Finite Group Databasebeta

← #44559944

Subgroups — #44559944

Order 32128.

520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22548734C211yesmaximal, characteristic, complemented
2C22×C8×D251C211yesmaximal, characteristic, complemented
3#30632197C212yesmaximal, complemented
4#30631929C211yesmaximal, characteristic, complemented
5#30631928C211yesmaximal, characteristic, complemented
6#30631920C211yesmaximal, characteristic, complemented
7#30631915C212yesmaximal, complemented
8#30631914C212yesmaximal, complemented
9#30631724C211yesmaximal, characteristic, complemented
10#30631680C212yesmaximal, complemented
11#466×C251C211yesmaximal, characteristic, complemented
12C2×#16860809C2212yes—
13C22×C4×D251C2211yescharacteristic
14#22548734—24no—
15C2×Dic1004C2211yescharacteristic, complemented
16C2×#16860798C2212yescomplemented
17C22×C251⋊C8C2211yescharacteristic
18C2×C8×D251—24no—
19C2×C8×D251C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21C2×C8×D251C2211yescharacteristic
22C2×C8×D251C2211yescharacteristic
23#22548697—22no—
24#22548695C2212yes—
25#22548694C2212yes—
26#22548686C2212yes—
27#22548683—22no—
28D4×Dic251—22no—
29(C22⋊D251)⋊C4—22no—
30#22548673C2212yes—
31#22548649C2211yescharacteristic, complemented
32#22548646C2212yescomplemented
33#22548640C2212yescomplemented
34#22548638C2212yescomplemented
35#22548636C2212yescomplemented
36C2×D8×C251C2211yescharacteristic, complemented
37C22×C8×C251C2211yescharacteristic
38#121×C251C2212yes—
39#107×C251C2211yescharacteristic, complemented
40#102×C251C2212yescomplemented
41C23×D251D411yescharacteristic
42#16860809—24no—
43#16860809—24no—
44C2×Dic502C2312yes—
45C2×C22⋊D251—22no—
46C2×C22⋊D251—22no—
47C22×Dic251—22no—
48C22×Dic251D411yescharacteristic
49C2×C4×D251—22no—
50C2×C4×D251—22no—
51C2×C4×D251D411yescharacteristic
52C2×C4×D251D411yescharacteristic
53C2×C4×D251C2311yescharacteristic
54C2×C4×D251C2311yescharacteristic
55Dic1004—22no—
56#16860798—24no—
57C2×C251⋊C8—22no—
58C2×C251⋊C8C2311yescharacteristic
59C2×C251⋊C8C2311yescharacteristic
60C8×D251—22no—
61C8×D251—22no—
62C8×D251—22no—
63C8×D251D414yescomplemented
64C4⋊Dic251—22no—
65C4⋊Dic251C2312yes—
66Dic251⋊C4—22no—
67C22⋊Dic251—22no—
68C22⋊Dic251—22no—
69(C2×D251)⋊C4—22no—
70(C2×D251)⋊C4—22no—
71C4×Dic251—22no—
72C2×D4×C251—22no—
73C2×D4×C251C2312yes—
74C22×C4×C251C2311yescharacteristic
75D8×C251—22no—
76C2×C8×C251—22no—
77C2×C8×C251C2311yescharacteristic
78C2×C8×C251C2311yescharacteristic
79C4⋊C4×C251C2312yes—
80C22⋊C4×C251—22no—
81C22×D251—22no—
82C22×D251—22no—
83C22×D251—22no—
84C22×D251C2×D411yescharacteristic
85C22×D251C2×D411yescharacteristic
86Dic502—22no—
87Dic502—22no—
88C22⋊D251—44no—
89C22⋊D251—44no—
90C2×Dic251—22no—
91C2×Dic251—22no—
92C2×Dic251—22no—
93C2×Dic251—44no—
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C4×D251—22no—
97C4×D251—21no—
98C4×D251—21no—
99C4×D251—22no—
100C4×D251C2×D412yes—
101C4×D251#4112yes—
102C251⋊C8—21no—
103C251⋊C8C2×D412yes—
104C23×C251—22no—
105C23×C251C2×D411yescharacteristic
106D4×C251—22no—
107D4×C251—22no—
108D4×C251—44no—
109C2×C4×C251—21no—
110C2×C4×C251—21no—
111C2×C4×C251—22no—
112C2×C4×C251C2×D411yescharacteristic
113C2×C4×C251C2411yescharacteristic
114C8×C251—21no—
115C8×C251C2×D412yes—
116C22×C251—21no—
117C22×C251—21no—
118C22×C251—42no—
119C22×C251—42no—
120C22×C251—42no—
121C22×C251#13512yescomplemented
122C22×C251C22×D411yescharacteristic
123C2×D251—28no—
124C2×D251—21no—
125C2×D251—21no—
126C2×D251—21no—
127C2×D251—21no—
128C2×D251#12112yes—
129C2×D251#13512yescomplemented
130C4×C251—21no—
131C4×C251—42no—
132C4×C251C22×D411yescharacteristic
133C4×C251C2×#4111yescharacteristic
134Dic251—21no—
135Dic251—42no—
136Dic251—42no—
137Dic251#12112yes—
138C2×C251—22no—
139C2×C251—82no—
140C2×C251C2×#12111yescharacteristic
141C2×C251C2×#13511yescharacteristic
142C2×C251C2×#13511yescharacteristic
143D251—22no—
144D251—22no—
145C251#305011yescharacteristic, complemented, Sylow
146#3050—2511nomaximal, Sylow
147C2×#135—2511no—
148C23×C8—2511no—
149#525—2512no—
150#467—2511no—
151#467—2511no—
152#466—2511no—
153#466D25111yescharacteristic, complemented
154#465—2512no—
155#465—2512no—
156#427—2511no—
157#416—2512no—
158C2×#41—2512no—
159C23×C4—2511no—
160#135—5024no—
161C2×Q16—2511no—
162C2×SD16—2512no—
163C2×D8C2×D25111yescharacteristic, complemented
164C22×C8—5024no—
165C22×C8—2511no—
166C22×C8—2511no—
167C22×C8—2511no—
168C22×C8—2511no—
169C22×C8—2511no—
170C22×C8C2×D25111yescharacteristic
171#123—5022no—
172C22⋊Q8—2512no—
173C22⋊Q8—2512no—
174#121—2512no—
175#121C2×D25112yes—
176#120—5022no—
177C4×D4—5022no—
178C4×D4—5022no—
179#117—2512no—
180#107—2511no—
181#107C2×D25111yescharacteristic, complemented
182#106—2512no—
183#103—2512no—
184#103—2512no—
185#102—2512no—
186#102C2×D25112yescomplemented
187C24—2511no—
188#41—5024no—
189#41—5024no—
190C2×Q8—2512no—
191C2×D4—22no—
192C2×D4—5022no—
193C2×D4—5022no—
194C2×D4C22×D25112yes—
195C22×C4—5022no—
196C22×C4—5022no—
197C22×C4—5022no—
198C22×C4—2511no—
199C22×C4—2511no—
200C22×C4—2511no—
201C22×C4—2511no—
202C22×C4—2511no—
203C22×C4C22×D25111yescharacteristic
204Q16—5022no—
205SD16—5024no—
206D8—22no—
207C2×C8—22no—
208C2×C8—5022no—
209C2×C8—5022no—
210C2×C8—5022no—
211C2×C8—5022no—
212C2×C8—2511no—
213C2×C8—2511no—
214C2×C8—2514no—
215C2×C8C22×D25111yescharacteristic
216C2×C8C22×D25111yescharacteristic
217C4⋊C4—5022no—
218C4⋊C4—5022no—
219C4⋊C4—2512no—
220C4⋊C4C22×D25112yes—
221C22⋊C4—22no—
222C22⋊C4—5022no—
223C22⋊C4—5022no—
224C22⋊C4—5022no—
225C22⋊C4—5022no—
226C42—5022no—
227C23—22no—
228C23—5022no—
229C23—5022no—
230C23—5022no—
231C23—2511no—
232C23—2511no—
233C23D4×D25111yescharacteristic
234Q8—5022no—
235Q8—5022no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—10044no—
240D4—10044no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—10044no—
245C2×C4—5022no—
246C2×C4—5022no—
247C2×C4—5022no—
248C2×C4—5022no—
249C2×C4—5022no—
250C2×C4—5021no—
251C2×C4—5021no—
252C2×C4—2511no—
253C2×C4—2511no—
254C2×C4—2512no—
255C2×C4—2512no—
256C2×C4D4×D25111yescharacteristic
257C2×C4C23×D25111yescharacteristic
258C8—21no—
259C8—5021no—
260C8—2512no—
261C8D4×D25112yes—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—5028no—
268C22—5021no—
269C22—5021no—
270C22—5021no—
271C22—5021no—
272C22—2512no—
273C22—2512no—
274C22#2254873412yescomplemented
275C22C2×D4×D25111yescharacteristic
276C4—21no—
277C4—42no—
278C4—10042no—
279C4—10042no—
280C4—5021no—
281C4—2512no—
282C4C2×D4×D25111yescharacteristic
283C4#41×D25111yescharacteristic
284C2—22no—
285C2—82no—
286C2—5022no—
287C2—5022no—
288C2#121×D25111yescharacteristic
289C2#135×D25111yescharacteristic
290C2C2×#2254873411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#466×C251C2
socleC22×C251C22×D4
hypercenter#466D251
hypocenterC251#3050
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#22548734C2semidirect
2. C22×C8×D251C2semidirect
3. #30632197C2semidirect
4. #30631929C2semidirect
5. #30631928C2semidirect
6. #30631920C2semidirect
7. #30631915C2semidirect
8. #30631914C2semidirect
9. #30631724C2semidirect
10. #30631680C2semidirect
11. #466×C251C2semidirect
15. C2×Dic1004C22semidirect
16. C2×#16860798C22semidirect
31. #22548649C22semidirect
32. #22548646C22semidirect
33. #22548640C22semidirect
34. #22548638C22semidirect
35. #22548636C22semidirect
36. C2×D8×C251C22semidirect
39. #107×C251C22semidirect
40. #102×C251C22semidirect
63. C8×D251D4semidirect
121. C22×C251#135semidirect
129. C2×D251#135semidirect
145. C251#3050semidirect
153. #466D251semidirect
163. C2×D8C2×D251semidirect
181. #107C2×D251semidirect
186. #102C2×D251semidirect
274. C22#22548734semidirect

Maximal subgroups

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

subgroupindex
C2×#225487342
C22×C8×D2512
#306321972
#306319292
#306319282
#306319202
#306319152
#306319142
#306317242
#306316802
#466×C2512
#3050251