Finite Group Databasebeta

← #4462691

Subgroups — #4462691

Order 512.

2121 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2643C211yesmaximal, characteristic
2C22×#401C211yesmaximal, characteristic, complemented
3#17230C212yesmaximal, complemented
4#17215C216yesmaximal, complemented
5#17213C212yesmaximal, complemented
6#17182C216yesmaximal, complemented
7#14935C213yesmaximal, complemented
8#14927C213yesmaximal
9#14921C213yesmaximal
10#14920C213yesmaximal, complemented
11#14353C211yesmaximal, characteristic, complemented
12C22×#123C2211yescharacteristic
13#2643—28no—
14C2×#401—28no—
15C2×#401C2211yescharacteristic
16C2×#401C2216yes—
17C2×#401C2212yes—
18C2×#400C2213yes—
19C2×#397C2213yes—
20C2×#396C2211yescharacteristic
21C2×#386C2213yes—
22C2×#386C2213yes—
23C2×#385C2213yes—
24C2×#385C2213yes—
25C2×#381C2213yes—
26C2×#380C2213yes—
27C2×#380C2213yes—
28C22×#95C2213yes—
29C22×#95C2213yes—
30C22×#95C2211yescharacteristic
31#1445C2216yes—
32#1444C2216yes—
33#1444C2212yes—
34#1443C2216yes—
35#1443C2216yes—
36#1442C2216yes—
37#1442C2216yes—
38#1441C2216yes—
39#1441C2216yes—
40#1441C2216yes—
41#1441C2216yes—
42#1441C2216yes—
43#1441C2216yes—
44#1441C2216yes—
45#1440C2216yes—
46#1439C2216yes—
47#1439C2216yes—
48#1439C2216yes—
49C22⋊#95C2213yes—
50C22⋊#95C2213yes—
51C23×D4C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C24×C4C2313yes—
54C24×C4C2313yes—
55C2×#123—46no—
56C22×C4⋊C4C2313yes—
57C22×C4⋊C4C2313yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2316yes—
60C22×C4⋊C4C2316yes—
61C22×C4⋊C4C2316yes—
62C22×C22⋊C4C2311yescharacteristic
63C22×C22⋊C4C2313yes—
64C22×C22⋊C4C2313yes—
65C22×C22⋊C4C2313yes—
66C22×C22⋊C4C2313yes—
67C22×C22⋊C4C2312yes—
68C22×C22⋊C4C2316yes—
69#401—24no—
70#401—28no—
71#401—224no—
72#400—212no—
73#397—212no—
74#396—24no—
75#386—212no—
76#386—46no—
77#385—212no—
78#385—46no—
79#381—46no—
80#380—46no—
81#380—46no—
82C2×#95—212no—
83C2×#95—212no—
84C2×#95D418yes—
85C2×#95C2313yes—
86C2×#95C2313yes—
87C2×#95C2313yes—
88C2×#95C2313yes—
89C2×#95C2316yes—
90C2×#95C2316yes—
91C2×#95C2316yes—
92C2×#95C2316yes—
93C2×#95C2316yes—
94C2×#95C2316yes—
95C2×#95C2316yes—
96C2×#95C2316yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2316yes—
100C2×#95C2316yes—
101C2×#95C2316yes—
102C2×#95C2316yes—
103C2×#95C2311yescharacteristic
104C2×#95C2313yes—
105C25C2411yescharacteristic
106C25C2411yescharacteristic
107C22×D4—22no—
108C22×D4—26no—
109C22×D4—26no—
110C23×C4—26no—
111C23×C4—26no—
112C23×C4—26no—
113C23×C4—22no—
114C23×C4—26no—
115C23×C4—26no—
116C23×C4—26no—
117C23×C4—26no—
118C23×C4—46no—
119C23×C4—46no—
120C23×C4—46no—
121C23×C4—46no—
122C23×C4C2×D4112yes—
123C23×C4C2416yes—
124C23×C4C2412yes—
125C23×C4C2413yes—
126C23×C4C2413yes—
127C23×C4C2412yes—
128C23×C4C2416yes—
129C23×C4C2411yescharacteristic
130C23×C4C2413yes—
131C23×C4C2413yes—
132#123—416no—
133C2×C4⋊C4—26no—
134C2×C4⋊C4—26no—
135C2×C4⋊C4—46no—
136C2×C4⋊C4—46no—
137C2×C4⋊C4—46no—
138C2×C4⋊C4—46no—
139C2×C4⋊C4—46no—
140C2×C4⋊C4—46no—
141C2×C4⋊C4—46no—
142C2×C4⋊C4—46no—
143C2×C4⋊C4—46no—
144C2×C4⋊C4—46no—
145C2×C4⋊C4—46no—
146C2×C4⋊C4—46no—
147C2×C22⋊C4—26no—
148C2×C22⋊C4—26no—
149C2×C22⋊C4—26no—
150C2×C22⋊C4—26no—
151C2×C22⋊C4—26no—
152C2×C22⋊C4—26no—
153C2×C22⋊C4—26no—
154C2×C22⋊C4—26no—
155C2×C22⋊C4—26no—
156C2×C22⋊C4—46no—
157C2×C22⋊C4—46no—
158C2×C22⋊C4—46no—
159C2×C22⋊C4—46no—
160C2×C22⋊C4—46no—
161C2×C22⋊C4—46no—
162#95—212no—
163#95—212no—
164#95—212no—
165#95—212no—
166#95—26no—
167#95—26no—
168#95—26no—
169#95—26no—
170#95—46no—
171#95—46no—
172#95—46no—
173#95—46no—
174#95—46no—
175#95—46no—
176#95—46no—
177#95—46no—
178#95—46no—
179#95—46no—
180#95C2×D414yes—
181#95C2×D4112yes—
182C24—26no—
183C24—22no—
184C24—21no—
185C24—23no—
186C24—23no—
187C24—26no—
188C24—22no—
189C24C22×D412yes—
190C24H(5,𝔽2)16yes—
191C242−1+416yes—
192C24C2511yescharacteristic
193C2×D4—412no—
194C2×D4—44no—
195C2×D4—412no—
196C22×C4—23no—
197C22×C4—23no—
198C22×C4—23no—
199C22×C4—23no—
200C22×C4—26no—
201C22×C4—26no—
202C22×C4—26no—
203C22×C4—26no—
204C22×C4—224no—
205C22×C4—22no—
206C22×C4—26no—
207C22×C4—21no—
208C22×C4—23no—
209C22×C4—23no—
210C22×C4—23no—
211C22×C4—26no—
212C22×C4—26no—
213C22×C4—23no—
214C22×C4—26no—
215C22×C4—26no—
216C22×C4—26no—
217C22×C4—23no—
218C22×C4—26no—
219C22×C4—23no—
220C22×C4—23no—
221C22×C4—26no—
222C22×C4—26no—
223C22×C4—23no—
224C22×C4—22no—
225C22×C4—26no—
226C22×C4—26no—
227C22×C4—26no—
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4—26no—
235C22×C4—26no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—412no—
239C22×C4—412no—
240C22×C4—412no—
241C22×C4—412no—
242C22×C4—412no—
243C22×C4—412no—
244C22×C4—44no—
245C22×C4—412no—
246C22×C4—812no—
247C22×C4—812no—
248C22×C4—812no—
249C22×C4—812no—
250C22×C4C22×D416yes—
251C22×C4C22×D416yes—
252C4⋊C4—48no—
253C4⋊C4—86no—
254C4⋊C4—86no—
255C4⋊C4—812no—
256C4⋊C4—812no—
257C4⋊C4—812no—
258C22⋊C4—48no—
259C22⋊C4—412no—
260C22⋊C4—412no—
261C22⋊C4—44no—
262C22⋊C4—86no—
263C22⋊C4—86no—
264C22⋊C4—812no—
265C23—26no—
266C23—22no—
267C23—212no—
268C23—212no—
269C23—212no—
270C23—212no—
271C23—43no—
272C23—43no—
273C23—41no—
274C23—41no—
275C23—43no—
276C23—46no—
277C23—46no—
278C23—46no—
279C23—46no—
280C23#537112yes—
281C23#56014yes—
282C23#563112yes—
283C23C23×D411yescharacteristic
284C23C2×H(5,𝔽2)13yes—
285C23C2×2−1+413yes—
286C23#58516yes—
287C23#58512yes—
288D4—88no—
289C2×C4—26no—
290C2×C4—26no—
291C2×C4—26no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—26no—
295C2×C4—46no—
296C2×C4—46no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—43no—
300C2×C4—43no—
301C2×C4—46no—
302C2×C4—46no—
303C2×C4—43no—
304C2×C4—43no—
305C2×C4—43no—
306C2×C4—43no—
307C2×C4—46no—
308C2×C4—43no—
309C2×C4—46no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—46no—
313C2×C4—41no—
314C2×C4—46no—
315C2×C4—43no—
316C2×C4—46no—
317C2×C4—46no—
318C2×C4—46no—
319C2×C4—46no—
320C2×C4—46no—
321C2×C4—46no—
322C2×C4—46no—
323C2×C4—46no—
324C2×C4—46no—
325C2×C4—46no—
326C2×C4—46no—
327C2×C4—46no—
328C2×C4—46no—
329C2×C4—46no—
330C2×C4—46no—
331C2×C4—46no—
332C2×C4—824no—
333C2×C4—824no—
334C2×C4—88no—
335C22—28no—
336C22—224no—
337C22—212no—
338C22—212no—
339C22—82no—
340C22—86no—
341C22—81no—
342C22—83no—
343C22—83no—
344C22#282318yes—
345C22C2×#53713yes—
346C22#348513yes—
347C22#348711yescharacteristic
348C22C2×#56011yescharacteristic
349C22C2×#56313yes—
350C22#356016yes—
351C22#356016yes—
352C22#356416yes—
353C22#356416yes—
354C4—46no—
355C4—46no—
356C4—82no—
357C4—86no—
358C4—82no—
359C4—81no—
360C4—83no—
361C4—83no—
362C4—86no—
363C2—28no—
364C2—161no—
365C2C2×#282311yescharacteristic
366C2#3696013yes—
367C2#3697113yes—
368C2#5276512yes—
369C2#5276616yes—
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
2. C22×#401C2semidirect
3. #17230C2semidirect
4. #17215C2semidirect
5. #17213C2semidirect
6. #17182C2semidirect
7. #14935C2semidirect
10. #14920C2semidirect
11. #14353C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26432
C22×#4012
#172302
#172152
#172132
#171822
#149352
#149272
#149212
#149202
#143532