Finite Group Databasebeta

← #485×D251 #44560253

Subgroups — #485×D251 #44560253

Order 32128.

416 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D251C211yesmaximal, characteristic
2#117×D251C211yesmaximal, characteristic
3#30632031C211yesmaximal, characteristic, complemented
4#30632030C211yesmaximal, characteristic, complemented
5#30632029C211yesmaximal, characteristic, complemented
6#30632028C211yesmaximal, characteristic, complemented
7#30632027C211yesmaximal, characteristic, complemented
8#30632026C211yesmaximal, characteristic, complemented
9#30632025C211yesmaximal, characteristic, complemented
10#107×D251C211yesmaximal, characteristic, complemented
11#106×D251C211yesmaximal, characteristic, complemented
12#103×D251C211yesmaximal, characteristic, complemented
13#103×D251C211yesmaximal, characteristic, complemented
14C22⋊C8×D251C211yesmaximal, characteristic
15#485×C251C211yesmaximal, characteristic, complemented
16C2×Q8×D251C2211yescharacteristic
17C22×C4×D251C2211yescharacteristic
18C2×C8×D251C2211yescharacteristic
19C2×C8×D251C2211yescharacteristic
20#22548695C2211yescharacteristic
21#22548694C2211yescharacteristic
22#22548693—21no—
23#22548692—21no—
24#22548691C2211yescharacteristic
25#22548674C2211yescharacteristic
26#22548673C2211yescharacteristic
27#22548673C2211yescharacteristic
28#22548672—21no—
29#22548671—21no—
30C4⋊C4×D251—21no—
31C4⋊C4×D251C2211yescharacteristic
32C4⋊C4×D251C2211yescharacteristic
33C4⋊C4×D251C2211yescharacteristic
34C22⋊C4×D251—21no—
35C22⋊C4×D251—21no—
36C42×D251—21no—
37#22548649C2211yescharacteristic, complemented
38#22548648C2211yescharacteristic, complemented
39#22548648C2211yescharacteristic, complemented
40#22548647C2211yescharacteristic, complemented
41#22548646C2211yescharacteristic, complemented
42#22548646C2211yescharacteristic, complemented
43#22548640C2211yescharacteristic, complemented
44#22548640C2211yescharacteristic, complemented
45#22548639C2211yescharacteristic, complemented
46#22548639C2211yescharacteristic, complemented
47#22548638C2211yescharacteristic, complemented
48#22548638C2211yescharacteristic, complemented
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51(C2×D251)⋊C8C2211yescharacteristic
52C22⋊Q8×C251C2211yescharacteristic
53#117×C251C2211yescharacteristic
54#107×C251C2211yescharacteristic, complemented
55#106×C251C2211yescharacteristic, complemented
56#103×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58C22⋊C8×C251C2211yescharacteristic
59C23×D251D411yescharacteristic
60C2×Dic502—21no—
61C2×Dic502C2311yescharacteristic
62Q8×D251—22no—
63Q8×D251—22no—
64C22×Dic251D411yescharacteristic
65C2×C4×D251—22no—
66C2×C4×D251—22no—
67C2×C4×D251—21no—
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251D411yescharacteristic
72C2×C4×D251D411yescharacteristic
73C2×C4×D251C2311yescharacteristic
74C2×C4×D251C2311yescharacteristic
75C2×C251⋊C8C2311yescharacteristic
76C2×C251⋊C8C2311yescharacteristic
77C8×D251—22no—
78C8×D251—22no—
79C4⋊Dic251—21no—
80C4⋊Dic251C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88C22⋊Dic251—21no—
89C22⋊Dic251—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94C4×Dic251—21no—
95C4×Dic251—21no—
96C4×Dic251—21no—
97C2×Q8×C251C2311yescharacteristic
98C22×C4×C251C2311yescharacteristic
99C2×C8×C251C2311yescharacteristic
100C2×C8×C251C2311yescharacteristic
101C4⋊C4×C251—21no—
102C4⋊C4×C251C2311yescharacteristic
103C4⋊C4×C251C2311yescharacteristic
104C4⋊C4×C251C2311yescharacteristic
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C42×C251—21no—
108C22×D251—22no—
109C22×D251—22no—
110C22×D251—22no—
111C22×D251C2×D411yescharacteristic
112C22×D251C2×D411yescharacteristic
113Dic502—21no—
114Dic502—21no—
115Dic502—42no—
116C2×Dic251—22no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251C2×D411yescharacteristic
122C2×Dic251C2×D411yescharacteristic
123C4×D251—22no—
124C4×D251—21no—
125C4×D251—21no—
126C4×D251—22no—
127C4×D251—42no—
128C4×D251—42no—
129C4×D251—42no—
130C4×D251—42no—
131C4×D251#4112yes—
132C4×D251#4112yes—
133C251⋊C8—21no—
134C251⋊C8—21no—
135C23×C251C2×D411yescharacteristic
136Q8×C251—21no—
137Q8×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2411yescharacteristic
146C8×C251—21no—
147C8×C251—21no—
148C22×C251—21no—
149C22×C251—21no—
150C22×C251—21no—
151C22×C251C22×D411yescharacteristic
152C2×D251—21no—
153C2×D251—21no—
154C2×D251—21no—
155C2×D251—44no—
156C2×D251#12312yes—
157C2×D251#13512yes—
158C2×D251#13712yes—
159C4×C251—21no—
160C4×C251—41no—
161C4×C251—41no—
162C4×C251—41no—
163C4×C251—41no—
164C4×C251C2×#4111yescharacteristic
165C4×C251C2×#4111yescharacteristic
166Dic251—21no—
167Dic251—41no—
168Dic251—41no—
169Dic251—41no—
170Dic251—41no—
171Dic251#12312yes—
172C2×C251—41no—
173C2×C251C2×#12311yescharacteristic
174C2×C251C2×#13511yescharacteristic
175C2×C251C2×#13711yescharacteristic
176D251—41no—
177D251#48514yesdirect product factor
178C251C2×#48511yescharacteristic, complemented, Sylow
179C2×#485—2511nomaximal, Sylow
180C2×C22⋊Q8—2511no—
181C2×#117—2511no—
182#485—2511no—
183#485—2511no—
184#485—2511no—
185#485—2511no—
186#485—2511no—
187#485—2511no—
188#485—2511no—
189#485D25111yescharacteristic, direct product factor
190C2×#107—2511no—
191C2×#106—2511no—
192C2×#103—2511no—
193C2×#103—2511no—
194C2×C22⋊C8—2511no—
195C22×Q8—2511no—
196C23×C4—2511no—
197C22×C8—2511no—
198C22×C8—2511no—
199C22⋊Q8—5021no—
200C22⋊Q8—5021no—
201C22⋊Q8—2511no—
202C22⋊Q8—2511no—
203C22⋊Q8—2511no—
204C22⋊Q8C2×D25111yescharacteristic
205#117—5021no—
206#117—5021no—
207#117—2511no—
208#117—2511no—
209#117—2511no—
210#117C2×D25111yescharacteristic
211C2×C4⋊C4—5021no—
212C2×C4⋊C4—2511no—
213C2×C4⋊C4—2511no—
214C2×C4⋊C4—2511no—
215C2×C22⋊C4—5021no—
216C2×C22⋊C4—5021no—
217C2×C42—5021no—
218#107—2511no—
219#107—2511no—
220#107—2511no—
221#107C2×D25111yescharacteristic, complemented
222#106—2511no—
223#106—2511no—
224#106—2511no—
225#106C2×D25111yescharacteristic, complemented
226#103—2511no—
227#103—2511no—
228#103—2511no—
229#103—2511no—
230#103—2511no—
231#103—2511no—
232#103C2×D25111yescharacteristic, complemented
233#103C2×D25111yescharacteristic, complemented
234C22⋊C8—2511no—
235C22⋊C8—2511no—
236C22⋊C8—2511no—
237C22⋊C8C2×D25111yescharacteristic
238C24—2511no—
239C2×Q8—5021no—
240C2×Q8—5022no—
241C2×Q8—5022no—
242C2×Q8—2511no—
243C2×Q8C22×D25111yescharacteristic
244C22×C4—5021no—
245C22×C4—5021no—
246C22×C4—5021no—
247C22×C4—5021no—
248C22×C4—5022no—
249C22×C4—5022no—
250C22×C4—2511no—
251C22×C4—2511no—
252C22×C4—2511no—
253C22×C4—2511no—
254C22×C4—2511no—
255C22×C4C22×D25111yescharacteristic
256C2×C8—5022no—
257C2×C8—5022no—
258C2×C8—2511no—
259C2×C8—2511no—
260C2×C8C22×D25111yescharacteristic
261C2×C8C22×D25111yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—5021no—
264C4⋊C4—5021no—
265C4⋊C4—5021no—
266C4⋊C4—5021no—
267C4⋊C4—5021no—
268C4⋊C4—5021no—
269C4⋊C4—2511no—
270C4⋊C4—2511no—
271C4⋊C4—2511no—
272C4⋊C4C22×D25111yescharacteristic
273C4⋊C4C22×D25111yescharacteristic
274C4⋊C4C22×D25111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—5021no—
278C22⋊C4—5021no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C42—21no—
284C42—5021no—
285C42—5021no—
286C42—5021no—
287C23—5022no—
288C23—5022no—
289C23—5022no—
290C23—2511no—
291C23—2511no—
292C23D4×D25111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—10042no—
296Q8—5021no—
297Q8—5021no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—10042no—
305C2×C4—10042no—
306C2×C4—10042no—
307C2×C4—10042no—
308C2×C4—5022no—
309C2×C4—5022no—
310C2×C4—5021no—
311C2×C4—5021no—
312C2×C4—5021no—
313C2×C4—5021no—
314C2×C4—5022no—
315C2×C4—5021no—
316C2×C4—5021no—
317C2×C4—2511no—
318C2×C4—2511no—
319C2×C4—2512no—
320C2×C4—2512no—
321C2×C4D4×D25111yescharacteristic
322C2×C4C23×D25111yescharacteristic
323C8—21no—
324C8—21no—
325C8—5021no—
326C8—5021no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—10044no—
331C22—5021no—
332C22—5021no—
333C22—5021no—
334C22—2512no—
335C22—2512no—
336C22—2512no—
337C22C2×D4×D25111yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—10041no—
344C4—10041no—
345C4—10041no—
346C4—10041no—
347C4—5021no—
348C4—2512no—
349C4#41×D25111yescharacteristic
350C4#41×D25111yescharacteristic
351C2—41no—
352C2—10041no—
353C2—2514no—
354C2#123×D25111yescharacteristic
355C2#135×D25111yescharacteristic
356C2#137×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#485×C251C2
socleC22×C251C22×D4
prime radicalC23×D251D4
hypercenter#485D251
hypocenterC251C2×#485
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #30632031C2semidirect
4. #30632030C2semidirect
5. #30632029C2semidirect
6. #30632028C2semidirect
7. #30632027C2semidirect
8. #30632026C2semidirect
9. #30632025C2semidirect
10. #107×D251C2semidirect
11. #106×D251C2semidirect
12. #103×D251C2semidirect
13. #103×D251C2semidirect
15. #485×C251C2semidirect
37. #22548649C22semidirect
38. #22548648C22semidirect
39. #22548648C22semidirect
40. #22548647C22semidirect
41. #22548646C22semidirect
42. #22548646C22semidirect
43. #22548640C22semidirect
44. #22548640C22semidirect
45. #22548639C22semidirect
46. #22548639C22semidirect
47. #22548638C22semidirect
48. #22548638C22semidirect
54. #107×C251C22semidirect
55. #106×C251C22semidirect
56. #103×C251C22semidirect
57. #103×C251C22semidirect
177. D251#485direct (N × H)
178. C251C2×#485semidirect
189. #485D251direct (N × H)
221. #107C2×D251semidirect
225. #106C2×D251semidirect
232. #103C2×D251semidirect
233. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2512
#117×D2512
#306320312
#306320302
#306320292
#306320282
#306320272
#306320262
#306320252
#107×D2512
#106×D2512
#103×D2512
C22⋊C8×D2512
#485×C2512
C2×#485251