Finite Group Databasebeta

← #2418×C251 #44547005

Subgroups — #2418×C251 #44547005

Order 32128.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#123×C251C211yesmaximal, characteristic, complemented
2C2×#120×C251C211yesmaximal, characteristic
3C2×C4×D4×C251C211yesmaximal, characteristic, complemented
4C22×C22⋊C4×C251C211yesmaximal, characteristic, complemented
5#399×C251C211yesmaximal, characteristic, complemented
6#397×C251C211yesmaximal, characteristic, complemented
7#394×C251C211yesmaximal, characteristic, complemented
8#394×C251C211yesmaximal, characteristic, complemented
9#388×C251C211yesmaximal, characteristic, complemented
10#388×C251C211yesmaximal, characteristic, complemented
11#386×C251C211yesmaximal, characteristic, complemented
12#386×C251C211yesmaximal, characteristic, complemented
13#385×C251C211yesmaximal, characteristic, complemented
14#380×C251C211yesmaximal, characteristic, complemented
15#379×C251C211yesmaximal, characteristic, complemented
16C25×C251C2211yescharacteristic
17C22×D4×C251C2211yescharacteristic
18C22×D4×C251C2211yescharacteristic
19C22×D4×C251C2211yescharacteristic
20C23×C4×C251C2211yescharacteristic
21C23×C4×C251C2211yescharacteristic
22C23×C4×C251C2211yescharacteristic, complemented
23#123×C251—24no—
24#120×C251—24no—
25C4×D4×C251—24no—
26C2×C4⋊C4×C251C2211yescharacteristic, complemented
27C2×C4⋊C4×C251C2211yescharacteristic, complemented
28C2×C4⋊C4×C251C2211yescharacteristic, complemented
29C2×C4⋊C4×C251C2211yescharacteristic, complemented
30C2×C4⋊C4×C251C2211yescharacteristic, complemented
31C2×C22⋊C4×C251—24no—
32C2×C22⋊C4×C251C2211yescharacteristic
33C2×C22⋊C4×C251C2211yescharacteristic
34C2×C22⋊C4×C251C2211yescharacteristic
35C2×C22⋊C4×C251C2211yescharacteristic
36C2×C22⋊C4×C251C2211yescharacteristic
37C2×C22⋊C4×C251C2211yescharacteristic
38C2×C22⋊C4×C251C2211yescharacteristic
39C2×C22⋊C4×C251C2211yescharacteristic
40C2×C22⋊C4×C251C2211yescharacteristic
41C2×C22⋊C4×C251C2211yescharacteristic
42C2×C22⋊C4×C251C2211yescharacteristic
43C2×C22⋊C4×C251C2211yescharacteristic
44C2×C22⋊C4×C251C2211yescharacteristic
45C2×C22⋊C4×C251C2211yescharacteristic, complemented
46C2×C22⋊C4×C251C2211yescharacteristic, complemented
47C2×C22⋊C4×C251C2211yescharacteristic, complemented
48C2×C42×C251C2211yescharacteristic, complemented
49#95×C251C2211yescharacteristic, complemented
50#95×C251C2211yescharacteristic, complemented
51#95×C251C2211yescharacteristic, complemented
52#95×C251C2211yescharacteristic, complemented
53#95×C251C2211yescharacteristic, complemented
54#95×C251C2211yescharacteristic, complemented
55C24×C251—22no—
56C24×C251—22no—
57C24×C251—22no—
58C24×C251—22no—
59C24×C251—22no—
60C24×C251—41no—
61C24×C251—41no—
62C24×C251C2311yescharacteristic
63C24×C251C2311yescharacteristic
64C24×C251C2311yescharacteristic
65C24×C251C2311yescharacteristic
66C2×D4×C251—22no—
67C2×D4×C251—22no—
68C2×D4×C251—22no—
69C2×D4×C251—22no—
70C2×D4×C251—22no—
71C2×D4×C251—22no—
72C2×D4×C251—41no—
73C2×D4×C251—41no—
74C2×D4×C251D414yes—
75C22×C4×C251—22no—
76C22×C4×C251—22no—
77C22×C4×C251—22no—
78C22×C4×C251—22no—
79C22×C4×C251—22no—
80C22×C4×C251—22no—
81C22×C4×C251—41no—
82C22×C4×C251—41no—
83C22×C4×C251—41no—
84C22×C4×C251C2311yescharacteristic
85C22×C4×C251C2311yescharacteristic
86C22×C4×C251C2311yescharacteristic
87C22×C4×C251C2311yescharacteristic
88C22×C4×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C22×C4×C251C2311yescharacteristic
95C4⋊C4×C251—22no—
96C4⋊C4×C251—22no—
97C4⋊C4×C251—22no—
98C4⋊C4×C251—41no—
99C4⋊C4×C251—41no—
100C22⋊C4×C251—22no—
101C22⋊C4×C251—22no—
102C22⋊C4×C251—22no—
103C22⋊C4×C251—22no—
104C22⋊C4×C251—22no—
105C22⋊C4×C251—22no—
106C22⋊C4×C251—22no—
107C22⋊C4×C251—22no—
108C22⋊C4×C251—22no—
109C22⋊C4×C251—22no—
110C22⋊C4×C251—22no—
111C22⋊C4×C251—22no—
112C22⋊C4×C251—41no—
113C22⋊C4×C251—41no—
114C22⋊C4×C251—41no—
115C22⋊C4×C251D414yescomplemented
116C42×C251—22no—
117C23×C251—21no—
118C23×C251—21no—
119C23×C251—21no—
120C23×C251—21no—
121C23×C251—28no—
122C23×C251—21no—
123C23×C251—21no—
124C23×C251—21no—
125C23×C251—21no—
126C23×C251—21no—
127C23×C251—21no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—21no—
134C23×C251—21no—
135C23×C251—21no—
136C23×C251—21no—
137C23×C251—21no—
138C23×C251—21no—
139C23×C251—21no—
140C23×C251—44no—
141C23×C251—44no—
142C23×C251—44no—
143C23×C251—44no—
144C23×C251—44no—
145C23×C251—44no—
146C23×C251C2×D412yes—
147C23×C251C2×D412yes—
148C23×C251C2×D412yes—
149C23×C251#4112yes—
150C23×C251#4112yes—
151C23×C251#4112yes—
152C23×C251C2411yescharacteristic
153D4×C251—28no—
154D4×C251—44no—
155D4×C251—44no—
156C2×C4×C251—21no—
157C2×C4×C251—21no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—21no—
171C2×C4×C251—21no—
172C2×C4×C251—21no—
173C2×C4×C251—21no—
174C2×C4×C251—21no—
175C2×C4×C251—21no—
176C2×C4×C251—21no—
177C2×C4×C251—21no—
178C2×C4×C251—21no—
179C2×C4×C251—21no—
180C2×C4×C251—21no—
181C2×C4×C251—21no—
182C2×C4×C251—21no—
183C2×C4×C251—21no—
184C2×C4×C251—21no—
185C2×C4×C251—21no—
186C2×C4×C251—44no—
187C2×C4×C251—44no—
188C2×C4×C251—44no—
189C2×C4×C251C2×D412yes—
190C2×C4×C251C2×D412yes—
191C2×C4×C251C2×D412yes—
192C22×C251—22no—
193C22×C251—22no—
194C22×C251—22no—
195C22×C251—22no—
196C22×C251—22no—
197C22×C251—22no—
198C22×C251—22no—
199C22×C251—22no—
200C22×C251—22no—
201C22×C251—22no—
202C22×C251—22no—
203C22×C251—22no—
204C22×C251—22no—
205C22×C251—22no—
206C22×C251—22no—
207C22×C251—416no—
208C22×C251—41no—
209C22×C251—41no—
210C22×C251—41no—
211C22×C251—41no—
212C22×C251—41no—
213C22×C251—41no—
214C22×C251—41no—
215C22×C251—41no—
216C22×C251—41no—
217C22×C251—41no—
218C22×C251—41no—
219C22×C251—41no—
220C22×C251#12314yescomplemented
221C22×C251C22×D411yescharacteristic
222C22×C251C22×D411yescharacteristic
223C22×C251C2×#4111yescharacteristic
224C22×C251C2×#4111yescharacteristic
225C22×C251C2×#4111yescharacteristic
226C22×C251H(5,𝔽2)11yescharacteristic
227C22×C251H(5,𝔽2)11yescharacteristic
228C4×C251—22no—
229C4×C251—22no—
230C4×C251—22no—
231C4×C251—41no—
232C4×C251—41no—
233C4×C251—41no—
234C4×C251—41no—
235C4×C251—41no—
236C4×C251—41no—
237C4×C251—41no—
238C4×C251—41no—
239C2×C251—24no—
240C2×C251—42no—
241C2×C251—42no—
242C2×C251—42no—
243C2×C251C2×#12311yescharacteristic
244C2×C251#52511yescharacteristic
245C2×C251#53811yescharacteristic
246C2×C251D4211yescharacteristic
247C2×C251#54611yescharacteristic
248C2×C251#54611yescharacteristic
249C2×C251#55111yescharacteristic
250C251#241811yescharacteristic, direct product factor, Sylow
251#2418C25111yesmaximal, characteristic, direct product factor, Sylow
252C2×#123C2×C25111yescharacteristic, complemented
253C2×#120C2×C25111yescharacteristic
254C2×C4×D4C2×C25111yescharacteristic, complemented
255C22×C22⋊C4C2×C25111yescharacteristic, complemented
256#399C2×C25111yescharacteristic, complemented
257#397C2×C25111yescharacteristic, complemented
258#394C2×C25111yescharacteristic, complemented
259#394C2×C25111yescharacteristic, complemented
260#388C2×C25111yescharacteristic, complemented
261#388C2×C25111yescharacteristic, complemented
262#386C2×C25111yescharacteristic, complemented
263#386C2×C25111yescharacteristic, complemented
264#385C2×C25111yescharacteristic, complemented
265#380C2×C25111yescharacteristic, complemented
266#379C2×C25111yescharacteristic, complemented
267C25C22×C25111yescharacteristic
268C22×D4C22×C25111yescharacteristic
269C22×D4C22×C25111yescharacteristic
270C22×D4C22×C25111yescharacteristic
271C23×C4C22×C25111yescharacteristic
272C23×C4C22×C25111yescharacteristic
273C23×C4C22×C25111yescharacteristic, complemented
274#123—24no—
275#120—24no—
276C4×D4—24no—
277C2×C4⋊C4C22×C25111yescharacteristic, complemented
278C2×C4⋊C4C22×C25111yescharacteristic, complemented
279C2×C4⋊C4C22×C25111yescharacteristic, complemented
280C2×C4⋊C4C22×C25111yescharacteristic, complemented
281C2×C4⋊C4C22×C25111yescharacteristic, complemented
282C2×C22⋊C4—24no—
283C2×C22⋊C4C22×C25111yescharacteristic
284C2×C22⋊C4C22×C25111yescharacteristic
285C2×C22⋊C4C22×C25111yescharacteristic
286C2×C22⋊C4C22×C25111yescharacteristic
287C2×C22⋊C4C22×C25111yescharacteristic
288C2×C22⋊C4C22×C25111yescharacteristic
289C2×C22⋊C4C22×C25111yescharacteristic
290C2×C22⋊C4C22×C25111yescharacteristic
291C2×C22⋊C4C22×C25111yescharacteristic
292C2×C22⋊C4C22×C25111yescharacteristic
293C2×C22⋊C4C22×C25111yescharacteristic
294C2×C22⋊C4C22×C25111yescharacteristic
295C2×C22⋊C4C22×C25111yescharacteristic
296C2×C22⋊C4C22×C25111yescharacteristic, complemented
297C2×C22⋊C4C22×C25111yescharacteristic, complemented
298C2×C22⋊C4C22×C25111yescharacteristic, complemented
299C2×C42C22×C25111yescharacteristic, complemented
300#95C22×C25111yescharacteristic, complemented
301#95C22×C25111yescharacteristic, complemented
302#95C22×C25111yescharacteristic, complemented
303#95C22×C25111yescharacteristic, complemented
304#95C22×C25111yescharacteristic, complemented
305#95C22×C25111yescharacteristic, complemented
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—41no—
312C24—41no—
313C24C23×C25111yescharacteristic
314C24C23×C25111yescharacteristic
315C24C23×C25111yescharacteristic
316C24C23×C25111yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4D4×C25114yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4C23×C25111yescharacteristic
336C22×C4C23×C25111yescharacteristic
337C22×C4C23×C25111yescharacteristic
338C22×C4C23×C25111yescharacteristic
339C22×C4C23×C25111yescharacteristic
340C22×C4C23×C25111yescharacteristic
341C22×C4C23×C25111yescharacteristic
342C22×C4C23×C25111yescharacteristic
343C22×C4C23×C25111yescharacteristic
344C22×C4C23×C25111yescharacteristic
345C22×C4C23×C25111yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4D4×C25114yescomplemented
367C42—22no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—28no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23C2×D4×C25112yes—
398C23C2×D4×C25112yes—
399C23C2×D4×C25112yes—
400C23#41×C25112yes—
401C23#41×C25112yes—
402C23#41×C25112yes—
403C23C24×C25111yescharacteristic
404D4—28no—
405D4—44no—
406D4—44no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4C2×D4×C25112yes—
441C2×C4C2×D4×C25112yes—
442C2×C4C2×D4×C25112yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—416no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22#123×C25114yescomplemented
472C22C22×D4×C25111yescharacteristic
473C22C22×D4×C25111yescharacteristic
474C22C2×#41×C25111yescharacteristic
475C22C2×#41×C25111yescharacteristic
476C22C2×#41×C25111yescharacteristic
477C22H(5,𝔽2)×C25111yescharacteristic
478C22H(5,𝔽2)×C25111yescharacteristic
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—41no—
490C2—24no—
491C2—42no—
492C2—42no—
493C2—42no—
494C2C2×#123×C25111yescharacteristic
495C2#525×C25111yescharacteristic
496C2#538×C25111yescharacteristic
497C2D42×C25111yescharacteristic
498C2#546×C25111yescharacteristic
499C2#546×C25111yescharacteristic
500C2#551×C25111yescharacteristic
subgroupG/H
center, socleC23×C251C24
commutator, Frattini, cosocleC23C24×C251
prime radicalC2×#120×C251C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#123×C251C2semidirect
3. C2×C4×D4×C251C2semidirect
4. C22×C22⋊C4×C251C2semidirect
5. #399×C251C2semidirect
6. #397×C251C2semidirect
7. #394×C251C2semidirect
8. #394×C251C2semidirect
9. #388×C251C2semidirect
10. #388×C251C2semidirect
11. #386×C251C2semidirect
12. #386×C251C2semidirect
13. #385×C251C2semidirect
14. #380×C251C2semidirect
15. #379×C251C2semidirect
22. C23×C4×C251C22semidirect
26. C2×C4⋊C4×C251C22semidirect
27. C2×C4⋊C4×C251C22semidirect
28. C2×C4⋊C4×C251C22semidirect
29. C2×C4⋊C4×C251C22semidirect
30. C2×C4⋊C4×C251C22semidirect
45. C2×C22⋊C4×C251C22semidirect
46. C2×C22⋊C4×C251C22semidirect
47. C2×C22⋊C4×C251C22semidirect
48. C2×C42×C251C22semidirect
49. #95×C251C22semidirect
50. #95×C251C22semidirect
51. #95×C251C22semidirect
52. #95×C251C22semidirect
53. #95×C251C22semidirect
54. #95×C251C22semidirect
115. C22⋊C4×C251D4semidirect
220. C22×C251#123semidirect
250. C251#2418direct (N × H)
251. #2418C251direct (N × H)
252. C2×#123C2×C251semidirect
254. C2×C4×D4C2×C251semidirect
255. C22×C22⋊C4C2×C251semidirect
256. #399C2×C251semidirect
257. #397C2×C251semidirect
258. #394C2×C251semidirect
259. #394C2×C251semidirect
260. #388C2×C251semidirect
261. #388C2×C251semidirect
262. #386C2×C251semidirect
263. #386C2×C251semidirect
264. #385C2×C251semidirect
265. #380C2×C251semidirect
266. #379C2×C251semidirect
273. C23×C4C22×C251semidirect
277. C2×C4⋊C4C22×C251semidirect
278. C2×C4⋊C4C22×C251semidirect
279. C2×C4⋊C4C22×C251semidirect
280. C2×C4⋊C4C22×C251semidirect
281. C2×C4⋊C4C22×C251semidirect
296. C2×C22⋊C4C22×C251semidirect
297. C2×C22⋊C4C22×C251semidirect
298. C2×C22⋊C4C22×C251semidirect
299. C2×C42C22×C251semidirect
300. #95C22×C251semidirect
301. #95C22×C251semidirect
302. #95C22×C251semidirect
303. #95C22×C251semidirect
304. #95C22×C251semidirect
305. #95C22×C251semidirect
366. C22⋊C4D4×C251semidirect
471. C22#123×C251semidirect

Maximal subgroups

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

subgroupindex
C2×#123×C2512
C2×#120×C2512
C2×C4×D4×C2512
C22×C22⋊C4×C2512
#399×C2512
#397×C2512
#394×C2512
#388×C2512
#386×C2512
#385×C2512
#380×C2512
#379×C2512
#2418251