no—
308C2×C4—46no—
309C2×C4—43no—
310C2×C4—43no—
311C2×C4—46no—
312C2×C4—43no—
313C2×C4—46no—
314C2×C4—46no—
315C2×C4—46no—
316C2×C4—46no—
317C2×C4—43no—
318C2×C4—42no—
319C2×C4—46no—
320C2×C4—46no—
321C2×C4—43no—
322C2×C4—43no—
323C2×C4—43no—
324C2×C4—46no—
325C2×C4—46no—
326C2×C4—43no—
327C2×C4—46no—
328C2×C4—46no—
329C2×C4—46no—
330C2×C4—46no—
331C2×C4—46no—
332C2×C4—46no—
333C2×C4—46no—
334C2×C4—46no—
335C2×C4—46no—
336C2×C4—46no—
337C2×C4—46no—
338C2×C4—46no—
339C2×C4—88no—
340C2×C4#56014yes—
341C22—46no—
342C22—46no—
343C22—42no—
344C22—81no—
345C22—81no—
346C22—83no—
347C22—83no—
348C22—82no—
349C22—86no—
350C22—86no—
351C22—82no—
352C22—86no—
353C22—86no—
354C22—82no—
355C22#346013yes—
356C22#351413yes—
357C22#352311yescharacteristic
358C22C2×#56011yescharacteristic
359C22#352913yes—
360C22#353313yes—
361C22#353513yes—
362C22#353613yes—
363C22#353713yes—
364C22#354713yes—
365C22#355116yes—
366C22#355313yes—
367C4—24no—
368C4—42no—
369C4—46no—
370C4—46no—
371C4—83no—
372C4—83no—
373C4—86no—
374C4—86no—
375C4—82no—
376C2—82no—
377C2—162no—
378C2#3669611yescharacteristic
379C2#3703513yes—
380C2#3715713yes—
381C2#5274211yescharacteristic
382C2#5276113yes—
383C2#5290413yes—
384C2#5298811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17243C2semidirect
2. #17234C2semidirect
3. #17204C2semidirect
4. #17135C2semidirect
5. #17125C2semidirect
6. #17114C2semidirect
7. #15074C2semidirect
8. #15017C2semidirect
10. #14331C2semidirect
11. C4×#399C2semidirect

Maximal subgroups

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

subgroupindex
#172432
#172342
#172042
#171352
#171252
#171142
#150742
#150172
#149962
#143312
C4×#3992