Finite Group Databasebeta

← #44562320

Subgroups — #44562320

Order 32128.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632333C211yesmaximal, characteristic, complemented
2#30632049C211yesmaximal, characteristic, complemented
3#30632017C211yesmaximal, characteristic, complemented
4#30632005C211yesmaximal, characteristic, complemented
5#30631974C211yesmaximal, characteristic, complemented
6#30631938C211yesmaximal, characteristic, complemented
7#30631914C211yesmaximal, characteristic, complemented
8#30631902C211yesmaximal, characteristic, complemented
9#30631830C211yesmaximal, characteristic, complemented
10#30631820C211yesmaximal, characteristic, complemented
11#30631797C211yesmaximal, characteristic, complemented
12#30631759C211yesmaximal, characteristic, complemented
13#30631686C211yesmaximal, characteristic, complemented
14C2×#22548635C211yesmaximal, characteristic, complemented
15#547×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C2×#16860798C2211yescharacteristic, complemented
19C2×#16860795C2211yescharacteristic
20C2×#16860795C2211yescharacteristic
21C2×#16860794C2211yescharacteristic, complemented
22C22×C251⋊C8C2211yescharacteristic
23#22548703C2211yescharacteristic
24#22548700—21no—
25#22548694C2211yescharacteristic
26#22548689C2211yescharacteristic
27#22548688—21no—
28#22548683—21no—
29C4×D1004C2211yescharacteristic
30C22⋊C4×D251—21no—
31#22548646C2211yescharacteristic, complemented
32#22548646C2211yescharacteristic, complemented
33#22548646C2211yescharacteristic, complemented
34C4⋊(C251⋊C8)C2211yescharacteristic, complemented
35#22548638C2211yescharacteristic, complemented
36#22548638C2211yescharacteristic, complemented
37#22548638C2211yescharacteristic, complemented
38#22548636C2211yescharacteristic, complemented
39#22548635C2211yescharacteristic
40#22548635C2211yescharacteristic
41#22548635C2211yescharacteristic
42#22548635C2211yescharacteristic
43#22548635C2211yescharacteristic
44#22548635C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46C22⋊(C251⋊C8)C2211yescharacteristic, complemented
47#22548626C2211yescharacteristic, complemented
48C2×#41×C251—21no—
49C4⋊Q8×C251C2211yescharacteristic, complemented
50#123×C251—21no—
51#123×C251—21no—
52C22⋊Q8×C251—21no—
53#121×C251C2211yescharacteristic
54#121×C251C2211yescharacteristic, complemented
55C4×D4×C251C2211yescharacteristic
56C4×D4×C251C2211yescharacteristic, complemented
57C2×C4⋊C4×C251C2211yescharacteristic
58C2×C4⋊C4×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860810—22no—
61#16860810—41no—
62C2×Dic502C2311yescharacteristic
63C2×D1004—22no—
64C2×D1004—21no—
65C2×D1004—41no—
66C2×D1004C2311yescharacteristic
67C2×D1004C2311yescharacteristic
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72#16860798—41no—
73#16860795—22no—
74#16860795—41no—
75#16860794—41no—
76C2×C251⋊C8—41no—
77C2×C251⋊C8C2311yescharacteristic
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C2×C251⋊C8C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82Dic251⋊C4—21no—
83C22⋊Dic251—21no—
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—21no—
88#41×C251—44no—
89C2×Q8×C251—21no—
90C2×D4×C251—21no—
91C2×D4×C251—21no—
92C2×D4×C251D411yescharacteristic
93C2×D4×C251C2311yescharacteristic
94C22×C4×C251—21no—
95C22×C4×C251—21no—
96C22×C4×C251—21no—
97C22×C4×C251C2311yescharacteristic
98C22×C4×C251C2311yescharacteristic
99C4⋊C4×C251—21no—
100C4⋊C4×C251—21no—
101C4⋊C4×C251—21no—
102C4⋊C4×C251D411yescharacteristic
103C4⋊C4×C251C2311yescharacteristic
104C4⋊C4×C251C2311yescharacteristic
105C4⋊C4×C251C2311yescharacteristic
106C22⋊C4×C251—21no—
107C22⋊C4×C251—21no—
108C22⋊C4×C251—21no—
109C22⋊C4×C251D411yescharacteristic
110C22⋊C4×C251D411yescharacteristic
111C42×C251C2311yescharacteristic
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—42no—
115C22×D251—42no—
116C22×D251—81no—
117Dic502—21no—
118Dic502—21no—
119D1004—21no—
120D1004—21no—
121D1004—21no—
122D1004—42no—
123D1004#4112yescomplemented
124C22⋊D251—42no—
125C22⋊D251—81no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C4×D251—42no—
129C4×D251—81no—
130C251⋊C8—21no—
131C251⋊C8—21no—
132C251⋊C8—21no—
133C251⋊C8—21no—
134C23×C251—21no—
135C23×C251C2×D411yescharacteristic
136C23×C251C2×D411yescharacteristic
137Q8×C251—42no—
138D4×C251—22no—
139D4×C251—21no—
140D4×C251—21no—
141D4×C251—42no—
142D4×C251—42no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—42no—
151C2×C4×C251—42no—
152C2×C4×C251—42no—
153C2×C4×C251C2×D411yescharacteristic
154C2×C4×C251C2×D411yescharacteristic
155C2×C4×C251C2×D411yescharacteristic
156C2×C4×C251C2×D411yescharacteristic
157C2×C4×C251C2411yescharacteristic
158C22×C251—21no—
159C22×C251—21no—
160C22×C251—21no—
161C22×C251—21no—
162C22×C251—21no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251—41no—
166C22×C251AGL(1,ℤ/8)12yes—
167C22×C251C22×D411yescharacteristic
168C2×D251—22no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—41no—
172C2×D251—41no—
173C2×D251—41no—
174C2×D251—84no—
175C4×C251—21no—
176C4×C251—21no—
177C4×C251—21no—
178C4×C251—21no—
179C4×C251—41no—
180C4×C251—41no—
181C4×C251—41no—
182C4×C251C2×#4111yescharacteristic
183C4×C251H(5,𝔽2)11yescharacteristic
184Dic251—41no—
185Dic251—41no—
186C2×C251—22no—
187C2×C251—41no—
188C2×C251—81no—
189C2×C251#54611yescharacteristic
190C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C251#57711yescharacteristic
192D251—42no—
193D251—81no—
194C251#330611yescharacteristic, complemented, Sylow
195#3306—2511nomaximal, Sylow
196#547D25111yescharacteristic, complemented
197#546—2511no—
198#489—2511no—
199#483—2511no—
200#481—2511no—
201#476—2511no—
202#469—2511no—
203#465—2511no—
204#463—2511no—
205#449—2511no—
206#447—2511no—
207#442—2511no—
208#435—2511no—
209#417—2511no—
210C2×#102—2511no—
211C2×#41—21no—
212C2×#41—2511no—
213C22×D4—2511no—
214C2×SD16—2511no—
215C2×D8—2511no—
216C2×D8—2511no—
217C2×M16—2511no—
218C22×C8—2511no—
219C4⋊Q8C2×D25111yescharacteristic, complemented
220#124—2511no—
221#123—21no—
222#123—21no—
223#123—5021no—
224C22⋊Q8—21no—
225C22⋊Q8—2511no—
226#121—5021no—
227#121—2511no—
228#121C2×D25111yescharacteristic
229#121C2×D25111yescharacteristic, complemented
230#120—5021no—
231C4×D4—2511no—
232C4×D4C2×D25111yescharacteristic
233C4×D4C2×D25111yescharacteristic, complemented
234C2×C4⋊C4C2×D25111yescharacteristic
235C2×C4⋊C4C2×D25111yescharacteristic, complemented
236C2×C22⋊C4—5021no—
237#106—2511no—
238#106—2511no—
239#106—2511no—
240C4⋊C8—2511no—
241#103—2511no—
242#103—2511no—
243#103—2511no—
244#102—2511no—
245#102—2511no—
246#102—2511no—
247#102—2511no—
248#102—2511no—
249#102—2511no—
250#102—2511no—
251C22⋊C8—2511no—
252C22⋊C8—2511no—
253#97—2511no—
254C24—5021no—
255#41—44no—
256#41—10041no—
257#41—5022no—
258C2×Q8—21no—
259C2×Q8—2511no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—10041no—
263C2×D4—5022no—
264C2×D4—5021no—
265C2×D4—5021no—
266C2×D4—5021no—
267C2×D4—2511no—
268C2×D4—2511no—
269C2×D4C22⋊D25111yescharacteristic
270C2×D4C22×D25111yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—5021no—
275C22×C4—5021no—
276C22×C4C22×D25111yescharacteristic
277C22×C4C22×D25111yescharacteristic
278SD16—10041no—
279D8—10041no—
280D8—5022no—
281M16—10041no—
282C2×C8—10041no—
283C2×C8—2511no—
284C2×C8—2511no—
285C2×C8—2511no—
286C2×C8—2511no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—5021no—
291C4⋊C4—2511no—
292C4⋊C4C22⋊D25111yescharacteristic
293C4⋊C4C22×D25111yescharacteristic
294C4⋊C4C22×D25111yescharacteristic
295C4⋊C4C22×D25111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—5021no—
300C22⋊C4—5021no—
301C22⋊C4—5021no—
302C22⋊C4—5021no—
303C22⋊C4—5021no—
304C22⋊C4C22⋊D25111yescharacteristic
305C22⋊C4C22⋊D25111yescharacteristic
306C42C22×D25111yescharacteristic
307C23—21no—
308C23—20081no—
309C23—10042no—
310C23—10042no—
311C23—5021no—
312C23—5021no—
313C23C2×C22⋊D25111yescharacteristic
314C23C2×C22⋊D25111yescharacteristic
315Q8—42no—
316Q8—5021no—
317Q8—5021no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—20081no—
324D4—10042no—
325D4—10042no—
326D4—5021no—
327D4—5021no—
328D4—5021no—
329D4—2512no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—20081no—
341C2×C4—10042no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4C2×C22⋊D25111yescharacteristic
345C2×C4C2×C22⋊D25111yescharacteristic
346C2×C4C2×C22⋊D25111yescharacteristic
347C2×C4C2×C22⋊D25111yescharacteristic
348C2×C4C23×D25111yescharacteristic
349C8—5021no—
350C8—5021no—
351C8—5021no—
352C8—5021no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—20084no—
362C22—10041no—
363C22—10041no—
364C22—10041no—
365C22—10041no—
366C22—10041no—
367C22—5022no—
368C22#2254874112yes—
369C22C22×C22⋊D25111yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—10041no—
378C4—10041no—
379C4#41×D25111yescharacteristic
380C4#2254876111yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—20081no—
385C2—10042no—
386C2#3063232511yescharacteristic
387C2C2×#2254874111yescharacteristic
388C2#3063249511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#547×C251C2
socleC22×C251C22×D4
hypercenter#547D251
hypocenterC251#3306
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #30632333C2semidirect
2. #30632049C2semidirect
3. #30632017C2semidirect
4. #30632005C2semidirect
5. #30631974C2semidirect
6. #30631938C2semidirect
7. #30631914C2semidirect
8. #30631902C2semidirect
9. #30631830C2semidirect
10. #30631820C2semidirect
11. #30631797C2semidirect
12. #30631759C2semidirect
13. #30631686C2semidirect
14. C2×#22548635C2semidirect
15. #547×C251C2semidirect
18. C2×#16860798C22semidirect
21. C2×#16860794C22semidirect
31. #22548646C22semidirect
32. #22548646C22semidirect
33. #22548646C22semidirect
34. C4⋊(C251⋊C8)C22semidirect
35. #22548638C22semidirect
36. #22548638C22semidirect
37. #22548638C22semidirect
38. #22548636C22semidirect
46. C22⋊(C251⋊C8)C22semidirect
47. #22548626C22semidirect
49. C4⋊Q8×C251C22semidirect
54. #121×C251C22semidirect
56. C4×D4×C251C22semidirect
58. C2×C4⋊C4×C251C22semidirect
123. D1004#41semidirect
194. C251#3306semidirect
196. #547D251semidirect
219. C4⋊Q8C2×D251semidirect
229. #121C2×D251semidirect
233. C4×D4C2×D251semidirect
235. C2×C4⋊C4C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323332
#306320492
#306320172
#306320052
#306319742
#306319382
#306319142
#306319022
#306318302
#306318202
#306317972
#306317592
#306316862
C2×#225486352
#547×C2512
#3306251