Finite Group Databasebeta

← #44560046

Subgroups — #44560046

Order 32128.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548743C211yesmaximal, characteristic, complemented
2C2×#22548723C211yesmaximal, characteristic, complemented
3#30632268C211yesmaximal, characteristic, complemented
4#30632256C211yesmaximal, characteristic, complemented
5#30631941C212yesmaximal, complemented
6#30631934C212yesmaximal, complemented
7#30631924C212yesmaximal, complemented
8#30631917C212yesmaximal, complemented
9#30631681C211yesmaximal, characteristic, complemented
10C2×#22548635C211yesmaximal, characteristic, complemented
11#428×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C2×#16860810C2211yescharacteristic
14C22×D1004C2211yescharacteristic
15#22548743—24no—
16C2×#16860799C2212yes—
17C2×D2008C2212yes—
18#22548723—24no—
19C2×#16860793C2212yescomplemented
20C22×C251⋊C8C2211yescharacteristic
21C2×C8×D251C2212yescomplemented
22#22548712—21no—
23#22548703—21no—
24#22548695—21no—
25#22548695C2212yescomplemented
26#22548690—21no—
27#22548690C2212yes—
28#22548690C2212yes—
29#22548690C2212yescomplemented
30#22548689—22no—
31#22548685—22no—
32#22548638C2211yescharacteristic, complemented
33#22548638C2211yescharacteristic, complemented
34#22548635C2212yes—
35#22548635C2212yes—
36#22548635C2211yescharacteristic, complemented
37#22548635C2211yescharacteristic, complemented
38C2×M16×C251C2211yescharacteristic
39#117×C251C2211yescharacteristic
40C2×C4⋊C4×C251C2211yescharacteristic
41#107×C251C2212yescomplemented
42#106×C251C2212yescomplemented
43C23×D251—21no—
44#16860810—22no—
45#16860810—22no—
46#16860810—22no—
47#16860810—22no—
48C2×Dic502D411yescharacteristic
49C2×Dic502C2311yescharacteristic
50C2×D1004—22no—
51C2×D1004—22no—
52C2×D1004—22no—
53C2×D1004—22no—
54C2×D1004—21no—
55C2×D1004—22no—
56C2×D1004D411yescharacteristic
57C2×D1004C2312yes—
58C2×D1004C2311yescharacteristic
59C2×C22⋊D251—21no—
60C2×C22⋊D251D412yes—
61C2×C4×D251—21no—
62C2×C4×D251C2312yes—
63#16860799—24no—
64D2008—24no—
65#16860793—24no—
66C2×C251⋊C8—22no—
67C2×C251⋊C8C2311yescharacteristic
68C2×C251⋊C8C2311yescharacteristic
69C8×D251—24no—
70C4⋊Dic251—22no—
71(C2×D251)⋊C4—22no—
72(C2×D251)⋊C4—22no—
73(C2×D251)⋊C4—22no—
74(C2×D251)⋊C4—22no—
75C22×C4×C251—21no—
76C22×C4×C251C2311yescharacteristic
77M16×C251D414yescomplemented
78C2×C8×C251C2312yes—
79C4⋊C4×C251—21no—
80C4⋊C4×C251C2311yescharacteristic
81C4⋊C4×C251C2312yes—
82C4⋊C4×C251C2311yescharacteristic
83C22⋊C4×C251—21no—
84C42×C251—21no—
85C22×D251—21no—
86C22×D251—22no—
87C22×D251—42no—
88C22×D251—81no—
89C22×D251—81no—
90C22×D251C2×D412yes—
91Dic502—22no—
92Dic502—21no—
93Dic502—21no—
94D1004—22no—
95D1004—21no—
96D1004—22no—
97D1004—22no—
98D1004—21no—
99D1004—42no—
100D1004—82no—
101D1004—82no—
102D1004—82no—
103C22⋊D251—24no—
104C22⋊D251—42no—
105C2×Dic251—21no—
106C2×Dic251C2×D412yes—
107C4×D251—22no—
108C4×D251—22no—
109C4×D251—42no—
110C251⋊C8—21no—
111C251⋊C8—21no—
112C23×C251C2×D411yescharacteristic
113C2×C4×C251—21no—
114C2×C4×C251—21no—
115C2×C4×C251—22no—
116C2×C4×C251—81no—
117C2×C4×C251C2×D411yescharacteristic
118C2×C4×C251C2×D412yes—
119C2×C4×C251#4112yes—
120C2×C4×C251C2411yescharacteristic
121C8×C251C2×D414yes—
122C22×C251—21no—
123C22×C251—21no—
124C22×C251#12112yes—
125C22×C251C22×D411yescharacteristic
126C2×D251—22no—
127C2×D251—22no—
128C2×D251—22no—
129C2×D251—41no—
130C2×D251—42no—
131C2×D251—41no—
132C2×D251—42no—
133C2×D251—42no—
134C2×D251—41no—
135C2×D251—84no—
136C4×C251—42no—
137C4×C251—41no—
138C4×C251—41no—
139C4×C251#12112yes—
140C4×C251C22×D411yescharacteristic
141C4×C251C2×#4111yescharacteristic
142Dic251—22no—
143Dic251—41no—
144C2×C251—22no—
145C2×C251C2×#12111yescharacteristic
146C2×C251#57611yescharacteristic
147C2×C251#57711yescharacteristic
148D251—42no—
149D251—81no—
150D251—82no—
151C251#306211yescharacteristic, complemented, Sylow
152#3062—2511nomaximal, Sylow
153C2×AGL(1,ℤ/8)—2511no—
154C2×#131—2511no—
155#537—2511no—
156#535—2511no—
157#469—2512no—
158#468—2512no—
159#466—2512no—
160#465—2512no—
161#428D25111yescharacteristic, complemented
162#416—2511no—
163C2×#102—2511no—
164C2×#41—2511no—
165C2×#41—2511no—
166C22×D4—2511no—
167AGL(1,ℤ/8)—5024no—
168C2×SD16—2512no—
169C2×D8—2512no—
170#131—5024no—
171C2×M16—2512no—
172C2×M16C2×D25111yescharacteristic
173C22×C8—2511no—
174C22×C8—2512no—
175#127—5021no—
176#124—5021no—
177C22⋊Q8—5021no—
178C22⋊Q8—2512no—
179#121—5021no—
180#121—5022no—
181#121—2512no—
182#121—2512no—
183#121—2512no—
184#120—5022no—
185#117C2×D25111yescharacteristic
186C2×C4⋊C4C2×D25111yescharacteristic
187#107C2×D25112yescomplemented
188#106C2×D25112yescomplemented
189#103—2511no—
190#103—2511no—
191#102—2512no—
192#102—2512no—
193#102—2511no—
194#102—2511no—
195C24—5021no—
196#41—5022no—
197#41—5022no—
198#41—5022no—
199#41—5022no—
200C2×Q8—2511no—
201C2×Q8—2511no—
202C2×D4—5022no—
203C2×D4—5022no—
204C2×D4—5022no—
205C2×D4—5021no—
206C2×D4—5021no—
207C2×D4—5022no—
208C2×D4—5022no—
209C2×D4—2512no—
210C2×D4—2511no—
211C2×D4—2511no—
212C2×D4—2512no—
213C22×C4—21no—
214C22×C4—5021no—
215C22×C4—2512no—
216C22×C4C22×D25111yescharacteristic
217SD16—5024no—
218D8—5024no—
219M16—5024no—
220M16D100414yescomplemented
221C2×C8—5022no—
222C2×C8—5024no—
223C2×C8—2511no—
224C2×C8—2511no—
225C2×C8C22×D25112yes—
226C4⋊C4—21no—
227C4⋊C4—5022no—
228C4⋊C4C22×D25112yes—
229C4⋊C4C22×D25111yescharacteristic
230C4⋊C4C22×D25111yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—5022no—
233C22⋊C4—5022no—
234C22⋊C4—5022no—
235C22⋊C4—5022no—
236C42—21no—
237C23—20081no—
238C23—20081no—
239C23—10042no—
240C23—5021no—
241C23—5022no—
242C23—2512no—
243C23D4×D25111yescharacteristic
244Q8—5021no—
245Q8—5021no—
246Q8—5022no—
247D4—20082no—
248D4—20082no—
249D4—20082no—
250D4—10042no—
251D4—10042no—
252D4—5024no—
253D4—5021no—
254D4—5022no—
255D4—5022no—
256D4—5021no—
257D4—5022no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—81no—
262C2×C4—10042no—
263C2×C4—5021no—
264C2×C4—5022no—
265C2×C4—5022no—
266C2×C4—2512no—
267C2×C4D4×D25111yescharacteristic
268C2×C4C2×D100412yes—
269C2×C4#1686081112yes—
270C2×C4C23×D25111yescharacteristic
271C8—5021no—
272C8—5021no—
273C8C2×D100414yes—
274C22—21no—
275C22—21no—
276C22—20084no—
277C22—10041no—
278C22—10042no—
279C22—10042no—
280C22—10041no—
281C22—10041no—
282C22—10042no—
283C22—5022no—
284C22—5022no—
285C22—5022no—
286C22#2254869012yes—
287C22C2×D4×D25111yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—10041no—
292C4—5022no—
293C4#2254869012yes—
294C4C22×D100411yescharacteristic
295C4C2×#1686081111yescharacteristic
296C2—22no—
297C2—20082no—
298C2—20081no—
299C2—10042no—
300C2C2×#2254869011yescharacteristic
301C2#3063249111yescharacteristic
302C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#428×C251C2
socleC22×C251C22×D4
hypercenter#428D251
hypocenterC251#3062
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. C2×#22548723C2semidirect
3. #30632268C2semidirect
4. #30632256C2semidirect
5. #30631941C2semidirect
6. #30631934C2semidirect
7. #30631924C2semidirect
8. #30631917C2semidirect
9. #30631681C2semidirect
10. C2×#22548635C2semidirect
11. #428×C251C2semidirect
19. C2×#16860793C22semidirect
21. C2×C8×D251C22semidirect
25. #22548695C22semidirect
29. #22548690C22semidirect
32. #22548638C22semidirect
33. #22548638C22semidirect
36. #22548635C22semidirect
37. #22548635C22semidirect
41. #107×C251C22semidirect
42. #106×C251C22semidirect
77. M16×C251D4semidirect
151. C251#3062semidirect
161. #428D251semidirect
187. #107C2×D251semidirect
188. #106C2×D251semidirect
220. M16D1004semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C2×#225487232
#306322682
#306322562
#306319412
#306319342
#306319242
#306319172
#306316812
C2×#225486352
#428×C2512
#3062251