Finite Group Databasebeta

← #29718359

Subgroups — #29718359

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840092C211yesmaximal, characteristic, complemented
2C22×#14519697C211yesmaximal, characteristic
3#21449470C211yesmaximal, characteristic
4#21447190C212yesmaximal, complemented
5#21447060C212yesmaximal, complemented
6#21446751C212yesmaximal, complemented
7#21446446C212yesmaximal, complemented
8#21446240C211yesmaximal, characteristic, complemented
9#21445999C211yesmaximal, characteristic, complemented
10#21445749C211yesmaximal, characteristic
11C61×#2344C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519765C2212yes—
15C2×#14519756C2211yescharacteristic
16#16840092—24no—
17C2×#14519708C2211yescharacteristic, complemented
18C2×#14519708C2211yescharacteristic, complemented
19C2×C4⋊#92418C2212yescomplemented
20C2×#14519700C2211yescharacteristic, complemented
21C2×#14519700C2211yescharacteristic, complemented
22C2×#14519700C2212yescomplemented
23C2×#14519697—24no—
24C2×#14519697C2212yes—
25C2×#14519697C2212yes—
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28C2×C22⋊#92418C2211yescharacteristic
29#16839680—22no—
30#16839664C2212yes—
31#16839617—21no—
32#16839617—21no—
33#16839524C2212yescomplemented
34#16839517C2212yescomplemented
35C2×#117×C61—21no—
36C22×C4⋊C4×C61C2211yescharacteristic
37C61×#389C2212yescomplemented
38C61×#387—22no—
39C61×#381—21no—
40C61×#380C2211yescharacteristic
41C4×C4⋊C4×C61C2212yescomplemented
42C4×C22⋊C4×C61C2211yescharacteristic
43C24×D61—21no—
44C22×Dic122C2311yescharacteristic
45C22×D244—22no—
46C22×D244—24no—
47C22×D244—22no—
48C22×D244—21no—
49C22×D244—42no—
50C22×D244C2312yes—
51C22×#92418—22no—
52C22×#92418—42no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55C22×#92418C2312yes—
56#14519765—24no—
57#14519765—42no—
58#14519756—22no—
59#14519756—22no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—21no—
62C2×C22⋊Dic61—21no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67#14519708—22no—
68#14519708—22no—
69C4⋊#92418—42no—
70#14519700—22no—
71#14519700—22no—
72#14519700—42no—
73#14519697—24no—
74#14519697—24no—
75#14519697—24no—
76#14519697—42no—
77C22⋊#92418—22no—
78C23×C4×C61—21no—
79C23×C4×C61C2311yescharacteristic
80#117×C61—44no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61D414yes—
85C2×C4⋊C4×C61C2312yes—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C22⋊C4×C61—21no—
89C2×C22⋊C4×C61D411yescharacteristic
90C2×C22⋊C4×C61D411yescharacteristic
91C2×C42×C61—22no—
92C2×C42×C61C2312yes—
93#95×C61—22no—
94#95×C61—21no—
95#95×C61—22no—
96#95×C61D412yes—
97C24×C61C2×D411yescharacteristic
98C23×D61—22no—
99C23×D61—44no—
100C23×D61—42no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61—82no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—24no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D412yes—
116C22×C4×C61#4112yes—
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—21no—
119C22×Dic61—21no—
120C2×D244—28no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—48no—
127C2×D244—44no—
128C2×D244—44no—
129C2×D244—42no—
130C2×D244#4114yes—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—44no—
141C4⋊C4×C61C2×D412yes—
142C4⋊C4×C61C2×D412yes—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—22no—
145C22⋊C4×C61—42no—
146C42×C61—22no—
147C42×C61—22no—
148C42×C61—44no—
149C22⋊Dic61—42no—
150#11758198—44no—
151#11758198—44no—
152#11758198—82no—
153#11758198—82no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156#11758196—42no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164C2×#92418—44no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61#12112yes—
168C23×C61#13512yes—
169C23×C61AGL(1,ℤ/8)12yes—
170C23×C61C22×D411yescharacteristic
171C22×D61—24no—
172C22×D61—48no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—88no—
180C22×D61—88no—
181C22×D61—84no—
182C22×D61—84no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—48no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61#12112yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4112yes—
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61H(5,𝔽2)12yes—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212D244—24no—
213D244—24no—
214D244—48no—
215D244—42no—
216D244—42no—
217Dic122—41no—
218Dic122—41no—
219Dic122—42no—
220#92418—41no—
221#92418—41no—
222#92418—42no—
223#24477—611nomaximal, Sylow
224C22×C61—22no—
225C22×C61—24no—
226C22×C61—22no—
227C22×C61—24no—
228C22×C61#46314yes—
229C22×C61C2×#12111yescharacteristic
230C22×C61#54612yes—
231C22×C61C2×#13511yescharacteristic
232C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
233C22×C61#57712yes—
234C2×D61—44no—
235C2×D61—44no—
236C2×D61—44no—
237C2×D61—816no—
238C2×D61—82no—
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C4×C61—22no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—82no—
248C4×C61#52511yescharacteristic
249C4×C61#53511yescharacteristic
250C4×C61#55112yes—
251Dic61—81no—
252Dic61—81no—
253C2×#483—611no—
254C22×#102—611no—
255#2435—611no—
256#2344D6111yescharacteristic, complemented
257#2065—612no—
258#2042—612no—
259#1983—612no—
260#1926—612no—
261#1892—611no—
262#1850—611no—
263#1786—611no—
264C2×C61—24no—
265C2×C61#241111yescharacteristic
266C2×C61C2×#46311yescharacteristic
267C2×C61#304311yescharacteristic
268C2×C61#329612yes—
269C2×C61#330612yes—
270D61—84no—
271C23×D4—611no—
272C23×C8—611no—
273C2×#124—612no—
274C2×C22⋊Q8—611no—
275C2×#117—21no—
276C22×C4⋊C4C2×D6111yescharacteristic
277#483—1224no—
278C2×#106—611no—
279C2×#106—611no—
280C2×C4⋊C8—612no—
281C2×#103—611no—
282C2×#103—611no—
283C2×#103—612no—
284C2×#102—612no—
285C2×#102—612no—
286C2×#102—612no—
287C2×#102—612no—
288C2×#102—1224no—
289C2×C22⋊C8—611no—
290#394—1222no—
291#390—612no—
292#389C2×D6112yescomplemented
293#387—22no—
294#381—21no—
295#380C2×D6111yescharacteristic
296#379—1221no—
297#379—1221no—
298C4×C4⋊C4C2×D6112yescomplemented
299C4×C22⋊C4C2×D6111yescharacteristic
300#340—612no—
301#336—612no—
302C61#2447711yescharacteristic, complemented, Sylow
303C25—1221no—
304C22×Q8—611no—
305C22×D4—612no—
306C22×D4—1224no—
307C22×D4—1222no—
308C22×D4—1222no—
309C22×D4—1221no—
310C22×D4—2442no—
311C23×C4—21no—
312C23×C4C22×D6111yescharacteristic
313C22×C8—612no—
314C22×C8—611no—
315C22×C8—611no—
316C22×C8—1222no—
317C22×C8—2442no—
318#124—1224no—
319#124—2442no—
320C22⋊Q8—1222no—
321C22⋊Q8—1222no—
322#117—44no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4D24414yes—
329C2×C4⋊C4C22×D6112yes—
330C2×C4⋊C4C22×D6111yescharacteristic
331C2×C4⋊C4C22×D6111yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1222no—
335C2×C22⋊C4—1221no—
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4C22⋊D6111yescharacteristic
339C2×C22⋊C4C22⋊D6111yescharacteristic
340C2×C42—22no—
341C2×C42C22×D6112yes—
342#106—1222no—
343#106—1222no—
344C4⋊C8—2442no—
345#103—1222no—
346#103—1222no—
347#103—2442no—
348#102—1224no—
349#102—1224no—
350#102—1224no—
351#102—2442no—
352C22⋊C8—1222no—
353#95—22no—
354#95—21no—
355#95—22no—
356#95C22⋊D6112yes—
357C24—1222no—
358C24—4882no—
359C24—2444no—
360C24—2442no—
361C24—2442no—
362C24—2442no—
363C24C2×C22⋊D6111yescharacteristic
364C2×Q8—1221no—
365C2×Q8—1222no—
366C2×Q8—1221no—
367C2×Q8—1222no—
368C2×D4—614no—
369C2×D4—1228no—
370C2×D4—1222no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—2448no—
376C2×D4—2444no—
377C2×D4—2444no—
378C2×D4—2442no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4C2×D24412yes—
390C22×C4#1175822212yes—
391C22×C4C2×C22⋊D6112yes—
392C22×C4C2×C22⋊D6112yes—
393C22×C4C2×C22⋊D6111yescharacteristic
394C22×C4C23×D6111yescharacteristic
395C2×C8—1221no—
396C2×C8—1221no—
397C2×C8—1222no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—2444no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—44no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—2442no—
412C4⋊C4C2×D24412yes—
413C4⋊C4C2×D24412yes—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—42no—
417C22⋊C4—4882no—
418C22⋊C4—4882no—
419C22⋊C4—2444no—
420C22⋊C4—2444no—
421C22⋊C4—2442no—
422C42—22no—
423C42—22no—
424C42—44no—
425C23—22no—
426C23—22no—
427C23—1224no—
428C23—4884no—
429C23—4888no—
430C23—4888no—
431C23—4884no—
432C23—2448no—
433C23—2442no—
434C23—2442no—
435C23—2442no—
436C23—2442no—
437C23—2442no—
438C23—2442no—
439C23#1451975112yes—
440C23#1451979712yes—
441C23#1451980312yes—
442C23C22×C22⋊D6111yescharacteristic
443Q8—2441no—
444Q8—2441no—
445Q8—2442no—
446D4—1224no—
447D4—1224no—
448D4—2448no—
449D4—2442no—
450D4—2442no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—48no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4#1451975112yes—
476C2×C4C22×D24411yescharacteristic
477C2×C4#41×D6112yes—
478C2×C4C2×#1175822211yescharacteristic
479C2×C4#1451982312yes—
480C8—2441no—
481C8—2441no—
482C8—2442no—
483C22—24no—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—48816no—
488C22—4882no—
489C22—4882no—
490C22—4882no—
491C22—4882no—
492C22—2444no—
493C22—2444no—
494C22—2444no—
495C22#1683997714yes—
496C22C2×#1451975111yescharacteristic
497C22#1684040012yes—
498C22C2×#1451979711yescharacteristic
499C22C2×#1451980311yescharacteristic
500C22#1684057012yes—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—82no—
507C4—4881no—
508C4—4881no—
509C4#1684027411yescharacteristic
510C4#1684033111yescharacteristic
511C4#1684043612yes—
512C2—24no—
513C2—4884no—
514C2#2144919811yescharacteristic
515C2C2×#1683997711yescharacteristic
516C2#2145534011yescharacteristic
517C2#2145764812yes—
518C2#2145777412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2344C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenter#2344D61
hypocenterC61#24477
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
1. C2×#16840092C2semidirect
4. #21447190C2semidirect
5. #21447060C2semidirect
6. #21446751C2semidirect
7. #21446446C2semidirect
8. #21446240C2semidirect
9. #21445999C2semidirect
11. C61×#2344C2semidirect
17. C2×#14519708C22semidirect
18. C2×#14519708C22semidirect
19. C2×C4⋊#92418C22semidirect
20. C2×#14519700C22semidirect
21. C2×#14519700C22semidirect
22. C2×#14519700C22semidirect
33. #16839524C22semidirect
34. #16839517C22semidirect
37. C61×#389C22semidirect
41. C4×C4⋊C4×C61C22semidirect
256. #2344D61semidirect
292. #389C2×D61semidirect
298. C4×C4⋊C4C2×D61semidirect
302. C61#24477semidirect

Maximal subgroups

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

subgroupindex
C2×#168400922
C22×#145196972
#214494702
#214471902
#214470602
#214467512
#214464462
#214462402
#214459992
#214457492
C61×#23442
#2447761