Finite Group Databasebeta

← #44559589

Subgroups — #44559589

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×#22548739C212yesmaximal, complemented
3#30632197C211yesmaximal, characteristic, complemented
4#30631840C211yesmaximal, characteristic, complemented
5#30631839C212yesmaximal, complemented
6#30631838C211yesmaximal, characteristic, complemented
7#30631820C211yesmaximal, characteristic, complemented
8#30631819C212yesmaximal, complemented
9#30631690C212yesmaximal, complemented
10#30631649C211yesmaximal, characteristic, complemented
11#447×C251C211yesmaximal, characteristic, complemented
12C24×D251—21no—
13C2×#16860809C2211yescharacteristic, 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#22548739—28no—
23C2×#16860799C2212yescomplemented
24C2×#16860798C2212yescomplemented
25C2×#16860795C2212yescomplemented
26C2×#16860793C2212yescomplemented
27#22548697—21no—
28#22548695C2211yescharacteristic, complemented
29#22548694C2211yescharacteristic, complemented
30#22548686C2211yescharacteristic, complemented
31#22548683—21no—
32D4×Dic251—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×D8×C251C2212yescomplemented
42#121×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#16860809—24no—
53C2×Dic502C2311yescharacteristic, complemented
54C2×D1004—24no—
55C2×D1004—21no—
56C2×D1004C2312yes—
57C2×C22⋊D251—28no—
58C2×C22⋊D251—21no—
59C2×C22⋊D251—21no—
60C2×C22⋊D251—22no—
61C2×C22⋊D251—22no—
62D4×D251—28no—
63D4×D251—28no—
64D4×D251—24no—
65D4×D251—24no—
66D4×D251—24no—
67D4×D251D418yescomplemented
68C22×Dic251—21no—
69C22×Dic251D411yescharacteristic
70C2×C4×D251—24no—
71C2×C4×D251D411yescharacteristic
72C2×C4×D251D411yescharacteristic
73C2×C4×D251C2311yescharacteristic
74C2×C4×D251C2311yescharacteristic
75#16860799—24no—
76#16860798—24no—
77#16860795—24no—
78#16860793—24no—
79C2×C251⋊C8C2312yescomplemented
80C4⋊Dic251—21no—
81C4⋊Dic251C2311yescharacteristic, complemented
82Dic251⋊C4—21no—
83C22⋊Dic251—21no—
84C22⋊Dic251—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87C4×Dic251—21no—
88C24×C251—21no—
89C2×D4×C251—24no—
90C2×D4×C251—21no—
91C2×D4×C251—21no—
92C2×D4×C251C2312yes—
93C2×D4×C251C2311yescharacteristic, complemented
94C22×C4×C251C2311yescharacteristic
95D8×C251—24no—
96C2×C8×C251C2312yescomplemented
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—432no—
107C22×D251—44no—
108C22×D251—44no—
109C22×D251—44no—
110C22×D251—42no—
111C22×D251—42no—
112C22×D251—44no—
113C22×D251C2×D411yescharacteristic
114C22×D251C2×D411yescharacteristic
115Dic502—22no—
116D1004—24no—
117D1004—22no—
118D1004C2×D414yes—
119C22⋊D251—28no—
120C22⋊D251—28no—
121C22⋊D251—42no—
122C22⋊D251—42no—
123C2×Dic251—22no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—42no—
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C4×D251—22no—
130C4×D251—22no—
131C4×D251—22no—
132C4×D251C2×D414yes—
133C251⋊C8—22no—
134C23×C251—24no—
135C23×C251—22no—
136C23×C251—21no—
137C23×C251—42no—
138C23×C251—42no—
139C23×C251C2×D411yescharacteristic
140D4×C251—24no—
141D4×C251—22no—
142D4×C251—22no—
143D4×C251—42no—
144D4×C251C2×D414yes—
145C2×C4×C251—22no—
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×C251—41no—
157C22×C251—41no—
158C22×C251—41no—
159C22×C251AGL(1,ℤ/8)12yescomplemented
160C22×C251C22×D411yescharacteristic
161C2×D251—28no—
162C2×D251—21no—
163C2×D251—21no—
164C2×D251—21no—
165C2×D251—21no—
166C2×D251—24no—
167C2×D251—416no—
168C2×D251—416no—
169C2×D251—48no—
170C2×D251—42no—
171C2×D251—42no—
172C2×D251#12012yes—
173C2×D251AGL(1,ℤ/8)12yescomplemented
174C4×C251—21no—
175C4×C251—41no—
176C4×C251C22×D412yes—
177Dic251—21no—
178Dic251—41no—
179Dic251—41no—
180Dic251#12012yes—
181C2×C251—22no—
182C2×C251—44no—
183C2×C251—81no—
184C2×C251C2×#12011yescharacteristic
185C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
186C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
187D251—22no—
188D251—22no—
189D251—44no—
190C251#300311yescharacteristic, complemented, Sylow
191#3003—2511nomaximal, Sylow
192C23×D4—2511no—
193C2×AGL(1,ℤ/8)—2512no—
194#525—2511no—
195#450—2511no—
196#450—2512no—
197#450—2511no—
198#447—2512no—
199#447—2511no—
200#447D25111yescharacteristic, complemented
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×SD16—2512no—
217C2×D8—2512no—
218C2×D8C2×D25112yescomplemented
219C2×M16—2512no—
220#123—5021no—
221C22⋊Q8—2511no—
222C22⋊Q8—2511no—
223#121—2511no—
224#121C2×D25111yescharacteristic, complemented
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×Q8—2511no—
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—5024no—
250C2×D4—5028no—
251C2×D4—5028no—
252C2×D4—5024no—
253C2×D4—5021no—
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—2512no—
263C2×D4C22×D25112yes—
264C2×D4C22×D25111yescharacteristic, complemented
265C22×C4—5024no—
266C22×C4—5021no—
267C22×C4—2511no—
268C22×C4—2511no—
269C22×C4—2511no—
270C22×C4—2511no—
271C22×C4—2511no—
272C22×C4C22×D25111yescharacteristic
273SD16—5024no—
274SD16—5024no—
275D8—24no—
276D8—5024no—
277M16—5024no—
278C2×C8—2512no—
279C2×C8C22×D25112yescomplemented
280C4⋊C4—5021no—
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—21no—
293C23—42no—
294C23—42no—
295C23—10042no—
296C23—100432no—
297C23—10044no—
298C23—10044no—
299C23—10044no—
300C23—10044no—
301C23—10042no—
302C23—5024no—
303C23—5028no—
304C23—5028no—
305C23—5022no—
306C23—5022no—
307C23—5022no—
308C23—5022no—
309C23—2511no—
310C23—2511no—
311C23D4×D25111yescharacteristic
312Q8—5022no—
313D4—24no—
314D4—22no—
315D4—22no—
316D4—42no—
317D4—10042no—
318D4—10042no—
319D4—5024no—
320D4—5028no—
321D4—5028no—
322D4—5022no—
323D4—2514no—
324D4D4×D25114yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—10042no—
328C2×C4—5022no—
329C2×C4—5022no—
330C2×C4—5021no—
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—41no—
348C22—41no—
349C22—41no—
350C22—10048no—
351C22—100416no—
352C22—100416no—
353C22—10042no—
354C22—10042no—
355C22—5028no—
356C22—5024no—
357C22—5021no—
358C22—5021no—
359C22—5021no—
360C22—5021no—
361C22—2512no—
362C22—2512no—
363C22#2254873912yescomplemented
364C22C2×D4×D25111yescharacteristic
365C4—21no—
366C4—41no—
367C4—10041no—
368C4—10041no—
369C4—5021no—
370C4—2512no—
371C4C2×D4×D25112yes—
372C2—22no—
373C2—44no—
374C2—81no—
375C2—10044no—
376C2—5022no—
377C2—5022no—
378C2#120×D25111yescharacteristic
379C2AGL(1,ℤ/8)×D25111yescharacteristic
380C2C2×#2254873911yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#447×C251C2
socleC22×C251C22×D4
hypercenter#447D251
hypocenterC251#3003
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×D4×D251C2semidirect
2. C2×#22548739C2semidirect
3. #30632197C2semidirect
4. #30631840C2semidirect
5. #30631839C2semidirect
6. #30631838C2semidirect
7. #30631820C2semidirect
8. #30631819C2semidirect
9. #30631690C2semidirect
10. #30631649C2semidirect
11. #447×C251C2semidirect
13. C2×#16860809C22semidirect
23. C2×#16860799C22semidirect
24. C2×#16860798C22semidirect
25. C2×#16860795C22semidirect
26. C2×#16860793C22semidirect
28. #22548695C22semidirect
29. #22548694C22semidirect
30. #22548686C22semidirect
34. #22548673C22semidirect
35. #22548637C22semidirect
36. #22548636C22semidirect
37. #22548635C22semidirect
38. C22⋊(C251⋊C8)C22semidirect
39. (C2×D251)⋊C8C22semidirect
41. C2×D8×C251C22semidirect
42. #121×C251C22semidirect
43. #102×C251C22semidirect
44. C22⋊C8×C251C22semidirect
53. C2×Dic502C23semidirect
67. D4×D251D4semidirect
79. C2×C251⋊C8C23semidirect
81. C4⋊Dic251C23semidirect
93. C2×D4×C251C23semidirect
96. C2×C8×C251C23semidirect
97. C4⋊C4×C251C23semidirect
159. C22×C251AGL(1,ℤ/8)semidirect
173. C2×D251AGL(1,ℤ/8)semidirect
190. C251#3003semidirect
200. #447D251semidirect
218. C2×D8C2×D251semidirect
224. #121C2×D251semidirect
232. #102C2×D251semidirect
235. C22⋊C8C2×D251semidirect
264. C2×D4C22×D251semidirect
279. C2×C8C22×D251semidirect
283. C4⋊C4C22×D251semidirect
363. C22#22548739semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2512
C2×#225487392
#306321972
#306318402
#306318392
#306318382
#306318202
#306318192
#306316902
#306316492
#447×C2512
#3003251