Finite Group Databasebeta

← #44560287

Subgroups — #44560287

Order 32128.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#30632197C211yesmaximal, characteristic
2C2×C4⋊C4×D251C211yesmaximal, characteristic, complemented
3#30632023C211yesmaximal, characteristic, complemented
4#30632022C211yesmaximal, characteristic, complemented
5#30632019C212yesmaximal, complemented
6#30632003C211yesmaximal, characteristic, complemented
7#30631999C212yesmaximal, complemented
8#30631726C212yesmaximal, complemented
9#30631685C212yesmaximal, complemented
10#30631649C211yesmaximal, characteristic, complemented
11#481×C251C211yesmaximal, characteristic, complemented
12C2×#16860809C2211yescharacteristic
13C22×C4×D251—21no—
14C22×C4×D251C2211yescharacteristic
15C2×#16860793C2212yes—
16#22548697—21no—
17#22548695C2211yescharacteristic
18#22548694C2211yescharacteristic
19#22548686C2211yescharacteristic
20#22548683—21no—
21D4×Dic251—21no—
22(C22⋊D251)⋊C4—21no—
23#22548673C2211yescharacteristic
24C2×C4⋊Dic251C2211yescharacteristic
25C2×Dic251⋊C4—21no—
26C4⋊C4×D251—21no—
27C4⋊C4×D251—21no—
28C4⋊C4×D251C2212yes—
29C4⋊C4×D251C2212yes—
30#22548649C2212yescomplemented
31#22548648C2212yescomplemented
32#22548646C2212yescomplemented
33#22548640C2212yescomplemented
34#22548638C2212yescomplemented
35#22548636C2212yescomplemented
36C22⋊(C251⋊C8)C2211yescharacteristic
37(C2×D251)⋊C8C2212yes—
38#121×C251C2211yescharacteristic
39C2×C4⋊C4×C251C2211yescharacteristic
40#106×C251C2212yescomplemented
41#102×C251C2212yescomplemented
42C22⋊C8×C251C2211yescharacteristic
43C23×D251D411yescharacteristic
44#16860809—24no—
45C2×Dic502C2311yescharacteristic
46C2×C22⋊D251—21no—
47C2×C22⋊D251—21no—
48C22×Dic251—21no—
49C22×Dic251—21no—
50C22×Dic251D411yescharacteristic
51C2×C4×D251—24no—
52C2×C4×D251—22no—
53C2×C4×D251—22no—
54C2×C4×D251—44no—
55C2×C4×D251D411yescharacteristic
56C2×C4×D251D411yescharacteristic
57C2×C4×D251C2311yescharacteristic
58C2×C4×D251C2311yescharacteristic
59#16860793—24no—
60C2×C251⋊C8C2312yes—
61C4⋊Dic251—21no—
62C4⋊Dic251—21no—
63C4⋊Dic251C2312yes—
64C4⋊Dic251C2311yescharacteristic
65Dic251⋊C4—22no—
66Dic251⋊C4—22no—
67Dic251⋊C4—21no—
68C22⋊Dic251—21no—
69C22⋊Dic251—21no—
70(C2×D251)⋊C4—21no—
71(C2×D251)⋊C4—21no—
72C4×Dic251—21no—
73C2×D4×C251—21no—
74C2×D4×C251C2311yescharacteristic
75C22×C4×C251—21no—
76C22×C4×C251C2311yescharacteristic
77C2×C8×C251C2312yes—
78C4⋊C4×C251—21no—
79C4⋊C4×C251C2312yes—
80C4⋊C4×C251C2311yescharacteristic
81C22⋊C4×C251—21no—
82C22×D251—22no—
83C22×D251—22no—
84C22×D251—22no—
85C22×D251C2×D411yescharacteristic
86C22×D251C2×D411yescharacteristic
87Dic502—22no—
88C22⋊D251—42no—
89C22⋊D251—42no—
90C2×Dic251—22no—
91C2×Dic251—21no—
92C2×Dic251—21no—
93C2×Dic251—22no—
94C2×Dic251—42no—
95C2×Dic251—42no—
96C2×Dic251C2×D411yescharacteristic
97C2×Dic251C2×D411yescharacteristic
98C4×D251—22no—
99C4×D251—22no—
100C4×D251—22no—
101C4×D251—44no—
102C4×D251—44no—
103C4×D251#4114yes—
104C251⋊C8—22no—
105C23×C251—21no—
106C23×C251C2×D411yescharacteristic
107D4×C251—22no—
108D4×C251—42no—
109C2×C4×C251—22no—
110C2×C4×C251—21no—
111C2×C4×C251—22no—
112C2×C4×C251—42no—
113C2×C4×C251C2×D411yescharacteristic
114C2×C4×C251C2411yescharacteristic
115C8×C251—22no—
116C22×C251—21no—
117C22×C251—21no—
118C22×C251—41no—
119C22×C251—41no—
120C22×C251—41no—
121C22×C251#13712yes—
122C22×C251C22×D411yescharacteristic
123C2×D251—28no—
124C2×D251—21no—
125C2×D251—21no—
126C2×D251—21no—
127C2×D251—21no—
128C2×D251#12312yes—
129C2×D251AGL(1,ℤ/8)12yes—
130C4×C251—21no—
131C4×C251—42no—
132C4×C251—41no—
133C4×C251C2×#4112yes—
134Dic251—21no—
135Dic251—41no—
136Dic251—42no—
137Dic251—41no—
138Dic251#12312yes—
139C2×C251—22no—
140C2×C251—81no—
141C2×C251C2×#12311yescharacteristic
142C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C251C2×#13711yescharacteristic
144D251—22no—
145D251—22no—
146C251#309211yescharacteristic, complemented, Sylow
147#3092—2511nomaximal, Sylow
148#525—2511no—
149C22×C4⋊C4—2511no—
150#484—2511no—
151#484—2511no—
152#483—2512no—
153#481—2511no—
154#481D25111yescharacteristic, complemented
155#480—2512no—
156#428—2512no—
157#417—2512no—
158#407—2511no—
159C2×#41—2511no—
160C23×C4—5021no—
161C23×C4—2511no—
162C2×M16—2512no—
163#123—5021no—
164C22⋊Q8—2511no—
165C22⋊Q8—2511no—
166#121—2511no—
167#121C2×D25111yescharacteristic
168#120—5021no—
169C4×D4—5021no—
170C4×D4—5021no—
171#117—2511no—
172C2×C4⋊C4—5021no—
173C2×C4⋊C4—5021no—
174C2×C4⋊C4—5021no—
175C2×C4⋊C4—2511no—
176C2×C4⋊C4—2512no—
177C2×C4⋊C4—2512no—
178C2×C4⋊C4C2×D25111yescharacteristic
179#107—2512no—
180#107—2512no—
181#106—2512no—
182#106C2×D25112yescomplemented
183#103—2512no—
184#103—2512no—
185#102—2512no—
186#102C2×D25112yescomplemented
187C22⋊C8—2511no—
188C22⋊C8—2512no—
189C22⋊C8C2×D25111yescharacteristic
190C24—2511no—
191#41—5024no—
192C2×Q8—2511no—
193C2×D4—21no—
194C2×D4—5021no—
195C2×D4—5021no—
196C2×D4C22×D25111yescharacteristic
197C22×C4—21no—
198C22×C4—10044no—
199C22×C4—5022no—
200C22×C4—5022no—
201C22×C4—5024no—
202C22×C4—5021no—
203C22×C4—5021no—
204C22×C4—2511no—
205C22×C4—2511no—
206C22×C4—2511no—
207C22×C4—2511no—
208C22×C4—2511no—
209C22×C4C22×D25111yescharacteristic
210M16—5024no—
211C2×C8—2512no—
212C2×C8C22×D25112yes—
213C4⋊C4—21no—
214C4⋊C4—5022no—
215C4⋊C4—5022no—
216C4⋊C4—5021no—
217C4⋊C4—5021no—
218C4⋊C4—5021no—
219C4⋊C4—2512no—
220C4⋊C4—2511no—
221C4⋊C4C22×D25112yes—
222C4⋊C4C22×D25111yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—5021no—
225C22⋊C4—5021no—
226C22⋊C4—5021no—
227C22⋊C4—5021no—
228C42—5021no—
229C23—21no—
230C23—5022no—
231C23—5022no—
232C23—5022no—
233C23—2511no—
234C23—2511no—
235C23D4×D25111yescharacteristic
236Q8—5022no—
237D4—22no—
238D4—42no—
239D4—10042no—
240D4—10042no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—10044no—
246C2×C4—10044no—
247C2×C4—10042no—
248C2×C4—10042no—
249C2×C4—5022no—
250C2×C4—5022no—
251C2×C4—5021no—
252C2×C4—5021no—
253C2×C4—5022no—
254C2×C4—5022no—
255C2×C4—5022no—
256C2×C4—2511no—
257C2×C4—2511no—
258C2×C4—2514no—
259C2×C4D4×D25111yescharacteristic
260C2×C4C23×D25111yescharacteristic
261C8—22no—
262C8—5022no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—5028no—
269C22—5021no—
270C22—5021no—
271C22—5021no—
272C22—5021no—
273C22—2512no—
274C22—2512no—
275C22#2254874412yes—
276C22C2×D4×D25111yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—10042no—
281C4—10041no—
282C4—10041no—
283C4—5021no—
284C4—2512no—
285C4#41×D25112yes—
286C2—22no—
287C2—81no—
288C2—5022no—
289C2—5022no—
290C2#123×D25111yescharacteristic
291C2AGL(1,ℤ/8)×D25111yescharacteristic
292C2C2×#2254874411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#481×C251C2
socleC22×C251C22×D4
prime radical#30632197C2
hypercenter#481D251
hypocenterC251#3092
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
2. C2×C4⋊C4×D251C2semidirect
3. #30632023C2semidirect
4. #30632022C2semidirect
5. #30632019C2semidirect
6. #30632003C2semidirect
7. #30631999C2semidirect
8. #30631726C2semidirect
9. #30631685C2semidirect
10. #30631649C2semidirect
11. #481×C251C2semidirect
30. #22548649C22semidirect
31. #22548648C22semidirect
32. #22548646C22semidirect
33. #22548640C22semidirect
34. #22548638C22semidirect
35. #22548636C22semidirect
40. #106×C251C22semidirect
41. #102×C251C22semidirect
146. C251#3092semidirect
154. #481D251semidirect
182. #106C2×D251semidirect
186. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306321972
C2×C4⋊C4×D2512
#306320232
#306320222
#306320192
#306320032
#306319992
#306317262
#306316852
#306316492
#481×C2512
#3092251