Finite Group Databasebeta

← #44560042

Subgroups — #44560042

Order 32128.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548739C211yesmaximal, characteristic, complemented
2C2×#22548723C211yesmaximal, characteristic, complemented
3#30632266C211yesmaximal, characteristic, complemented
4#30632253C211yesmaximal, characteristic, complemented
5#30631939C211yesmaximal, characteristic, complemented
6#30631933C211yesmaximal, characteristic, complemented
7#30631932C211yesmaximal, characteristic, complemented
8#30631921C211yesmaximal, characteristic, complemented
9#30631920C211yesmaximal, characteristic, complemented
10#30631917C211yesmaximal, characteristic, complemented
11#30631914C211yesmaximal, characteristic, complemented
12#30631726C211yesmaximal, characteristic, complemented
13#30631680C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#469×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×#16860809C2211yescharacteristic
18C2×D4×D251C2211yescharacteristic
19#22548739—24no—
20C2×#16860799C2211yescharacteristic
21C2×#16860798C2211yescharacteristic
22C2×#16860795C2211yescharacteristic
23#22548723—24no—
24C2×#16860793C2211yescharacteristic
25C2×#16860793C2211yescharacteristic, complemented
26C22×C251⋊C8C2211yescharacteristic, complemented
27C2×C8×D251C2211yescharacteristic
28C2×C8×D251C2211yescharacteristic, complemented
29#22548711—21no—
30#22548701—21no—
31#22548695C2211yescharacteristic, complemented
32#22548694C2211yescharacteristic, complemented
33#22548692—21no—
34#22548690C2211yescharacteristic
35#22548689C2211yescharacteristic
36#22548688—21no—
37#22548687—21no—
38#22548686—21no—
39#22548686C2211yescharacteristic
40#22548686C2211yescharacteristic, complemented
41#22548685—21no—
42#22548683—21no—
43#22548673C2211yescharacteristic
44C4⋊C4×D251C2211yescharacteristic
45#22548649C2211yescharacteristic, complemented
46#22548648C2211yescharacteristic, complemented
47#22548646C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548638C2211yescharacteristic, complemented
50#22548637C2211yescharacteristic
51#22548636C2211yescharacteristic
52#22548636C2211yescharacteristic, complemented
53#22548635C2211yescharacteristic
54C2×D8×C251C2211yescharacteristic
55C2×M16×C251C2211yescharacteristic, complemented
56#121×C251C2211yescharacteristic
57#121×C251C2211yescharacteristic, complemented
58#106×C251C2211yescharacteristic, complemented
59#102×C251C2211yescharacteristic
60#102×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860810—22no—
63#16860810—22no—
64#16860809—22no—
65#16860809—22no—
66C2×Dic502D411yescharacteristic
67C2×Dic502C2311yescharacteristic
68C2×D1004—21no—
69C2×D1004D411yescharacteristic
70C2×D1004C2311yescharacteristic
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C2×C22⋊D251—21no—
74C2×C22⋊D251—21no—
75C2×C22⋊D251—21no—
76C2×C22⋊D251D411yescharacteristic
77C2×C22⋊D251D411yescharacteristic
78D4×D251—22no—
79D4×D251—22no—
80C22×Dic251—21no—
81C2×C4×D251—21no—
82C2×C4×D251C2311yescharacteristic
83C2×C4×D251C2311yescharacteristic
84#16860799—22no—
85#16860798—22no—
86#16860795—22no—
87#16860793—22no—
88#16860793D414yescomplemented
89C2×C251⋊C8—22no—
90C2×C251⋊C8C2311yescharacteristic
91C2×C251⋊C8C2311yescharacteristic
92C8×D251—22no—
93C8×D251—22no—
94C4⋊Dic251—21no—
95C4⋊Dic251C2311yescharacteristic
96C4⋊Dic251C2311yescharacteristic
97Dic251⋊C4—21no—
98Dic251⋊C4—21no—
99C22⋊Dic251—21no—
100C22⋊Dic251—21no—
101C22⋊Dic251—21no—
102(C2×D251)⋊C4—21no—
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4—21no—
106C4×Dic251—21no—
107C2×D4×C251—21no—
108C2×D4×C251—21no—
109C2×D4×C251C2311yescharacteristic
110C2×D4×C251C2311yescharacteristic
111C22×C4×C251C2311yescharacteristic
112D8×C251—22no—
113M16×C251—22no—
114C2×C8×C251C2311yescharacteristic
115C2×C8×C251C2311yescharacteristic
116C4⋊C4×C251C2311yescharacteristic
117C4⋊C4×C251C2311yescharacteristic
118C22⋊C4×C251—21no—
119C22⋊C4×C251—21no—
120C22×D251—21no—
121C22×D251—42no—
122C22×D251—81no—
123C22×D251—81no—
124C22×D251C2×D411yescharacteristic
125C22×D251C2×D411yescharacteristic
126Dic502—22no—
127Dic502—21no—
128Dic502—21no—
129D1004—22no—
130D1004—21no—
131D1004—21no—
132D1004—81no—
133C22⋊D251—22no—
134C22⋊D251—22no—
135C22⋊D251—42no—
136C22⋊D251—42no—
137C22⋊D251—81no—
138C22⋊D251—81no—
139C22⋊D251—81no—
140C2×Dic251—21no—
141C2×Dic251—21no—
142C2×Dic251—21no—
143C2×Dic251—42no—
144C2×Dic251C2×D411yescharacteristic
145C2×Dic251C2×D411yescharacteristic
146C4×D251—21no—
147C4×D251—21no—
148C4×D251—81no—
149C4×D251C2×D412yes—
150C4×D251#4112yes—
151C251⋊C8—21no—
152C251⋊C8C2×D412yes—
153C23×C251—21no—
154C23×C251—21no—
155C23×C251C2×D411yescharacteristic
156D4×C251—21no—
157D4×C251—21no—
158D4×C251—21no—
159D4×C251—21no—
160D4×C251—81no—
161D4×C251—81no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251C2×D411yescharacteristic
167C2×C4×C251C2411yescharacteristic
168C8×C251—21no—
169C8×C251C2×D412yes—
170C22×C251—21no—
171C22×C251—21no—
172C22×C251—21no—
173C22×C251—41no—
174C22×C251—41no—
175C22×C251—41no—
176C22×C251—41no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251C22×D411yescharacteristic
180C2×D251—21no—
181C2×D251—21no—
182C2×D251—21no—
183C2×D251—21no—
184C2×D251—21no—
185C2×D251—41no—
186C2×D251—41no—
187C2×D251—41no—
188C2×D251—84no—
189C2×D251#12112yes—
190C4×C251—21no—
191C4×C251—41no—
192C4×C251—41no—
193C4×C251C22×D411yescharacteristic
194C4×C251C2×#4111yescharacteristic
195Dic251—21no—
196Dic251—41no—
197Dic251—41no—
198Dic251—41no—
199Dic251#12112yes—
200C2×C251—41no—
201C2×C251—81no—
202C2×C251—81no—
203C2×C251C2×#12111yescharacteristic
204C2×C251#57611yescharacteristic
205C2×C251#57711yescharacteristic
206D251—22no—
207D251—41no—
208D251—81no—
209C251#306211yescharacteristic, complemented, Sylow
210#3062—2511nomaximal, Sylow
211C2×AGL(1,ℤ/8)—2511no—
212C2×#131—2511no—
213#537—2511no—
214#535—2511no—
215#469—2511no—
216#469D25111yescharacteristic, complemented
217#468—2511no—
218#468—2511no—
219#466—2511no—
220#466—2511no—
221#465—2511no—
222#465—2511no—
223#428—2511no—
224#416—2511no—
225C2×#102—2511no—
226C2×#41—2511no—
227C2×#41—2511no—
228C22×D4—2511no—
229AGL(1,ℤ/8)—5024no—
230C2×SD16—2511no—
231C2×SD16—2511no—
232C2×D8—2511no—
233C2×D8C2×D25111yescharacteristic
234#131—5024no—
235C2×M16—2511no—
236C2×M16—2511no—
237C2×M16C2×D25111yescharacteristic, complemented
238C22×C8—2511no—
239C22×C8—2511no—
240C22×C8—2511no—
241#127—5021no—
242#124—5021no—
243C22⋊Q8—5021no—
244C22⋊Q8—2511no—
245C22⋊Q8—2511no—
246#121—5021no—
247#121—5021no—
248#121—5021no—
249#121—2511no—
250#121—2511no—
251#121—2511no—
252#121—2511no—
253#121C2×D25111yescharacteristic
254#121C2×D25111yescharacteristic, complemented
255#120—5021no—
256#120—5021no—
257#117—2511no—
258C2×C4⋊C4—2511no—
259#107—2511no—
260#107—2511no—
261#106—2511no—
262#106C2×D25111yescharacteristic, complemented
263#103—2511no—
264#103—2511no—
265#102—2511no—
266#102—2511no—
267#102—2511no—
268#102—2511no—
269#102C2×D25111yescharacteristic
270#102C2×D25111yescharacteristic, complemented
271C24—5021no—
272#41—5022no—
273#41—5022no—
274#41—5022no—
275#41—5022no—
276C2×Q8—2511no—
277C2×Q8—2511no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—5021no—
281C2×D4—5021no—
282C2×D4—5021no—
283C2×D4—5021no—
284C2×D4—5021no—
285C2×D4—5022no—
286C2×D4—5022no—
287C2×D4—5021no—
288C2×D4—2511no—
289C2×D4—2511no—
290C2×D4—2511no—
291C2×D4—2511no—
292C2×D4C22×D25111yescharacteristic
293C2×D4C22×D25111yescharacteristic
294C22×C4—5021no—
295C22×C4—5021no—
296C22×C4—2511no—
297C22×C4—2511no—
298C22×C4C22×D25111yescharacteristic
299SD16—5022no—
300SD16—5022no—
301D8—22no—
302D8—5022no—
303M16—22no—
304M16—5022no—
305M16—2514no—
306C2×C8—5022no—
307C2×C8—5022no—
308C2×C8—5022no—
309C2×C8—2511no—
310C2×C8—2511no—
311C2×C8C22×D25111yescharacteristic
312C2×C8C22×D25111yescharacteristic
313C4⋊C4—5021no—
314C4⋊C4—5021no—
315C4⋊C4—5021no—
316C4⋊C4—2511no—
317C4⋊C4—2511no—
318C4⋊C4C22×D25111yescharacteristic
319C4⋊C4C22×D25111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—5021no—
323C22⋊C4—5021no—
324C22⋊C4—5021no—
325C22⋊C4—5021no—
326C22⋊C4—5021no—
327C22⋊C4—5021no—
328C22⋊C4—5021no—
329C42—5021no—
330C23—21no—
331C23—21no—
332C23—20081no—
333C23—20081no—
334C23—10042no—
335C23—5021no—
336C23—2511no—
337C23—2511no—
338C23D4×D25111yescharacteristic
339Q8—5022no—
340Q8—5021no—
341Q8—5021no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—81no—
347D4—81no—
348D4—20081no—
349D4—20081no—
350D4—20081no—
351D4—20081no—
352D4—10042no—
353D4—10042no—
354D4—5022no—
355D4—5022no—
356D4—5022no—
357D4—5021no—
358D4—5021no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—20081no—
364C2×C4—10042no—
365C2×C4—5021no—
366C2×C4—5021no—
367C2×C4—5021no—
368C2×C4—5021no—
369C2×C4—5021no—
370C2×C4—2511no—
371C2×C4—2511no—
372C2×C4—2512no—
373C2×C4—2512no—
374C2×C4D4×D25111yescharacteristic
375C2×C4C23×D25111yescharacteristic
376C8—21no—
377C8—5021no—
378C8—2512no—
379C8D4×D25112yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—20084no—
390C22—10041no—
391C22—10041no—
392C22—10041no—
393C22—5021no—
394C22—5021no—
395C22—5021no—
396C22—5021no—
397C22—5021no—
398C22—2512no—
399C22C2×D4×D25111yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—10041no—
404C4—10041no—
405C4—10041no—
406C4—5021no—
407C4—2512no—
408C4C2×D4×D25111yescharacteristic
409C4#41×D25111yescharacteristic
410C2—41no—
411C2—81no—
412C2—81no—
413C2—20081no—
414C2—10041no—
415C2—5022no—
416C2#121×D25111yescharacteristic
417C2#3063249111yescharacteristic
418C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#469×C251C2
socleC22×C251C22×D4
hypercenter#469D251
hypocenterC251#3062
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#22548739C2semidirect
2. C2×#22548723C2semidirect
3. #30632266C2semidirect
4. #30632253C2semidirect
5. #30631939C2semidirect
6. #30631933C2semidirect
7. #30631932C2semidirect
8. #30631921C2semidirect
9. #30631920C2semidirect
10. #30631917C2semidirect
11. #30631914C2semidirect
12. #30631726C2semidirect
13. #30631680C2semidirect
14. #102×D251C2semidirect
15. #469×C251C2semidirect
25. C2×#16860793C22semidirect
26. C22×C251⋊C8C22semidirect
28. C2×C8×D251C22semidirect
31. #22548695C22semidirect
32. #22548694C22semidirect
40. #22548686C22semidirect
45. #22548649C22semidirect
46. #22548648C22semidirect
47. #22548646C22semidirect
48. #22548640C22semidirect
49. #22548638C22semidirect
52. #22548636C22semidirect
55. C2×M16×C251C22semidirect
57. #121×C251C22semidirect
58. #106×C251C22semidirect
60. #102×C251C22semidirect
88. #16860793D4semidirect
209. C251#3062semidirect
216. #469D251semidirect
237. C2×M16C2×D251semidirect
254. #121C2×D251semidirect
262. #106C2×D251semidirect
270. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487392
C2×#225487232
#306322662
#306322532
#306319392
#306319332
#306319322
#306319212
#306319202
#306319172
#306319142
#306317262
#306316802
#102×D2512
#469×C2512
#3062251