Finite Group Databasebeta

← #4417802

Subgroups — #4417802

Order 512.

3776 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic, complemented
2C2×#2847C211yesmaximal, characteristic
3#16885C2112yesmaximal, complemented
4#15236C213yesmaximal, complemented
5#15233C213yesmaximal, complemented
6#15219C213yesmaximal, complemented
7#15215C213yesmaximal, complemented
8#14657C214yesmaximal, complemented
9C4×#379C211yesmaximal, characteristic, complemented
10C25×C4C2211yescharacteristic
11C22×#124C2211yescharacteristic, complemented
12C22×C22⋊Q8C2213yes—
13C22×#121C2213yes—
14C22×#120C2212yes—
15#2848—28no—
16#2847—28no—
17C2×#391C2213yescomplemented
18C2×#390C2213yescomplemented
19C2×#386C2216yes—
20C2×#385C2216yes—
21C2×#382C2213yescomplemented
22C2×#380C2213yes—
23C2×#380C2213yes—
24C2×#379C2212yes—
25C2×#379C2212yes—
26C2×C4×C22⋊C4C2213yes—
27C2×C4×C22⋊C4C2213yes—
28#1443C22112yescomplemented
29#1442C22112yescomplemented
30#1440C22112yes—
31#1440C22112yes—
32#1440C22112yescomplemented
33#1439C22112yescomplemented
34#1438C22112yes—
35#1438C22112yes—
36C4⋊#95C2213yescomplemented
37C4⋊#95C2213yescomplemented
38C22⋊#95C2216yes—
39C26C2311yescharacteristic
40C23×Q8C2311yescharacteristic
41C23×D4C2311yescharacteristic
42C23×D4C2316yes—
43C24×C4—46no—
44C24×C4—46no—
45C24×C4D418yes—
46C24×C4C2313yes—
47C24×C4C2313yes—
48C2×#124—46no—
49C2×C22⋊Q8—212no—
50C2×C22⋊Q8—46no—
51C2×C22⋊Q8—46no—
52C2×#121—212no—
53C2×#121—46no—
54C2×#121—46no—
55C2×#120—28no—
56C2×#120—216no—
57C22×C4⋊C4C2313yes—
58C22×C4⋊C4C2313yes—
59C22×C4⋊C4C2316yes—
60C22×C22⋊C4C2316yes—
61C22×C22⋊C4C2316yes—
62C22×C22⋊C4C2316yes—
63C22×C22⋊C4C2316yes—
64C22×C22⋊C4C2316yes—
65C22×C22⋊C4C2316yes—
66C22×C22⋊C4C2314yes—
67C22×C22⋊C4C23112yes—
68C22×C42C2311yescharacteristic
69C22×C42C2313yes—
70#391—46no—
71#390—46no—
72#386—224no—
73#385—224no—
74#382—46no—
75#380—212no—
76#380—212no—
77#379—28no—
78#379—28no—
79C4×C22⋊C4—46no—
80C4×C22⋊C4—46no—
81C2×#95C2316yes—
82C2×#95C2316yes—
83C2×#95C2316yes—
84C2×#95C2316yes—
85C2×#95C23112yes—
86C2×#95C23112yes—
87C2×#95C23112yes—
88C2×#95C23112yes—
89C25—43no—
90C25—43no—
91C25—46no—
92C25C2×D414yes—
93C25#4118yes—
94C25C2412yes—
95C25C2413yes—
96C22×Q8—22no—
97C22×Q8—26no—
98C22×Q8—26no—
99C22×D4—212no—
100C22×D4—224no—
101C22×D4—22no—
102C22×D4—26no—
103C22×D4—26no—
104C22×D4—46no—
105C22×D4—46no—
106C22×D4—412no—
107C23×C4—26no—
108C23×C4—26no—
109C23×C4—26no—
110C23×C4—26no—
111C23×C4—26no—
112C23×C4—26no—
113C23×C4—448no—
114C23×C4—43no—
115C23×C4—43no—
116C23×C4—43no—
117C23×C4—43no—
118C23×C4—43no—
119C23×C4—43no—
120C23×C4—448no—
121C23×C4—416no—
122C23×C4C2×D4112yes—
123C23×C4C2×D4112yes—
124C23×C4C2412yes—
125C23×C4C2416yes—
126C23×C4C2416yes—
127C23×C4C2412yes—
128C23×C4C2416yes—
129C23×C4C2413yes—
130C23×C4C2411yescharacteristic
131#124—416no—
132C22⋊Q8—448no—
133#121—448no—
134#120—432no—
135C2×C4⋊C4—26no—
136C2×C4⋊C4—26no—
137C2×C4⋊C4—212no—
138C2×C4⋊C4—26no—
139C2×C4⋊C4—26no—
140C2×C4⋊C4—26no—
141C2×C4⋊C4—26no—
142C2×C4⋊C4—46no—
143C2×C4⋊C4—46no—
144C2×C22⋊C4—212no—
145C2×C22⋊C4—212no—
146C2×C22⋊C4—212no—
147C2×C22⋊C4—224no—
148C2×C22⋊C4—212no—
149C2×C22⋊C4—212no—
150C2×C22⋊C4—212no—
151C2×C22⋊C4—46no—
152C2×C22⋊C4—46no—
153C2×C22⋊C4—412no—
154C2×C22⋊C4—46no—
155C2×C22⋊C4—46no—
156C2×C22⋊C4—46no—
157C2×C22⋊C4—46no—
158C2×C22⋊C4—412no—
159C2×C22⋊C4—46no—
160C2×C22⋊C4—46no—
161C2×C22⋊C4—412no—
162C2×C22⋊C4—412no—
163C2×C22⋊C4—412no—
164C2×C42—26no—
165C2×C42—26no—
166C2×C42—26no—
167C2×C42—26no—
168#95—212no—
169#95—212no—
170#95—412no—
171#95—412no—
172#95—46no—
173#95—46no—
174#95—412no—
175#95—412no—
176C24—23no—
177C24—23no—
178C24—23no—
179C24—23no—
180C24—23no—
181C24—23no—
182C24—26no—
183C24—26no—
184C24—26no—
185C24—22no—
186C24—24no—
187C24—26no—
188C24—26no—
189C24—212no—
190C24—412no—
191C24—412no—
192C24—424no—
193C24—424no—
194C24—412no—
195C24—44no—
196C24—424no—
197C24—424no—
198C24#124116yescomplemented
199C24C22×D416yes—
200C24C2×#41112yes—
201C24C2511yescharacteristic
202C2×Q8—412no—
203C2×Q8—412no—
204C2×Q8—44no—
205C2×D4—424no—
206C2×D4—424no—
207C2×D4—424no—
208C2×D4—412no—
209C2×D4—412no—
210C2×D4—44no—
211C2×D4—824no—
212C2×D4—824no—
213C22×C4—23no—
214C22×C4—23no—
215C22×C4—23no—
216C22×C4—23no—
217C22×C4—23no—
218C22×C4—23no—
219C22×C4—224no—
220C22×C4—224no—
221C22×C4—224no—
222C22×C4—26no—
223C22×C4—26no—
224C22×C4—22no—
225C22×C4—26no—
226C22×C4—22no—
227C22×C4—212no—
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—26no—
239C22×C4—26no—
240C22×C4—212no—
241C22×C4—212no—
242C22×C4—212no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—412no—
246C22×C4—412no—
247C22×C4—412no—
248C22×C4—412no—
249C22×C4—412no—
250C22×C4—412no—
251C22×C4—412no—
252C22×C4—4128no—
253C22×C4—412no—
254C22×C4—412no—
255C22×C4—412no—
256C22×C4—412no—
257C22×C4C22×D416yes—
258C22×C4C22×D412yes—
259C22×C4H(5,𝔽2)16yes—
260C22×C4H(5,𝔽2)16yes—
261C4⋊C4—424no—
262C4⋊C4—412no—
263C4⋊C4—412no—
264C22⋊C4—424no—
265C22⋊C4—424no—
266C22⋊C4—424no—
267C22⋊C4—424no—
268C22⋊C4—424no—
269C22⋊C4—416no—
270C22⋊C4—424no—
271C22⋊C4—824no—
272C42—412no—
273C42—44no—
274C23—26no—
275C23—26no—
276C23—26no—
277C23—26no—
278C23—26no—
279C23—26no—
280C23—212no—
281C23—212no—
282C23—212no—
283C23—212no—
284C23—212no—
285C23—212no—
286C23—448no—
287C23—416no—
288C23—448no—
289C23—496no—
290C23—432no—
291C23—43no—
292C23—43no—
293C23—43no—
294C23—43no—
295C23—46no—
296C23—46no—
297C23—46no—
298C23—46no—
299C23—46no—
300C23—46no—
301C23—42no—
302C23—46no—
303C23—46no—
304C23—412no—
305C23—412no—
306C23—46no—
307C23—42no—
308C23—412no—
309C23—412no—
310C23C2×#124112yes—
311C23C23×D411yescharacteristic
312C23C22×#4112yes—
313C23C2×H(5,𝔽2)13yes—
314C23C2×H(5,𝔽2)13yes—
315C23#58516yes—
316Q8—88no—
317D4—848no—
318D4—88no—
319C2×C4—26no—
320C2×C4—26no—
321C2×C4—26no—
322C2×C4—448no—
323C2×C4—43no—
324C2×C4—43no—
325C2×C4—43no—
326C2×C4—43no—
327C2×C4—448no—
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—412no—
340C2×C4—412no—
341C2×C4—46no—
342C2×C4—46no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—412no—
346C2×C4—46no—
347C2×C4—46no—
348C2×C4—46no—
349C2×C4—412no—
350C2×C4—46no—
351C2×C4#534112yes—
352C2×C4#560112yes—
353C2×C4H(3,𝔽4)14yes—
354C22—212no—
355C22—212no—
356C22—212no—
357C22—464no—
358C22—46no—
359C22—46no—
360C22—46no—
361C22—46no—
362C22—412no—
363C22—412no—
364C22—412no—
365C22—412no—
366C22—82no—
367C22—86no—
368C22—812no—
369C22—84no—
370C22—86no—
371C22C22×#12411yescharacteristic
372C22C2×#53413yes—
373C22#344913yes—
374C22#350516yes—
375C22#350516yes—
376C22C2×#56013yes—
377C22C2×H(3,𝔽4)11yescharacteristic
378C22#353916yes—
379C22#354016yes—
380C4—46no—
381C4—82no—
382C4—86no—
383C4—86no—
384C4—82no—
385C4—86no—
386C4#281218yes—
387C2—412no—
388C2—162no—
389C2#3501913yes—
390C2C2×#281211yescharacteristic
391C2#3745513yes—
392C2#4550816yes—
393C2#4550912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2847C2
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×#2848C2semidirect
3. #16885C2semidirect
4. #15236C2semidirect
5. #15233C2semidirect
6. #15219C2semidirect
7. #15215C2semidirect
8. #14657C2semidirect
9. C4×#379C2semidirect
11. C22×#124C22semidirect
17. C2×#391C22semidirect
18. C2×#390C22semidirect
21. C2×#382C22semidirect
28. #1443C22semidirect
29. #1442C22semidirect
32. #1440C22semidirect
33. #1439C22semidirect
36. C4⋊#95C22semidirect
37. C4⋊#95C22semidirect
198. C24#124semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#28472
#168852
#152362
#152332
#152192
#152152
#146572
C4×#3792