Finite Group Databasebeta

← #1007240

Subgroups — #1007240

Order 512.

3426 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2413C216yesmaximal, complemented
2#17135C211yesmaximal, characteristic, complemented
3#17114C211yesmaximal, characteristic, complemented
4#16330C216yesmaximal, complemented
5#15417C213yesmaximal, complemented
6#15363C213yesmaximal, complemented
7#14926C213yesmaximal
8#14748C212yesmaximal, complemented
9#14562C212yesmaximal, complemented
10#14128C213yesmaximal, complemented
11C2×#1444C211yesmaximal, characteristic, complemented
12C22×#123C2216yes—
13C22×#121C2211yescharacteristic
14C22×#117C2213yes—
15C23×C4⋊C4C2213yes—
16#2413—248no—
17C2×#398C2216yes—
18C2×#398C2216yes—
19C2×#396C2216yes—
20C2×#396C2212yes—
21C2×#396C2216yes—
22C2×#389C2216yes—
23C2×#389C2216yes—
24C2×#388C2216yes—
25C2×#388C2216yes—
26C2×#386C2216yes—
27C2×#385C2216yes—
28C2×#381C2213yes—
29C22×#95C2211yescharacteristic
30C22×#95C2213yes—
31#1444C2213yes—
32#1444C2213yes—
33#1444C2213yes—
34#1444C2213yes—
35#1444C2213yes—
36#1444C2211yescharacteristic
37#1444C2211yescharacteristic
38#1444C2213yes—
39#1443C2212yes—
40#1442C2213yes—
41#1442C2213yes—
42#1441C2216yes—
43#1439C2216yes—
44#1439C2216yes—
45#1439C2216yes—
46#1438C2213yes—
47#1438C2213yes—
48#1436C2216yes—
49#1436C2216yes—
50C22⋊#95C2213yes—
51C23×D4C2312yes—
52C23×D4C2311yescharacteristic
53C24×C4C2313yes—
54C24×C4C2313yes—
55C24×C4C2311yescharacteristic
56C2×#123—224no—
57C2×#123—224no—
58C2×#123—412no—
59C2×#121—46no—
60C2×#117—224no—
61C2×#117—46no—
62C22×C4⋊C4—224no—
63C22×C4⋊C4—46no—
64C22×C4⋊C4C2313yes—
65C22×C4⋊C4C2313yes—
66C22×C4⋊C4C2313yes—
67C22×C4⋊C4C2316yes—
68C22×C4⋊C4C2316yes—
69C22×C4⋊C4C2316yes—
70C22×C4⋊C4C2316yes—
71C22×C4⋊C4C2316yes—
72C22×C4⋊C4C2313yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2316yes—
76C22×C22⋊C4C2316yes—
77C22×C22⋊C4C2316yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2312yes—
80C22×C42C2316yes—
81#398—224no—
82#398—224no—
83#396—224no—
84#396—224no—
85#396—28no—
86#389—224no—
87#389—224no—
88#388—224no—
89#388—224no—
90#386—224no—
91#385—224no—
92#381D4124yescomplemented
93C2×#95—24no—
94C2×#95—46no—
95C2×#95C2316yes—
96C2×#95C2316yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2313yes—
100C2×#95C2313yes—
101C2×#95C2313yes—
102C2×#95C2316yes—
103C2×#95C2313yes—
104C2×#95C2313yes—
105C2×#95C2313yes—
106C2×#95C2313yes—
107C2×#95C2313yes—
108C2×#95C2316yes—
109C2×#95C2313yes—
110C2×#95C2311yescharacteristic
111C25C2412yes—
112C25C2411yescharacteristic
113C22×D4—212no—
114C22×D4—212no—
115C22×D4—24no—
116C22×D4—22no—
117C22×D4—43no—
118C22×D4—43no—
119C23×C4—212no—
120C23×C4—212no—
121C23×C4—26no—
122C23×C4—26no—
123C23×C4—26no—
124C23×C4—26no—
125C23×C4—26no—
126C23×C4—212no—
127C23×C4—26no—
128C23×C4—22no—
129C23×C4—46no—
130C23×C4—43no—
131C23×C4C2×D4112yes—
132C23×C4C2411yescharacteristic
133C23×C4C2412yes—
134C23×C4C2416yes—
135C23×C4C2416yes—
136C23×C4C2413yes—
137C23×C4C2413yes—
138C23×C4C2413yes—
139C23×C4C2413yes—
140C23×C4C2411yescharacteristic
141#123—496no—
142#121—416no—
143#117—448no—
144C2×C4⋊C4—212no—
145C2×C4⋊C4—212no—
146C2×C4⋊C4—212no—
147C2×C4⋊C4—212no—
148C2×C4⋊C4—212no—
149C2×C4⋊C4—212no—
150C2×C4⋊C4—212no—
151C2×C4⋊C4—212no—
152C2×C4⋊C4—212no—
153C2×C4⋊C4—212no—
154C2×C4⋊C4—212no—
155C2×C4⋊C4—212no—
156C2×C4⋊C4—212no—
157C2×C4⋊C4—212no—
158C2×C4⋊C4—43no—
159C2×C4⋊C4—46no—
160C2×C4⋊C4—43no—
161C2×C4⋊C4—46no—
162C2×C4⋊C4—46no—
163C2×C4⋊C4—43no—
164C2×C4⋊C4—43no—
165C2×C4⋊C4—43no—
166C2×C4⋊C4—46no—
167C2×C4⋊C4—46no—
168C2×C4⋊C4—448no—
169C2×C22⋊C4—212no—
170C2×C22⋊C4—212no—
171C2×C22⋊C4—212no—
172C2×C22⋊C4—212no—
173C2×C22⋊C4—212no—
174C2×C22⋊C4—212no—
175C2×C22⋊C4—212no—
176C2×C22⋊C4—212no—
177C2×C22⋊C4—212no—
178C2×C22⋊C4—212no—
179C2×C22⋊C4—46no—
180C2×C22⋊C4—46no—
181C2×C22⋊C4—46no—
182C2×C22⋊C4—46no—
183C2×C22⋊C4—46no—
184C2×C22⋊C4C2×D4124yes—
185C2×C42—212no—
186C2×C42—212no—
187C2×C42—46no—
188#95—212no—
189#95—212no—
190#95—212no—
191#95—212no—
192#95—26no—
193#95—26no—
194#95—43no—
195#95—43no—
196#95—43no—
197#95—43no—
198#95—46no—
199#95—43no—
200#95C2×D4112yes—
201#95C2×D4124yes—
202#95C2×D4112yes—
203#95#4114yes—
204C24—21no—
205C24—23no—
206C24—21no—
207C24—23no—
208C24—26no—
209C24—22no—
210C24—26no—
211C24—26no—
212C24—22no—
213C24—22no—
214C24—26no—
215C24C22×D416yes—
216C24H(5,𝔽2)16yes—
217C242−1+412yes—
218C24C2511yescharacteristic
219C2×D4—44no—
220C2×D4—424no—
221C2×D4—424no—
222C2×D4—48no—
223C2×D4—86no—
224C2×D4—86no—
225C22×C4—224no—
226C22×C4—224no—
227C22×C4—248no—
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4—23no—
235C22×C4—26no—
236C22×C4—23no—
237C22×C4—23no—
238C22×C4—23no—
239C22×C4—23no—
240C22×C4—23no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—26no—
248C22×C4—26no—
249C22×C4—23no—
250C22×C4—23no—
251C22×C4—23no—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4—26no—
255C22×C4—23no—
256C22×C4—26no—
257C22×C4—26no—
258C22×C4—26no—
259C22×C4—23no—
260C22×C4—26no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—23no—
264C22×C4—424no—
265C22×C4—424no—
266C22×C4—412no—
267C22×C4—412no—
268C22×C4—424no—
269C22×C4—412no—
270C22×C4—412no—
271C22×C4—412no—
272C22×C4—44no—
273C22×C4—86no—
274C22×C4C22×D4112yes—
275C22×C4C22×D4112yes—
276C22×C4C22×D416yes—
277C22×C4C22×D416yes—
278C22×C4C2×#4116yes—
279C22×C42−1+412yes—
280C22×C42−1+416yes—
281C4⋊C4—424no—
282C4⋊C4—424no—
283C4⋊C4—412no—
284C4⋊C4—424no—
285C4⋊C4—412no—
286C4⋊C4—412no—
287C4⋊C4—412no—
288C4⋊C4—424no—
289C4⋊C4—424no—
290C4⋊C4—44no—
291C22⋊C4—248no—
292C22⋊C4—424no—
293C22⋊C4—424no—
294C22⋊C4—424no—
295C22⋊C4—424no—
296C22⋊C4—48no—
297C42—424no—
298C23—26no—
299C23—212no—
300C23—26no—
301C23—26no—
302C23—22no—
303C23—26no—
304C23—26no—
305C23—26no—
306C23—26no—
307C23—46no—
308C23—46no—
309C23—46no—
310C23—46no—
311C23—46no—
312C23—46no—
313C23—46no—
314C23—42no—
315C23—42no—
316C23—46no—
317C23—46no—
318C23—46no—
319C23—46no—
320C23#537112yes—
321C23D42112yes—
322C23#56014yes—
323C23C23×D413yes—
324C23C22×#4111yescharacteristic
325C23C2×H(5,𝔽2)13yes—
326C23C2×2−1+411yescharacteristic
327C23C2×2−1+413yes—
328C23C2×2−1+411yescharacteristic
329C23#58513yes—
330D4—88no—
331D4—816no—
332C2×C4—26no—
333C2×C4—26no—
334C2×C4—212no—
335C2×C4—212no—
336C2×C4—212no—
337C2×C4—212no—
338C2×C4—212no—
339C2×C4—212no—
340C2×C4—212no—
341C2×C4—212no—
342C2×C4—212no—
343C2×C4—26no—
344C2×C4—26no—
345C2×C4—212no—
346C2×C4—26no—
347C2×C4—448no—
348C2×C4—448no—
349C2×C4—46no—
350C2×C4—46no—
351C2×C4—43no—
352C2×C4—43no—
353C2×C4—43no—
354C2×C4—43no—
355C2×C4—46no—
356C2×C4—46no—
357C2×C4—46no—
358C2×C4—46no—
359C2×C4—46no—
360C2×C4—46no—
361C2×C4—46no—
362C2×C4—41no—
363C2×C4—43no—
364C2×C4—43no—
365C2×C4—46no—
366C2×C4—43no—
367C2×C4—43no—
368C2×C4—43no—
369C2×C4—46no—
370C2×C4—46no—
371C2×C4—46no—
372C2×C4—46no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—43no—
376C2×C4—43no—
377C2×C4—88no—
378C22—24no—
379C22—212no—
380C22—212no—
381C22—212no—
382C22—212no—
383C22—24no—
384C22—82no—
385C22—86no—
386C22—86no—
387C22—82no—
388C22—86no—
389C22—86no—
390C22—82no—
391C22#267518yes—
392C22C2×#53713yes—
393C22C2×D4213yes—
394C22D4×#4113yes—
395C22#348916yes—
396C22#349213yes—
397C22#349913yes—
398C22#352311yescharacteristic
399C22C2×#56011yescharacteristic
400C22#353313yes—
401C22#353513yes—
402C22#354413yes—
403C22#356313yes—
404C4—42no—
405C4—412no—
406C4—46no—
407C4—46no—
408C4—46no—
409C4—46no—
410C4—412no—
411C4—81no—
412C4—82no—
413C2—28no—
414C2—162no—
415C2#3842713yes—
416C2#4216713yes—
417C2#4475813yes—
418C2#4632711yescharacteristic
419C2C2×#267511yescharacteristic
420C2#5274211yescharacteristic
421C2#5899313yes—
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×#2413C2semidirect
2. #17135C2semidirect
3. #17114C2semidirect
4. #16330C2semidirect
5. #15417C2semidirect
6. #15363C2semidirect
8. #14748C2semidirect
9. #14562C2semidirect
10. #14128C2semidirect
11. C2×#1444C2semidirect
92. #381D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24132
#171352
#171142
#163302
#154172
#153632
#149262
#147482
#145622
#141282
C2×#14442