Finite Group Databasebeta

← #45667298

Subgroups — #45667298

Order 32896.

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×#22551562C211yesmaximal, characteristic
2C2×C4×D1028C211yesmaximal, characteristic, complemented
3C2×#22551554C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D257C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D257C211yesmaximal, characteristic, complemented
6#30656637C211yesmaximal, characteristic, complemented
7#30656630C211yesmaximal, characteristic, complemented
8#30656615C211yesmaximal, characteristic, complemented
9#30656615C211yesmaximal, characteristic, complemented
10#30656613C211yesmaximal, characteristic, complemented
11#30656611C211yesmaximal, characteristic, complemented
12#30656608C211yesmaximal, characteristic, complemented
13C4×#16861595C211yesmaximal, characteristic, complemented
14#30656584C211yesmaximal, characteristic, complemented
15#388×C257C211yesmaximal, characteristic, complemented
16C24×D257C2211yescharacteristic
17C22×D1028C2211yescharacteristic
18C22×D1028C2211yescharacteristic
19C22×C22⋊D257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21C22×C4×D257C2211yescharacteristic
22C22×C4×D257C2211yescharacteristic
23C22×C4×D257C2211yescharacteristic
24C22×C4×D257C2211yescharacteristic, complemented
25#22551562C418yescomplemented
26C4×D1028—24no—
27#22551554—24no—
28C2×C4⋊Dic257C2211yescharacteristic, complemented
29C2×#16861597C2211yescharacteristic, complemented
30C2×C22⋊Dic257C2211yescharacteristic
31C2×C22⋊Dic257C2211yescharacteristic
32C2×#16861595C2211yescharacteristic
33C2×#16861595C2211yescharacteristic
34C2×#16861595C2211yescharacteristic
35C2×#16861595C2211yescharacteristic
36C2×#16861595C2211yescharacteristic
37C2×#16861595C2211yescharacteristic
38C2×#16861595C2211yescharacteristic, complemented
39C2×#16861595C2211yescharacteristic, complemented
40C2×#16861595C2211yescharacteristic, complemented
41C22⋊C4×D257—24no—
42C22⋊C4×D257—24no—
43C2×C4×Dic257C2211yescharacteristic, complemented
44C2×C4×Dic257C2211yescharacteristic, complemented
45#22551499C2211yescharacteristic, complemented
46#22551499C2211yescharacteristic, complemented
47#22551499C2211yescharacteristic, complemented
48#22551499C2211yescharacteristic, complemented
49C2×C4⋊C4×C257C2211yescharacteristic, complemented
50C2×C22⋊C4×C257C2211yescharacteristic
51C2×C22⋊C4×C257C2211yescharacteristic
52C2×C22⋊C4×C257C2211yescharacteristic
53C2×C42×C257C2211yescharacteristic, complemented
54#95×C257C2211yescharacteristic, complemented
55#95×C257C2211yescharacteristic, complemented
56C23×D257—22no—
57C23×D257—22no—
58C23×D257—22no—
59C23×D257—22no—
60C23×D257—41no—
61C23×D257—41no—
62C23×D257C2×C414yescomplemented
63C23×D257C2311yescharacteristic
64C23×D257C2311yescharacteristic
65C23×D257C2311yescharacteristic
66C2×D1028—22no—
67C2×D1028—22no—
68C2×D1028—22no—
69C2×D1028—22no—
70C2×D1028C2×C414yescomplemented
71C2×D1028C2×C414yescomplemented
72C2×C22⋊D257—41no—
73C2×C22⋊D257—41no—
74C2×C22⋊D257C2×C414yescomplemented
75C22×Dic257C2311yescharacteristic
76C22×Dic257C2311yescharacteristic
77C22×Dic257C2311yescharacteristic
78C22×Dic257C2311yescharacteristic
79C22×Dic257C2311yescharacteristic
80C2×C4×D257—22no—
81C2×C4×D257—22no—
82C2×C4×D257—22no—
83C2×C4×D257—22no—
84C2×C4×D257—22no—
85C2×C4×D257—22no—
86C2×C4×D257—22no—
87C2×C4×D257—22no—
88C2×C4×D257—22no—
89C2×C4×D257—22no—
90C2×C4×D257—41no—
91C2×C4×D257—41no—
92C2×C4×D257—41no—
93C2×C4×D257—41no—
94C2×C4×D257—41no—
95C4⋊Dic257—22no—
96#16861597—41no—
97C22⋊Dic257—22no—
98C22⋊Dic257—22no—
99#16861595—22no—
100#16861595—22no—
101#16861595—22no—
102#16861595—22no—
103#16861595—41no—
104#16861595C2×C414yescomplemented
105#16861595C2×C414yescomplemented
106#16861595D414yescomplemented
107#16861595D414yescomplemented
108C4×Dic257—22no—
109C4×Dic257—41no—
110C24×C257C2311yescharacteristic
111C22×C4×C257C2311yescharacteristic
112C22×C4×C257C2311yescharacteristic
113C22×C4×C257C2311yescharacteristic
114C22×C4×C257C2311yescharacteristic
115C22×C4×C257C2311yescharacteristic
116C22×C4×C257C2311yescharacteristic
117C4⋊C4×C257—22no—
118C22⋊C4×C257—22no—
119C22⋊C4×C257—22no—
120C22⋊C4×C257C2×C414yescomplemented
121C42×C257—22no—
122C22×D257—21no—
123C22×D257—21no—
124C22×D257—21no—
125C22×D257—21no—
126C22×D257—21no—
127C22×D257—21no—
128C22×D257—21no—
129C22×D257—21no—
130C22×D257—21no—
131C22×D257—21no—
132C22×D257—21no—
133C22×D257—21no—
134C22×D257—21no—
135C22×D257—21no—
136C22×D257—28no—
137C22×D257—28no—
138C22×D257—44no—
139C22×D257—44no—
140C22×D257—44no—
141C22×D257—44no—
142C22×D257—44no—
143C22×D257C22×C412yes—
144C22×D257C22×C412yes—
145C22×D257C22×C412yes—
146C22×D257C2×D412yes—
147C22×D257C2×D412yes—
148C22×D257#4112yes—
149C22×D257#4112yes—
150D1028—28no—
151D1028—28no—
152C22⋊D257—44no—
153C2×Dic257—21no—
154C2×Dic257—21no—
155C2×Dic257—21no—
156C2×Dic257—21no—
157C2×Dic257—21no—
158C2×Dic257—21no—
159C2×Dic257—21no—
160C2×Dic257—21no—
161C2×Dic257—21no—
162C2×Dic257—21no—
163C2×Dic257—21no—
164C2×Dic257—21no—
165C2×Dic257C22×C412yes—
166C2×Dic257C2×D412yes—
167C2×Dic257C2×D412yes—
168C4×D257—44no—
169C4×D257—44no—
170C4×D257—44no—
171C4×D257—44no—
172C4×D257—44no—
173C23×C257—21no—
174C23×C257—21no—
175C23×C257—21no—
176C23×C257—21no—
177C23×C257—21no—
178C23×C257—21no—
179C23×C257C22×C412yes—
180C23×C257C2411yescharacteristic
181C2×C4×C257—21no—
182C2×C4×C257—21no—
183C2×C4×C257—21no—
184C2×C4×C257—21no—
185C2×C4×C257—21no—
186C2×C4×C257—21no—
187C2×C4×C257—21no—
188C2×C4×C257—21no—
189C2×C4×C257—21no—
190C2×C4×C257—21no—
191C2×C4×C257—21no—
192C2×C4×C257—21no—
193C2×C4×C257—21no—
194C2×C4×C257—21no—
195C2×C4×C257C22×C412yes—
196C2×C4×C257C22×C412yes—
197C2×C4×C257C2×D412yes—
198C2×C4×C257C2×D412yes—
199C22×C257—22no—
200C22×C257—22no—
201C22×C257—22no—
202C22×C257—41no—
203C22×C257—41no—
204C22×C257—41no—
205C22×C257—41no—
206C22×C257C23×C411yescharacteristic
207C22×C257C22×D411yescharacteristic
208C22×C257C22×D411yescharacteristic
209C22×C257C2×#4111yescharacteristic
210C22×C257C2×#4111yescharacteristic
211C22×C257H(5,𝔽2)11yescharacteristic
212C22×C257H(5,𝔽2)11yescharacteristic
213C2×D257—22no—
214C2×D257—22no—
215C2×D257—22no—
216C2×D257—22no—
217C2×D257—22no—
218C2×D257—22no—
219C2×D257—22no—
220C2×D257—22no—
221C2×D257—22no—
222C2×D257—22no—
223C2×D257—22no—
224C2×D257—22no—
225C2×D257—22no—
226C2×D257—22no—
227C2×D257—22no—
228C2×D257—416no—
229C2×D257—41no—
230C2×D257—41no—
231C2×D257—41no—
232C2×D257—41no—
233C2×D257C4×D414yescomplemented
234C2×D257C4×D414yescomplemented
235C4×C257—22no—
236C4×C257—22no—
237C4×C257—22no—
238C4×C257—22no—
239C4×C257—41no—
240C4×C257—41no—
241Dic257—22no—
242Dic257—22no—
243Dic257—22no—
244Dic257—41no—
245Dic257—41no—
246C2×C257—42no—
247C2×C257C2×C4×D411yescharacteristic
248C2×C257C2×C4×D411yescharacteristic
249C2×C257#51811yescharacteristic
250C2×C257D4211yescharacteristic
251C2×C257#54611yescharacteristic
252C2×C257#54611yescharacteristic
253C2×C257#55111yescharacteristic
254D257—24no—
255D257—24no—
256D257—42no—
257C257#235811yescharacteristic, complemented, Sylow
258#2358—2571nomaximal, Sylow
259C2×#120—2571no—
260C2×C4×D4—2571no—
261C2×C4×D4—2571no—
262C22×C22⋊C4—2571no—
263C22×C22⋊C4—2571no—
264#390—2571no—
265#388—2571no—
266#388D25711yescharacteristic, complemented
267#386—2571no—
268#386—2571no—
269#386—2571no—
270#385—2571no—
271#385—2571no—
272C4×C22⋊C4—2571no—
273C4×C22⋊C4—2571no—
274C25—2571no—
275C22×D4—2571no—
276C22×D4—2571no—
277C22×D4—2571no—
278C23×C4—2571no—
279C23×C4—2571no—
280C23×C4—2571no—
281C23×C4—2571no—
282C23×C4—2571no—
283#120—2578no—
284C4×D4—5144no—
285C4×D4—5144no—
286C2×C4⋊C4—2571no—
287C2×C4⋊C4—2571no—
288C2×C4⋊C4C2×D25711yescharacteristic, complemented
289C2×C22⋊C4—2571no—
290C2×C22⋊C4—2571no—
291C2×C22⋊C4—2571no—
292C2×C22⋊C4—2571no—
293C2×C22⋊C4—2571no—
294C2×C22⋊C4—2571no—
295C2×C22⋊C4—2571no—
296C2×C22⋊C4—2571no—
297C2×C22⋊C4—2571no—
298C2×C22⋊C4—2571no—
299C2×C22⋊C4—2571no—
300C2×C22⋊C4—5144no—
301C2×C22⋊C4—5144no—
302C2×C22⋊C4C2×D25711yescharacteristic
303C2×C22⋊C4C2×D25711yescharacteristic
304C2×C22⋊C4C2×D25711yescharacteristic
305C2×C42—2571no—
306C2×C42—2571no—
307C2×C42C2×D25711yescharacteristic, complemented
308#95—2571no—
309#95—2571no—
310#95—2571no—
311#95—2571no—
312#95C2×D25711yescharacteristic, complemented
313#95C2×D25711yescharacteristic, complemented
314C24—2571no—
315C24—2571no—
316C24—2571no—
317C24—2574no—
318C24—5142no—
319C24—5142no—
320C24—5142no—
321C24—5142no—
322C24—10281no—
323C24—10281no—
324C24C22×D25711yescharacteristic
325C2×D4—2574no—
326C2×D4—2574no—
327C2×D4—2574no—
328C2×D4—5142no—
329C2×D4—5142no—
330C2×D4—5142no—
331C2×D4—5142no—
332C2×D4—10281no—
333C2×D4—10281no—
334C22×C4—2571no—
335C22×C4—2571no—
336C22×C4—2571no—
337C22×C4—2571no—
338C22×C4—2571no—
339C22×C4—5142no—
340C22×C4—5142no—
341C22×C4—5142no—
342C22×C4—5142no—
343C22×C4—5142no—
344C22×C4—5142no—
345C22×C4—5142no—
346C22×C4—5142no—
347C22×C4—5142no—
348C22×C4—5142no—
349C22×C4—10281no—
350C22×C4—10281no—
351C22×C4—10281no—
352C22×C4—10281no—
353C22×C4—10281no—
354C22×C4C22×D25711yescharacteristic
355C22×C4C22×D25711yescharacteristic
356C22×C4C22×D25711yescharacteristic
357C22×C4C22×D25711yescharacteristic
358C22×C4C22×D25711yescharacteristic
359C22×C4C22×D25711yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—5142no—
362C4⋊C4—10281no—
363C22⋊C4—2574no—
364C22⋊C4—2574no—
365C22⋊C4—2574no—
366C22⋊C4—2574no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—5142no—
370C22⋊C4—5142no—
371C22⋊C4—5142no—
372C22⋊C4—5142no—
373C22⋊C4—5142no—
374C22⋊C4—5142no—
375C22⋊C4—10281no—
376C22⋊C4C4×D25714yescomplemented
377C42—22no—
378C42—5142no—
379C42—10281no—
380C23—2572no—
381C23—2572no—
382C23—2572no—
383C23—2572no—
384C23—2572no—
385C23—2572no—
386C23—2572no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—5148no—
394C23—5148no—
395C23—5141no—
396C23—5141no—
397C23—5141no—
398C23—5141no—
399C23—5141no—
400C23—5141no—
401C23—5141no—
402C23—5141no—
403C23—5141no—
404C23—5141no—
405C23—5141no—
406C23—5141no—
407C23—5141no—
408C23—5141no—
409C23—10284no—
410C23—10284no—
411C23—10284no—
412C23—10284no—
413C23—10284no—
414C23C2×C4×D25712yes—
415C23C23×D25711yescharacteristic
416D4—5148no—
417D4—5148no—
418D4—10284no—
419C2×C4—2572no—
420C2×C4—2572no—
421C2×C4—2572no—
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—5141no—
437C2×C4—5141no—
438C2×C4—5141no—
439C2×C4—5141no—
440C2×C4—5141no—
441C2×C4—5141no—
442C2×C4—5141no—
443C2×C4—5141no—
444C2×C4—5141no—
445C2×C4—5141no—
446C2×C4—5141no—
447C2×C4—5141no—
448C2×C4—10284no—
449C2×C4—10284no—
450C2×C4—10284no—
451C2×C4—10284no—
452C2×C4—10284no—
453C2×C4C2×C4×D25712yes—
454C2×C4C2×C4×D25712yes—
455C2×C4D4×D25712yes—
456C2×C4D4×D25712yes—
457C22—2574no—
458C22—2574no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—5142no—
463C22—5142no—
464C22—5142no—
465C22—5142no—
466C22—5142no—
467C22—5142no—
468C22—5142no—
469C22—5142no—
470C22—5142no—
471C22—5142no—
472C22—5142no—
473C22—5142no—
474C22—5142no—
475C22—5142no—
476C22—5142no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—102816no—
482C22—10281no—
483C22—10281no—
484C22—10281no—
485C22—10281no—
486C22C22×C4×D25711yescharacteristic
487C22C2×D4×D25711yescharacteristic
488C22C2×D4×D25711yescharacteristic
489C22#41×D25711yescharacteristic
490C22#41×D25711yescharacteristic
491C22#2255163811yescharacteristic
492C22#2255163811yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—5142no—
498C4—5142no—
499C4—5142no—
500C4—41no—
501C4—41no—
502C4—10281no—
503C4—10281no—
504C2—5144no—
505C2—5144no—
506C2—42no—
507C2—10282no—
508C2C4×D4×D25711yescharacteristic
509C2C4×D4×D25711yescharacteristic
510C2#3065721111yescharacteristic
511C2#3065737111yescharacteristic
512C2#3065737911yescharacteristic
513C2#3065738011yescharacteristic
514C2#3065741211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#388×C257C2
socle, cosocleC23×C257C24
prime radicalC2×#22551562C2
hypercenter#388D257
hypocenterC257#2358
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D1028C2semidirect
3. C2×#22551554C2semidirect
4. C2×C22⋊C4×D257C2semidirect
5. C2×C22⋊C4×D257C2semidirect
6. #30656637C2semidirect
7. #30656630C2semidirect
8. #30656615C2semidirect
9. #30656615C2semidirect
10. #30656613C2semidirect
11. #30656611C2semidirect
12. #30656608C2semidirect
13. C4×#16861595C2semidirect
14. #30656584C2semidirect
15. #388×C257C2semidirect
24. C22×C4×D257C22semidirect
25. #22551562C4semidirect
28. C2×C4⋊Dic257C22semidirect
29. C2×#16861597C22semidirect
38. C2×#16861595C22semidirect
39. C2×#16861595C22semidirect
40. C2×#16861595C22semidirect
43. C2×C4×Dic257C22semidirect
44. C2×C4×Dic257C22semidirect
45. #22551499C22semidirect
46. #22551499C22semidirect
47. #22551499C22semidirect
48. #22551499C22semidirect
49. C2×C4⋊C4×C257C22semidirect
53. C2×C42×C257C22semidirect
54. #95×C257C22semidirect
55. #95×C257C22semidirect
62. C23×D257C2×C4semidirect
70. C2×D1028C2×C4semidirect
71. C2×D1028C2×C4semidirect
74. C2×C22⋊D257C2×C4semidirect
104. #16861595C2×C4semidirect
105. #16861595C2×C4semidirect
106. #16861595D4semidirect
107. #16861595D4semidirect
120. C22⋊C4×C257C2×C4semidirect
233. C2×D257C4×D4semidirect
234. C2×D257C4×D4semidirect
257. C257#2358semidirect
266. #388D257semidirect
288. C2×C4⋊C4C2×D257semidirect
307. C2×C42C2×D257semidirect
312. #95C2×D257semidirect
313. #95C2×D257semidirect
376. C22⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515622
C2×C4×D10282
C2×#225515542
C2×C22⋊C4×D2572
#306566372
#306566302
#306566152
#306566132
#306566112
#306566082
C4×#168615952
#306565842
#388×C2572
#2358257