Finite Group Databasebeta

← #44558823

Subgroups — #44558823

Order 32128.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#22548760C211yesmaximal, characteristic
2#30632160C212yesmaximal, complemented
3#30631659C214yesmaximal, complemented
4C2×#22548630C211yesmaximal, characteristic, complemented
5C2×#22548629C212yesmaximal, complemented
6#99×D251C214yesmaximal, complemented
7C2×#99×C251C211yesmaximal, characteristic, complemented
8#22548760—24no—
9#22548760—24no—
10C2×#16860810C412yescomplemented
11C2×#16860809C412yescomplemented
12C2×#16860809C2212yes—
13C22×C22⋊D251C412yescomplemented
14C22×C22⋊D251C2211yescharacteristic
15C22×C22⋊D251C2211yescharacteristic
16C2×D4×D251C412yescomplemented
17C2×D4×D251C2212yes—
18C4×C22⋊D251—24no—
19(C22⋊D251)⋊C4—24no—
20#22548671C2214yescomplemented
21C2×C22⋊Dic251C2212yescomplemented
22C22⋊C4×D251C2214yescomplemented
23#22548630C2214yescomplemented
24#22548629C2218yescomplemented
25C22×D4×C251C2211yescharacteristic
26C2×C22⋊C4×C251C2212yescomplemented
27#99×C251C2214yescomplemented
28C23×D251C2×C411yescharacteristic, complemented
29C23×D251C2×C412yescomplemented
30C23×D251C2311yescharacteristic
31#16860810—24no—
32#16860810—24no—
33#16860809—28no—
34#16860809—28no—
35C2×Dic502C2×C412yescomplemented
36C2×D1004C2×C412yescomplemented
37C2×C22⋊D251—24no—
38C2×C22⋊D251—24no—
39C2×C22⋊D251—22no—
40C2×C22⋊D251—22no—
41C2×C22⋊D251C2×C412yes—
42C2×C22⋊D251C2×C412yes—
43C2×C22⋊D251C2×C412yescomplemented
44C2×C22⋊D251C2×C412yescomplemented
45C2×C22⋊D251C2×C412yescomplemented
46C2×C22⋊D251D414yes—
47C2×C22⋊D251D414yes—
48C2×C22⋊D251C2312yes—
49D4×D251—28no—
50D4×D251—28no—
51C22×Dic251—22no—
52C22×Dic251C2×C412yescomplemented
53C22×Dic251C2×C411yescharacteristic, complemented
54C22×Dic251C2311yescharacteristic
55C2×C4×D251—24no—
56C2×C4×D251C2×C412yescomplemented
57C2×C4×D251C2×C412yescomplemented
58Dic251⋊C4—24no—
59C22⋊Dic251—22no—
60C22⋊Dic251C2314yes—
61(C2×D251)⋊C4—24no—
62C4×Dic251—24no—
63C24×C251C2×C411yescharacteristic, complemented
64C24×C251C2311yescharacteristic
65C2×D4×C251—22no—
66C2×D4×C251—22no—
67C2×D4×C251C2×C412yescomplemented
68C2×D4×C251C2312yes—
69C22×C4×C251—22no—
70C22×C4×C251C2×C411yescharacteristic, complemented
71C22⋊C4×C251—22no—
72C22⋊C4×C251C2314yes—
73C22×D251—22no—
74C22×D251—24no—
75C22×D251—81no—
76C22×D251—82no—
77C22×D251—81no—
78C22×D251C22×C412yes—
79C22×D251C22×C412yes—
80C22×D251C2×D414yes—
81Dic502—24no—
82D1004—24no—
83C22⋊D251—22no—
84C22⋊D251—22no—
85C22⋊D251—24no—
86C22⋊D251—24no—
87C22⋊D251—24no—
88C22⋊D251C22⋊C418yes—
89C22⋊D251C22⋊C414yes—
90C22⋊D251C22⋊C414yes—
91C2×Dic251—22no—
92C2×Dic251—24no—
93C2×Dic251—24no—
94C2×Dic251—82no—
95C2×Dic251C22×C412yes—
96C2×Dic251C22×C412yes—
97C2×Dic251C2×D414yes—
98C4×D251—24no—
99C4×D251—24no—
100C4×D251—84no—
101C23×C251—21no—
102C23×C251—21no—
103C23×C251—81no—
104C23×C251—81no—
105C23×C251C22×C412yes—
106C23×C251C22×C411yescharacteristic
107C23×C251C22×C411yescharacteristic
108C23×C251C2×D412yes—
109C23×C251C2×D412yes—
110C23×C251C2411yescharacteristic
111D4×C251—24no—
112D4×C251—24no—
113C2×C4×C251—21no—
114C2×C4×C251—21no—
115C2×C4×C251—24no—
116C2×C4×C251—82no—
117C2×C4×C251C22×C412yes—
118C22×C251—22no—
119C22×C251—42no—
120C22×C251—41no—
121C22×C251—41no—
122C22×C251—42no—
123C22×C251—41no—
124C22×C251—42no—
125C22×C251—81no—
126C22×C251—81no—
127C22×C251C2×C22⋊C411yescharacteristic
128C22×C251C2×C22⋊C411yescharacteristic
129C22×C251C2×C22⋊C412yes—
130C22×C251C23×C411yescharacteristic
131C22×C251C22×D412yes—
132C2×D251—22no—
133C2×D251—44no—
134C2×D251—42no—
135C2×D251—42no—
136C2×D251—84no—
137C2×D251—82no—
138C2×D251C2×C22⋊C414yes—
139C4×C251—22no—
140C4×C251—44no—
141Dic251—22no—
142Dic251—44no—
143Dic251C2×C22⋊C414yes—
144C2×C251—22no—
145C2×C251—21no—
146C2×C251—22no—
147C2×C251—21no—
148C2×C251—21no—
149C2×C251—42no—
150C2×C251C22×C22⋊C411yescharacteristic
151D251—24no—
152D251—42no—
153C251#288311yescharacteristic, complemented, Sylow
154#2883—2511nomaximal, Sylow
155C2×H(5,𝔽2)—2511no—
156#518—2512no—
157#410—2514no—
158C2×#99—2511no—
159C2×#99—2512no—
160C2×#99—2514no—
161C2×#99D25111yescharacteristic, complemented
162H(5,𝔽2)—5024no—
163H(5,𝔽2)—5024no—
164C2×#41—2512no—
165C2×#41—2512no—
166C2×#41—2512no—
167C22×D4—2512no—
168C22×D4—2511no—
169C22×D4—2512no—
170C22×D4—2512no—
171C22×D4—2511no—
172C22×D4C2×D25111yescharacteristic
173C4×D4—5024no—
174C4×D4—5024no—
175#117—2514no—
176C2×C22⋊C4—2512no—
177C2×C22⋊C4—2514no—
178C2×C22⋊C4C2×D25112yescomplemented
179#99—2514no—
180#99—2518no—
181#99C2×D25114yescomplemented
182C24—2512no—
183C24—2511no—
184C24—2511no—
185C24C4×D25111yescharacteristic, complemented
186C24C22×D25111yescharacteristic
187#41—5024no—
188#41—5024no—
189#41—5028no—
190#41—5028no—
191C2×Q8—2512no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—5028no—
195C2×D4—5028no—
196C2×D4—5022no—
197C2×D4—5022no—
198C2×D4—5024no—
199C2×D4—5024no—
200C2×D4—2512no—
201C2×D4—2512no—
202C2×D4—2512no—
203C2×D4—2512no—
204C2×D4—2512no—
205C2×D4—2512no—
206C2×D4—2514no—
207C2×D4—2514no—
208C2×D4—2512no—
209C2×D4C4×D25112yescomplemented
210C2×D4C22×D25112yes—
211C22×C4—22no—
212C22×C4—5024no—
213C22×C4—5022no—
214C22×C4—2511no—
215C22×C4—2512no—
216C22×C4—2512no—
217C22×C4—2512no—
218C22×C4—2511no—
219C22×C4C4×D25111yescharacteristic, complemented
220C4⋊C4—5024no—
221C22⋊C4—22no—
222C22⋊C4—5024no—
223C22⋊C4—5022no—
224C22⋊C4—2514no—
225C22⋊C4C22×D25114yes—
226C42—5024no—
227C23—21no—
228C23—21no—
229C23—81no—
230C23—81no—
231C23—20081no—
232C23—20082no—
233C23—20081no—
234C23—5024no—
235C23—5022no—
236C23—2512no—
237C23—2512no—
238C23—2514no—
239C23C2×C4×D25112yes—
240C23C2×C4×D25111yescharacteristic
241C23C2×C4×D25111yescharacteristic
242C23D4×D25112yes—
243C23D4×D25112yes—
244C23C23×D25111yescharacteristic
245Q8—5024no—
246D4—24no—
247D4—24no—
248D4—5024no—
249D4—5024no—
250D4—5024no—
251D4—5022no—
252D4—5022no—
253D4—5024no—
254D4—2514no—
255D4—2514no—
256D4—2518no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—82no—
261C2×C4—20084no—
262C2×C4—20082no—
263C2×C4—5024no—
264C2×C4—5024no—
265C2×C4—5024no—
266C2×C4—5022no—
267C2×C4—5024no—
268C2×C4—2512no—
269C2×C4—2512no—
270C2×C4—2514no—
271C2×C4C2×C4×D25112yes—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—81no—
280C22—81no—
281C22—20082no—
282C22—20084no—
283C22—10042no—
284C22—10044no—
285C22—10042no—
286C22—5022no—
287C22—2514no—
288C22C22⋊C4×D25111yescharacteristic
289C22C22⋊C4×D25111yescharacteristic
290C22C22⋊C4×D25112yes—
291C22C22×C4×D25111yescharacteristic
292C22C2×D4×D25112yes—
293C4—22no—
294C4—44no—
295C4—10044no—
296C4—5022no—
297C4—2514no—
298C2—22no—
299C2—21no—
300C2—21no—
301C2—21no—
302C2—22no—
303C2—42no—
304C2—10042no—
305C2—5024no—
306C2C2×C22⋊C4×D25111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D251
commutatorC22×C251C23×C4
FrattiniC23C23×D251
FittingC2×#99×C251C2
socleC2×C251C22×C22⋊C4
prime radicalC2×#22548760C2
cosocleC23×C251C24
hypercenterC2×#99D251
hypocenterC251#2883
LCSG ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #30632160C2semidirect
3. #30631659C2semidirect
4. C2×#22548630C2semidirect
5. C2×#22548629C2semidirect
6. #99×D251C2semidirect
7. C2×#99×C251C2semidirect
10. C2×#16860810C4semidirect
11. C2×#16860809C4semidirect
13. C22×C22⋊D251C4semidirect
16. C2×D4×D251C4semidirect
20. #22548671C22semidirect
21. C2×C22⋊Dic251C22semidirect
22. C22⋊C4×D251C22semidirect
23. #22548630C22semidirect
24. #22548629C22semidirect
26. C2×C22⋊C4×C251C22semidirect
27. #99×C251C22semidirect
28. C23×D251C2×C4semidirect
29. C23×D251C2×C4semidirect
35. C2×Dic502C2×C4semidirect
36. C2×D1004C2×C4semidirect
43. C2×C22⋊D251C2×C4semidirect
44. C2×C22⋊D251C2×C4semidirect
45. C2×C22⋊D251C2×C4semidirect
52. C22×Dic251C2×C4semidirect
53. C22×Dic251C2×C4semidirect
56. C2×C4×D251C2×C4semidirect
57. C2×C4×D251C2×C4semidirect
63. C24×C251C2×C4semidirect
67. C2×D4×C251C2×C4semidirect
70. C22×C4×C251C2×C4semidirect
153. C251#2883semidirect
161. C2×#99D251semidirect
178. C2×C22⋊C4C2×D251semidirect
181. #99C2×D251semidirect
185. C24C4×D251semidirect
209. C2×D4C4×D251semidirect
219. C22×C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487602
#306321602
#306316592
C2×#225486302
C2×#225486292
#99×D2512
C2×#99×C2512
#2883251