Finite Group Databasebeta

← #44553343

Subgroups — #44553343

Order 32128.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548685C211yesmaximal, characteristic
2C2×C4×D1004C211yesmaximal, characteristic, complemented
3C2×#22548677C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D251C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D251C211yesmaximal, characteristic, complemented
6#30631587C211yesmaximal, characteristic, complemented
7#30631580C211yesmaximal, characteristic, complemented
8#30631565C211yesmaximal, characteristic, complemented
9#30631565C211yesmaximal, characteristic, complemented
10#30631563C211yesmaximal, characteristic, complemented
11#30631561C211yesmaximal, characteristic, complemented
12#30631558C211yesmaximal, characteristic, complemented
13C4×(C2×D251)⋊C4C211yesmaximal, characteristic, complemented
14#30631534C211yesmaximal, characteristic, complemented
15#388×C251C211yesmaximal, characteristic, complemented
16C24×D251C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C22×D1004C2211yescharacteristic
19C22×C22⋊D251C2211yescharacteristic
20C22×C4×D251C2211yescharacteristic
21C22×C4×D251C2211yescharacteristic
22C22×C4×D251C2211yescharacteristic
23C22×C4×D251C2211yescharacteristic
24C22×C4×D251C2211yescharacteristic, complemented
25#22548685C418yescomplemented
26C4×D1004—24no—
27#22548677—24no—
28C2×C4⋊Dic251C2211yescharacteristic, complemented
29C2×Dic251⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic251C2211yescharacteristic
31C2×C22⋊Dic251C2211yescharacteristic
32C2×(C2×D251)⋊C4C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C2×(C2×D251)⋊C4C2211yescharacteristic
37C2×(C2×D251)⋊C4C2211yescharacteristic
38C2×(C2×D251)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D251)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D251)⋊C4C2211yescharacteristic, complemented
41C22⋊C4×D251—24no—
42C22⋊C4×D251—24no—
43C2×C4×Dic251C2211yescharacteristic, complemented
44C2×C4×Dic251C2211yescharacteristic, complemented
45C251⋊#95C2211yescharacteristic, complemented
46C251⋊#95C2211yescharacteristic, complemented
47C251⋊#95C2211yescharacteristic, complemented
48C251⋊#95C2211yescharacteristic, complemented
49C2×C4⋊C4×C251C2211yescharacteristic, complemented
50C2×C22⋊C4×C251C2211yescharacteristic
51C2×C22⋊C4×C251C2211yescharacteristic
52C2×C22⋊C4×C251C2211yescharacteristic
53C2×C42×C251C2211yescharacteristic, complemented
54#95×C251C2211yescharacteristic, complemented
55#95×C251C2211yescharacteristic, complemented
56C23×D251—22no—
57C23×D251—22no—
58C23×D251—22no—
59C23×D251—22no—
60C23×D251—41no—
61C23×D251—41no—
62C23×D251C2×C414yescomplemented
63C23×D251C2311yescharacteristic
64C23×D251C2311yescharacteristic
65C23×D251C2311yescharacteristic
66C2×D1004—22no—
67C2×D1004—22no—
68C2×D1004—22no—
69C2×D1004—22no—
70C2×D1004C2×C414yescomplemented
71C2×D1004C2×C414yescomplemented
72C2×C22⋊D251—41no—
73C2×C22⋊D251—41no—
74C2×C22⋊D251C2×C414yescomplemented
75C22×Dic251C2311yescharacteristic
76C22×Dic251C2311yescharacteristic
77C22×Dic251C2311yescharacteristic
78C22×Dic251C2311yescharacteristic
79C22×Dic251C2311yescharacteristic
80C2×C4×D251—22no—
81C2×C4×D251—22no—
82C2×C4×D251—22no—
83C2×C4×D251—22no—
84C2×C4×D251—22no—
85C2×C4×D251—22no—
86C2×C4×D251—22no—
87C2×C4×D251—22no—
88C2×C4×D251—22no—
89C2×C4×D251—22no—
90C2×C4×D251—41no—
91C2×C4×D251—41no—
92C2×C4×D251—41no—
93C2×C4×D251—41no—
94C2×C4×D251—41no—
95C4⋊Dic251—22no—
96Dic251⋊C4—41no—
97C22⋊Dic251—22no—
98C22⋊Dic251—22no—
99(C2×D251)⋊C4—22no—
100(C2×D251)⋊C4—22no—
101(C2×D251)⋊C4—22no—
102(C2×D251)⋊C4—22no—
103(C2×D251)⋊C4—41no—
104(C2×D251)⋊C4C2×C414yescomplemented
105(C2×D251)⋊C4C2×C414yescomplemented
106(C2×D251)⋊C4D414yescomplemented
107(C2×D251)⋊C4D414yescomplemented
108C4×Dic251—22no—
109C4×Dic251—41no—
110C24×C251C2311yescharacteristic
111C22×C4×C251C2311yescharacteristic
112C22×C4×C251C2311yescharacteristic
113C22×C4×C251C2311yescharacteristic
114C22×C4×C251C2311yescharacteristic
115C22×C4×C251C2311yescharacteristic
116C22×C4×C251C2311yescharacteristic
117C4⋊C4×C251—22no—
118C22⋊C4×C251—22no—
119C22⋊C4×C251—22no—
120C22⋊C4×C251C2×C414yescomplemented
121C42×C251—22no—
122C22×D251—21no—
123C22×D251—21no—
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—21no—
127C22×D251—21no—
128C22×D251—21no—
129C22×D251—21no—
130C22×D251—21no—
131C22×D251—21no—
132C22×D251—21no—
133C22×D251—21no—
134C22×D251—21no—
135C22×D251—21no—
136C22×D251—28no—
137C22×D251—28no—
138C22×D251—44no—
139C22×D251—44no—
140C22×D251—44no—
141C22×D251—44no—
142C22×D251—44no—
143C22×D251C22×C412yes—
144C22×D251C22×C412yes—
145C22×D251C22×C412yes—
146C22×D251C2×D412yes—
147C22×D251C2×D412yes—
148C22×D251#4112yes—
149C22×D251#4112yes—
150D1004—28no—
151D1004—28no—
152C22⋊D251—44no—
153C2×Dic251—21no—
154C2×Dic251—21no—
155C2×Dic251—21no—
156C2×Dic251—21no—
157C2×Dic251—21no—
158C2×Dic251—21no—
159C2×Dic251—21no—
160C2×Dic251—21no—
161C2×Dic251—21no—
162C2×Dic251—21no—
163C2×Dic251—21no—
164C2×Dic251—21no—
165C2×Dic251C22×C412yes—
166C2×Dic251C2×D412yes—
167C2×Dic251C2×D412yes—
168C4×D251—44no—
169C4×D251—44no—
170C4×D251—44no—
171C4×D251—44no—
172C4×D251—44no—
173C23×C251—21no—
174C23×C251—21no—
175C23×C251—21no—
176C23×C251—21no—
177C23×C251—21no—
178C23×C251—21no—
179C23×C251C22×C412yes—
180C23×C251C2411yescharacteristic
181C2×C4×C251—21no—
182C2×C4×C251—21no—
183C2×C4×C251—21no—
184C2×C4×C251—21no—
185C2×C4×C251—21no—
186C2×C4×C251—21no—
187C2×C4×C251—21no—
188C2×C4×C251—21no—
189C2×C4×C251—21no—
190C2×C4×C251—21no—
191C2×C4×C251—21no—
192C2×C4×C251—21no—
193C2×C4×C251—21no—
194C2×C4×C251—21no—
195C2×C4×C251C22×C412yes—
196C2×C4×C251C22×C412yes—
197C2×C4×C251C2×D412yes—
198C2×C4×C251C2×D412yes—
199C22×C251—22no—
200C22×C251—22no—
201C22×C251—22no—
202C22×C251—41no—
203C22×C251—41no—
204C22×C251—41no—
205C22×C251—41no—
206C22×C251C23×C411yescharacteristic
207C22×C251C22×D411yescharacteristic
208C22×C251C22×D411yescharacteristic
209C22×C251C2×#4111yescharacteristic
210C22×C251C2×#4111yescharacteristic
211C22×C251H(5,𝔽2)11yescharacteristic
212C22×C251H(5,𝔽2)11yescharacteristic
213C2×D251—22no—
214C2×D251—22no—
215C2×D251—22no—
216C2×D251—22no—
217C2×D251—22no—
218C2×D251—22no—
219C2×D251—22no—
220C2×D251—22no—
221C2×D251—22no—
222C2×D251—22no—
223C2×D251—22no—
224C2×D251—22no—
225C2×D251—22no—
226C2×D251—22no—
227C2×D251—22no—
228C2×D251—416no—
229C2×D251—41no—
230C2×D251—41no—
231C2×D251—41no—
232C2×D251—41no—
233C2×D251C4×D414yescomplemented
234C2×D251C4×D414yescomplemented
235C4×C251—22no—
236C4×C251—22no—
237C4×C251—22no—
238C4×C251—22no—
239C4×C251—41no—
240C4×C251—41no—
241Dic251—22no—
242Dic251—22no—
243Dic251—22no—
244Dic251—41no—
245Dic251—41no—
246C2×C251—42no—
247C2×C251C2×C4×D411yescharacteristic
248C2×C251C2×C4×D411yescharacteristic
249C2×C251#51811yescharacteristic
250C2×C251D4211yescharacteristic
251C2×C251#54611yescharacteristic
252C2×C251#54611yescharacteristic
253C2×C251#55111yescharacteristic
254D251—24no—
255D251—24no—
256D251—42no—
257C251#235811yescharacteristic, complemented, Sylow
258#2358—2511nomaximal, Sylow
259C2×#120—2511no—
260C2×C4×D4—2511no—
261C2×C4×D4—2511no—
262C22×C22⋊C4—2511no—
263C22×C22⋊C4—2511no—
264#390—2511no—
265#388—2511no—
266#388D25111yescharacteristic, complemented
267#386—2511no—
268#386—2511no—
269#386—2511no—
270#385—2511no—
271#385—2511no—
272C4×C22⋊C4—2511no—
273C4×C22⋊C4—2511no—
274C25—2511no—
275C22×D4—2511no—
276C22×D4—2511no—
277C22×D4—2511no—
278C23×C4—2511no—
279C23×C4—2511no—
280C23×C4—2511no—
281C23×C4—2511no—
282C23×C4—2511no—
283#120—2518no—
284C4×D4—5024no—
285C4×D4—5024no—
286C2×C4⋊C4—2511no—
287C2×C4⋊C4—2511no—
288C2×C4⋊C4C2×D25111yescharacteristic, complemented
289C2×C22⋊C4—5024no—
290C2×C22⋊C4—5024no—
291C2×C22⋊C4—2511no—
292C2×C22⋊C4—2511no—
293C2×C22⋊C4—2511no—
294C2×C22⋊C4—2511no—
295C2×C22⋊C4—2511no—
296C2×C22⋊C4—2511no—
297C2×C22⋊C4—2511no—
298C2×C22⋊C4—2511no—
299C2×C22⋊C4—2511no—
300C2×C22⋊C4—2511no—
301C2×C22⋊C4—2511no—
302C2×C22⋊C4C2×D25111yescharacteristic
303C2×C22⋊C4C2×D25111yescharacteristic
304C2×C22⋊C4C2×D25111yescharacteristic
305C2×C42—2511no—
306C2×C42—2511no—
307C2×C42C2×D25111yescharacteristic, complemented
308#95—2511no—
309#95—2511no—
310#95—2511no—
311#95—2511no—
312#95C2×D25111yescharacteristic, complemented
313#95C2×D25111yescharacteristic, complemented
314C24—10041no—
315C24—10041no—
316C24—5022no—
317C24—5022no—
318C24—5022no—
319C24—5022no—
320C24—2514no—
321C24—2511no—
322C24—2511no—
323C24—2511no—
324C24C22×D25111yescharacteristic
325C2×D4—10041no—
326C2×D4—10041no—
327C2×D4—5022no—
328C2×D4—5022no—
329C2×D4—5022no—
330C2×D4—5022no—
331C2×D4—2514no—
332C2×D4—2514no—
333C2×D4—2514no—
334C22×C4—10041no—
335C22×C4—10041no—
336C22×C4—10041no—
337C22×C4—10041no—
338C22×C4—10041no—
339C22×C4—5022no—
340C22×C4—5022no—
341C22×C4—5022no—
342C22×C4—5022no—
343C22×C4—5022no—
344C22×C4—5022no—
345C22×C4—5022no—
346C22×C4—5022no—
347C22×C4—5022no—
348C22×C4—5022no—
349C22×C4—2511no—
350C22×C4—2511no—
351C22×C4—2511no—
352C22×C4—2511no—
353C22×C4—2511no—
354C22×C4C22×D25111yescharacteristic
355C22×C4C22×D25111yescharacteristic
356C22×C4C22×D25111yescharacteristic
357C22×C4C22×D25111yescharacteristic
358C22×C4C22×D25111yescharacteristic
359C22×C4C22×D25111yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—10041no—
362C4⋊C4—5022no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—10041no—
366C22⋊C4—5022no—
367C22⋊C4—5022no—
368C22⋊C4—5022no—
369C22⋊C4—5022no—
370C22⋊C4—5022no—
371C22⋊C4—5022no—
372C22⋊C4—2514no—
373C22⋊C4—2514no—
374C22⋊C4—2514no—
375C22⋊C4—2514no—
376C22⋊C4C4×D25114yescomplemented
377C42—22no—
378C42—10041no—
379C42—5022no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—10044no—
387C23—10044no—
388C23—10044no—
389C23—10044no—
390C23—10044no—
391C23—5028no—
392C23—5028no—
393C23—5021no—
394C23—5021no—
395C23—5021no—
396C23—5021no—
397C23—5021no—
398C23—5021no—
399C23—5021no—
400C23—5021no—
401C23—5021no—
402C23—5021no—
403C23—5021no—
404C23—5021no—
405C23—5021no—
406C23—5021no—
407C23—2512no—
408C23—2512no—
409C23—2512no—
410C23—2512no—
411C23—2512no—
412C23—2512no—
413C23—2512no—
414C23C2×C4×D25112yes—
415C23C23×D25111yescharacteristic
416D4—10044no—
417D4—5028no—
418D4—5028no—
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—10044no—
434C2×C4—10044no—
435C2×C4—10044no—
436C2×C4—10044no—
437C2×C4—10044no—
438C2×C4—5021no—
439C2×C4—5021no—
440C2×C4—5021no—
441C2×C4—5021no—
442C2×C4—5021no—
443C2×C4—5021no—
444C2×C4—5021no—
445C2×C4—5021no—
446C2×C4—5021no—
447C2×C4—5021no—
448C2×C4—5021no—
449C2×C4—5021no—
450C2×C4—2512no—
451C2×C4—2512no—
452C2×C4—2512no—
453C2×C4C2×C4×D25112yes—
454C2×C4C2×C4×D25112yes—
455C2×C4D4×D25112yes—
456C2×C4D4×D25112yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—100416no—
465C22—10041no—
466C22—10041no—
467C22—10041no—
468C22—10041no—
469C22—5022no—
470C22—5022no—
471C22—5022no—
472C22—5022no—
473C22—5022no—
474C22—5022no—
475C22—5022no—
476C22—5022no—
477C22—5022no—
478C22—5022no—
479C22—5022no—
480C22—5022no—
481C22—5022no—
482C22—5022no—
483C22—5022no—
484C22—2514no—
485C22—2514no—
486C22C22×C4×D25111yescharacteristic
487C22C2×D4×D25111yescharacteristic
488C22C2×D4×D25111yescharacteristic
489C22#41×D25111yescharacteristic
490C22#41×D25111yescharacteristic
491C22#2254876111yescharacteristic
492C22#2254876111yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—41no—
498C4—41no—
499C4—10041no—
500C4—10041no—
501C4—5022no—
502C4—5022no—
503C4—5022no—
504C2—42no—
505C2—10042no—
506C2—5024no—
507C2—5024no—
508C2C4×D4×D25111yescharacteristic
509C2C4×D4×D25111yescharacteristic
510C2#3063216111yescharacteristic
511C2#3063232111yescharacteristic
512C2#3063232911yescharacteristic
513C2#3063233011yescharacteristic
514C2#3063236211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#388×C251C2
socle, cosocleC23×C251C24
prime radicalC2×#22548685C2
hypercenter#388D251
hypocenterC251#2358
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D1004C2semidirect
3. C2×#22548677C2semidirect
4. C2×C22⋊C4×D251C2semidirect
5. C2×C22⋊C4×D251C2semidirect
6. #30631587C2semidirect
7. #30631580C2semidirect
8. #30631565C2semidirect
9. #30631565C2semidirect
10. #30631563C2semidirect
11. #30631561C2semidirect
12. #30631558C2semidirect
13. C4×(C2×D251)⋊C4C2semidirect
14. #30631534C2semidirect
15. #388×C251C2semidirect
24. C22×C4×D251C22semidirect
25. #22548685C4semidirect
28. C2×C4⋊Dic251C22semidirect
29. C2×Dic251⋊C4C22semidirect
38. C2×(C2×D251)⋊C4C22semidirect
39. C2×(C2×D251)⋊C4C22semidirect
40. C2×(C2×D251)⋊C4C22semidirect
43. C2×C4×Dic251C22semidirect
44. C2×C4×Dic251C22semidirect
45. C251⋊#95C22semidirect
46. C251⋊#95C22semidirect
47. C251⋊#95C22semidirect
48. C251⋊#95C22semidirect
49. C2×C4⋊C4×C251C22semidirect
53. C2×C42×C251C22semidirect
54. #95×C251C22semidirect
55. #95×C251C22semidirect
62. C23×D251C2×C4semidirect
70. C2×D1004C2×C4semidirect
71. C2×D1004C2×C4semidirect
74. C2×C22⋊D251C2×C4semidirect
104. (C2×D251)⋊C4C2×C4semidirect
105. (C2×D251)⋊C4C2×C4semidirect
106. (C2×D251)⋊C4D4semidirect
107. (C2×D251)⋊C4D4semidirect
120. C22⋊C4×C251C2×C4semidirect
233. C2×D251C4×D4semidirect
234. C2×D251C4×D4semidirect
257. C251#2358semidirect
266. #388D251semidirect
288. C2×C4⋊C4C2×D251semidirect
307. C2×C42C2×D251semidirect
312. #95C2×D251semidirect
313. #95C2×D251semidirect
376. C22⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486852
C2×C4×D10042
C2×#225486772
C2×C22⋊C4×D2512
#306315872
#306315802
#306315652
#306315632
#306315612
#306315582
C4×(C2×D251)⋊C42
#306315342
#388×C2512
#2358251