Finite Group Databasebeta

← #44562329

Subgroups — #44562329

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#30632338C211yesmaximal, characteristic, complemented
2#30632334C211yesmaximal, characteristic, complemented
3#30632045C211yesmaximal, characteristic, complemented
4#30632019C211yesmaximal, characteristic, complemented
5#30631976C211yesmaximal, characteristic, complemented
6#30631938C211yesmaximal, characteristic, complemented
7#30631917C211yesmaximal, characteristic, complemented
8#30631901C211yesmaximal, characteristic, complemented
9#30631834C211yesmaximal, characteristic, complemented
10#30631822C211yesmaximal, characteristic, complemented
11#30631798C211yesmaximal, characteristic, complemented
12#30631761C211yesmaximal, characteristic, complemented
13#30631686C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#481×C251C211yesmaximal, characteristic, complemented
16C2×#16860810—21no—
17C2×#16860810C2211yescharacteristic
18C2×D4×D251C2211yescharacteristic
19C2×#16860799C2211yescharacteristic, complemented
20C2×D2008C2211yescharacteristic
21C2×#16860795C2211yescharacteristic
22C2×#16860794C2211yescharacteristic, complemented
23C2×C8×D251C2211yescharacteristic
24#22548714C2211yescharacteristic, complemented
25#22548702C2211yescharacteristic
26#22548700—21no—
27#22548698—21no—
28#22548697—21no—
29#22548695C2211yescharacteristic
30#22548693—21no—
31#22548690C2211yescharacteristic
32#22548689C2211yescharacteristic, complemented
33#22548686—21no—
34#22548683—21no—
35#22548677C2211yescharacteristic
36#22548677C2211yescharacteristic, complemented
37C4×C22⋊D251C2211yescharacteristic
38C4⋊C4×D251C2211yescharacteristic
39C22⋊C4×D251—21no—
40#22548646C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42#22548640C2211yescharacteristic, complemented
43#22548638C2211yescharacteristic, complemented
44#22548638C2211yescharacteristic, complemented
45#22548637C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49#22548635C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic
51#22548625C2211yescharacteristic, complemented
52#121×C251C2211yescharacteristic
53C2×C4⋊C4×C251C2211yescharacteristic, complemented
54#106×C251C2211yescharacteristic, complemented
55#106×C251C2211yescharacteristic, complemented
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860810—22no—
61#16860810—41no—
62#16860810—44no—
63C2×Dic502—21no—
64C2×Dic502C2311yescharacteristic
65C2×D1004—21no—
66C2×D1004C2311yescharacteristic
67C2×D1004C2311yescharacteristic
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C22⋊D251—21no—
71C2×C22⋊D251D411yescharacteristic
72D4×D251—22no—
73D4×D251—41no—
74C2×C4×D251—21no—
75C2×C4×D251—21no—
76C2×C4×D251—21no—
77C2×C4×D251—21no—
78C2×C4×D251C2311yescharacteristic
79#16860799—41no—
80D2008—22no—
81#16860795—41no—
82#16860794—41no—
83C2×C251⋊C8C2311yescharacteristic
84C2×C251⋊C8C2311yescharacteristic
85C8×D251—41no—
86C4⋊Dic251—21no—
87C4⋊Dic251C2311yescharacteristic
88Dic251⋊C4—21no—
89Dic251⋊C4—21no—
90Dic251⋊C4D411yescharacteristic
91C22⋊Dic251—21no—
92C22⋊Dic251—21no—
93C22⋊Dic251D411yescharacteristic
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4—21no—
97(C2×D251)⋊C4—21no—
98(C2×D251)⋊C4—21no—
99(C2×D251)⋊C4D411yescharacteristic
100C4×Dic251C2311yescharacteristic
101C2×D4×C251—21no—
102C2×D4×C251C2311yescharacteristic
103C22×C4×C251—21no—
104C22×C4×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251—21no—
108C4⋊C4×C251C2311yescharacteristic
109C4⋊C4×C251C2311yescharacteristic
110C4⋊C4×C251C2311yescharacteristic
111C22⋊C4×C251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—42no—
115C22×D251—42no—
116C22×D251—81no—
117C22×D251C2×D411yescharacteristic
118Dic502—21no—
119Dic502—21no—
120Dic502—42no—
121D1004—21no—
122D1004—21no—
123D1004—21no—
124D1004—42no—
125D1004#4112yescomplemented
126C22⋊D251—22no—
127C22⋊D251—42no—
128C22⋊D251—42no—
129C22⋊D251—42no—
130C2×Dic251—21no—
131C2×Dic251—21no—
132C2×Dic251C2×D411yescharacteristic
133C2×Dic251C2×D411yescharacteristic
134C2×Dic251C2×D411yescharacteristic
135C4×D251—21no—
136C4×D251—21no—
137C4×D251—42no—
138C4×D251—42no—
139C4×D251—42no—
140C4×D251—81no—
141C251⋊C8—21no—
142C251⋊C8—21no—
143C23×C251—21no—
144C23×C251C2×D411yescharacteristic
145D4×C251—21no—
146D4×C251—21no—
147D4×C251—81no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251—21no—
153C2×C4×C251—42no—
154C2×C4×C251C2×D411yescharacteristic
155C2×C4×C251C2411yescharacteristic
156C8×C251—21no—
157C8×C251—21no—
158C22×C251—21no—
159C22×C251—21no—
160C22×C251—21no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251C22×D411yescharacteristic
165C2×D251—21no—
166C2×D251—21no—
167C2×D251—22no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—41no—
172C2×D251—41no—
173C2×D251—84no—
174C2×D251AGL(1,ℤ/8)12yes—
175C4×C251—21no—
176C4×C251—41no—
177C4×C251—41no—
178C4×C251—41no—
179C4×C251C2×#4111yescharacteristic
180C4×C251H(5,𝔽2)11yescharacteristic
181Dic251—21no—
182Dic251—21no—
183Dic251—21no—
184Dic251—41no—
185Dic251—41no—
186C2×C251—41no—
187C2×C251—81no—
188C2×C251#54611yescharacteristic
189C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C251#57711yescharacteristic
191D251—22no—
192D251—42no—
193D251—81no—
194C251#330611yescharacteristic, complemented, Sylow
195#3306—2511nomaximal, Sylow
196#547—2511no—
197#546—2511no—
198#489—2511no—
199#483—2511no—
200#481D25111yescharacteristic, complemented
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—5021no—
212C2×#41—2511no—
213C22×D4—2511no—
214C2×SD16—2511no—
215C2×D8—2511no—
216C2×D8—2511no—
217C2×M16—2511no—
218C22×C8—2511no—
219C4⋊Q8—2511no—
220#124—2511no—
221#123—5021no—
222#123—5021no—
223#123—5021no—
224C22⋊Q8—5021no—
225C22⋊Q8—2511no—
226#121—5021no—
227#121—2511no—
228#121—2511no—
229#121C2×D25111yescharacteristic
230#120—5021no—
231C4×D4—2511no—
232C4×D4—2511no—
233C4×D4—2511no—
234C2×C4⋊C4—2511no—
235C2×C4⋊C4C2×D25111yescharacteristic, complemented
236C2×C22⋊C4—5021no—
237#106—2511no—
238#106C2×D25111yescharacteristic, complemented
239#106C2×D25111yescharacteristic, complemented
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#102C2×D25111yescharacteristic
250#102C2×D25111yescharacteristic
251C22⋊C8—2511no—
252C22⋊C8C2×D25111yescharacteristic, complemented
253#97—2511no—
254C24—5021no—
255#41—10041no—
256#41—10044no—
257#41—5022no—
258C2×Q8—5021no—
259C2×Q8—2511no—
260C2×D4—21no—
261C2×D4—10041no—
262C2×D4—5021no—
263C2×D4—5022no—
264C2×D4—5021no—
265C2×D4—5021no—
266C2×D4—5021no—
267C2×D4—2511no—
268C2×D4—2511no—
269C2×D4—2511no—
270C2×D4C22×D25111yescharacteristic
271C22×C4—21no—
272C22×C4—5021no—
273C22×C4—5021no—
274C22×C4—5021no—
275C22×C4—5021no—
276C22×C4—2511no—
277C22×C4C22×D25111yescharacteristic
278SD16—10041no—
279D8—10041no—
280D8—5022no—
281M16—10041no—
282C2×C8—10041no—
283C2×C8—2511no—
284C2×C8—2511no—
285C2×C8C22×D25111yescharacteristic
286C2×C8C22×D25111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—5021no—
289C4⋊C4—5021no—
290C4⋊C4—5021no—
291C4⋊C4—2511no—
292C4⋊C4—2511no—
293C4⋊C4C22×D25111yescharacteristic
294C4⋊C4C22×D25111yescharacteristic
295C4⋊C4C22×D25111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—5021no—
298C22⋊C4—5021no—
299C22⋊C4—5021no—
300C22⋊C4—5021no—
301C22⋊C4—5021no—
302C22⋊C4—5021no—
303C22⋊C4—5021no—
304C22⋊C4—2511no—
305C22⋊C4—2511no—
306C42—2511no—
307C23—21no—
308C23—20081no—
309C23—10042no—
310C23—10042no—
311C23—5021no—
312C23—5021no—
313C23—2511no—
314C23D4×D25111yescharacteristic
315Q8—10042no—
316Q8—5021no—
317Q8—5021no—
318D4—21no—
319D4—21no—
320D4—81no—
321D4—10042no—
322D4—10042no—
323D4—10042no—
324D4—10042no—
325D4—5022no—
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—42no—
336C2×C4—20081no—
337C2×C4—10042no—
338C2×C4—10042no—
339C2×C4—10042no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4—2511no—
345C2×C4—2511no—
346C2×C4—2511no—
347C2×C4D4×D25111yescharacteristic
348C2×C4C23×D25111yescharacteristic
349C8—21no—
350C8—21no—
351C8—5021no—
352C8—5021no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—20084no—
360C22—10041no—
361C22—10041no—
362C22—10041no—
363C22—10041no—
364C22—10041no—
365C22—5022no—
366C22—5021no—
367C22—5021no—
368C22—2512no—
369C22C2×D4×D25111yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—10041no—
375C4—10041no—
376C4—5021no—
377C4—5021no—
378C4—5021no—
379C4#41×D25111yescharacteristic
380C4#2254876111yescharacteristic
381C2—41no—
382C2—81no—
383C2—20081no—
384C2—10042no—
385C2—5022no—
386C2#3063232911yescharacteristic
387C2AGL(1,ℤ/8)×D25111yescharacteristic
388C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#481×C251C2
socleC22×C251C22×D4
hypercenter#481D251
hypocenterC251#3306
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #30632338C2semidirect
2. #30632334C2semidirect
3. #30632045C2semidirect
4. #30632019C2semidirect
5. #30631976C2semidirect
6. #30631938C2semidirect
7. #30631917C2semidirect
8. #30631901C2semidirect
9. #30631834C2semidirect
10. #30631822C2semidirect
11. #30631798C2semidirect
12. #30631761C2semidirect
13. #30631686C2semidirect
14. #102×D251C2semidirect
15. #481×C251C2semidirect
19. C2×#16860799C22semidirect
22. C2×#16860794C22semidirect
24. #22548714C22semidirect
32. #22548689C22semidirect
36. #22548677C22semidirect
40. #22548646C22semidirect
41. Dic251⋊C8C22semidirect
42. #22548640C22semidirect
43. #22548638C22semidirect
44. #22548638C22semidirect
49. #22548635C22semidirect
51. #22548625C22semidirect
53. C2×C4⋊C4×C251C22semidirect
54. #106×C251C22semidirect
55. #106×C251C22semidirect
58. C22⋊C8×C251C22semidirect
125. D1004#41semidirect
194. C251#3306semidirect
200. #481D251semidirect
235. C2×C4⋊C4C2×D251semidirect
238. #106C2×D251semidirect
239. #106C2×D251semidirect
252. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323382
#306323342
#306320452
#306320192
#306319762
#306319382
#306319172
#306319012
#306318342
#306318222
#306317982
#306317612
#306316862
#102×D2512
#481×C2512
#3306251