Finite Group Databasebeta

← #44562077

Subgroups — #44562077

Order 32128.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548737C211yesmaximal, characteristic, complemented
2C22×#16860799C211yesmaximal, characteristic, complemented
3#30632346C211yesmaximal, characteristic, complemented
4#30632333C211yesmaximal, characteristic, complemented
5#30632071C211yesmaximal, characteristic, complemented
6#30631916C211yesmaximal, characteristic, complemented
7#30631916C211yesmaximal, characteristic, complemented
8#30631910C211yesmaximal, characteristic, complemented
9#30631903C211yesmaximal, characteristic, complemented
10#30631851C211yesmaximal, characteristic, complemented
11#30631845C211yesmaximal, characteristic, complemented
12#30631834C211yesmaximal, characteristic, complemented
13#30631823C211yesmaximal, characteristic, complemented
14C4×#16860799C211yesmaximal, characteristic, complemented
15C8×D4×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×#16860810C2211yescharacteristic, complemented
18C22×Dic502C2211yescharacteristic
19C22×D1004C2211yescharacteristic
20#22548737—24no—
21C2×Dic1004C2211yescharacteristic, complemented
22C2×Dic1004C2211yescharacteristic, complemented
23C2×#16860799—24no—
24C2×#16860799C2211yescharacteristic
25C2×#16860799C2211yescharacteristic
26C2×#16860799C2211yescharacteristic
27C2×#16860799C2211yescharacteristic
28C2×#16860799C2211yescharacteristic, complemented
29C2×D2008C2211yescharacteristic
30C2×D2008C2211yescharacteristic
31#22548703C2211yescharacteristic
32#22548703C2211yescharacteristic, complemented
33#22548701—21no—
34#22548700—21no—
35#22548694C2211yescharacteristic
36#22548694C2211yescharacteristic, complemented
37#22548693—21no—
38#22548689C2211yescharacteristic
39#22548689C2211yescharacteristic
40#22548688—21no—
41#22548687—21no—
42#22548683—21no—
43C4×Dic502C2211yescharacteristic, complemented
44C4×D1004C2211yescharacteristic
45(C22⋊D251)⋊C4—21no—
46C22⋊C4×D251—21no—
47#22548647C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548640C2211yescharacteristic, complemented
50#22548640C2211yescharacteristic, complemented
51#22548637C2211yescharacteristic
52#22548637C2211yescharacteristic
53#22548637C2211yescharacteristic, complemented
54C22×C8×C251C2211yescharacteristic
55C22×C8×C251C2211yescharacteristic, complemented
56C4×D4×C251C2211yescharacteristic
57C4⋊C8×C251C2211yescharacteristic, complemented
58C22⋊C8×C251C2211yescharacteristic
59C22⋊C8×C251C2211yescharacteristic, complemented
60C4×C8×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860810—22no—
63#16860810—22no—
64#16860810—41no—
65#16860810—41no—
66C2×Dic502—22no—
67C2×Dic502—21no—
68C2×Dic502—41no—
69C2×Dic502C2311yescharacteristic
70C2×Dic502C2311yescharacteristic
71C2×Dic502C2311yescharacteristic
72C2×D1004—22no—
73C2×D1004—21no—
74C2×D1004—41no—
75C2×D1004C2311yescharacteristic
76C2×D1004C2311yescharacteristic
77C2×D1004C2311yescharacteristic
78C2×C22⋊D251—21no—
79C2×C22⋊D251—21no—
80C2×C22⋊D251—21no—
81C2×C22⋊D251—21no—
82C22×Dic251—21no—
83C2×C4×D251—21no—
84C2×C4×D251—21no—
85C2×C4×D251—21no—
86Dic1004—22no—
87Dic1004—41no—
88#16860799—22no—
89#16860799—22no—
90#16860799—22no—
91#16860799—22no—
92#16860799D414yescomplemented
93D2008—22no—
94D2008—41no—
95C4⋊Dic251C2311yescharacteristic
96C4⋊Dic251C2311yescharacteristic
97Dic251⋊C4—21no—
98Dic251⋊C4—21no—
99Dic251⋊C4—21no—
100C22⋊Dic251—21no—
101C22⋊Dic251—21no—
102(C2×D251)⋊C4—21no—
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4—21no—
107(C2×D251)⋊C4—21no—
108(C2×D251)⋊C4—21no—
109C4×Dic251—21no—
110C2×D4×C251D411yescharacteristic
111C22×C4×C251C2311yescharacteristic
112C22×C4×C251C2311yescharacteristic
113C2×C8×C251—22no—
114C2×C8×C251—22no—
115C2×C8×C251C2311yescharacteristic
116C2×C8×C251C2311yescharacteristic
117C2×C8×C251C2311yescharacteristic
118C2×C8×C251C2311yescharacteristic
119C4⋊C4×C251D411yescharacteristic
120C22⋊C4×C251D411yescharacteristic
121C22⋊C4×C251D411yescharacteristic
122C42×C251C2311yescharacteristic
123C22×D251—21no—
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—42no—
127C22×D251—42no—
128C22×D251—81no—
129Dic502—21no—
130Dic502—21no—
131Dic502—21no—
132Dic502—21no—
133Dic502—21no—
134Dic502—42no—
135Dic502C2×D412yes—
136D1004—21no—
137D1004—21no—
138D1004—21no—
139D1004—21no—
140D1004—21no—
141D1004—42no—
142D1004C2×D412yes—
143C22⋊D251—42no—
144C22⋊D251—42no—
145C22⋊D251—42no—
146C22⋊D251—81no—
147C2×Dic251—21no—
148C2×Dic251—21no—
149C2×Dic251—21no—
150C2×Dic251—21no—
151C2×Dic251—21no—
152C2×Dic251—42no—
153C4×D251—42no—
154C4×D251—42no—
155C4×D251—81no—
156C23×C251C2×D411yescharacteristic
157C23×C251C2×D411yescharacteristic
158D4×C251—22no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251C2×D411yescharacteristic
164C2×C4×C251C2×D411yescharacteristic
165C2×C4×C251C2×D411yescharacteristic
166C2×C4×C251C2×D411yescharacteristic
167C2×C4×C251C2411yescharacteristic
168C8×C251—21no—
169C8×C251—21no—
170C8×C251—21no—
171C8×C251C2×D412yes—
172C22×C251—21no—
173C22×C251—21no—
174C22×C251—21no—
175C22×C251—21no—
176C22×C251—21no—
177C22×C251#13512yescomplemented
178C22×C251C22×D411yescharacteristic
179C2×D251—22no—
180C2×D251—41no—
181C2×D251—41no—
182C2×D251—41no—
183C2×D251—41no—
184C2×D251—41no—
185C2×D251—41no—
186C2×D251—41no—
187C2×D251—41no—
188C2×D251—84no—
189C4×C251—21no—
190C4×C251—21no—
191C4×C251—21no—
192C4×C251—21no—
193C4×C251C22×D411yescharacteristic
194C4×C251H(5,𝔽2)11yescharacteristic
195Dic251—22no—
196Dic251—41no—
197Dic251—41no—
198Dic251—41no—
199Dic251—41no—
200C2×C251—22no—
201C2×C251—41no—
202C2×C251D4211yescharacteristic
203C2×C251C2×#13511yescharacteristic
204C2×C251#57711yescharacteristic
205D251—42no—
206D251—81no—
207D251—81no—
208C251#328211yescharacteristic, complemented, Sylow
209#3282—2511nomaximal, Sylow
210C2×#135—2511no—
211C22×SD16—2511no—
212#548—2511no—
213#546—2511no—
214#495—2511no—
215#465—2511no—
216#465—2511no—
217#464—2511no—
218#463—2511no—
219#452—2511no—
220#451—2511no—
221#449—2511no—
222#447—2511no—
223C4×SD16—2511no—
224C8×D4D25111yescharacteristic, complemented
225C2×#41—2511no—
226C2×#41—2511no—
227C22×Q8—2511no—
228C22×D4—2511no—
229#135—5024no—
230C2×Q16—2511no—
231C2×Q16—2511no—
232C2×SD16—5024no—
233C2×SD16—2511no—
234C2×SD16—2511no—
235C2×SD16—2511no—
236C2×SD16—2511no—
237C2×SD16—2511no—
238C2×D8—2511no—
239C2×D8—2511no—
240C22×C8C2×D25111yescharacteristic
241C22×C8C2×D25111yescharacteristic, complemented
242#124—5021no—
243#124—2511no—
244#124—2511no—
245#123—5021no—
246C22⋊Q8—5021no—
247C22⋊Q8—2511no—
248C22⋊Q8—2511no—
249#121—5021no—
250#121—5021no—
251#121—2511no—
252#121—2511no—
253#120—5021no—
254C4×Q8—2511no—
255C4×D4—5021no—
256C4×D4—2511no—
257C4×D4C2×D25111yescharacteristic
258C2×C22⋊C4—5021no—
259#106—2511no—
260C4⋊C8C2×D25111yescharacteristic, complemented
261#103—2511no—
262#103—2511no—
263#103—2511no—
264#102—2511no—
265#102—2511no—
266#102—2511no—
267C22⋊C8C2×D25111yescharacteristic
268C22⋊C8C2×D25111yescharacteristic, complemented
269C4×C8C2×D25111yescharacteristic, complemented
270C24—5021no—
271#41—10041no—
272#41—10041no—
273#41—5022no—
274#41—5022no—
275C2×Q8—10041no—
276C2×Q8—5022no—
277C2×Q8—5021no—
278C2×Q8—2511no—
279C2×Q8—2511no—
280C2×Q8—2511no—
281C2×D4—10041no—
282C2×D4—5022no—
283C2×D4—5021no—
284C2×D4—5021no—
285C2×D4—5021no—
286C2×D4—5021no—
287C2×D4—5021no—
288C2×D4—2511no—
289C2×D4—2511no—
290C2×D4—2511no—
291C2×D4D100411yescharacteristic
292C22×C4—5021no—
293C22×C4—5021no—
294C22×C4—5021no—
295C22×C4—5021no—
296C22×C4C22×D25111yescharacteristic
297C22×C4C22×D25111yescharacteristic
298Q16—10041no—
299Q16—5022no—
300SD16—5022no—
301SD16—5022no—
302SD16—5022no—
303SD16—5022no—
304SD16—2514no—
305D8—10041no—
306D8—5022no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8C22×D25111yescharacteristic
310C2×C8C22×D25111yescharacteristic
311C2×C8C22×D25111yescharacteristic
312C2×C8C22×D25111yescharacteristic
313C4⋊C4—5021no—
314C4⋊C4—5021no—
315C4⋊C4—5021no—
316C4⋊C4—2511no—
317C4⋊C4—2511no—
318C4⋊C4D100411yescharacteristic
319C22⋊C4—5021no—
320C22⋊C4—5021no—
321C22⋊C4—5021no—
322C22⋊C4—5021no—
323C22⋊C4—5021no—
324C22⋊C4—5021no—
325C22⋊C4—5021no—
326C22⋊C4—5021no—
327C22⋊C4—5021no—
328C22⋊C4D100411yescharacteristic
329C22⋊C4D100411yescharacteristic
330C42—5021no—
331C42C22×D25111yescharacteristic
332C23—20081no—
333C23—10042no—
334C23—10042no—
335C23—5021no—
336C23—5021no—
337C23—5021no—
338C23C2×D100411yescharacteristic
339C23C2×D100411yescharacteristic
340Q8—10042no—
341Q8—5021no—
342Q8—5021no—
343Q8—5021no—
344Q8—5021no—
345Q8—5021no—
346Q8—2512no—
347D4—22no—
348D4—20081no—
349D4—10042no—
350D4—10042no—
351D4—10042no—
352D4—10042no—
353D4—5021no—
354D4—5021no—
355D4—5021no—
356D4—5021no—
357D4—5021no—
358D4—2512no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—20081no—
364C2×C4—10042no—
365C2×C4—10042no—
366C2×C4—10042no—
367C2×C4—5021no—
368C2×C4—5021no—
369C2×C4—5021no—
370C2×C4—5021no—
371C2×C4—5021no—
372C2×C4C2×D100411yescharacteristic
373C2×C4C2×D100411yescharacteristic
374C2×C4C2×D100411yescharacteristic
375C2×C4C2×D100411yescharacteristic
376C2×C4C23×D25111yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D25112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—20084no—
387C22—10041no—
388C22—10041no—
389C22—10041no—
390C22—10041no—
391C22—10041no—
392C22—10041no—
393C22—10041no—
394C22—10041no—
395C22—5022no—
396C22#2254873712yescomplemented
397C22C22×D100411yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—10041no—
403C4—10041no—
404C4—10041no—
405C4—10041no—
406C4—5022no—
407C4C2×D4×D25111yescharacteristic
408C4#2254876111yescharacteristic
409C2—22no—
410C2—41no—
411C2—20081no—
412C2—20081no—
413C2—10042no—
414C2D4×D100411yescharacteristic
415C2C2×#2254873711yescharacteristic
416C2#3063249911yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC8×D4×C251C2
socleC22×C251C22×D4
hypercenterC8×D4D251
hypocenterC251#3282
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#22548737C2semidirect
2. C22×#16860799C2semidirect
3. #30632346C2semidirect
4. #30632333C2semidirect
5. #30632071C2semidirect
6. #30631916C2semidirect
7. #30631916C2semidirect
8. #30631910C2semidirect
9. #30631903C2semidirect
10. #30631851C2semidirect
11. #30631845C2semidirect
12. #30631834C2semidirect
13. #30631823C2semidirect
14. C4×#16860799C2semidirect
15. C8×D4×C251C2semidirect
17. C2×#16860810C22semidirect
21. C2×Dic1004C22semidirect
22. C2×Dic1004C22semidirect
28. C2×#16860799C22semidirect
32. #22548703C22semidirect
36. #22548694C22semidirect
43. C4×Dic502C22semidirect
47. #22548647C22semidirect
48. #22548640C22semidirect
49. #22548640C22semidirect
50. #22548640C22semidirect
53. #22548637C22semidirect
55. C22×C8×C251C22semidirect
57. C4⋊C8×C251C22semidirect
59. C22⋊C8×C251C22semidirect
60. C4×C8×C251C22semidirect
92. #16860799D4semidirect
177. C22×C251#135semidirect
208. C251#3282semidirect
224. C8×D4D251semidirect
241. C22×C8C2×D251semidirect
260. C4⋊C8C2×D251semidirect
268. C22⋊C8C2×D251semidirect
269. C4×C8C2×D251semidirect
396. C22#22548737semidirect

Maximal subgroups

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

subgroupindex
C2×#225487372
C22×#168607992
#306323462
#306323332
#306320712
#306319162
#306319102
#306319032
#306318512
#306318452
#306318342
#306318232
C4×#168607992
C8×D4×C2512
#3282251