Finite Group Databasebeta

← #5022214

Subgroups — #5022214

Order 512.

3097 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15941C216yesmaximal, complemented
2#15763C213yesmaximal, complemented
3#15636C213yesmaximal, complemented
4#15317C216yesmaximal, complemented
5#15267C213yesmaximal, complemented
6#15259C213yesmaximal, complemented
7#14708C213yesmaximal, complemented
8#14651C213yesmaximal, complemented
9#13827C211yesmaximal, characteristic, complemented
10C22×#124C2213yes—
11C22×#124C2213yescomplemented
12C22×#120C2213yes—
13C2×#394C2216yes—
14C2×#393C2216yes—
15C2×#392C2213yes—
16C2×#392C2216yes—
17C2×#392C2211yescharacteristic, complemented
18C2×#391C2216yescomplemented
19C2×#390C2216yes—
20C2×#382C2213yescomplemented
21C2×#379C2213yes—
22C2×#379C2211yescharacteristic
23C2×C43C2211yescharacteristic, complemented
24#1446C2213yescomplemented
25#1443C2216yescomplemented
26#1443C2212yescomplemented
27#1440C2213yes—
28#1440C2213yes—
29#1440C2216yescomplemented
30#1439C2216yes—
31#1439C2216yescomplemented
32#1438C2216yes—
33#1438C2216yes—
34#1438C2216yes—
35#1438C2213yes—
36#1438C2216yescomplemented
37#1437C2213yes—
38#1437C2213yes—
39#1437C2216yes—
40#1437C2213yes—
41#1437C2216yes—
42#1437C2213yescomplemented
43#1437C2213yescomplemented
44#1437C2216yescomplemented
45#1437C2213yescomplemented
46#1436C2216yescomplemented
47C26C2311yescharacteristic
48C23×Q8C2313yes—
49C23×D4C2313yes—
50C23×D4C2313yes—
51C23×D4C2316yes—
52C2×#124—212no—
53C2×#124—46no—
54C2×#124—46no—
55C2×#124—412no—
56C2×#124—46no—
57C2×#120—412no—
58C2×#120—46no—
59C22×C4⋊C4C2316yes—
60C22×C22⋊C4C2316yes—
61C22×C22⋊C4C2316yes—
62C22×C22⋊C4C2313yes—
63C22×C22⋊C4C2313yes—
64C22×C22⋊C4C2316yes—
65C22×C22⋊C4C2316yes—
66C22×C22⋊C4C2313yes—
67C22×C22⋊C4C2313yes—
68C22×C22⋊C4C2316yes—
69C22×C22⋊C4C2316yes—
70C22×C22⋊C4C2313yes—
71C22×C22⋊C4C2313yes—
72C22×C22⋊C4C2316yes—
73C22×C22⋊C4C2313yes—
74C22×C42C2313yes—
75C22×C42C2313yes—
76C22×C42C2313yes—
77C22×C42C2311yescharacteristic
78#394—412no—
79#393—412no—
80#392—212no—
81#392—412no—
82#392—42no—
83#391—412no—
84#390—412no—
85#382—46no—
86#379—212no—
87#379—42no—
88C2×#95C2316yes—
89C2×#95C2313yes—
90C2×#95C2316yes—
91C2×#95C2311yescharacteristic
92C2×#95C2316yes—
93C2×#95C2316yes—
94C2×#95C2316yes—
95C2×#95C2316yes—
96C2×#95C2313yes—
97C2×#95C2312yes—
98C2×#95C2316yes—
99C2×#95C2313yes—
100C2×#95C2316yes—
101C43—42no—
102C25—26no—
103C25—46no—
104C25—43no—
105C25—42no—
106C25—41no—
107C25C2413yes—
108C25C2413yes—
109C22×Q8—26no—
110C22×Q8—26no—
111C22×Q8—212no—
112C22×Q8—43no—
113C22×Q8—46no—
114C22×D4—26no—
115C22×D4—212no—
116C22×D4—26no—
117C22×D4—26no—
118C22×D4—212no—
119C22×D4—46no—
120C22×D4—46no—
121C22×D4—46no—
122C22×D4—43no—
123C22×D4—46no—
124C22×D4—43no—
125C22×D4—46no—
126C22×D4—46no—
127C22×D4—46no—
128C22×D4—43no—
129C22×D4—46no—
130C22×D4—46no—
131C23×C4C2416yes—
132C23×C4C2413yes—
133C23×C4C2416yes—
134C23×C4C2413yes—
135C23×C4C2413yes—
136C23×C4C2411yescharacteristic
137C23×C4C2413yes—
138#124—448no—
139#124—448no—
140#120—448no—
141C2×C4⋊C4—212no—
142C2×C4⋊C4—46no—
143C2×C4⋊C4—46no—
144C2×C22⋊C4—26no—
145C2×C22⋊C4—26no—
146C2×C22⋊C4—26no—
147C2×C22⋊C4—212no—
148C2×C22⋊C4—212no—
149C2×C22⋊C4—212no—
150C2×C22⋊C4—26no—
151C2×C22⋊C4—212no—
152C2×C22⋊C4—26no—
153C2×C22⋊C4—212no—
154C2×C22⋊C4—212no—
155C2×C22⋊C4—26no—
156C2×C22⋊C4—212no—
157C2×C22⋊C4—212no—
158C2×C22⋊C4—212no—
159C2×C22⋊C4—46no—
160C2×C22⋊C4—43no—
161C2×C22⋊C4—43no—
162C2×C22⋊C4—43no—
163C2×C22⋊C4—43no—
164C2×C22⋊C4—43no—
165C2×C22⋊C4—43no—
166C2×C22⋊C4—43no—
167C2×C22⋊C4—46no—
168C2×C22⋊C4—46no—
169C2×C22⋊C4—46no—
170C2×C22⋊C4—46no—
171C2×C22⋊C4—46no—
172C2×C22⋊C4—46no—
173C2×C22⋊C4—43no—
174C2×C22⋊C4—46no—
175C2×C22⋊C4—46no—
176C2×C22⋊C4—46no—
177C2×C22⋊C4—46no—
178C2×C22⋊C4—46no—
179C2×C22⋊C4—46no—
180C2×C22⋊C4—46no—
181C2×C22⋊C4—46no—
182C2×C22⋊C4#41112yes—
183C2×C42—26no—
184C2×C42—26no—
185C2×C42—26no—
186C2×C42—212no—
187C2×C42—46no—
188C2×C42—43no—
189C2×C42—43no—
190C2×C42—41no—
191C2×C42—42no—
192#95—26no—
193#95—212no—
194#95—212no—
195#95—42no—
196#95—43no—
197#95—46no—
198#95—43no—
199#95—46no—
200#95—46no—
201#95—41no—
202#95—46no—
203#95—46no—
204#95—46no—
205C24—28no—
206C24—26no—
207C24—23no—
208C24—26no—
209C24—26no—
210C24—26no—
211C24—23no—
212C24—26no—
213C24—26no—
214C24—23no—
215C24—26no—
216C24—26no—
217C24—26no—
218C24—23no—
219C24—26no—
220C24—23no—
221C24—26no—
222C24—26no—
223C24—412no—
224C24—44no—
225C24—86no—
226C24—86no—
227C24—812no—
228C24—812no—
229C24—86no—
230C24—86no—
231C24—812no—
232C24C2×#4116yes—
233C24C2511yescharacteristic
234C2×Q8—412no—
235C2×Q8—424no—
236C2×Q8—412no—
237C2×Q8—812no—
238C2×Q8—86no—
239C2×D4—412no—
240C2×D4—424no—
241C2×D4—424no—
242C2×D4—412no—
243C2×D4—412no—
244C2×D4—812no—
245C2×D4—86no—
246C2×D4—86no—
247C2×D4—812no—
248C2×D4—812no—
249C2×D4—812no—
250C2×D4—812no—
251C2×D4—812no—
252C2×D4—86no—
253C2×D4—812no—
254C2×D4—812no—
255C2×D4—812no—
256C22×C4—23no—
257C22×C4—23no—
258C22×C4—26no—
259C22×C4—23no—
260C22×C4—26no—
261C22×C4—23no—
262C22×C4—23no—
263C22×C4—26no—
264C22×C4—26no—
265C22×C4—23no—
266C22×C4—23no—
267C22×C4—23no—
268C22×C4—26no—
269C22×C4—26no—
270C22×C4—26no—
271C22×C4—23no—
272C22×C4—26no—
273C22×C4—26no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—26no—
277C22×C4—26no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—26no—
281C22×C4—26no—
282C22×C4—26no—
283C22×C4—23no—
284C22×C4—26no—
285C22×C4—23no—
286C22×C4—26no—
287C22×C4—26no—
288C22×C4C2×#4116yes—
289C22×C4C2×#4116yes—
290C22×C4H(5,𝔽2)12yes—
291C22×C4H(5,𝔽2)112yes—
292C22×C4H(5,𝔽2)16yes—
293C4⋊C4—812no—
294C22⋊C4—412no—
295C22⋊C4—412no—
296C22⋊C4—412no—
297C22⋊C4—412no—
298C22⋊C4—412no—
299C22⋊C4—424no—
300C22⋊C4—424no—
301C22⋊C4—424no—
302C22⋊C4—812no—
303C22⋊C4—812no—
304C22⋊C4—812no—
305C22⋊C4—86no—
306C22⋊C4—86no—
307C22⋊C4—812no—
308C42—224no—
309C42—412no—
310C42—44no—
311C42—412no—
312C23—26no—
313C23—26no—
314C23—26no—
315C23—212no—
316C23—212no—
317C23—46no—
318C23—46no—
319C23—46no—
320C23—46no—
321C23—46no—
322C23—46no—
323C23—43no—
324C23—46no—
325C23—43no—
326C23—43no—
327C23—46no—
328C23—43no—
329C23—46no—
330C23—46no—
331C23—46no—
332C23—46no—
333C23—43no—
334C23—43no—
335C23—46no—
336C23—46no—
337C23—43no—
338C23—46no—
339C23—46no—
340C23—46no—
341C23—46no—
342C23—46no—
343C23—46no—
344C23—46no—
345C23—43no—
346C23—43no—
347C23—46no—
348C23—43no—
349C23—46no—
350C23—43no—
351C23—46no—
352C23—46no—
353C23—46no—
354C23—88no—
355C23—816no—
356C23—824no—
357C23—824no—
358C23—848no—
359C23C22×#4113yes—
360C23C2×H(5,𝔽2)11yescharacteristic
361C23C2×H(5,𝔽2)13yes—
362C23C2×H(5,𝔽2)16yes—
363C23#58512yes—
364Q8—824no—
365D4—824no—
366D4—824no—
367D4—848no—
368C2×C4—26no—
369C2×C4—212no—
370C2×C4—212no—
371C2×C4—26no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—26no—
375C2×C4—212no—
376C2×C4—212no—
377C2×C4—41no—
378C2×C4—43no—
379C2×C4—43no—
380C2×C4—43no—
381C2×C4—43no—
382C2×C4—46no—
383C2×C4—43no—
384C2×C4—46no—
385C2×C4—43no—
386C2×C4—46no—
387C2×C4—46no—
388C2×C4—46no—
389C2×C4—46no—
390C2×C4—46no—
391C2×C4—46no—
392C2×C4—46no—
393C2×C4—46no—
394C2×C4—46no—
395C2×C4—43no—
396C2×C4—46no—
397C2×C4—46no—
398C2×C4—46no—
399C2×C4—43no—
400C2×C4—46no—
401C2×C4—46no—
402C2×C4—46no—
403C2×C4—46no—
404C2×C4H(3,𝔽4)112yes—
405C22—46no—
406C22—46no—
407C22—412no—
408C22—46no—
409C22—412no—
410C22—832no—
411C22—86no—
412C22—86no—
413C22—86no—
414C22—86no—
415C22—83no—
416C22—86no—
417C22—83no—
418C22—86no—
419C22—83no—
420C22—86no—
421C22—86no—
422C22—86no—
423C22—86no—
424C22#344411yescharacteristic
425C22#349013yes—
426C22#350813yes—
427C22#351416yes—
428C22#351916yes—
429C22C2×H(3,𝔽4)13yes—
430C22#352911yescharacteristic
431C22#353713yes—
432C22#354313yes—
433C22#354616yes—
434C4—212no—
435C4—46no—
436C4—42no—
437C4—46no—
438C4—83no—
439C4—86no—
440C4—86no—
441C2—86no—
442C2—163no—
443C2#3543811yescharacteristic
444C2#3548112yes—
445C2#4574113yes—
446C2#4605416yes—
447C2#4608913yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #15941C2semidirect
2. #15763C2semidirect
3. #15636C2semidirect
4. #15317C2semidirect
5. #15267C2semidirect
6. #15259C2semidirect
7. #14708C2semidirect
8. #14651C2semidirect
9. #13827C2semidirect
11. C22×#124C22semidirect
17. C2×#392C22semidirect
18. C2×#391C22semidirect
20. C2×#382C22semidirect
23. C2×C43C22semidirect
24. #1446C22semidirect
25. #1443C22semidirect
26. #1443C22semidirect
29. #1440C22semidirect
31. #1439C22semidirect
36. #1438C22semidirect
42. #1437C22semidirect
43. #1437C22semidirect
44. #1437C22semidirect
45. #1437C22semidirect
46. #1436C22semidirect

Maximal subgroups

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

subgroupindex
#159412
#157632
#156362
#153172
#152672
#152592
#147082
#146512
#138272