Finite Group Databasebeta

← #44561897

Subgroups — #44561897

Order 32128.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548743C211yesmaximal, characteristic, complemented
2C2×#22548738C211yesmaximal, characteristic, complemented
3#30632339C211yesmaximal, characteristic, complemented
4#30632328C211yesmaximal, characteristic, complemented
5#30632080C211yesmaximal, characteristic, complemented
6#30631917C211yesmaximal, characteristic, complemented
7#30631905C211yesmaximal, characteristic, complemented
8#30631891C211yesmaximal, characteristic, complemented
9#30631852C211yesmaximal, characteristic, complemented
10#30631847C211yesmaximal, characteristic, complemented
11#30631837C211yesmaximal, characteristic, complemented
12#30631835C211yesmaximal, characteristic, complemented
13#30631798C211yesmaximal, characteristic, complemented
14#30631756C211yesmaximal, characteristic, complemented
15#470×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860811C2211yescharacteristic, complemented
18C2×#16860810C2211yescharacteristic
19C22×D1004C2211yescharacteristic
20#22548743—24no—
21#22548738—24no—
22C2×#16860800C2211yescharacteristic, complemented
23C2×#16860799C2211yescharacteristic
24C2×#16860799C2211yescharacteristic
25C2×#16860797C2211yescharacteristic
26C2×#16860797C2211yescharacteristic
27C2×#16860797C2211yescharacteristic, complemented
28C2×D2008C2211yescharacteristic
29C2×D2008C2211yescharacteristic
30C2×C8×D251C2211yescharacteristic, complemented
31#22548713C2211yescharacteristic, complemented
32#22548704C2211yescharacteristic
33#22548700—21no—
34#22548700—21no—
35#22548697—21no—
36#22548695C2211yescharacteristic, complemented
37#22548690—21no—
38#22548690C2211yescharacteristic
39#22548688—21no—
40#22548685—21no—
41#22548677C2211yescharacteristic
42#22548677C2211yescharacteristic, complemented
43C4×C22⋊D251C2211yescharacteristic
44C4⋊C4×D251—21no—
45C22⋊C4×D251—21no—
46Dic251⋊C8C2211yescharacteristic, complemented
47#22548638C2211yescharacteristic, complemented
48#22548635C2211yescharacteristic
49#22548635C2211yescharacteristic
50#22548635C2211yescharacteristic, complemented
51C22⋊(C251⋊C8)C2211yescharacteristic
52(C2×D251)⋊C8C2211yescharacteristic, complemented
53#22548625C2211yescharacteristic, complemented
54C2×Q16×C251C2211yescharacteristic, complemented
55C2×M16×C251C2211yescharacteristic
56C22⋊Q8×C251C2211yescharacteristic
57C22⋊Q8×C251C2211yescharacteristic
58#106×C251C2211yescharacteristic, complemented
59#103×C251C2211yescharacteristic, complemented
60#103×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860811—22no—
63#16860811—22no—
64#16860811—41no—
65#16860811—41no—
66#16860810—22no—
67#16860810—41no—
68C2×Dic502C2311yescharacteristic
69C2×D1004—22no—
70C2×D1004—21no—
71C2×D1004—21no—
72C2×D1004—21no—
73C2×D1004—21no—
74C2×D1004—41no—
75C2×D1004C2311yescharacteristic
76C2×D1004C2311yescharacteristic
77C2×D1004C2311yescharacteristic
78C2×C22⋊D251—21no—
79C2×C22⋊D251D411yescharacteristic
80C2×C4×D251—21no—
81C2×C4×D251—21no—
82C2×C4×D251—21no—
83C2×C4×D251—21no—
84C2×C4×D251—21no—
85C2×C4×D251C2311yescharacteristic
86#16860800—41no—
87#16860799—22no—
88#16860799—22no—
89#16860797—22no—
90#16860797—22no—
91#16860797—41no—
92D2008—22no—
93D2008D414yescomplemented
94C2×C251⋊C8C2311yescharacteristic
95C2×C251⋊C8C2311yescharacteristic
96C8×D251—22no—
97C4⋊Dic251—21no—
98Dic251⋊C4—21no—
99Dic251⋊C4—21no—
100Dic251⋊C4D411yescharacteristic
101C22⋊Dic251D411yescharacteristic
102(C2×D251)⋊C4—21no—
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4—21no—
107(C2×D251)⋊C4—21no—
108(C2×D251)⋊C4D411yescharacteristic
109C4×Dic251C2311yescharacteristic
110C2×Q8×C251C2311yescharacteristic
111C2×Q8×C251C2311yescharacteristic
112C22×C4×C251C2311yescharacteristic
113Q16×C251—22no—
114M16×C251—22no—
115C2×C8×C251C2311yescharacteristic
116C2×C8×C251C2311yescharacteristic
117C4⋊C4×C251—21no—
118C4⋊C4×C251—21no—
119C4⋊C4×C251C2311yescharacteristic
120C4⋊C4×C251C2311yescharacteristic
121C22⋊C4×C251—21no—
122C22⋊C4×C251—21no—
123C22×D251—21no—
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—42no—
127C22×D251—42no—
128C22×D251—81no—
129C22×D251C2×D411yescharacteristic
130Dic502—21no—
131Dic502—21no—
132D1004—21no—
133D1004—21no—
134D1004—21no—
135D1004—21no—
136D1004—42no—
137D1004—42no—
138D1004—42no—
139D1004—81no—
140D1004C2×D412yes—
141D1004C2×D412yes—
142C22⋊D251—41no—
143C22⋊D251—42no—
144C2×Dic251—21no—
145C2×Dic251C2×D411yescharacteristic
146C2×Dic251C2×D411yescharacteristic
147C2×Dic251C2×D411yescharacteristic
148C4×D251—21no—
149C4×D251—21no—
150C4×D251—42no—
151C4×D251—42no—
152C4×D251—42no—
153C4×D251—42no—
154C4×D251—81no—
155C251⋊C8—21no—
156C251⋊C8—21no—
157C23×C251C2×D411yescharacteristic
158Q8×C251—21no—
159Q8×C251—21no—
160Q8×C251—21no—
161Q8×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251C2×D411yescharacteristic
169C2×C4×C251C2411yescharacteristic
170C8×C251—21no—
171C8×C251C2×D412yes—
172C22×C251—21no—
173C22×C251—21no—
174C22×C251—21no—
175C22×C251C22×D411yescharacteristic
176C2×D251—21no—
177C2×D251—22no—
178C2×D251—22no—
179C2×D251—21no—
180C2×D251—21no—
181C2×D251—41no—
182C2×D251—41no—
183C2×D251—41no—
184C2×D251—41no—
185C2×D251—41no—
186C2×D251—41no—
187C2×D251—41no—
188C2×D251—84no—
189C4×C251—21no—
190C4×C251—41no—
191C4×C251—41no—
192C4×C251—41no—
193C4×C251—41no—
194C4×C251C22×D411yescharacteristic
195C4×C251H(5,𝔽2)11yescharacteristic
196Dic251—21no—
197Dic251—21no—
198Dic251—21no—
199Dic251—41no—
200C2×C251—41no—
201C2×C251D4211yescharacteristic
202C2×C251#57511yescharacteristic
203C2×C251#57711yescharacteristic
204D251—41no—
205D251—42no—
206D251—42no—
207D251—81no—
208C251#326711yescharacteristic, complemented, Sylow
209#3267—2511nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2511no—
211C2×#135—2511no—
212#547—2511no—
213#546—2511no—
214#497—2511no—
215#470D25111yescharacteristic, complemented
216#465—2511no—
217#463—2511no—
218#461—2511no—
219#452—2511no—
220#452—2511no—
221#450—2511no—
222#449—2511no—
223#442—2511no—
224#435—2511no—
225C2×#41—2511no—
226C2×#41—2511no—
227C2×#41—2511no—
228C22×D4—2511no—
229AGL(1,ℤ/8)—5024no—
230#135—5024no—
231C2×Q16—2511no—
232C2×Q16C2×D25111yescharacteristic, complemented
233C2×SD16—2511no—
234C2×SD16—2511no—
235C2×SD16—2511no—
236C2×SD16—2511no—
237C2×SD16—2511no—
238C2×D8—2511no—
239C2×D8—2511no—
240C2×M16C2×D25111yescharacteristic
241C22×C8—2511no—
242C4⋊Q8—2511no—
243#124—2511no—
244#123—5021no—
245#123—5021no—
246#123—5021no—
247C22⋊Q8—2511no—
248C22⋊Q8C2×D25111yescharacteristic
249C22⋊Q8C2×D25111yescharacteristic
250#121—5021no—
251#121—5021no—
252#121—2511no—
253#120—5021no—
254C4×D4—2511no—
255C4×D4—2511no—
256C4×D4—2511no—
257C2×C4⋊C4—5021no—
258C2×C22⋊C4—5021no—
259#106C2×D25111yescharacteristic, complemented
260C4⋊C8—2511no—
261#103—2511no—
262#103C2×D25111yescharacteristic, complemented
263#103C2×D25111yescharacteristic, complemented
264#102—2511no—
265#102—2511no—
266#102—2511no—
267C22⋊C8—2511no—
268C22⋊C8—2511no—
269#97—2511no—
270C24—5021no—
271#41—10041no—
272#41—10041no—
273#41—10041no—
274#41—5022no—
275#41—5022no—
276#41—5022no—
277C2×Q8—2511no—
278C2×Q8C22×D25111yescharacteristic
279C2×Q8C22×D25111yescharacteristic
280C2×D4—10041no—
281C2×D4—5021no—
282C2×D4—5021no—
283C2×D4—5021no—
284C2×D4—5022no—
285C2×D4—5021no—
286C2×D4—5021no—
287C2×D4—2511no—
288C2×D4—2511no—
289C2×D4—2511no—
290C2×D4—2511no—
291C22×C4—5021no—
292C22×C4—5021no—
293C22×C4—5021no—
294C22×C4—5021no—
295C22×C4—5021no—
296C22×C4—2511no—
297C22×C4C22×D25111yescharacteristic
298Q16—22no—
299Q16—10041no—
300SD16—10041no—
301SD16—5022no—
302SD16—5022no—
303SD16—5022no—
304SD16—5022no—
305D8—5022no—
306D8—2514no—
307M16—22no—
308C2×C8—5022no—
309C2×C8—2511no—
310C2×C8—2511no—
311C2×C8C22×D25111yescharacteristic
312C2×C8C22×D25111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—5021no—
316C4⋊C4—5021no—
317C4⋊C4—5021no—
318C4⋊C4—2511no—
319C4⋊C4C22×D25111yescharacteristic
320C4⋊C4C22×D25111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—5021no—
324C22⋊C4—5021no—
325C22⋊C4—5021no—
326C22⋊C4—5021no—
327C22⋊C4—5021no—
328C22⋊C4—5021no—
329C22⋊C4—2511no—
330C22⋊C4—2511no—
331C42—2511no—
332C23—20081no—
333C23—10042no—
334C23—10042no—
335C23—5021no—
336C23—5021no—
337C23—5021no—
338C23—2511no—
339C23D4×D25111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—5021no—
345Q8—5021no—
346D4—20081no—
347D4—10042no—
348D4—10042no—
349D4—10042no—
350D4—10042no—
351D4—10041no—
352D4—5021no—
353D4—5021no—
354D4—5021no—
355D4—5021no—
356D4—2512no—
357D4—2512no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—20081no—
365C2×C4—10042no—
366C2×C4—10042no—
367C2×C4—10042no—
368C2×C4—10042no—
369C2×C4—5021no—
370C2×C4—5021no—
371C2×C4—5021no—
372C2×C4—2511no—
373C2×C4—2511no—
374C2×C4—2511no—
375C2×C4D4×D25111yescharacteristic
376C2×C4C23×D25111yescharacteristic
377C8—21no—
378C8—5021no—
379C8—5021no—
380C8D4×D25112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—20084no—
385C22—10041no—
386C22—10041no—
387C22—10041no—
388C22—10041no—
389C22—10041no—
390C22—10041no—
391C22—10041no—
392C22—5021no—
393C22—5021no—
394C22—5022no—
395C22—5022no—
396C22—5021no—
397C22C2×D4×D25111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—10041no—
404C4—5021no—
405C4—5021no—
406C4—5021no—
407C4C2×D4×D25111yescharacteristic
408C4#2254876111yescharacteristic
409C2—41no—
410C2—20081no—
411C2—10042no—
412C2—10042no—
413C2—10041no—
414C2#3063232111yescharacteristic
415C2#3063248611yescharacteristic
416C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#470×C251C2
socleC22×C251C22×D4
hypercenter#470D251
hypocenterC251#3267
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. C2×#22548738C2semidirect
3. #30632339C2semidirect
4. #30632328C2semidirect
5. #30632080C2semidirect
6. #30631917C2semidirect
7. #30631905C2semidirect
8. #30631891C2semidirect
9. #30631852C2semidirect
10. #30631847C2semidirect
11. #30631837C2semidirect
12. #30631835C2semidirect
13. #30631798C2semidirect
14. #30631756C2semidirect
15. #470×C251C2semidirect
17. C2×#16860811C22semidirect
22. C2×#16860800C22semidirect
27. C2×#16860797C22semidirect
30. C2×C8×D251C22semidirect
31. #22548713C22semidirect
36. #22548695C22semidirect
42. #22548677C22semidirect
46. Dic251⋊C8C22semidirect
47. #22548638C22semidirect
50. #22548635C22semidirect
52. (C2×D251)⋊C8C22semidirect
53. #22548625C22semidirect
54. C2×Q16×C251C22semidirect
58. #106×C251C22semidirect
59. #103×C251C22semidirect
60. #103×C251C22semidirect
93. D2008D4semidirect
208. C251#3267semidirect
215. #470D251semidirect
232. C2×Q16C2×D251semidirect
259. #106C2×D251semidirect
262. #103C2×D251semidirect
263. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C2×#225487382
#306323392
#306323282
#306320802
#306319172
#306319052
#306318912
#306318522
#306318472
#306318372
#306318352
#306317982
#306317562
#470×C2512
#3267251