Finite Group Databasebeta

← #1082666

Subgroups — #1082666

Order 512.

2248 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2801C212yesmaximal, complemented
2#17230C211yesmaximal, characteristic, complemented
3#17135C211yesmaximal, characteristic, complemented
4#16534C216yesmaximal, complemented
5#16047C213yesmaximal, complemented
6#16002C213yesmaximal, complemented
7#15171C212yesmaximal, complemented
8#14996C213yesmaximal
9#14225C213yesmaximal, complemented
10#14200C216yesmaximal, complemented
11C4×#399C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13#2801—216no—
14C2×#401C2212yes—
15C2×#399C2211yescharacteristic
16C2×#397C2213yes—
17C2×#396C2212yes—
18C2×#396C2216yes—
19C2×#394C2216yes—
20C2×#389C2216yes—
21C2×#388C2216yes—
22C2×#388C2216yes—
23C2×#388C2216yes—
24C2×C4×C4⋊C4C2213yes—
25C2×C4×C4⋊C4C2213yes—
26C2×C4×C22⋊C4C2213yes—
27#1445C2213yes—
28#1445C2213yes—
29#1444C2211yescharacteristic
30#1444C2213yes—
31#1444C2213yes—
32#1442C2216yes—
33#1442C2213yes—
34#1441C2216yes—
35#1441C2216yes—
36#1440C2216yes—
37#1439C2213yes—
38#1439C2216yes—
39#1439C2213yes—
40#1438C2216yes—
41#1438C2213yes—
42#1438C2213yes—
43#1437C2216yes—
44#1437C2216yes—
45#1436C2216yes—
46#1436C2216yes—
47#1434C2213yes—
48#1434C2216yes—
49C22⋊#95C2213yes—
50C4×#95C2211yescharacteristic
51C23×D4C2312yes—
52C23×D4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C2×#121—46no—
55C22×C4⋊C4C2316yes—
56C22×C4⋊C4C2313yes—
57C22×C4⋊C4C2313yes—
58C22×C4⋊C4C2316yes—
59C22×C4⋊C4C2313yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2313yes—
62C22×C22⋊C4C2312yes—
63C22×C22⋊C4C2316yes—
64C22×C22⋊C4C2316yes—
65C22×C22⋊C4C2316yes—
66C22×C22⋊C4C2316yes—
67C22×C22⋊C4C2313yes—
68C22×C22⋊C4C2316yes—
69C22×C22⋊C4C2313yes—
70C22×C42C2313yes—
71C22×C42C2313yes—
72C22×C42C2316yes—
73#401—28no—
74#399D418yescomplemented
75#397—212no—
76#396—28no—
77#396—224no—
78#394—224no—
79#389—224no—
80#388—224no—
81#388—224no—
82#388—412no—
83C4×C4⋊C4—46no—
84C4×C4⋊C4—46no—
85C4×C22⋊C4—46no—
86C2×#95C2316yes—
87C2×#95C2316yes—
88C2×#95C2316yes—
89C2×#95C2313yes—
90C2×#95C2316yes—
91C2×#95C2316yes—
92C2×#95C2313yes—
93C2×#95C2313yes—
94C2×#95C2313yes—
95C2×#95C2316yes—
96C2×#95C2313yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2316yes—
100C2×#95C2313yes—
101C2×#95C2311yescharacteristic
102C2×#95C2313yes—
103C25C2412yes—
104C25C2411yescharacteristic
105C22×D4—24no—
106C22×D4—22no—
107C22×D4—43no—
108C22×D4—46no—
109C22×D4—43no—
110C22×D4—46no—
111C23×C4—22no—
112C23×C4—26no—
113C23×C4—43no—
114C23×C4C2416yes—
115C23×C4C2413yes—
116C23×C4C2412yes—
117C23×C4C2413yes—
118C23×C4C2413yes—
119C23×C4C2416yes—
120C23×C4C2413yes—
121C23×C4C2411yescharacteristic
122C23×C4C2411yescharacteristic
123#121—416no—
124C2×C4⋊C4—212no—
125C2×C4⋊C4—212no—
126C2×C4⋊C4—212no—
127C2×C4⋊C4—212no—
128C2×C4⋊C4—26no—
129C2×C4⋊C4—212no—
130C2×C4⋊C4—43no—
131C2×C4⋊C4—46no—
132C2×C4⋊C4—43no—
133C2×C4⋊C4—43no—
134C2×C4⋊C4—43no—
135C2×C4⋊C4—43no—
136C2×C4⋊C4—43no—
137C2×C4⋊C4—43no—
138C2×C4⋊C4—46no—
139C2×C4⋊C4—46no—
140C2×C4⋊C4C2×D4112yes—
141C2×C22⋊C4—212no—
142C2×C22⋊C4—212no—
143C2×C22⋊C4—212no—
144C2×C22⋊C4—212no—
145C2×C22⋊C4—212no—
146C2×C22⋊C4—212no—
147C2×C22⋊C4—26no—
148C2×C22⋊C4—212no—
149C2×C22⋊C4—212no—
150C2×C22⋊C4—43no—
151C2×C22⋊C4—46no—
152C2×C22⋊C4—46no—
153C2×C22⋊C4—43no—
154C2×C22⋊C4—43no—
155C2×C22⋊C4—43no—
156C2×C22⋊C4—46no—
157C2×C22⋊C4—46no—
158C2×C22⋊C4—46no—
159C2×C22⋊C4—46no—
160C2×C22⋊C4—46no—
161C2×C22⋊C4—46no—
162C2×C22⋊C4C2×D4112yes—
163C2×C42—26no—
164C2×C42—26no—
165C2×C42—212no—
166C2×C42—212no—
167C2×C42—46no—
168C2×C42—46no—
169C2×C42—46no—
170#95—212no—
171#95—212no—
172#95—212no—
173#95—212no—
174#95—212no—
175#95—26no—
176#95—43no—
177#95—46no—
178#95—46no—
179#95—43no—
180#95—43no—
181#95—43no—
182#95—43no—
183#95—46no—
184#95—46no—
185#95C2×D414yes—
186#95#41112yes—
187C24—23no—
188C24—23no—
189C24—21no—
190C24—21no—
191C24—23no—
192C24—23no—
193C24—22no—
194C24—26no—
195C24—26no—
196C24—22no—
197C24—22no—
198C24—26no—
199C24—26no—
200C24C22×D412yes—
201C24C2511yescharacteristic
202C2×D4—44no—
203C2×D4—48no—
204C2×D4—86no—
205C2×D4—812no—
206C2×D4—86no—
207C2×D4—812no—
208C22×C4—21no—
209C22×C4—23no—
210C22×C4—26no—
211C22×C4—26no—
212C22×C4—26no—
213C22×C4—26no—
214C22×C4—26no—
215C22×C4—26no—
216C22×C4—23no—
217C22×C4—23no—
218C22×C4—23no—
219C22×C4—23no—
220C22×C4—26no—
221C22×C4—26no—
222C22×C4—26no—
223C22×C4—23no—
224C22×C4—23no—
225C22×C4—23no—
226C22×C4—23no—
227C22×C4—26no—
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—22no—
232C22×C4—23no—
233C22×C4—26no—
234C22×C4—26no—
235C22×C4—23no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—26no—
239C22×C4—26no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—26no—
244C22×C4—23no—
245C22×C4—412no—
246C22×C4—44no—
247C22×C4—86no—
248C22×C4C22×D416yes—
249C22×C4C22×D416yes—
250C22×C4C2×#41112yes—
251C22×C4C2×#4116yes—
252C22×C4H(5,𝔽2)16yes—
253C22×C42−1+412yes—
254C4⋊C4—412no—
255C4⋊C4—424no—
256C4⋊C4—412no—
257C4⋊C4—412no—
258C4⋊C4—412no—
259C4⋊C4—44no—
260C4⋊C4—812no—
261C22⋊C4—412no—
262C22⋊C4—424no—
263C22⋊C4—424no—
264C22⋊C4—424no—
265C22⋊C4—412no—
266C22⋊C4—48no—
267C22⋊C4—812no—
268C22⋊C4—812no—
269C42—412no—
270C42—412no—
271C42—812no—
272C23—26no—
273C23—26no—
274C23—22no—
275C23—43no—
276C23—43no—
277C23—46no—
278C23—43no—
279C23—41no—
280C23—43no—
281C23—43no—
282C23—43no—
283C23—43no—
284C23—46no—
285C23—46no—
286C23—46no—
287C23—46no—
288C23—42no—
289C23—412no—
290C23—46no—
291C23—42no—
292C23—46no—
293C23—46no—
294C23—46no—
295C23—46no—
296C23C23×D411yescharacteristic
297C23C22×#4113yes—
298C23C2×H(5,𝔽2)13yes—
299C23C2×2−1+411yescharacteristic
300C23#58511yescharacteristic
301C23#58513yes—
302C23#58513yes—
303D4—88no—
304D4—816no—
305C2×C4—26no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—212no—
309C2×C4—212no—
310C2×C4—212no—
311C2×C4—212no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—212no—
315C2×C4—212no—
316C2×C4—26no—
317C2×C4—46no—
318C2×C4—46no—
319C2×C4—43no—
320C2×C4—43no—
321C2×C4—46no—
322C2×C4—46no—
323C2×C4—46no—
324C2×C4—43no—
325C2×C4—43no—
326C2×C4—41no—
327C2×C4—43no—
328C2×C4—46no—
329C2×C4—43no—
330C2×C4—43no—
331C2×C4—46no—
332C2×C4—43no—
333C2×C4—46no—
334C2×C4—46no—
335C2×C4—46no—
336C2×C4—46no—
337C2×C4—46no—
338C2×C4—46no—
339C2×C4—46no—
340C2×C4—42no—
341C2×C4—43no—
342C2×C4—46no—
343C2×C4—43no—
344C2×C4—46no—
345C2×C4—46no—
346C2×C4—46no—
347C2×C4—88no—
348C2×C4#56014yes—
349C22—46no—
350C22—46no—
351C22—42no—
352C22—81no—
353C22—83no—
354C22—81no—
355C22—83no—
356C22—82no—
357C22—86no—
358C22—86no—
359C22—82no—
360C22—86no—
361C22—86no—
362C22—82no—
363C22#346013yes—
364C22D4×#4113yes—
365C22#348113yes—
366C22#349613yes—
367C22#349711yescharacteristic
368C22#351216yes—
369C22#351813yes—
370C22C2×#56011yescharacteristic
371C22#353313yes—
372C22#353513yes—
373C22#355113yes—
374C22#356813yes—
375C4—24no—
376C4—42no—
377C4—46no—
378C4—412no—
379C4—46no—
380C4—46no—
381C4—86no—
382C4—83no—
383C4—82no—
384C2—82no—
385C2—162no—
386C2#3945513yes—
387C2#4221713yes—
388C2#4483413yes—
389C2#4632811yescharacteristic
390C2#5229911yescharacteristic
391C2#5274211yescharacteristic
392C2#5984713yes—
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. C2×#2801C2semidirect
2. #17230C2semidirect
3. #17135C2semidirect
4. #16534C2semidirect
5. #16047C2semidirect
6. #16002C2semidirect
7. #15171C2semidirect
9. #14225C2semidirect
10. #14200C2semidirect
11. C4×#399C2semidirect
74. #399D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28012
#172302
#171352
#165342
#160472
#160022
#151712
#149962
#142252
#142002
C4×#3992