Finite Group Databasebeta

← #5037112

Subgroups — #5037112

Order 512.

3132 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2840C211yesmaximal, characteristic, complemented
2C2×#2614C211yesmaximal, characteristic, complemented
3#17223C212yesmaximal, complemented
4#17198C213yesmaximal, complemented
5#16375C216yesmaximal, complemented
6#15948C213yesmaximal
7#15668C213yesmaximal, complemented
8#15468C216yesmaximal, complemented
9#15467C213yesmaximal, complemented
10#14873C212yesmaximal, complemented
11#14609C211yesmaximal, characteristic, complemented
12C24×Q8C2211yescharacteristic, complemented
13C22×#123C2213yes—
14C22×#120C2211yescharacteristic
15#2840—28no—
16#2614—28no—
17C2×#395C2211yescharacteristic, complemented
18C2×#395C2213yescomplemented
19C2×#393C2212yes—
20C2×#393C2213yes—
21C2×#393C2213yes—
22C2×#393C2216yes—
23C2×#393C2216yes—
24C2×#393C2211yescharacteristic
25C2×#393C2213yescomplemented
26C2×#391C2213yescomplemented
27C2×#386C2216yes—
28C2×#381C2213yescomplemented
29C2×#379C2213yes—
30C2×#379C2213yes—
31#1446C2212yescomplemented
32#1446C2216yescomplemented
33#1446C2216yescomplemented
34#1445C2213yescomplemented
35#1443C2216yescomplemented
36#1443C2216yescomplemented
37#1442C2216yescomplemented
38#1440C2213yes—
39#1440C2213yes—
40#1440C2216yes—
41#1440C2216yes—
42#1440C2216yes—
43#1439C2216yes—
44#1439C2216yes—
45#1438C2213yes—
46#1438C2216yes—
47#1437C2213yes—
48#1437C2216yes—
49#1437C2213yescomplemented
50#1434C2216yescomplemented
51C22⋊#95C2216yescomplemented
52C26C2311yescharacteristic
53C23×Q8—24no—
54C23×Q8—46no—
55C23×Q8—46no—
56C23×Q8C2313yes—
57C23×Q8C2313yes—
58C23×Q8C2311yescharacteristic
59C23×D4C2311yescharacteristic
60C23×D4C2312yes—
61C24×C4C2313yes—
62C2×#123—212no—
63C2×#123—46no—
64C2×#123—46no—
65C2×#120—46no—
66C22×C4⋊C4C2313yes—
67C22×C4⋊C4C2313yes—
68C22×C22⋊C4C2313yes—
69C22×C22⋊C4C2313yes—
70C22×C22⋊C4C2316yes—
71C22×C22⋊C4C2316yes—
72C22×C22⋊C4C2316yes—
73C22×C22⋊C4C2313yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2316yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2311yescharacteristic
78C22×C22⋊C4C2313yes—
79C22×C22⋊C4C2313yes—
80C22×C42C2313yes—
81#395—24no—
82#395—212no—
83#393—28no—
84#393—212no—
85#393—212no—
86#393—224no—
87#393—412no—
88#393—46no—
89#393D418yescomplemented
90#391—46no—
91#386—412no—
92#381—212no—
93#379—212no—
94#379—46no—
95C2×#95C2316yes—
96C2×#95C2313yes—
97C2×#95C2316yes—
98C2×#95C2313yes—
99C2×#95C2316yes—
100C2×#95C2316yes—
101C2×#95C2316yes—
102C2×#95C2313yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2313yes—
111C25—22no—
112C25—24no—
113C25—43no—
114C25—46no—
115C25—43no—
116C25C2411yescharacteristic
117C25C2412yes—
118C25C2411yescharacteristic
119C22×Q8—26no—
120C22×Q8—26no—
121C22×Q8—212no—
122C22×Q8—43no—
123C22×Q8—46no—
124C22×Q8—43no—
125C22×Q8—43no—
126C22×Q8—43no—
127C22×Q8—46no—
128C22×Q8—43no—
129C22×Q8—46no—
130C22×Q8—43no—
131C22×Q8—448no—
132C22×Q8—448no—
133C22×Q8—416no—
134C22×Q8C2×D414yes—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—26no—
138C22×D4—212no—
139C22×D4—43no—
140C22×D4—46no—
141C23×C4—26no—
142C23×C4—26no—
143C23×C4—26no—
144C23×C4—212no—
145C23×C4—46no—
146C23×C4C2413yes—
147C23×C4C2416yes—
148C23×C4C2416yes—
149C23×C4C2411yescharacteristic
150C23×C4C2412yes—
151C23×C4C2413yes—
152C23×C4C2413yes—
153C23×C4C2413yes—
154#123—448no—
155#120—416no—
156C2×C4⋊C4—26no—
157C2×C4⋊C4—26no—
158C2×C4⋊C4—26no—
159C2×C4⋊C4—43no—
160C2×C4⋊C4—43no—
161C2×C4⋊C4—43no—
162C2×C22⋊C4—26no—
163C2×C22⋊C4—26no—
164C2×C22⋊C4—26no—
165C2×C22⋊C4—26no—
166C2×C22⋊C4—26no—
167C2×C22⋊C4—212no—
168C2×C22⋊C4—212no—
169C2×C22⋊C4—212no—
170C2×C22⋊C4—212no—
171C2×C22⋊C4—212no—
172C2×C22⋊C4—43no—
173C2×C22⋊C4—46no—
174C2×C22⋊C4—43no—
175C2×C22⋊C4—43no—
176C2×C22⋊C4—43no—
177C2×C22⋊C4—43no—
178C2×C22⋊C4—46no—
179C2×C22⋊C4—46no—
180C2×C22⋊C4—46no—
181C2×C22⋊C4—46no—
182C2×C22⋊C4—46no—
183C2×C22⋊C4—46no—
184C2×C22⋊C4—46no—
185C2×C22⋊C4—46no—
186C2×C22⋊C4—43no—
187C2×C22⋊C4—46no—
188C2×C22⋊C4—46no—
189C2×C22⋊C4—46no—
190C2×C22⋊C4—46no—
191C2×C22⋊C4C2×D4112yes—
192C2×C42—43no—
193C2×C42—46no—
194#95—212no—
195#95—26no—
196#95—26no—
197#95—212no—
198#95—212no—
199#95—46no—
200#95—46no—
201#95—46no—
202#95—46no—
203#95—46no—
204#95—46no—
205#95—43no—
206#95—46no—
207#95—46no—
208#95—46no—
209#95C2×D4112yes—
210C24—21no—
211C24—23no—
212C24—22no—
213C24—26no—
214C24—28no—
215C24—22no—
216C24—26no—
217C24—23no—
218C24—23no—
219C24—26no—
220C24—22no—
221C24—22no—
222C24—26no—
223C24—26no—
224C24—26no—
225C24—44no—
226C24—412no—
227C24—86no—
228C24—86no—
229C24—812no—
230C24—812no—
231C24—86no—
232C24—86no—
233C24—812no—
234C24C22×D412yes—
235C24H(5,𝔽2)16yes—
236C242−1+414yes—
237C24C2511yescharacteristic
238C2×Q8—412no—
239C2×Q8—424no—
240C2×Q8—412no—
241C2×Q8—412no—
242C2×Q8—424no—
243C2×Q8—4128no—
244C2×Q8—44no—
245C2×Q8—412no—
246C2×Q8—412no—
247C2×Q8—412no—
248C2×Q8—86no—
249C2×Q8—812no—
250C2×Q8—86no—
251C2×Q8—812no—
252C2×D4—412no—
253C2×D4—424no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—86no—
257C2×D4—812no—
258C22×C4—224no—
259C22×C4—23no—
260C22×C4—23no—
261C22×C4—26no—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—26no—
265C22×C4—23no—
266C22×C4—23no—
267C22×C4—23no—
268C22×C4—23no—
269C22×C4—26no—
270C22×C4—26no—
271C22×C4—23no—
272C22×C4—26no—
273C22×C4—21no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—26no—
277C22×C4—26no—
278C22×C4—23no—
279C22×C4—26no—
280C22×C4—26no—
281C22×C4—23no—
282C22×C4—23no—
283C22×C4—23no—
284C22×C4—26no—
285C22×C4—26no—
286C22×C4—22no—
287C22×C4—26no—
288C22×C4—26no—
289C22×C4—26no—
290C22×C4—26no—
291C22×C4—26no—
292C22×C4—26no—
293C22×C4—26no—
294C22×C4—26no—
295C22×C4—26no—
296C22×C4—424no—
297C22×C4—424no—
298C22×C4—412no—
299C22×C4—412no—
300C22×C4C22×D416yes—
301C22×C4C22×D416yes—
302C22×C4H(5,𝔽2)16yes—
303C4⋊C4—412no—
304C4⋊C4—412no—
305C22⋊C4—412no—
306C22⋊C4—412no—
307C22⋊C4—412no—
308C22⋊C4—48no—
309C22⋊C4—44no—
310C22⋊C4—812no—
311C22⋊C4—812no—
312C22⋊C4—812no—
313C22⋊C4—86no—
314C22⋊C4—812no—
315C22⋊C4—812no—
316C22⋊C4—86no—
317C42—86no—
318C23—26no—
319C23—26no—
320C23—212no—
321C23—212no—
322C23—26no—
323C23—22no—
324C23—24no—
325C23—26no—
326C23—212no—
327C23—212no—
328C23—41no—
329C23—43no—
330C23—43no—
331C23—43no—
332C23—46no—
333C23—46no—
334C23—46no—
335C23—41no—
336C23—42no—
337C23—46no—
338C23—43no—
339C23—46no—
340C23—46no—
341C23—43no—
342C23—43no—
343C23—46no—
344C23—46no—
345C23—46no—
346C23—46no—
347C23—46no—
348C23—46no—
349C23—46no—
350C23—824no—
351C23—824no—
352C23—848no—
353C23—88no—
354C23—816no—
355C23H(3,𝔽4)14yes—
356C23C23×D411yescharacteristic
357C23C2×H(5,𝔽2)13yes—
358C23C2×H(5,𝔽2)13yes—
359C23C2×2−1+412yes—
360C23#58516yes—
361Q8—416no—
362Q8—448no—
363Q8—824no—
364D4—88no—
365D4—816no—
366C2×C4—26no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—212no—
370C2×C4—212no—
371C2×C4—212no—
372C2×C4—448no—
373C2×C4—46no—
374C2×C4—43no—
375C2×C4—43no—
376C2×C4—46no—
377C2×C4—43no—
378C2×C4—43no—
379C2×C4—43no—
380C2×C4—43no—
381C2×C4—46no—
382C2×C4—43no—
383C2×C4—46no—
384C2×C4—43no—
385C2×C4—43no—
386C2×C4—46no—
387C2×C4—46no—
388C2×C4—43no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—46no—
392C2×C4—43no—
393C2×C4—46no—
394C2×C4—46no—
395C2×C4—46no—
396C2×C4—46no—
397C2×C4—46no—
398C2×C4—46no—
399C2×C4—46no—
400C2×C4—46no—
401C2×C4—46no—
402C2×C4—46no—
403C2×C4—46no—
404C2×C4—46no—
405C2×C4—46no—
406C2×C4—46no—
407C22—24no—
408C22—28no—
409C22—46no—
410C22—46no—
411C22—412no—
412C22—46no—
413C22—412no—
414C22—46no—
415C22—412no—
416C22—42no—
417C22—44no—
418C22—832no—
419C22—86no—
420C22—86no—
421C22—86no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22#348413yes—
426C22#348611yescharacteristic
427C22#350013yes—
428C22C2×H(3,𝔽4)11yescharacteristic
429C22#353213yes—
430C22#353516yes—
431C22#354013yes—
432C22#354013yes—
433C22#354516yes—
434C22#356116yes—
435C4—46no—
436C4—46no—
437C4—46no—
438C4—83no—
439C4—86no—
440C4—86no—
441C4—81no—
442C4—82no—
443C2—44no—
444C2—82no—
445C2—84no—
446C2#4543911yescharacteristic
447C2#4635713yes—
448C2#5174913yes—
449C2#5292512yes—
450C2#5815816yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
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×#2840C2semidirect
2. C2×#2614C2semidirect
3. #17223C2semidirect
4. #17198C2semidirect
5. #16375C2semidirect
7. #15668C2semidirect
8. #15468C2semidirect
9. #15467C2semidirect
10. #14873C2semidirect
11. #14609C2semidirect
12. C24×Q8C22semidirect
17. C2×#395C22semidirect
18. C2×#395C22semidirect
25. C2×#393C22semidirect
26. C2×#391C22semidirect
28. C2×#381C22semidirect
31. #1446C22semidirect
32. #1446C22semidirect
33. #1446C22semidirect
34. #1445C22semidirect
35. #1443C22semidirect
36. #1443C22semidirect
37. #1442C22semidirect
49. #1437C22semidirect
50. #1434C22semidirect
51. C22⋊#95C22semidirect
89. #393D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28402
C2×#26142
#172232
#171982
#163752
#159482
#156682
#154682
#154672
#148732
#146092