Finite Group Databasebeta

← #44559593

Subgroups — #44559593

Order 32128.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D251C211yesmaximal, characteristic, complemented
2C2×#22548742C212yesmaximal, complemented
3#30632200C211yesmaximal, characteristic, complemented
4#30631837C211yesmaximal, characteristic, complemented
5#30631836C212yesmaximal, complemented
6#30631823C211yesmaximal, characteristic, complemented
7#30631822C212yesmaximal, complemented
8#30631821C211yesmaximal, characteristic, complemented
9#30631690C212yesmaximal, complemented
10#30631649C211yesmaximal, characteristic, complemented
11#450×C251C211yesmaximal, characteristic, complemented
12C24×D251—21no—
13C2×#16860811C2211yescharacteristic, complemented
14C22×D1004C2211yescharacteristic
15C22×C22⋊D251—21no—
16C2×D4×D251—28no—
17C2×D4×D251—21no—
18C2×D4×D251—21no—
19C2×D4×D251C2212yes—
20C2×D4×D251C2212yes—
21C22×C4×D251C2211yescharacteristic
22#22548742—28no—
23C2×#16860797C2212yescomplemented
24C2×D2008C2212yescomplemented
25C2×#16860795C2212yescomplemented
26C2×#16860793C2212yescomplemented
27#22548700—21no—
28#22548691C2211yescharacteristic, complemented
29#22548690C2211yescharacteristic, complemented
30#22548689C2211yescharacteristic, complemented
31#22548685—21no—
32#22548677—21no—
33(C22⋊D251)⋊C4—21no—
34#22548673C2211yescharacteristic, complemented
35#22548637C2212yescomplemented
36#22548636C2212yescomplemented
37#22548635C2212yescomplemented
38C22⋊(C251⋊C8)C2211yescharacteristic, complemented
39(C2×D251)⋊C8C2212yescomplemented
40C22×D4×C251C2211yescharacteristic
41C2×SD16×C251C2212yescomplemented
42C22⋊Q8×C251C2211yescharacteristic, complemented
43#102×C251C2212yescomplemented
44C22⋊C8×C251C2211yescharacteristic, complemented
45C23×D251—28no—
46C23×D251—22no—
47C23×D251—22no—
48C23×D251—21no—
49C23×D251—44no—
50C23×D251—44no—
51C23×D251D411yescharacteristic
52#16860811—24no—
53C2×D1004—24no—
54C2×D1004—21no—
55C2×D1004—21no—
56C2×D1004—21no—
57C2×D1004C2312yes—
58C2×D1004C2311yescharacteristic, complemented
59C2×C22⋊D251—28no—
60C2×C22⋊D251—22no—
61C2×C22⋊D251—22no—
62C2×C22⋊D251—21no—
63D4×D251—28no—
64D4×D251—28no—
65D4×D251—24no—
66D4×D251—24no—
67D4×D251—24no—
68D4×D251D418yescomplemented
69C22×Dic251D411yescharacteristic
70C2×C4×D251—24no—
71C2×C4×D251—21no—
72C2×C4×D251D411yescharacteristic
73C2×C4×D251D411yescharacteristic
74C2×C4×D251C2311yescharacteristic
75C2×C4×D251C2311yescharacteristic
76#16860797—24no—
77D2008—24no—
78#16860795—24no—
79#16860793—24no—
80C2×C251⋊C8C2312yescomplemented
81C4⋊Dic251C2311yescharacteristic, complemented
82Dic251⋊C4—21no—
83(C2×D251)⋊C4—21no—
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87C4×Dic251—21no—
88C24×C251—21no—
89C2×Q8×C251C2311yescharacteristic, complemented
90C2×D4×C251—24no—
91C2×D4×C251—21no—
92C2×D4×C251C2312yes—
93C22×C4×C251C2311yescharacteristic
94SD16×C251—24no—
95C2×C8×C251C2312yescomplemented
96C4⋊C4×C251—21no—
97C4⋊C4×C251C2311yescharacteristic, complemented
98C22⋊C4×C251—21no—
99C22×D251—22no—
100C22×D251—22no—
101C22×D251—22no—
102C22×D251—28no—
103C22×D251—24no—
104C22×D251—22no—
105C22×D251—28no—
106C22×D251—21no—
107C22×D251—432no—
108C22×D251—44no—
109C22×D251—44no—
110C22×D251—44no—
111C22×D251—42no—
112C22×D251—42no—
113C22×D251—44no—
114C22×D251C2×D411yescharacteristic
115C22×D251C2×D411yescharacteristic
116D1004—24no—
117D1004—22no—
118D1004—22no—
119D1004—42no—
120D1004—42no—
121D1004C2×D414yes—
122C22⋊D251—28no—
123C22⋊D251—28no—
124C22⋊D251—42no—
125C2×Dic251—22no—
126C2×Dic251—21no—
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C4×D251—22no—
130C4×D251—22no—
131C4×D251—22no—
132C4×D251—42no—
133C4×D251C2×D414yes—
134C251⋊C8—22no—
135C23×C251—24no—
136C23×C251—22no—
137C23×C251—42no—
138C23×C251—42no—
139C23×C251C2×D411yescharacteristic
140Q8×C251—22no—
141D4×C251—24no—
142D4×C251—22no—
143D4×C251C2×D414yes—
144C2×C4×C251—22no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2411yescharacteristic
149C8×C251—22no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C251—24no—
153C22×C251—48no—
154C22×C251—42no—
155C22×C251—42no—
156C22×C251AGL(1,ℤ/8)12yescomplemented
157C22×C251C22×D411yescharacteristic
158C2×D251—28no—
159C2×D251—21no—
160C2×D251—21no—
161C2×D251—21no—
162C2×D251—21no—
163C2×D251—24no—
164C2×D251—416no—
165C2×D251—416no—
166C2×D251—48no—
167C2×D251—42no—
168C2×D251—42no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—41no—
172C2×D251#12012yes—
173C2×D251AGL(1,ℤ/8)12yescomplemented
174C4×C251—21no—
175C4×C251—41no—
176C4×C251—41no—
177C4×C251C22×D412yes—
178Dic251—21no—
179Dic251—41no—
180Dic251#12012yes—
181C2×C251—22no—
182C2×C251—44no—
183C2×C251C2×#12011yescharacteristic
184C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
186D251—22no—
187D251—22no—
188D251—44no—
189D251—81no—
190C251#300311yescharacteristic, complemented, Sylow
191#3003—2511nomaximal, Sylow
192C23×D4—2511no—
193C2×AGL(1,ℤ/8)—2512no—
194#525—2511no—
195#450—2512no—
196#450—2511no—
197#450D25111yescharacteristic, complemented
198#447—2511no—
199#447—2512no—
200#447—2511no—
201#418—2512no—
202#407—2511no—
203C25—5021no—
204C2×#41—2511no—
205C22×D4—5028no—
206C22×D4—5021no—
207C22×D4—5021no—
208C22×D4—5021no—
209C22×D4—2512no—
210C22×D4—2512no—
211C22×D4—2511no—
212C22×D4C2×D25111yescharacteristic
213C23×C4—2511no—
214AGL(1,ℤ/8)—5028no—
215C2×SD16—2512no—
216C2×SD16C2×D25112yescomplemented
217C2×D8—2512no—
218C2×D8—2512no—
219C2×M16—2512no—
220#123—5021no—
221C22⋊Q8—2511no—
222C22⋊Q8C2×D25111yescharacteristic, complemented
223#121—2511no—
224#121—2511no—
225#120—5021no—
226C4×D4—5021no—
227C4×D4—5021no—
228#117—2511no—
229#102—2512no—
230#102—2512no—
231#102—2512no—
232#102C2×D25112yescomplemented
233C22⋊C8—2511no—
234C22⋊C8—2512no—
235C22⋊C8C2×D25111yescharacteristic, complemented
236C24—21no—
237C24—10044no—
238C24—10044no—
239C24—5028no—
240C24—5022no—
241C24—5022no—
242C24—5021no—
243C24—2511no—
244#41—5024no—
245C2×Q8C22×D25111yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—5021no—
249C2×D4—5021no—
250C2×D4—5024no—
251C2×D4—5028no—
252C2×D4—5028no—
253C2×D4—5024no—
254C2×D4—5021no—
255C2×D4—5028no—
256C2×D4—5024no—
257C2×D4—5024no—
258C2×D4—5022no—
259C2×D4—5022no—
260C2×D4—5021no—
261C2×D4—2518no—
262C2×D4—2511no—
263C2×D4—2512no—
264C2×D4C22×D25112yes—
265C22×C4—5021no—
266C22×C4—5024no—
267C22×C4—2511no—
268C22×C4—2511no—
269C22×C4—2511no—
270C22×C4—2511no—
271C22×C4—2511no—
272C22×C4C22×D25111yescharacteristic
273SD16—24no—
274SD16—5024no—
275D8—5024no—
276D8—5024no—
277M16—5024no—
278C2×C8—2512no—
279C2×C8C22×D25112yescomplemented
280C4⋊C4—21no—
281C4⋊C4—5021no—
282C4⋊C4—2511no—
283C4⋊C4C22×D25111yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—5021no—
286C22⋊C4—5021no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C42—5021no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—10042no—
295C23—100432no—
296C23—10044no—
297C23—10044no—
298C23—10044no—
299C23—10044no—
300C23—10042no—
301C23—5024no—
302C23—5028no—
303C23—5028no—
304C23—5022no—
305C23—5022no—
306C23—5021no—
307C23—5022no—
308C23—5022no—
309C23—2511no—
310C23—2511no—
311C23D4×D25111yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—10042no—
316D4—10042no—
317D4—10042no—
318D4—5024no—
319D4—5028no—
320D4—5028no—
321D4—5022no—
322D4—5022no—
323D4—2514no—
324D4D4×D25114yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—10042no—
329C2×C4—5022no—
330C2×C4—5022no—
331C2×C4—5021no—
332C2×C4—5022no—
333C2×C4—5022no—
334C2×C4—2511no—
335C2×C4—2511no—
336C2×C4—2514no—
337C2×C4D4×D25111yescharacteristic
338C2×C4C23×D25111yescharacteristic
339C8—22no—
340C8—5022no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—10048no—
348C22—100416no—
349C22—100416no—
350C22—10041no—
351C22—10041no—
352C22—10041no—
353C22—10042no—
354C22—10042no—
355C22—5028no—
356C22—5024no—
357C22—5021no—
358C22—5021no—
359C22—5021no—
360C22—5021no—
361C22—2512no—
362C22—2512no—
363C22#2254874212yescomplemented
364C22C2×D4×D25111yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—10041no—
369C4—5021no—
370C4—2512no—
371C4C2×D4×D25112yes—
372C2—22no—
373C2—44no—
374C2—20081no—
375C2—10044no—
376C2—5022no—
377C2—5022no—
378C2#120×D25111yescharacteristic
379C2AGL(1,ℤ/8)×D25111yescharacteristic
380C2C2×#2254874211yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#450×C251C2
socleC22×C251C22×D4
hypercenter#450D251
hypocenterC251#3003
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D251C2semidirect
2. C2×#22548742C2semidirect
3. #30632200C2semidirect
4. #30631837C2semidirect
5. #30631836C2semidirect
6. #30631823C2semidirect
7. #30631822C2semidirect
8. #30631821C2semidirect
9. #30631690C2semidirect
10. #30631649C2semidirect
11. #450×C251C2semidirect
13. C2×#16860811C22semidirect
23. C2×#16860797C22semidirect
24. C2×D2008C22semidirect
25. C2×#16860795C22semidirect
26. C2×#16860793C22semidirect
28. #22548691C22semidirect
29. #22548690C22semidirect
30. #22548689C22semidirect
34. #22548673C22semidirect
35. #22548637C22semidirect
36. #22548636C22semidirect
37. #22548635C22semidirect
38. C22⋊(C251⋊C8)C22semidirect
39. (C2×D251)⋊C8C22semidirect
41. C2×SD16×C251C22semidirect
42. C22⋊Q8×C251C22semidirect
43. #102×C251C22semidirect
44. C22⋊C8×C251C22semidirect
58. C2×D1004C23semidirect
68. D4×D251D4semidirect
80. C2×C251⋊C8C23semidirect
81. C4⋊Dic251C23semidirect
89. C2×Q8×C251C23semidirect
95. C2×C8×C251C23semidirect
97. C4⋊C4×C251C23semidirect
156. C22×C251AGL(1,ℤ/8)semidirect
173. C2×D251AGL(1,ℤ/8)semidirect
190. C251#3003semidirect
197. #450D251semidirect
216. C2×SD16C2×D251semidirect
222. C22⋊Q8C2×D251semidirect
232. #102C2×D251semidirect
235. C22⋊C8C2×D251semidirect
245. C2×Q8C22×D251semidirect
279. C2×C8C22×D251semidirect
283. C4⋊C4C22×D251semidirect
363. C22#22548742semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2512
C2×#225487422
#306322002
#306318372
#306318362
#306318232
#306318222
#306318212
#306316902
#306316492
#450×C2512
#3003251