Finite Group Databasebeta

← #44559586

Subgroups — #44559586

Order 32128.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D251C211yesmaximal, characteristic, complemented
2C2×#22548738C211yesmaximal, characteristic, complemented
3C2×#22548736C211yesmaximal, characteristic, complemented
4#30632200C211yesmaximal, characteristic, complemented
5#30631847C211yesmaximal, characteristic, complemented
6#30631841C211yesmaximal, characteristic, complemented
7#30631836C211yesmaximal, characteristic, complemented
8#30631834C211yesmaximal, characteristic, complemented
9#30631832C211yesmaximal, characteristic, complemented
10#30631827C211yesmaximal, characteristic, complemented
11#30631822C211yesmaximal, characteristic, complemented
12#30631682C211yesmaximal, characteristic, complemented
13#30631680C211yesmaximal, characteristic, complemented
14C22⋊C8×D251C211yesmaximal, characteristic, complemented
15#452×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860811C2211yescharacteristic, complemented
18C2×#16860810—21no—
19C2×#16860810C2211yescharacteristic
20C2×#16860809—21no—
21C2×#16860809C2211yescharacteristic
22#41×D251—24no—
23#41×D251—24no—
24C2×Q8×D251C2211yescharacteristic
25C2×D4×D251—21no—
26C2×D4×D251C2211yescharacteristic
27C22×C4×D251—21no—
28C22×C4×D251C2211yescharacteristic
29#22548738—24no—
30#22548736—24no—
31C2×#16860800C2211yescharacteristic, complemented
32C2×#16860799C2211yescharacteristic, complemented
33C2×#16860797C2211yescharacteristic, complemented
34C2×#16860797C2211yescharacteristic, complemented
35C2×D2008C2211yescharacteristic, complemented
36C2×#16860795C2211yescharacteristic, complemented
37C2×C8×D251C2211yescharacteristic, complemented
38C2×C8×D251C2211yescharacteristic, complemented
39#22548700—21no—
40#22548691C2211yescharacteristic
41#22548690C2211yescharacteristic
42#22548689C2211yescharacteristic, complemented
43#22548685—21no—
44#22548677—21no—
45(C22⋊D251)⋊C4—21no—
46#22548673C2211yescharacteristic, complemented
47#22548640C2211yescharacteristic, complemented
48#22548639C2211yescharacteristic, complemented
49#22548638C2211yescharacteristic, complemented
50#22548637C2211yescharacteristic, complemented
51#22548636C2211yescharacteristic, complemented
52#22548635C2211yescharacteristic, complemented
53C22⋊(C251⋊C8)C2211yescharacteristic, complemented
54(C2×D251)⋊C8C2211yescharacteristic
55(C2×D251)⋊C8C2211yescharacteristic
56C2×#41×C251C2211yescharacteristic
57C2×Q16×C251C2211yescharacteristic, complemented
58C2×SD16×C251C2211yescharacteristic, complemented
59C22⋊Q8×C251C2211yescharacteristic, complemented
60#103×C251C2211yescharacteristic, complemented
61#102×C251C2211yescharacteristic, complemented
62C22⋊C8×C251C2211yescharacteristic, complemented
63C23×D251—21no—
64C23×D251D411yescharacteristic
65#16860811—22no—
66#16860811—22no—
67#16860811—22no—
68#16860811D414yescomplemented
69#16860810—28no—
70#16860810—22no—
71#16860810—22no—
72#16860809—28no—
73#16860809—22no—
74#16860809D414yescomplemented
75C2×Dic502—21no—
76C2×Dic502C2311yescharacteristic
77Q8×D251—22no—
78Q8×D251—22no—
79C2×D1004—21no—
80C2×D1004—21no—
81C2×D1004—21no—
82C2×D1004C2311yescharacteristic
83C2×D1004C2311yescharacteristic
84C2×C22⋊D251—21no—
85C2×C22⋊D251—21no—
86C2×C22⋊D251—21no—
87C2×C22⋊D251—21no—
88D4×D251—28no—
89D4×D251—22no—
90D4×D251—22no—
91C22×Dic251—21no—
92C22×Dic251D411yescharacteristic
93C2×C4×D251—28no—
94C2×C4×D251—21no—
95C2×C4×D251—21no—
96C2×C4×D251—22no—
97C2×C4×D251—22no—
98C2×C4×D251—21no—
99C2×C4×D251—21no—
100C2×C4×D251D411yescharacteristic
101C2×C4×D251D411yescharacteristic
102C2×C4×D251C2311yescharacteristic
103C2×C4×D251C2311yescharacteristic
104#16860800—22no—
105#16860799—22no—
106#16860797—22no—
107#16860797—22no—
108D2008—22no—
109#16860795—22no—
110C2×C251⋊C8C2311yescharacteristic
111C2×C251⋊C8C2311yescharacteristic
112C8×D251—22no—
113C8×D251—22no—
114C4⋊Dic251C2311yescharacteristic
115Dic251⋊C4—21no—
116(C2×D251)⋊C4—21no—
117(C2×D251)⋊C4—21no—
118(C2×D251)⋊C4—21no—
119(C2×D251)⋊C4—21no—
120C4×Dic251—21no—
121#41×C251—22no—
122#41×C251—22no—
123C2×Q8×C251C2311yescharacteristic
124C2×Q8×C251C2311yescharacteristic
125C2×D4×C251—21no—
126C2×D4×C251C2311yescharacteristic
127C22×C4×C251—21no—
128C22×C4×C251C2311yescharacteristic
129Q16×C251—22no—
130SD16×C251—22no—
131C2×C8×C251C2311yescharacteristic
132C2×C8×C251C2311yescharacteristic
133C4⋊C4×C251—21no—
134C4⋊C4×C251C2311yescharacteristic
135C22⋊C4×C251—21no—
136C22×D251—24no—
137C22×D251—21no—
138C22×D251—22no—
139C22×D251—22no—
140C22×D251—21no—
141C22×D251—41no—
142C22×D251—42no—
143C22×D251—42no—
144C22×D251C2×D411yescharacteristic
145C22×D251C2×D411yescharacteristic
146Dic502—24no—
147Dic502—21no—
148Dic502C2×D412yes—
149D1004—24no—
150D1004—21no—
151D1004—21no—
152D1004—21no—
153D1004—42no—
154D1004—42no—
155D1004C2×D412yes—
156C22⋊D251—24no—
157C22⋊D251—24no—
158C22⋊D251—24no—
159C22⋊D251—42no—
160C2×Dic251—21no—
161C2×Dic251—24no—
162C2×Dic251—22no—
163C2×Dic251—21no—
164C2×Dic251C2×D411yescharacteristic
165C2×Dic251C2×D411yescharacteristic
166C4×D251—24no—
167C4×D251—24no—
168C4×D251—22no—
169C4×D251—21no—
170C4×D251—21no—
171C4×D251—22no—
172C4×D251—24no—
173C4×D251—42no—
174C4×D251C2×D412yes—
175C4×D251C2×D412yes—
176C251⋊C8—21no—
177C251⋊C8—21no—
178C23×C251—21no—
179C23×C251C2×D411yescharacteristic
180Q8×C251—21no—
181Q8×C251—21no—
182Q8×C251—21no—
183Q8×C251C2×D412yes—
184D4×C251—24no—
185D4×C251—21no—
186D4×C251C2×D412yes—
187C2×C4×C251—24no—
188C2×C4×C251—21no—
189C2×C4×C251—21no—
190C2×C4×C251—21no—
191C2×C4×C251—21no—
192C2×C4×C251—21no—
193C2×C4×C251C2×D411yescharacteristic
194C2×C4×C251C2411yescharacteristic
195C8×C251—21no—
196C8×C251—21no—
197C22×C251—21no—
198C22×C251—22no—
199C22×C251—21no—
200C22×C251—21no—
201C22×C251—41no—
202C22×C251—41no—
203C22×C251C22×D411yescharacteristic
204C2×D251—21no—
205C2×D251—21no—
206C2×D251—21no—
207C2×D251—21no—
208C2×D251—22no—
209C2×D251—44no—
210C2×D251—48no—
211C2×D251—41no—
212C2×D251—41no—
213C2×D251—41no—
214C2×D251—41no—
215C2×D251—41no—
216C2×D251#12012yes—
217C2×D251#13512yescomplemented
218C4×C251—21no—
219C4×C251—22no—
220C4×C251—41no—
221C4×C251—41no—
222C4×C251C22×D411yescharacteristic
223C4×C251C22×D411yescharacteristic
224Dic251—22no—
225Dic251—21no—
226Dic251—41no—
227Dic251#12012yes—
228C2×C251—41no—
229C2×C251—42no—
230C2×C251C2×#12011yescharacteristic
231C2×C251C2×#13511yescharacteristic
232C2×C251#57511yescharacteristic
233D251—22no—
234D251—22no—
235D251—42no—
236D251—81no—
237C251#300211yescharacteristic, complemented, Sylow
238#3002—2511nomaximal, Sylow
239C22×#41—2511no—
240C2×#135—2511no—
241C2×#135—2511no—
242#525—2511no—
243#452—2511no—
244#452D25111yescharacteristic, complemented
245#451—2511no—
246#450—2511no—
247#449—2511no—
248#449—2511no—
249#448—2511no—
250#447—2511no—
251#416—2511no—
252#416—2511no—
253C2×C22⋊C8—2511no—
254C2×#41—5021no—
255C2×#41—5024no—
256C2×#41—5024no—
257C2×#41—5021no—
258C2×#41—2511no—
259C2×#41—2511no—
260C2×#41—2511no—
261C2×#41—2511no—
262C2×#41C2×D25111yescharacteristic
263C22×Q8—2511no—
264C22×D4—5021no—
265C22×D4—2511no—
266C23×C4—5021no—
267C23×C4—2511no—
268#135—5024no—
269#135—5024no—
270C2×Q16—2511no—
271C2×Q16C2×D25111yescharacteristic, complemented
272C2×SD16—2511no—
273C2×SD16—2511no—
274C2×SD16—2511no—
275C2×SD16C2×D25111yescharacteristic, complemented
276C2×D8—2511no—
277C2×D8—2511no—
278C22×C8—2511no—
279C22×C8—2511no—
280#123—5021no—
281C22⋊Q8—2511no—
282C22⋊Q8C2×D25111yescharacteristic, complemented
283#121—2511no—
284#121—2511no—
285#120—5021no—
286C4×D4—5021no—
287C4×D4—5021no—
288#117—2511no—
289#103—2511no—
290#103—2511no—
291#103—2511no—
292#103C2×D25111yescharacteristic, complemented
293#102—2511no—
294#102—2511no—
295#102—2511no—
296#102C2×D25111yescharacteristic, complemented
297C22⋊C8—2511no—
298C22⋊C8—2511no—
299C22⋊C8—2511no—
300C22⋊C8C2×D25111yescharacteristic, complemented
301C24—5021no—
302C24—2511no—
303#41—22no—
304#41—22no—
305#41—5028no—
306#41—5022no—
307#41—5022no—
308#41—5022no—
309#41—5022no—
310#41—5028no—
311#41—5022no—
312#41—5022no—
313#41—2514no—
314#41—2514no—
315C2×Q8—5021no—
316C2×Q8—5022no—
317C2×Q8—5022no—
318C2×Q8—2511no—
319C2×Q8C22×D25111yescharacteristic
320C2×Q8C22×D25111yescharacteristic
321C2×D4—21no—
322C2×D4—5021no—
323C2×D4—5021no—
324C2×D4—5021no—
325C2×D4—5028no—
326C2×D4—5022no—
327C2×D4—5021no—
328C2×D4—5021no—
329C2×D4—5021no—
330C2×D4—5022no—
331C2×D4—5021no—
332C2×D4—2511no—
333C2×D4—2511no—
334C2×D4C22×D25111yescharacteristic
335C22×C4—21no—
336C22×C4—5021no—
337C22×C4—5028no—
338C22×C4—5021no—
339C22×C4—5021no—
340C22×C4—5022no—
341C22×C4—5022no—
342C22×C4—5021no—
343C22×C4—5021no—
344C22×C4—2511no—
345C22×C4—2511no—
346C22×C4—2511no—
347C22×C4—2511no—
348C22×C4—2511no—
349C22×C4C22×D25111yescharacteristic
350Q16—22no—
351Q16—5022no—
352SD16—22no—
353SD16—5022no—
354SD16—5022no—
355SD16—5022no—
356D8—5022no—
357D8—5022no—
358C2×C8—5022no—
359C2×C8—5022no—
360C2×C8—2511no—
361C2×C8—2511no—
362C2×C8C22×D25111yescharacteristic
363C2×C8C22×D25111yescharacteristic
364C4⋊C4—21no—
365C4⋊C4—5021no—
366C4⋊C4—2511no—
367C4⋊C4C22×D25111yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—5021no—
370C22⋊C4—5021no—
371C22⋊C4—5021no—
372C22⋊C4—5021no—
373C42—5021no—
374C23—21no—
375C23—10041no—
376C23—10042no—
377C23—10042no—
378C23—5024no—
379C23—5022no—
380C23—5022no—
381C23—5021no—
382C23—5021no—
383C23—2511no—
384C23—2511no—
385C23D4×D25111yescharacteristic
386Q8—21no—
387Q8—21no—
388Q8—21no—
389Q8—5024no—
390Q8—5021no—
391Q8—2512no—
392Q8D4×D25112yes—
393D4—24no—
394D4—21no—
395D4—10042no—
396D4—10042no—
397D4—10042no—
398D4—5024no—
399D4—5024no—
400D4—5024no—
401D4—5024no—
402D4—5021no—
403D4—5021no—
404D4—5021no—
405D4—2512no—
406D4D4×D25112yes—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—10042no—
414C2×C4—5024no—
415C2×C4—5024no—
416C2×C4—5022no—
417C2×C4—5022no—
418C2×C4—5024no—
419C2×C4—5021no—
420C2×C4—5021no—
421C2×C4—5024no—
422C2×C4—5022no—
423C2×C4—5021no—
424C2×C4—5021no—
425C2×C4—2511no—
426C2×C4—2511no—
427C2×C4—2512no—
428C2×C4—2512no—
429C2×C4D4×D25111yescharacteristic
430C2×C4C23×D25111yescharacteristic
431C8—21no—
432C8—21no—
433C8—5021no—
434C8—5021no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—41no—
440C22—41no—
441C22—10048no—
442C22—10044no—
443C22—10041no—
444C22—10041no—
445C22—10041no—
446C22—10041no—
447C22—10041no—
448C22—5022no—
449C22—5021no—
450C22—5021no—
451C22—5021no—
452C22—5021no—
453C22—2512no—
454C22—2512no—
455C22C2×D4×D25111yescharacteristic
456C4—21no—
457C4—22no—
458C4—41no—
459C4—41no—
460C4—10041no—
461C4—5022no—
462C4—5021no—
463C4—2512no—
464C4C2×D4×D25111yescharacteristic
465C4C2×D4×D25111yescharacteristic
466C2—41no—
467C2—42no—
468C2—20081no—
469C2—10042no—
470C2—5022no—
471C2—5022no—
472C2#120×D25111yescharacteristic
473C2#135×D25111yescharacteristic
474C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#452×C251C2
socleC22×C251C22×D4
hypercenter#452D251
hypocenterC251#3002
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#41×D251C2semidirect
2. C2×#22548738C2semidirect
3. C2×#22548736C2semidirect
4. #30632200C2semidirect
5. #30631847C2semidirect
6. #30631841C2semidirect
7. #30631836C2semidirect
8. #30631834C2semidirect
9. #30631832C2semidirect
10. #30631827C2semidirect
11. #30631822C2semidirect
12. #30631682C2semidirect
13. #30631680C2semidirect
14. C22⋊C8×D251C2semidirect
15. #452×C251C2semidirect
17. C2×#16860811C22semidirect
31. C2×#16860800C22semidirect
32. C2×#16860799C22semidirect
33. C2×#16860797C22semidirect
34. C2×#16860797C22semidirect
35. C2×D2008C22semidirect
36. C2×#16860795C22semidirect
37. C2×C8×D251C22semidirect
38. C2×C8×D251C22semidirect
42. #22548689C22semidirect
46. #22548673C22semidirect
47. #22548640C22semidirect
48. #22548639C22semidirect
49. #22548638C22semidirect
50. #22548637C22semidirect
51. #22548636C22semidirect
52. #22548635C22semidirect
53. C22⋊(C251⋊C8)C22semidirect
57. C2×Q16×C251C22semidirect
58. C2×SD16×C251C22semidirect
59. C22⋊Q8×C251C22semidirect
60. #103×C251C22semidirect
61. #102×C251C22semidirect
62. C22⋊C8×C251C22semidirect
68. #16860811D4semidirect
74. #16860809D4semidirect
217. C2×D251#135semidirect
237. C251#3002semidirect
244. #452D251semidirect
271. C2×Q16C2×D251semidirect
275. C2×SD16C2×D251semidirect
282. C22⋊Q8C2×D251semidirect
292. #103C2×D251semidirect
296. #102C2×D251semidirect
300. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2512
C2×#225487382
C2×#225487362
#306322002
#306318472
#306318412
#306318362
#306318342
#306318322
#306318272
#306318222
#306316822
#306316802
C22⋊C8×D2512
#452×C2512
#3002251