Finite Group Databasebeta

← #447×D251 #44559538

Subgroups — #447×D251 #44559538

Order 32128.

1048 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 155 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×D8×D251C212yesmaximal, complemented
3#121×D251C211yesmaximal, characteristic, complemented
4#30631823C211yesmaximal, characteristic, complemented
5#30631822C212yesmaximal, complemented
6#30631821C211yesmaximal, characteristic, complemented
7#30631820C211yesmaximal, characteristic, complemented
8#30631819C212yesmaximal, complemented
9#102×D251C212yesmaximal, complemented
10C22⋊C8×D251C211yesmaximal, characteristic, complemented
11#447×C251C211yesmaximal, characteristic, complemented
12C24×D251—21no—
13C22×D1004C2211yescharacteristic, complemented
14C22×C22⋊D251—21no—
15C2×D4×D251—28no—
16C2×D4×D251—21no—
17C2×D4×D251—21no—
18C2×D4×D251—21no—
19C2×D4×D251C2212yes—
20C2×D4×D251C2212yescomplemented
21C2×D4×D251C2211yescharacteristic, complemented
22C22×C4×D251C2211yescharacteristic
23C2×D2008C2212yescomplemented
24C2×#16860795C2212yescomplemented
25C2×#16860795C2212yescomplemented
26D8×D251—28no—
27C2×C8×D251C2212yescomplemented
28#22548690C2211yescharacteristic, complemented
29#22548689C2211yescharacteristic, complemented
30#22548688—21no—
31#22548687—21no—
32#22548686C2211yescharacteristic, complemented
33C4⋊C4×D251C2211yescharacteristic, complemented
34C22⋊C4×D251—21no—
35#22548637C2212yescomplemented
36#22548636C2212yescomplemented
37#22548635C2212yescomplemented
38C22⋊(C251⋊C8)C2211yescharacteristic, complemented
39(C2×D251)⋊C8C2212yescomplemented
40C22×D4×C251C2211yescharacteristic, complemented
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—21no—
50C23×D251—44no—
51C23×D251—44no—
52C23×D251D411yescharacteristic
53C2×D1004—24no—
54C2×D1004—21no—
55C2×D1004—21no—
56C2×D1004C2312yes—
57C2×D1004C2311yescharacteristic, complemented
58C2×C22⋊D251—28no—
59C2×C22⋊D251—21no—
60C2×C22⋊D251—22no—
61C2×C22⋊D251—22no—
62C2×C22⋊D251—21no—
63C2×C22⋊D251—21no—
64D4×D251—28no—
65D4×D251—28no—
66D4×D251—24no—
67D4×D251—24no—
68D4×D251—24no—
69D4×D251—24no—
70D4×D251—44no—
71D4×D251D418yescomplemented
72C22×Dic251D411yescharacteristic
73C2×C4×D251—24no—
74C2×C4×D251—21no—
75C2×C4×D251D411yescharacteristic
76C2×C4×D251D411yescharacteristic
77C2×C4×D251C2311yescharacteristic
78C2×C4×D251C2311yescharacteristic
79D2008—24no—
80#16860795—24no—
81#16860795—24no—
82C2×C251⋊C8C2312yescomplemented
83C8×D251—24no—
84C4⋊Dic251C2311yescharacteristic, complemented
85Dic251⋊C4—21no—
86C22⋊Dic251—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89C24×C251—21no—
90C2×D4×C251—24no—
91C2×D4×C251—21no—
92C2×D4×C251—21no—
93C2×D4×C251C2312yes—
94C2×D4×C251C2311yescharacteristic, complemented
95C22×C4×C251C2311yescharacteristic
96D8×C251—24no—
97C2×C8×C251C2312yescomplemented
98C4⋊C4×C251C2311yescharacteristic, complemented
99C22⋊C4×C251—21no—
100C22×D251—22no—
101C22×D251—22no—
102C22×D251—28no—
103C22×D251—24no—
104C22×D251—22no—
105C22×D251—28no—
106C22×D251—21no—
107C22×D251—44no—
108C22×D251—432no—
109C22×D251—42no—
110C22×D251—42no—
111C22×D251—44no—
112C22×D251—44no—
113C22×D251—44no—
114C22×D251—42no—
115C22×D251—42no—
116C22×D251—42no—
117C22×D251D814yescomplemented
118C22×D251C2×D411yescharacteristic
119C22×D251C2×D411yescharacteristic
120D1004—24no—
121D1004—22no—
122D1004—22no—
123D1004—42no—
124D1004C2×D414yescomplemented
125C22⋊D251—28no—
126C22⋊D251—28no—
127C22⋊D251—42no—
128C22⋊D251—42no—
129C22⋊D251—42no—
130C2×Dic251—22no—
131C2×Dic251—21no—
132C2×Dic251C2×D411yescharacteristic
133C2×Dic251C2×D411yescharacteristic
134C4×D251—22no—
135C4×D251—22no—
136C4×D251—22no—
137C4×D251—42no—
138C4×D251C2×D414yes—
139C251⋊C8—22no—
140C23×C251—24no—
141C23×C251—22no—
142C23×C251—21no—
143C23×C251—42no—
144C23×C251—42no—
145C23×C251C2×D411yescharacteristic
146D4×C251—24no—
147D4×C251—22no—
148D4×C251—22no—
149D4×C251—42no—
150D4×C251C2×D414yescomplemented
151C2×C4×C251—22no—
152C2×C4×C251—21no—
153C2×C4×C251C2×D411yescharacteristic
154C2×C4×C251C2411yescharacteristic
155C8×C251—22no—
156C22×C251—21no—
157C22×C251—21no—
158C22×C251—24no—
159C22×C251—48no—
160C22×C251—42no—
161C22×C251—42no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251C2×D812yescomplemented
166C22×C251C22×D411yescharacteristic
167C2×D251—21no—
168C2×D251—21no—
169C2×D251—28no—
170C2×D251—24no—
171C2×D251—416no—
172C2×D251—48no—
173C2×D251—42no—
174C2×D251—42no—
175C2×D251—416no—
176C2×D251—41no—
177C2×D251—41no—
178C2×D251—41no—
179C2×D251—84no—
180C2×D251#12012yes—
181C2×D251C2×D812yes—
182C2×D251C2×D812yescomplemented
183C2×D251AGL(1,ℤ/8)12yes—
184C4×C251—21no—
185C4×C251—41no—
186C4×C251C22×D412yes—
187Dic251—21no—
188Dic251—41no—
189Dic251#12012yes—
190C2×C251—22no—
191C2×C251—44no—
192C2×C251—81no—
193C2×C251C2×#12011yescharacteristic
194C2×C251C22×D811yescharacteristic
195C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
196D251—22no—
197D251—44no—
198D251—81no—
199D251#44714yesdirect product factor
200C251C2×#44711yescharacteristic, complemented, Sylow
201C2×#447—2511nomaximal, Sylow
202C23×D4—2511no—
203C22×D8—2512no—
204C2×#121—2511no—
205#447—2511no—
206#447—2512no—
207#447—2511no—
208#447—2512no—
209#447—2511no—
210#447D25111yescharacteristic, direct product factor
211C2×#102—2512no—
212C2×C22⋊C8—2511no—
213C25—5021no—
214C22×D4—5021no—
215C22×D4—5028no—
216C22×D4—5021no—
217C22×D4—5021no—
218C22×D4—5021no—
219C22×D4—2512no—
220C22×D4—2511no—
221C22×D4—2512no—
222C22×D4—2511no—
223C22×D4C2×D25111yescharacteristic, complemented
224C23×C4—2511no—
225C2×D8—5028no—
226C2×D8—2512no—
227C2×D8—2512no—
228C2×D8—2512no—
229C2×D8C2×D25112yescomplemented
230C22×C8—2512no—
231#121—5021no—
232#121—5021no—
233#121—2511no—
234#121—2511no—
235#121—2511no—
236#121C2×D25111yescharacteristic, complemented
237C2×C4⋊C4—2511no—
238C2×C22⋊C4—5021no—
239#102—2512no—
240#102—2512no—
241#102—2512no—
242#102C2×D25112yescomplemented
243C22⋊C8—2511no—
244C22⋊C8—2512no—
245C22⋊C8C2×D25111yescharacteristic, complemented
246C24—21no—
247C24—10044no—
248C24—10044no—
249C24—5021no—
250C24—5028no—
251C24—5022no—
252C24—5022no—
253C24—5021no—
254C24—2511no—
255C2×D4—24no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—10044no—
259C2×D4—5021no—
260C2×D4—5024no—
261C2×D4—5028no—
262C2×D4—5028no—
263C2×D4—5024no—
264C2×D4—5021no—
265C2×D4—5021no—
266C2×D4—5028no—
267C2×D4—5024no—
268C2×D4—5024no—
269C2×D4—5024no—
270C2×D4—5021no—
271C2×D4—5022no—
272C2×D4—5022no—
273C2×D4—5021no—
274C2×D4—2518no—
275C2×D4—2511no—
276C2×D4—2512no—
277C2×D4C22×D25112yes—
278C2×D4C22×D25111yescharacteristic, complemented
279C22×C4—5021no—
280C22×C4—5024no—
281C22×C4—2511no—
282C22×C4—2511no—
283C22×C4—2511no—
284C22×C4—2511no—
285C22×C4—2511no—
286C22×C4C22×D25111yescharacteristic
287D8—24no—
288D8—5024no—
289D8—5024no—
290D8—5024no—
291C2×C8—5024no—
292C2×C8—2512no—
293C2×C8C22×D25112yescomplemented
294C4⋊C4—5021no—
295C4⋊C4—2511no—
296C4⋊C4C22×D25111yescharacteristic, complemented
297C22⋊C4—21no—
298C22⋊C4—5021no—
299C22⋊C4—5021no—
300C22⋊C4—5021no—
301C23—24no—
302C23—22no—
303C23—21no—
304C23—42no—
305C23—42no—
306C23—10042no—
307C23—10042no—
308C23—10042no—
309C23—10042no—
310C23—100432no—
311C23—10044no—
312C23—10044no—
313C23—10044no—
314C23—10044no—
315C23—10042no—
316C23—5024no—
317C23—5028no—
318C23—5028no—
319C23—5022no—
320C23—5022no—
321C23—5021no—
322C23—5022no—
323C23—2511no—
324C23—2514no—
325C23—2511no—
326C23D4×D25111yescharacteristic
327D4—24no—
328D4—22no—
329D4—22no—
330D4—42no—
331D4—10042no—
332D4—10042no—
333D4—10042no—
334D4—10042no—
335D4—5024no—
336D4—5028no—
337D4—5028no—
338D4—5022no—
339D4—5022no—
340D4—2514no—
341D4D4×D25114yescomplemented
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—10042no—
345C2×C4—5022no—
346C2×C4—5022no—
347C2×C4—5021no—
348C2×C4—5022no—
349C2×C4—5022no—
350C2×C4—2511no—
351C2×C4—2511no—
352C2×C4—2514no—
353C2×C4D4×D25111yescharacteristic
354C2×C4C23×D25111yescharacteristic
355C8—22no—
356C8—5022no—
357C22—21no—
358C22—21no—
359C22—24no—
360C22—48no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—20084no—
367C22—10048no—
368C22—100416no—
369C22—100416no—
370C22—10041no—
371C22—10041no—
372C22—10041no—
373C22—10042no—
374C22—10042no—
375C22—5028no—
376C22—5024no—
377C22—5021no—
378C22—5021no—
379C22—2512no—
380C22—2512no—
381C22—2512no—
382C22—2512no—
383C22D8×D25112yescomplemented
384C22C2×D4×D25111yescharacteristic
385C4—21no—
386C4—41no—
387C4—10041no—
388C4—5021no—
389C4—2512no—
390C4C2×D4×D25112yes—
391C2—22no—
392C2—44no—
393C2—81no—
394C2—20081no—
395C2—10044no—
396C2—5022no—
397C2—2514no—
398C2#120×D25111yescharacteristic
399C2C2×D8×D25111yescharacteristic
400C2AGL(1,ℤ/8)×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#447×C251C2
socleC22×C251C22×D4
hypercenter#447D251
hypocenterC251C2×#447
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×D8×D251C2semidirect
3. #121×D251C2semidirect
4. #30631823C2semidirect
5. #30631822C2semidirect
6. #30631821C2semidirect
7. #30631820C2semidirect
8. #30631819C2semidirect
9. #102×D251C2semidirect
10. C22⋊C8×D251C2semidirect
11. #447×C251C2semidirect
13. C22×D1004C22semidirect
20. C2×D4×D251C22semidirect
21. C2×D4×D251C22semidirect
23. C2×D2008C22semidirect
24. C2×#16860795C22semidirect
25. C2×#16860795C22semidirect
27. C2×C8×D251C22semidirect
28. #22548690C22semidirect
29. #22548689C22semidirect
32. #22548686C22semidirect
33. C4⋊C4×D251C22semidirect
35. #22548637C22semidirect
36. #22548636C22semidirect
37. #22548635C22semidirect
38. C22⋊(C251⋊C8)C22semidirect
39. (C2×D251)⋊C8C22semidirect
40. C22×D4×C251C22semidirect
41. C2×D8×C251C22semidirect
42. #121×C251C22semidirect
43. #102×C251C22semidirect
44. C22⋊C8×C251C22semidirect
57. C2×D1004C23semidirect
71. D4×D251D4semidirect
82. C2×C251⋊C8C23semidirect
84. C4⋊Dic251C23semidirect
94. C2×D4×C251C23semidirect
97. C2×C8×C251C23semidirect
98. C4⋊C4×C251C23semidirect
117. C22×D251D8semidirect
124. D1004C2×D4semidirect
150. D4×C251C2×D4semidirect
165. C22×C251C2×D8semidirect
182. C2×D251C2×D8semidirect
199. D251#447direct (N × H)
200. C251C2×#447semidirect
210. #447D251direct (N × H)
223. C22×D4C2×D251semidirect
229. C2×D8C2×D251semidirect
236. #121C2×D251semidirect
242. #102C2×D251semidirect
245. C22⋊C8C2×D251semidirect
278. C2×D4C22×D251semidirect
293. C2×C8C22×D251semidirect
296. C4⋊C4C22×D251semidirect
341. D4D4×D251semidirect
383. C22D8×D251semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2512
C2×D8×D2512
#121×D2512
#306318232
#306318222
#306318212
#306318202
#306318192
#102×D2512
C22⋊C8×D2512
#447×C2512
C2×#447251