Finite Group Databasebeta

← #44559637

Subgroups — #44559637

Order 32128.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548760C211yesmaximal, characteristic
2C2×#22548744C211yesmaximal, characteristic, complemented
3C2×SD16×D251C211yesmaximal, characteristic, complemented
4#30632260C211yesmaximal, characteristic, complemented
5#30631851C211yesmaximal, characteristic, complemented
6#30631850C211yesmaximal, characteristic, complemented
7#30631847C211yesmaximal, characteristic, complemented
8#30631846C211yesmaximal, characteristic, complemented
9#30631839C211yesmaximal, characteristic, complemented
10#30631838C211yesmaximal, characteristic, complemented
11#30631836C211yesmaximal, characteristic, complemented
12#30631690C211yesmaximal, characteristic, complemented
13#30631680C211yesmaximal, characteristic, complemented
14#30631652C211yesmaximal, characteristic, complemented
15#450×C251C211yesmaximal, characteristic, complemented
16#22548760—24no—
17#22548760—24no—
18C2×#16860810C2211yescharacteristic
19C2×#16860810C2211yescharacteristic
20C2×#16860809—21no—
21C2×#16860809C2211yescharacteristic
22C2×#16860809C2211yescharacteristic
23C2×Q8×D251C2211yescharacteristic, complemented
24C22×C22⋊D251—21no—
25C22×C22⋊D251—21no—
26C2×D4×D251—21no—
27C2×D4×D251C2211yescharacteristic
28C2×D4×D251C2211yescharacteristic
29#22548744—24no—
30C2×Dic1004C2211yescharacteristic, complemented
31C2×#16860800C2211yescharacteristic, complemented
32C2×#16860799C2211yescharacteristic, complemented
33C2×#16860798C2211yescharacteristic, complemented
34C2×#16860798C2211yescharacteristic, complemented
35C2×#16860797C2211yescharacteristic, complemented
36SD16×D251—24no—
37C2×#16860793C2211yescharacteristic, complemented
38C2×C8×D251C2211yescharacteristic, complemented
39#22548714—21no—
40#22548701—21no—
41#22548700—21no—
42#22548697—21no—
43#22548695C2211yescharacteristic, complemented
44#22548694C2211yescharacteristic, complemented
45#22548691C2211yescharacteristic, complemented
46#22548673C2211yescharacteristic, complemented
47#22548640C2211yescharacteristic, complemented
48#22548638C2211yescharacteristic, complemented
49#22548637C2211yescharacteristic, complemented
50#22548636C2211yescharacteristic, complemented
51#22548636C2211yescharacteristic, complemented
52#22548635C2211yescharacteristic, complemented
53C22⋊(C251⋊C8)C2211yescharacteristic, complemented
54(C2×D251)⋊C8C2211yescharacteristic, complemented
55(C2×D251)⋊C8C2211yescharacteristic, complemented
56C22×D4×C251C2211yescharacteristic
57C2×SD16×C251C2211yescharacteristic, complemented
58C2×SD16×C251C2211yescharacteristic, complemented
59C22⋊Q8×C251C2211yescharacteristic, complemented
60#102×C251C2211yescharacteristic, complemented
61#102×C251C2211yescharacteristic, complemented
62C22⋊C8×C251C2211yescharacteristic, complemented
63C23×D251—21no—
64C23×D251—21no—
65#16860810—22no—
66#16860810—22no—
67#16860810—22no—
68#16860810—22no—
69#16860809—28no—
70#16860809—22no—
71#16860809—22no—
72#16860809—22no—
73#16860809D414yes—
74C2×Dic502—21no—
75C2×Dic502D411yescharacteristic
76C2×Dic502C2311yescharacteristic
77C2×Dic502C2311yescharacteristic
78Q8×D251—22no—
79Q8×D251—22no—
80C2×D1004D411yescharacteristic
81C2×D1004C2311yescharacteristic
82C2×C22⋊D251—28no—
83C2×C22⋊D251—28no—
84C2×C22⋊D251—21no—
85C2×C22⋊D251—21no—
86C2×C22⋊D251—21no—
87C2×C22⋊D251—21no—
88C2×C22⋊D251—21no—
89C2×C22⋊D251D411yescharacteristic
90C2×C22⋊D251D411yescharacteristic
91D4×D251—28no—
92D4×D251—22no—
93D4×D251—22no—
94D4×D251—22no—
95D4×D251D414yes—
96C22×Dic251—21no—
97C22×Dic251—21no—
98C2×C4×D251—21no—
99C2×C4×D251—21no—
100C2×C4×D251C2311yescharacteristic
101C2×C4×D251C2311yescharacteristic
102Dic1004—22no—
103#16860800—22no—
104#16860799—22no—
105#16860798—22no—
106#16860798—22no—
107#16860797—22no—
108#16860793—22no—
109C2×C251⋊C8C2311yescharacteristic
110C2×C251⋊C8C2311yescharacteristic
111C8×D251—22no—
112C4⋊Dic251—21no—
113C4⋊Dic251C2311yescharacteristic
114Dic251⋊C4—21no—
115Dic251⋊C4—21no—
116C22⋊Dic251—21no—
117(C2×D251)⋊C4—21no—
118(C2×D251)⋊C4—21no—
119(C2×D251)⋊C4—21no—
120C4×Dic251—21no—
121C24×C251—21no—
122C2×Q8×C251C2311yescharacteristic
123C2×D4×C251—22no—
124C2×D4×C251—22no—
125C2×D4×C251—21no—
126C2×D4×C251C2311yescharacteristic
127C2×D4×C251C2311yescharacteristic
128C22×C4×C251C2311yescharacteristic
129SD16×C251—22no—
130SD16×C251—22no—
131C2×C8×C251C2311yescharacteristic
132C2×C8×C251C2311yescharacteristic
133C4⋊C4×C251—21no—
134C4⋊C4×C251C2311yescharacteristic
135C22⋊C4×C251—21no—
136C22×D251—21no—
137C22×D251—24no—
138C22×D251—24no—
139C22×D251—81no—
140C22×D251—81no—
141C22×D251—81no—
142C22×D251—81no—
143C22×D251C2×D411yescharacteristic
144C22×D251C2×D411yescharacteristic
145Dic502—21no—
146Dic502—22no—
147Dic502—21no—
148Dic502—21no—
149Dic502—42no—
150Dic502C2×D412yes—
151D1004—22no—
152D1004—21no—
153D1004C2×D412yes—
154C22⋊D251—24no—
155C22⋊D251—24no—
156C22⋊D251—24no—
157C22⋊D251—24no—
158C22⋊D251—22no—
159C22⋊D251—24no—
160C22⋊D251—22no—
161C2×Dic251—24no—
162C2×Dic251—21no—
163C2×Dic251—24no—
164C2×Dic251—21no—
165C2×Dic251—21no—
166C2×Dic251C2×D411yescharacteristic
167C2×Dic251C2×D411yescharacteristic
168C4×D251—24no—
169C4×D251—21no—
170C4×D251—21no—
171C4×D251—42no—
172C4×D251C2×D412yes—
173C4×D251C2×D412yes—
174C251⋊C8—21no—
175C251⋊C8—21no—
176C23×C251—24no—
177C23×C251—21no—
178C23×C251—21no—
179C23×C251—81no—
180C23×C251—81no—
181C23×C251C2×D411yescharacteristic
182Q8×C251—21no—
183Q8×C251—21no—
184D4×C251—24no—
185D4×C251—21no—
186D4×C251—21no—
187D4×C251C2×D412yes—
188D4×C251C2×D412yes—
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—22no—
200C22×C251—21no—
201C22×C251—21no—
202C22×C251—41no—
203C22×C251—41no—
204C22×C251—41no—
205C22×C251—41no—
206C22×C251—84no—
207C22×C251C22×D411yescharacteristic
208C2×D251—21no—
209C2×D251—21no—
210C2×D251—21no—
211C2×D251—22no—
212C2×D251—21no—
213C2×D251—21no—
214C2×D251—41no—
215C2×D251—41no—
216C2×D251—84no—
217C2×D251—84no—
218C2×D251#12012yes—
219C4×C251—21no—
220C4×C251—41no—
221C4×C251—41no—
222C4×C251C22×D411yescharacteristic
223C4×C251C22×D411yescharacteristic
224Dic251—22no—
225Dic251—21no—
226Dic251—41no—
227Dic251—41no—
228Dic251#12012yes—
229C2×C251—42no—
230C2×C251—42no—
231C2×C251—41no—
232C2×C251C2×#12011yescharacteristic
233C2×C251#57711yescharacteristic
234C2×C251#57711yescharacteristic
235D251—22no—
236D251—41no—
237D251—42no—
238C251#300911yescharacteristic, complemented, Sylow
239#3009—2511nomaximal, Sylow
240C2×H(5,𝔽2)—2511no—
241C2×#137—2511no—
242C22×SD16—2511no—
243#536—2511no—
244#452—2511no—
245#452—2511no—
246#452—2511no—
247#452—2511no—
248#450—2511no—
249#450—2511no—
250#450—2511no—
251#450D25111yescharacteristic, complemented
252#418—2511no—
253#416—2511no—
254#408—2511no—
255H(5,𝔽2)—5024no—
256H(5,𝔽2)—5024no—
257C2×#41—5021no—
258C2×#41—2511no—
259C2×#41—2511no—
260C2×#41—2511no—
261C2×#41—2511no—
262C22×Q8—2511no—
263C22×D4—5021no—
264C22×D4—5021no—
265C22×D4—5021no—
266C22×D4—2511no—
267C22×D4—2511no—
268C22×D4C2×D25111yescharacteristic
269#137—5024no—
270C2×Q16—2511no—
271C2×Q16—2511no—
272C2×SD16—5024no—
273C2×SD16—2511no—
274C2×SD16—2511no—
275C2×SD16—2511no—
276C2×SD16—2511no—
277C2×SD16C2×D25111yescharacteristic, complemented
278C2×SD16C2×D25111yescharacteristic, complemented
279C2×M16—2511no—
280C22×C8—2511no—
281C4⋊Q8—5021no—
282#124—5021no—
283#123—5021no—
284#123—5021no—
285C22⋊Q8—2511no—
286C22⋊Q8—2511no—
287C22⋊Q8—2511no—
288C22⋊Q8C2×D25111yescharacteristic, complemented
289#117—2511no—
290#103—2511no—
291#103—2511no—
292#102—2511no—
293#102—2511no—
294#102—2511no—
295#102—2511no—
296#102C2×D25111yescharacteristic, complemented
297#102C2×D25111yescharacteristic, complemented
298C22⋊C8—2511no—
299C22⋊C8—2511no—
300C22⋊C8—2511no—
301C22⋊C8C2×D25111yescharacteristic, complemented
302C24—21no—
303C24—5021no—
304C24—5021no—
305#41—5022no—
306#41—5022no—
307#41—5022no—
308#41—5022no—
309#41—5028no—
310#41—5022no—
311#41—5022no—
312#41—5022no—
313#41—2514no—
314C2×Q8—5021no—
315C2×Q8—5022no—
316C2×Q8—5022no—
317C2×Q8—2511no—
318C2×Q8—2511no—
319C2×Q8—2511no—
320C2×Q8C22×D25111yescharacteristic
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—5028no—
325C2×D4—5022no—
326C2×D4—5028no—
327C2×D4—5028no—
328C2×D4—5021no—
329C2×D4—5021no—
330C2×D4—5021no—
331C2×D4—5022no—
332C2×D4—5022no—
333C2×D4—5021no—
334C2×D4—5021no—
335C2×D4—2511no—
336C2×D4—2511no—
337C2×D4—2514no—
338C2×D4—2511no—
339C2×D4—2511no—
340C2×D4C22×D25111yescharacteristic
341C2×D4C22×D25111yescharacteristic
342C22×C4—5021no—
343C22×C4—5021no—
344C22×C4—5021no—
345C22×C4—5021no—
346C22×C4—2511no—
347C22×C4—2511no—
348C22×C4C22×D25111yescharacteristic
349Q16—5022no—
350Q16—5022no—
351SD16—22no—
352SD16—22no—
353SD16—5022no—
354SD16—5022no—
355SD16—5022no—
356SD16—5022no—
357M16—5022no—
358C2×C8—5022no—
359C2×C8—2511no—
360C2×C8—2511no—
361C2×C8C22×D25111yescharacteristic
362C2×C8C22×D25111yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—5021no—
365C4⋊C4—5021no—
366C4⋊C4—5021no—
367C4⋊C4—2511no—
368C4⋊C4C22×D25111yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—5021no—
371C22⋊C4—5021no—
372C22⋊C4—5021no—
373C22⋊C4—5021no—
374C42—5021no—
375C23—24no—
376C23—21no—
377C23—21no—
378C23—81no—
379C23—81no—
380C23—20081no—
381C23—20081no—
382C23—20081no—
383C23—20081no—
384C23—5024no—
385C23—5024no—
386C23—5021no—
387C23—2511no—
388C23—2511no—
389C23D4×D25111yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—10042no—
393Q8—5022no—
394Q8—5021no—
395Q8—5021no—
396Q8—5021no—
397Q8—2512no—
398D4—24no—
399D4—21no—
400D4—21no—
401D4—5022no—
402D4—5024no—
403D4—5024no—
404D4—5024no—
405D4—5024no—
406D4—5022no—
407D4—5022no—
408D4—5024no—
409D4—5021no—
410D4—2512no—
411D4D4×D25112yes—
412D4D4×D25112yes—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—10042no—
418C2×C4—5024no—
419C2×C4—5021no—
420C2×C4—5021no—
421C2×C4—5024no—
422C2×C4—5021no—
423C2×C4—5024no—
424C2×C4—5021no—
425C2×C4—5021no—
426C2×C4—2511no—
427C2×C4—2511no—
428C2×C4—2512no—
429C2×C4—2512no—
430C2×C4D4×D25111yescharacteristic
431C2×C4C23×D25111yescharacteristic
432C8—21no—
433C8—21no—
434C8—5021no—
435C8—5021no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—22no—
440C22—22no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—84no—
446C22—20084no—
447C22—20084no—
448C22—10041no—
449C22—10041no—
450C22—5021no—
451C22—5021no—
452C22—5021no—
453C22—5022no—
454C22—5021no—
455C22—5021no—
456C22—2512no—
457C22C2×D4×D25111yescharacteristic
458C4—21no—
459C4—41no—
460C4—41no—
461C4—10041no—
462C4—10041no—
463C4—5022no—
464C4—5021no—
465C4—2512no—
466C4C2×D4×D25111yescharacteristic
467C4C2×D4×D25111yescharacteristic
468C2—41no—
469C2—42no—
470C2—42no—
471C2—10042no—
472C2—10041no—
473C2—5022no—
474C2#120×D25111yescharacteristic
475C2#3063249311yescharacteristic
476C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#450×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548760C2
hypercenter#450D251
hypocenterC251#3009
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
2. C2×#22548744C2semidirect
3. C2×SD16×D251C2semidirect
4. #30632260C2semidirect
5. #30631851C2semidirect
6. #30631850C2semidirect
7. #30631847C2semidirect
8. #30631846C2semidirect
9. #30631839C2semidirect
10. #30631838C2semidirect
11. #30631836C2semidirect
12. #30631690C2semidirect
13. #30631680C2semidirect
14. #30631652C2semidirect
15. #450×C251C2semidirect
23. C2×Q8×D251C22semidirect
30. C2×Dic1004C22semidirect
31. C2×#16860800C22semidirect
32. C2×#16860799C22semidirect
33. C2×#16860798C22semidirect
34. C2×#16860798C22semidirect
35. C2×#16860797C22semidirect
37. C2×#16860793C22semidirect
38. C2×C8×D251C22semidirect
43. #22548695C22semidirect
44. #22548694C22semidirect
45. #22548691C22semidirect
46. #22548673C22semidirect
47. #22548640C22semidirect
48. #22548638C22semidirect
49. #22548637C22semidirect
50. #22548636C22semidirect
51. #22548636C22semidirect
52. #22548635C22semidirect
53. C22⋊(C251⋊C8)C22semidirect
54. (C2×D251)⋊C8C22semidirect
55. (C2×D251)⋊C8C22semidirect
57. C2×SD16×C251C22semidirect
58. C2×SD16×C251C22semidirect
59. C22⋊Q8×C251C22semidirect
60. #102×C251C22semidirect
61. #102×C251C22semidirect
62. C22⋊C8×C251C22semidirect
238. C251#3009semidirect
251. #450D251semidirect
277. C2×SD16C2×D251semidirect
278. C2×SD16C2×D251semidirect
288. C22⋊Q8C2×D251semidirect
296. #102C2×D251semidirect
297. #102C2×D251semidirect
301. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487602
C2×#225487442
C2×SD16×D2512
#306322602
#306318512
#306318502
#306318472
#306318462
#306318392
#306318382
#306318362
#306316902
#306316802
#306316522
#450×C2512
#3009251