Finite Group Databasebeta

← #2746328

Subgroups — #2746328

Order 512.

2934 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2834C211yesmaximal, characteristic, complemented
2C22×#399C211yesmaximal, characteristic
3#17226C212yesmaximal, complemented
4#17130C216yesmaximal, complemented
5#16830C213yesmaximal, complemented
6#16321C213yesmaximal, complemented
7C22⋊#401C212yesmaximal, complemented
8#14820C213yesmaximal
9#14728C216yesmaximal, complemented
10#14636C211yesmaximal, characteristic, complemented
11#14127C213yesmaximal
12C23×C4⋊C4C2213yes—
13C23×C22⋊C4C2213yes—
14C23×C22⋊C4C2211yescharacteristic
15#2834—28no—
16C2×#401C2212yescomplemented
17C2×#399—28no—
18C2×#399C2211yescharacteristic
19C2×#399C2213yes—
20C2×#399C2213yes—
21C2×#399C2211yescharacteristic
22C2×#398C2216yescomplemented
23C2×#398C2216yescomplemented
24C2×#397C2213yes—
25C2×#397C2213yes—
26C2×#385C2213yes—
27C2×#385C2216yes—
28C2×#385C2213yes—
29C2×#380C2213yes—
30C2×#379C2213yes—
31C22×#95C2211yescharacteristic
32#1444C2216yescomplemented
33#1444C2216yescomplemented
34#1443C2212yescomplemented
35#1443C2216yescomplemented
36#1443C2216yescomplemented
37#1442C2216yescomplemented
38#1442C2216yescomplemented
39#1441C2216yescomplemented
40#1441C2216yescomplemented
41#1440C2216yes—
42#1440C2216yes—
43#1439C2213yes—
44#1439C2216yes—
45#1439C2216yes—
46#1439C2213yes—
47C22⋊#95C2213yes—
48C22⋊#95C2216yes—
49C22⋊#95C2213yes—
50C22⋊#95C2213yes—
51C22⋊#95C2216yes—
52C26C2311yescharacteristic
53C24×C4C2313yes—
54C24×C4C2313yes—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C22×C4⋊C4—212no—
58C22×C4⋊C4—46no—
59C22×C4⋊C4—46no—
60C22×C4⋊C4C2316yes—
61C22×C4⋊C4C2313yes—
62C22×C4⋊C4C2316yes—
63C22×C4⋊C4C2313yes—
64C22×C4⋊C4C2313yes—
65C22×C4⋊C4C2316yes—
66C22×C4⋊C4C2313yes—
67C22×C22⋊C4—212no—
68C22×C22⋊C4—46no—
69C22×C22⋊C4—46no—
70C22×C22⋊C4—46no—
71C22×C22⋊C4C2313yes—
72C22×C22⋊C4C2312yes—
73C22×C22⋊C4C2316yes—
74C22×C22⋊C4C2313yes—
75C22×C22⋊C4C2313yes—
76C22×C22⋊C4C2316yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C2313yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2313yes—
81#401—28no—
82#399—212no—
83#399—212no—
84#399—24no—
85#399D418yes—
86#398—224no—
87#398—412no—
88#397—212no—
89#397—46no—
90#385—212no—
91#385—46no—
92#385—412no—
93#380—46no—
94#379—212no—
95C2×#95—24no—
96C2×#95C2313yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2316yes—
100C2×#95C2316yes—
101C2×#95C2313yes—
102C2×#95C2313yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2311yescharacteristic
112C25—22no—
113C25—22no—
114C25—26no—
115C25—26no—
116C25—43no—
117C25—43no—
118C25#4114yes—
119C25C2411yescharacteristic
120C25C2411yescharacteristic
121C25C2411yescharacteristic
122C23×C4—26no—
123C23×C4—26no—
124C23×C4—26no—
125C23×C4—212no—
126C23×C4—26no—
127C23×C4—212no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—26no—
131C23×C4—46no—
132C23×C4—43no—
133C23×C4—43no—
134C23×C4—43no—
135C23×C4—46no—
136C23×C4—43no—
137C23×C4—43no—
138C23×C4C2416yes—
139C23×C4C2416yes—
140C23×C4C2412yes—
141C23×C4C2413yes—
142C23×C4C2412yes—
143C23×C4C2413yes—
144C23×C4C2413yes—
145C23×C4C2413yes—
146C2×C4⋊C4—212no—
147C2×C4⋊C4—26no—
148C2×C4⋊C4—212no—
149C2×C4⋊C4—212no—
150C2×C4⋊C4—26no—
151C2×C4⋊C4—43no—
152C2×C4⋊C4—43no—
153C2×C4⋊C4—46no—
154C2×C4⋊C4—43no—
155C2×C4⋊C4—43no—
156C2×C4⋊C4—448no—
157C2×C4⋊C4—43no—
158C2×C4⋊C4—43no—
159C2×C4⋊C4—46no—
160C2×C4⋊C4—46no—
161C2×C4⋊C4—46no—
162C2×C4⋊C4—43no—
163C2×C4⋊C4—46no—
164C2×C4⋊C4—46no—
165C2×C4⋊C4—46no—
166C2×C4⋊C4C2×D4112yes—
167C2×C22⋊C4—26no—
168C2×C22⋊C4—26no—
169C2×C22⋊C4—26no—
170C2×C22⋊C4—212no—
171C2×C22⋊C4—26no—
172C2×C22⋊C4—212no—
173C2×C22⋊C4—212no—
174C2×C22⋊C4—212no—
175C2×C22⋊C4—26no—
176C2×C22⋊C4—26no—
177C2×C22⋊C4—26no—
178C2×C22⋊C4—26no—
179C2×C22⋊C4—26no—
180C2×C22⋊C4—212no—
181C2×C22⋊C4—43no—
182C2×C22⋊C4—43no—
183C2×C22⋊C4—43no—
184C2×C22⋊C4—43no—
185C2×C22⋊C4—43no—
186C2×C22⋊C4—43no—
187C2×C22⋊C4—46no—
188C2×C22⋊C4—43no—
189C2×C22⋊C4—43no—
190C2×C22⋊C4—448no—
191C2×C22⋊C4—416no—
192C2×C22⋊C4—46no—
193C2×C22⋊C4—46no—
194C2×C22⋊C4C2×D4112yes—
195#95—26no—
196#95—212no—
197#95—26no—
198#95—212no—
199#95—212no—
200#95—212no—
201#95—46no—
202#95—46no—
203#95—46no—
204#95—46no—
205#95—46no—
206#95—46no—
207#95—46no—
208#95—43no—
209#95—46no—
210#95C2×D414yes—
211C24—21no—
212C24—23no—
213C24—21no—
214C24—23no—
215C24—23no—
216C24—23no—
217C24—23no—
218C24—23no—
219C24—28no—
220C24—21no—
221C24—23no—
222C24—23no—
223C24—23no—
224C24—21no—
225C24—23no—
226C24—224no—
227C24—412no—
228C24—412no—
229C24—412no—
230C24—412no—
231C24—412no—
232C24—44no—
233C24—412no—
234C24—86no—
235C24—812no—
236C24—86no—
237C24C22×D412yes—
238C24C2×#4112yes—
239C24C2×#4112yes—
240C24C2×#4112yes—
241C24H(5,𝔽2)16yes—
242C242−1+412yes—
243C242−1+416yes—
244C24C2511yescharacteristic
245C22×C4—224no—
246C22×C4—23no—
247C22×C4—23no—
248C22×C4—26no—
249C22×C4—23no—
250C22×C4—23no—
251C22×C4—23no—
252C22×C4—23no—
253C22×C4—26no—
254C22×C4—26no—
255C22×C4—26no—
256C22×C4—26no—
257C22×C4—26no—
258C22×C4—26no—
259C22×C4—22no—
260C22×C4—23no—
261C22×C4—26no—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—26no—
265C22×C4—26no—
266C22×C4—26no—
267C22×C4—26no—
268C22×C4—26no—
269C22×C4—26no—
270C22×C4—23no—
271C22×C4—26no—
272C22×C4—26no—
273C22×C4—23no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—26no—
277C22×C4—26no—
278C22×C4—22no—
279C22×C4—26no—
280C22×C4—26no—
281C22×C4—26no—
282C22×C4—26no—
283C22×C4—424no—
284C22×C4—424no—
285C22×C4—412no—
286C22×C4—424no—
287C22×C4—412no—
288C22×C4—412no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—412no—
292C22×C4—86no—
293C22×C4—812no—
294C22×C4—86no—
295C22×C4—86no—
296C22×C4—86no—
297C22×C4—86no—
298C22×C4C22×D416yes—
299C22×C4C22×D416yes—
300C22×C4H(5,𝔽2)16yes—
301C4⋊C4—412no—
302C4⋊C4—424no—
303C4⋊C4—412no—
304C4⋊C4—86no—
305C4⋊C4—812no—
306C4⋊C4—86no—
307C4⋊C4—812no—
308C22⋊C4—412no—
309C22⋊C4—412no—
310C22⋊C4—412no—
311C22⋊C4—412no—
312C22⋊C4—412no—
313C22⋊C4—48no—
314C22⋊C4—424no—
315C22⋊C4—412no—
316C22⋊C4—48no—
317C22⋊C4—812no—
318C23—22no—
319C23—26no—
320C23—26no—
321C23—26no—
322C23—26no—
323C23—212no—
324C23—26no—
325C23—26no—
326C23—26no—
327C23—26no—
328C23—26no—
329C23—26no—
330C23—26no—
331C23—26no—
332C23—26no—
333C23—26no—
334C23—22no—
335C23—26no—
336C23—448no—
337C23—41no—
338C23—43no—
339C23—43no—
340C23—46no—
341C23—43no—
342C23—416no—
343C23—43no—
344C23—43no—
345C23—41no—
346C23—43no—
347C23—43no—
348C23—43no—
349C23—46no—
350C23—43no—
351C23—43no—
352C23—824no—
353C23—88no—
354C23—824no—
355C23—824no—
356C23—88no—
357C23#539112yes—
358C23#54814yes—
359C23#56014yes—
360C23#563112yes—
361C23C23×D411yescharacteristic
362C23C22×#4111yescharacteristic
363C23C2×H(5,𝔽2)13yes—
364C23C2×H(5,𝔽2)13yes—
365C23C2×2−1+413yes—
366C23C2×2−1+411yescharacteristic
367C23#58513yes—
368C2×C4—26no—
369C2×C4—26no—
370C2×C4—26no—
371C2×C4—212no—
372C2×C4—212no—
373C2×C4—212no—
374C2×C4—448no—
375C2×C4—46no—
376C2×C4—43no—
377C2×C4—43no—
378C2×C4—43no—
379C2×C4—43no—
380C2×C4—43no—
381C2×C4—46no—
382C2×C4—43no—
383C2×C4—46no—
384C2×C4—46no—
385C2×C4—46no—
386C2×C4—46no—
387C2×C4—43no—
388C2×C4—46no—
389C2×C4—46no—
390C2×C4—46no—
391C2×C4—46no—
392C2×C4—46no—
393C2×C4—46no—
394C2×C4—46no—
395C2×C4—42no—
396C2×C4—42no—
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—46no—
406C2×C4—46no—
407C2×C4—46no—
408C2×C4—43no—
409C2×C4—824no—
410C2×C4—88no—
411C2×C4—88no—
412C22—24no—
413C22—212no—
414C22—212no—
415C22—212no—
416C22—212no—
417C22—24no—
418C22—212no—
419C22—42no—
420C22—42no—
421C22—46no—
422C22—46no—
423C22—46no—
424C22—42no—
425C22—42no—
426C22—46no—
427C22—46no—
428C22—832no—
429C22—81no—
430C22—83no—
431C22—83no—
432C22—81no—
433C22#283418yescomplemented
434C22C2×#53913yes—
435C22C2×#54811yescharacteristic
436C22#348413yes—
437C22#348913yes—
438C22#350513yes—
439C22C2×#56011yescharacteristic
440C22C2×#56313yes—
441C22#353313yes—
442C22#353513yes—
443C22#353516yes—
444C22#353813yes—
445C22#354313yes—
446C4—46no—
447C4—46no—
448C4—46no—
449C4—82no—
450C4—86no—
451C4—86no—
452C4—83no—
453C4—82no—
454C2—28no—
455C2—44no—
456C2—82no—
457C2C2×#283411yescharacteristic
458C2#3770213yes—
459C2#4066313yes—
460C2#4545613yes—
461C2#4632411yescharacteristic
462C2#5256713yes—
463C2#5274211yescharacteristic
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×#2834C2semidirect
3. #17226C2semidirect
4. #17130C2semidirect
5. #16830C2semidirect
6. #16321C2semidirect
7. C22⋊#401C2semidirect
9. #14728C2semidirect
10. #14636C2semidirect
16. C2×#401C22semidirect
22. C2×#398C22semidirect
23. C2×#398C22semidirect
32. #1444C22semidirect
33. #1444C22semidirect
34. #1443C22semidirect
35. #1443C22semidirect
36. #1443C22semidirect
37. #1442C22semidirect
38. #1442C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect
433. C22#2834semidirect

Maximal subgroups

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

subgroupindex
C2×#28342
C22×#3992
#172262
#171302
#168302
#163212
C22⋊#4012
#148202
#147282
#146362
#141272