Finite Group Databasebeta

← #2746325

Subgroups — #2746325

Order 512.

3254 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2835C211yesmaximal, characteristic, complemented
2C22×#393C211yesmaximal, characteristic
3#17223C212yesmaximal, complemented
4#17127C216yesmaximal, complemented
5#16831C213yesmaximal, complemented
6#16318C213yesmaximal, complemented
7#14874C212yesmaximal, complemented
8#14820C213yesmaximal
9#14767C216yesmaximal, complemented
10#14635C211yesmaximal, characteristic, complemented
11#14127C213yesmaximal
12C24×Q8C2211yescharacteristic
13C23×C22⋊C4C2213yes—
14C23×C22⋊C4C2211yescharacteristic
15#2835—28no—
16C2×#400C2216yescomplemented
17C2×#397C2213yes—
18C2×#397C2213yes—
19C2×#395C2212yescomplemented
20C2×#395C2216yescomplemented
21C2×#393—28no—
22C2×#393C2211yescharacteristic
23C2×#393C2213yes—
24C2×#393C2213yes—
25C2×#393C2211yescharacteristic
26C2×#385C2213yes—
27C2×#385C2216yes—
28C2×#385C2213yes—
29C2×#380C2213yes—
30C2×#379C2213yes—
31C22×#95C2213yes—
32#1446C2212yescomplemented
33#1446C2216yescomplemented
34#1445C2216yescomplemented
35#1443C2216yescomplemented
36#1443C2216yescomplemented
37#1442C2216yescomplemented
38#1442C2216yescomplemented
39#1442C2216yescomplemented
40#1441C2216yescomplemented
41#1440C2216yes—
42#1440C2216yes—
43#1439C2213yes—
44#1439C2216yes—
45#1439C2216yes—
46#1439C2213yes—
47C22⋊#95C2216yes—
48C22⋊#95C2213yes—
49C22⋊#95C2216yes—
50C22⋊#95C2213yes—
51C22⋊#95C2213yes—
52C26C2311yescharacteristic
53C23×Q8—24no—
54C23×Q8—46no—
55C23×Q8—46no—
56C23×Q8C2313yes—
57C23×Q8C2311yescharacteristic
58C24×C4C2313yes—
59C24×C4C2313yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C22×C4⋊C4C2316yes—
63C22×C4⋊C4C2316yes—
64C22×C4⋊C4C2313yes—
65C22×C4⋊C4C2313yes—
66C22×C22⋊C4—212no—
67C22×C22⋊C4—46no—
68C22×C22⋊C4—46no—
69C22×C22⋊C4—46no—
70C22×C22⋊C4C2313yes—
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2316yes—
73C22×C22⋊C4C2313yes—
74C22×C22⋊C4C2313yes—
75C22×C22⋊C4C2313yes—
76C22×C22⋊C4C2316yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2313yes—
80#400—224no—
81#397—212no—
82#397—46no—
83#395—28no—
84#395—412no—
85#393—24no—
86#393—212no—
87#393—212no—
88#393D418yes—
89#385—212no—
90#385—412no—
91#385—46no—
92#380—46no—
93#379—212no—
94C2×#95—212no—
95C2×#95C2316yes—
96C2×#95C2313yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2313yes—
100C2×#95C2316yes—
101C2×#95C2316yes—
102C2×#95C2316yes—
103C2×#95C2313yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2313yes—
112C25—22no—
113C25—22no—
114C25—26no—
115C25—26no—
116C25—43no—
117C25—43no—
118C25#4114yes—
119C25C2411yescharacteristic
120C25C2411yescharacteristic
121C25C2411yescharacteristic
122C22×Q8—26no—
123C22×Q8—212no—
124C22×Q8—43no—
125C22×Q8—46no—
126C22×Q8—43no—
127C22×Q8—43no—
128C22×Q8—43no—
129C22×Q8—448no—
130C22×Q8—448no—
131C22×Q8—416no—
132C22×Q8C2×D414yes—
133C23×C4—26no—
134C23×C4—26no—
135C23×C4—26no—
136C23×C4—212no—
137C23×C4—26no—
138C23×C4—212no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—26no—
142C23×C4—46no—
143C23×C4—43no—
144C23×C4—43no—
145C23×C4—43no—
146C23×C4—46no—
147C23×C4—43no—
148C23×C4—43no—
149C23×C4C2416yes—
150C23×C4C2416yes—
151C23×C4C2412yes—
152C23×C4C2413yes—
153C23×C4C2412yes—
154C23×C4C2413yes—
155C23×C4C2413yes—
156C23×C4C2413yes—
157C2×C4⋊C4—212no—
158C2×C4⋊C4—26no—
159C2×C4⋊C4—212no—
160C2×C4⋊C4—46no—
161C2×C4⋊C4—43no—
162C2×C4⋊C4—43no—
163C2×C4⋊C4—46no—
164C2×C4⋊C4—46no—
165C2×C4⋊C4—46no—
166C2×C4⋊C4—46no—
167C2×C4⋊C4—43no—
168C2×C22⋊C4—26no—
169C2×C22⋊C4—26no—
170C2×C22⋊C4—26no—
171C2×C22⋊C4—212no—
172C2×C22⋊C4—26no—
173C2×C22⋊C4—212no—
174C2×C22⋊C4—212no—
175C2×C22⋊C4—212no—
176C2×C22⋊C4—26no—
177C2×C22⋊C4—26no—
178C2×C22⋊C4—26no—
179C2×C22⋊C4—26no—
180C2×C22⋊C4—26no—
181C2×C22⋊C4—212no—
182C2×C22⋊C4—43no—
183C2×C22⋊C4—43no—
184C2×C22⋊C4—43no—
185C2×C22⋊C4—43no—
186C2×C22⋊C4—43no—
187C2×C22⋊C4—43no—
188C2×C22⋊C4—46no—
189C2×C22⋊C4—43no—
190C2×C22⋊C4—43no—
191C2×C22⋊C4—448no—
192C2×C22⋊C4—416no—
193C2×C22⋊C4—46no—
194C2×C22⋊C4—46no—
195C2×C22⋊C4C2×D4112yes—
196#95—212no—
197#95—26no—
198#95—212no—
199#95—212no—
200#95—26no—
201#95—212no—
202#95—46no—
203#95—46no—
204#95—46no—
205#95—46no—
206#95—46no—
207#95—46no—
208#95—46no—
209#95—46no—
210#95—46no—
211#95—43no—
212#95C2×D4112yes—
213C24—21no—
214C24—23no—
215C24—21no—
216C24—23no—
217C24—23no—
218C24—23no—
219C24—23no—
220C24—23no—
221C24—28no—
222C24—21no—
223C24—23no—
224C24—23no—
225C24—23no—
226C24—21no—
227C24—23no—
228C24—224no—
229C24—412no—
230C24—412no—
231C24—412no—
232C24—412no—
233C24—412no—
234C24—44no—
235C24—412no—
236C24—86no—
237C24—86no—
238C24—812no—
239C24C22×D412yes—
240C24C2×#4112yes—
241C24C2×#4112yes—
242C24C2×#4112yes—
243C24H(5,𝔽2)16yes—
244C242−1+412yes—
245C242−1+416yes—
246C24C2511yescharacteristic
247C2×Q8—412no—
248C2×Q8—424no—
249C2×Q8—412no—
250C2×Q8—412no—
251C2×Q8—424no—
252C2×Q8—4128no—
253C2×Q8—44no—
254C2×Q8—412no—
255C2×Q8—412no—
256C22×C4—224no—
257C22×C4—23no—
258C22×C4—23no—
259C22×C4—26no—
260C22×C4—26no—
261C22×C4—26no—
262C22×C4—26no—
263C22×C4—23no—
264C22×C4—23no—
265C22×C4—23no—
266C22×C4—23no—
267C22×C4—26no—
268C22×C4—26no—
269C22×C4—23no—
270C22×C4—26no—
271C22×C4—22no—
272C22×C4—26no—
273C22×C4—26no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—26no—
277C22×C4—23no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—26no—
281C22×C4—23no—
282C22×C4—26no—
283C22×C4—26no—
284C22×C4—26no—
285C22×C4—26no—
286C22×C4—26no—
287C22×C4—26no—
288C22×C4—22no—
289C22×C4—26no—
290C22×C4—26no—
291C22×C4—26no—
292C22×C4—26no—
293C22×C4—26no—
294C22×C4—424no—
295C22×C4—424no—
296C22×C4—412no—
297C22×C4—424no—
298C22×C4—412no—
299C22×C4—412no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—412no—
303C22×C4—86no—
304C22×C4—812no—
305C22×C4—86no—
306C22×C4—86no—
307C22×C4—86no—
308C22×C4—86no—
309C22×C4C22×D416yes—
310C22×C4C22×D416yes—
311C22×C4H(5,𝔽2)16yes—
312C4⋊C4—86no—
313C4⋊C4—812no—
314C4⋊C4—812no—
315C4⋊C4—86no—
316C22⋊C4—412no—
317C22⋊C4—412no—
318C22⋊C4—412no—
319C22⋊C4—412no—
320C22⋊C4—412no—
321C22⋊C4—48no—
322C22⋊C4—424no—
323C22⋊C4—412no—
324C22⋊C4—48no—
325C22⋊C4—812no—
326C23—22no—
327C23—26no—
328C23—26no—
329C23—26no—
330C23—26no—
331C23—26no—
332C23—26no—
333C23—26no—
334C23—212no—
335C23—26no—
336C23—26no—
337C23—26no—
338C23—26no—
339C23—26no—
340C23—26no—
341C23—26no—
342C23—22no—
343C23—26no—
344C23—448no—
345C23—41no—
346C23—43no—
347C23—43no—
348C23—43no—
349C23—46no—
350C23—416no—
351C23—43no—
352C23—43no—
353C23—41no—
354C23—43no—
355C23—43no—
356C23—43no—
357C23—43no—
358C23—43no—
359C23—46no—
360C23—824no—
361C23—824no—
362C23—88no—
363C23—824no—
364C23—88no—
365C23#539112yes—
366C23#54814yes—
367C23H(3,𝔽4)14yes—
368C23#563112yes—
369C23C23×D411yescharacteristic
370C23C22×#4111yescharacteristic
371C23C2×H(5,𝔽2)13yes—
372C23C2×H(5,𝔽2)13yes—
373C23C2×2−1+413yes—
374C23C2×2−1+411yescharacteristic
375C23#58513yes—
376Q8—416no—
377Q8—448no—
378C2×C4—26no—
379C2×C4—26no—
380C2×C4—26no—
381C2×C4—212no—
382C2×C4—212no—
383C2×C4—212no—
384C2×C4—448no—
385C2×C4—46no—
386C2×C4—43no—
387C2×C4—43no—
388C2×C4—43no—
389C2×C4—46no—
390C2×C4—43no—
391C2×C4—43no—
392C2×C4—43no—
393C2×C4—46no—
394C2×C4—46no—
395C2×C4—46no—
396C2×C4—46no—
397C2×C4—43no—
398C2×C4—46no—
399C2×C4—46no—
400C2×C4—46no—
401C2×C4—46no—
402C2×C4—46no—
403C2×C4—46no—
404C2×C4—46no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—43no—
408C2×C4—46no—
409C2×C4—46no—
410C2×C4—46no—
411C2×C4—46no—
412C2×C4—46no—
413C2×C4—46no—
414C2×C4—43no—
415C2×C4—46no—
416C2×C4—46no—
417C2×C4—46no—
418C2×C4—46no—
419C2×C4—824no—
420C2×C4—88no—
421C2×C4—88no—
422C22—212no—
423C22—212no—
424C22—24no—
425C22—212no—
426C22—24no—
427C22—212no—
428C22—212no—
429C22—42no—
430C22—46no—
431C22—42no—
432C22—46no—
433C22—42no—
434C22—46no—
435C22—42no—
436C22—46no—
437C22—46no—
438C22—832no—
439C22—83no—
440C22—81no—
441C22—83no—
442C22—81no—
443C22#283518yescomplemented
444C22C2×#53913yes—
445C22C2×#54811yescharacteristic
446C22#348413yes—
447C22#348913yes—
448C22#350513yes—
449C22C2×H(3,𝔽4)11yescharacteristic
450C22C2×#56313yes—
451C22#353213yes—
452C22#353513yes—
453C22#353813yes—
454C22#354513yes—
455C22#356116yes—
456C4—46no—
457C4—46no—
458C4—46no—
459C4—83no—
460C4—86no—
461C4—82no—
462C4—82no—
463C4—86no—
464C2—28no—
465C2—44no—
466C2—82no—
467C2C2×#283511yescharacteristic
468C2#3770213yes—
469C2#4065813yes—
470C2#4543911yescharacteristic
471C2#4613613yes—
472C2#5239613yes—
473C2#5292511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2835C2semidirect
3. #17223C2semidirect
4. #17127C2semidirect
5. #16831C2semidirect
6. #16318C2semidirect
7. #14874C2semidirect
9. #14767C2semidirect
10. #14635C2semidirect
16. C2×#400C22semidirect
19. C2×#395C22semidirect
20. C2×#395C22semidirect
32. #1446C22semidirect
33. #1446C22semidirect
34. #1445C22semidirect
35. #1443C22semidirect
36. #1443C22semidirect
37. #1442C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1441C22semidirect
443. C22#2835semidirect

Maximal subgroups

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

subgroupindex
C2×#28352
C22×#3932
#172232
#171272
#168312
#163182
#148742
#148202
#147672
#146352
#141272