Finite Group Databasebeta

← #44559645

Subgroups — #44559645

Order 32128.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22548761C211yesmaximal, characteristic
2C2×#22548745C211yesmaximal, characteristic, complemented
3C22×#16860799C211yesmaximal, characteristic, complemented
4#30632263C211yesmaximal, characteristic, complemented
5#30631852C214yesmaximal, complemented
6#30631839C214yesmaximal, complemented
7#30631691C211yesmaximal, characteristic, complemented
8#30631653C211yesmaximal, characteristic, complemented
9#416×C251C211yesmaximal, characteristic, complemented
10#22548761—24no—
11#22548761—24no—
12C2×#16860811C2212yes—
13C2×#16860810—21no—
14C2×#16860810C2211yescharacteristic
15C22×Dic502C2211yescharacteristic, complemented
16C22×D1004—21no—
17C22×D1004C2211yescharacteristic
18C2×D4×D251—22no—
19C2×D4×D251C2212yes—
20#22548745—24no—
21C2×#16860800C2212yescomplemented
22C2×#16860799—24no—
23C2×#16860799C2214yescomplemented
24C2×#16860798C2212yescomplemented
25C2×#16860794C2211yescharacteristic, complemented
26#22548715—21no—
27#22548703—21no—
28#22548699—22no—
29#22548695C2214yescomplemented
30#22548635C2214yescomplemented
31(C2×D251)⋊C8C2214yescomplemented
32C2×#41×C251C2211yescharacteristic
33C22×C8×C251C2211yescharacteristic, complemented
34#117×C251C2211yescharacteristic, complemented
35#103×C251C2212yescomplemented
36#102×C251C2212yescomplemented
37C23×D251—22no—
38C23×D251—21no—
39#16860811—24no—
40#16860811—24no—
41#16860810—28no—
42#16860810—22no—
43#16860810—22no—
44C2×Dic502—21no—
45C2×Dic502—22no—
46C2×Dic502—22no—
47C2×Dic502D411yescharacteristic
48C2×Dic502C2312yes—
49C2×D1004—28no—
50C2×D1004—22no—
51C2×D1004—21no—
52C2×D1004—22no—
53C2×D1004D411yescharacteristic
54C2×D1004D414yes—
55C2×D1004C2312yes—
56C2×C22⋊D251—22no—
57C2×C22⋊D251D412yes—
58D4×D251—216no—
59D4×D251—24no—
60D4×D251—24no—
61C22×Dic251—21no—
62C2×C4×D251—22no—
63C2×C4×D251C2312yes—
64#16860800—24no—
65#16860799—28no—
66#16860798—24no—
67#16860794—22no—
68C2×C251⋊C8C2312yes—
69C4⋊Dic251—24no—
70(C2×D251)⋊C4—24no—
71#41×C251—22no—
72#41×C251D414yes—
73C2×Q8×C251C2311yescharacteristic
74C2×D4×C251—21no—
75C2×D4×C251C2311yescharacteristic
76C22×C4×C251—21no—
77C22×C4×C251C2311yescharacteristic
78C2×C8×C251—22no—
79C2×C8×C251C2312yes—
80C4⋊C4×C251C2312yes—
81C22⋊C4×C251—21no—
82C42×C251—21no—
83C22×D251—22no—
84C22×D251—28no—
85C22×D251—24no—
86C22×D251—84no—
87C22×D251—82no—
88C22×D251C2×D412yes—
89Dic502—22no—
90Dic502—22no—
91Dic502—22no—
92Dic502—42no—
93D1004—28no—
94D1004—22no—
95D1004—24no—
96D1004—22no—
97D1004C2×D414yes—
98C22⋊D251—28no—
99C22⋊D251—24no—
100C2×Dic251—22no—
101C2×Dic251—42no—
102C2×Dic251C2×D412yes—
103C4×D251—28no—
104C4×D251—22no—
105C4×D251—22no—
106C251⋊C8—22no—
107C23×C251—21no—
108C23×C251C2×D411yescharacteristic
109Q8×C251—21no—
110Q8×C251C2×D412yes—
111D4×C251—24no—
112D4×C251—21no—
113D4×C251C2×D412yes—
114C2×C4×C251—21no—
115C2×C4×C251—24no—
116C2×C4×C251—22no—
117C2×C4×C251C2×D411yescharacteristic
118C2×C4×C251C2×D412yes—
119C2×C4×C251C2×D412yes—
120C2×C4×C251C2411yescharacteristic
121C8×C251—22no—
122C22×C251—22no—
123C22×C251—22no—
124C22×C251—42no—
125C22×C251#12012yes—
126C22×C251C22×D411yescharacteristic
127C2×D251—22no—
128C2×D251—24no—
129C2×D251—24no—
130C2×D251—44no—
131C2×D251—84no—
132C2×D251—88no—
133C4×C251—22no—
134C4×C251—42no—
135C4×C251#12012yes—
136C4×C251C22×D411yescharacteristic
137C4×C251C22×D411yescharacteristic
138Dic251—22no—
139Dic251—42no—
140C2×C251—22no—
141C2×C251—42no—
142C2×C251C2×#12011yescharacteristic
143C2×C251#57712yes—
144D251—42no—
145D251—44no—
146C251#300911yescharacteristic, complemented, Sylow
147#3009—2511nomaximal, Sylow
148C2×H(5,𝔽2)—2511no—
149C2×#137—2511no—
150C22×SD16—2511no—
151#536—2511no—
152#452—2514no—
153#450—2514no—
154#418—2511no—
155#416D25111yescharacteristic, complemented
156#408—2511no—
157H(5,𝔽2)—5024no—
158H(5,𝔽2)—5024no—
159C2×#41—5021no—
160C2×#41—2512no—
161C2×#41—2511no—
162C2×#41C2×D25111yescharacteristic
163C22×Q8—2511no—
164C22×D4—5021no—
165C22×D4—5022no—
166C22×D4—2512no—
167C22×D4—2511no—
168#137—5024no—
169C2×Q16—2512no—
170C2×SD16—5024no—
171C2×SD16—2512no—
172C2×SD16—2514no—
173C2×M16—2511no—
174C22×C8C2×D25111yescharacteristic, complemented
175C4⋊Q8—5021no—
176#124—5021no—
177#123—5022no—
178C22⋊Q8—2514no—
179#117C2×D25111yescharacteristic, complemented
180#103C2×D25112yescomplemented
181#102—2514no—
182#102C2×D25112yescomplemented
183C22⋊C8—2514no—
184C24—5022no—
185C24—5021no—
186#41—22no—
187#41—5028no—
188#41—5024no—
189#41—5024no—
190#41—5022no—
191#41—5022no—
192#41D100414yes—
193C2×Q8—5022no—
194C2×Q8—5022no—
195C2×Q8—5021no—
196C2×Q8—2512no—
197C2×Q8—2511no—
198C2×Q8C22×D25111yescharacteristic
199C2×D4—21no—
200C2×D4—5028no—
201C2×D4—5022no—
202C2×D4—50216no—
203C2×D4—5024no—
204C2×D4—5024no—
205C2×D4—5022no—
206C2×D4—5021no—
207C2×D4—5022no—
208C2×D4—2512no—
209C2×D4—2511no—
210C2×D4—2514no—
211C2×D4—2512no—
212C2×D4C22×D25111yescharacteristic
213C22×C4—21no—
214C22×C4—5022no—
215C22×C4—5021no—
216C22×C4—2512no—
217C22×C4C22×D25111yescharacteristic
218Q16—5024no—
219SD16—5028no—
220SD16—5024no—
221M16—5022no—
222C2×C8—22no—
223C2×C8—2512no—
224C2×C8C22×D25112yes—
225C4⋊C4—5024no—
226C4⋊C4C22×D25112yes—
227C22⋊C4—21no—
228C22⋊C4—5024no—
229C42—21no—
230C23—21no—
231C23—20082no—
232C23—20084no—
233C23—5028no—
234C23—5024no—
235C23—5022no—
236C23—2512no—
237C23D4×D25111yescharacteristic
238Q8—21no—
239Q8—10042no—
240Q8—5022no—
241Q8—5022no—
242Q8—5022no—
243Q8C2×D100412yes—
244D4—24no—
245D4—21no—
246D4—5028no—
247D4—5028no—
248D4—5024no—
249D4—5022no—
250D4—5022no—
251D4—5024no—
252D4—2514no—
253D4C2×D100412yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—10042no—
258C2×C4—5028no—
259C2×C4—5022no—
260C2×C4—5022no—
261C2×C4—5022no—
262C2×C4—2512no—
263C2×C4D4×D25112yes—
264C2×C4D4×D25111yescharacteristic
265C2×C4C2×D100412yes—
266C2×C4C23×D25111yescharacteristic
267C8—22no—
268C8—5022no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—20084no—
273C22—20088no—
274C22—10044no—
275C22—5024no—
276C22—5022no—
277C22—5024no—
278C22#2254868512yes—
279C22C2×D4×D25111yescharacteristic
280C4—22no—
281C4—42no—
282C4—10042no—
283C4—5022no—
284C4#2254868512yes—
285C4C2×D4×D25111yescharacteristic
286C4C22×D100411yescharacteristic
287C2—22no—
288C2—42no—
289C2—10044no—
290C2—10042no—
291C2C2×#2254868511yescharacteristic
292C2#3063249412yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#416×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548761C2
hypercenter#416D251
hypocenterC251#3009
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#22548745C2semidirect
3. C22×#16860799C2semidirect
4. #30632263C2semidirect
5. #30631852C2semidirect
6. #30631839C2semidirect
7. #30631691C2semidirect
8. #30631653C2semidirect
9. #416×C251C2semidirect
15. C22×Dic502C22semidirect
21. C2×#16860800C22semidirect
23. C2×#16860799C22semidirect
24. C2×#16860798C22semidirect
25. C2×#16860794C22semidirect
29. #22548695C22semidirect
30. #22548635C22semidirect
31. (C2×D251)⋊C8C22semidirect
33. C22×C8×C251C22semidirect
34. #117×C251C22semidirect
35. #103×C251C22semidirect
36. #102×C251C22semidirect
146. C251#3009semidirect
155. #416D251semidirect
174. C22×C8C2×D251semidirect
179. #117C2×D251semidirect
180. #103C2×D251semidirect
182. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
C2×#225487452
C22×#168607992
#306322632
#306318522
#306318392
#306316912
#306316532
#416×C2512
#3009251