Finite Group Databasebeta

← #29718361

Subgroups — #29718361

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×#16840090C211yesmaximal, characteristic, complemented
2C22×#14519698C211yesmaximal, characteristic
3#21448686C211yesmaximal, characteristic, complemented
4#21447193C212yesmaximal, complemented
5#21447055C212yesmaximal, complemented
6#21446752C212yesmaximal, complemented
7#21446443C212yesmaximal, complemented
8#21446243C211yesmaximal, characteristic, complemented
9#21445999C211yesmaximal, characteristic, complemented
10#21445750C211yesmaximal, characteristic
11C61×#2435C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C2×#14519733—21no—
14C22×C4⋊Dic61C2211yescharacteristic
15#16840090—24no—
16C2×#14519708C2211yescharacteristic, complemented
17C2×#14519708C2211yescharacteristic, complemented
18C2×C4⋊#92418C2212yescomplemented
19C2×#14519701C2211yescharacteristic, complemented
20C2×#14519701C2211yescharacteristic, complemented
21C2×#14519701C2212yescomplemented
22C2×#14519698—24no—
23C2×#14519698C2212yes—
24C2×#14519698C2212yes—
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27C2×C22⋊#92418C2211yescharacteristic
28#16839660C2212yescomplemented
29#16839647—22no—
30#16839623—21no—
31#16839622C2211yescharacteristic
32C4×C4⋊Dic61C2212yescomplemented
33#16839524C2212yescomplemented
34#16839517C2212yescomplemented
35C23×D4×C61C2211yescharacteristic
36C2×#124×C61C2212yes—
37C2×C22⋊Q8×C61C2211yescharacteristic
38C61×#394—22no—
39C61×#390C2212yes—
40C61×#379—21no—
41C61×#379—21no—
42C4×C22⋊C4×C61C2211yescharacteristic
43C23×Dic61—21no—
44C22×#92418—22no—
45C22×#92418—42no—
46C22×#92418C2311yescharacteristic
47C22×#92418C2311yescharacteristic
48C22×#92418C2312yes—
49#14519733—44no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61D414yes—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—22no—
57C2×C22⋊Dic61—21no—
58C2×C4×Dic61—22no—
59#14519708—22no—
60#14519708—22no—
61C4⋊#92418—42no—
62#14519701—22no—
63#14519701—22no—
64#14519701—42no—
65#14519698—24no—
66#14519698—24no—
67#14519698—24no—
68#14519698—42no—
69C22⋊#92418—22no—
70#14519684—22no—
71#14519684—21no—
72#14519684—22no—
73C25×C61—21no—
74C22×Q8×C61C2311yescharacteristic
75C22×D4×C61—24no—
76C22×D4×C61—22no—
77C22×D4×C61—21no—
78C22×D4×C61—22no—
79C22×D4×C61—42no—
80C22×D4×C61C2312yes—
81C23×C4×C61C2311yescharacteristic
82#124×C61—24no—
83#124×C61—42no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—22no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C22⋊C4×C61—21no—
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61D411yescharacteristic
94C2×C22⋊C4×C61D411yescharacteristic
95C2×C42×C61C2312yes—
96#95×C61D412yes—
97C24×C61—22no—
98C24×C61—44no—
99C24×C61—42no—
100C24×C61—42no—
101C24×C61—42no—
102C24×C61—82no—
103C24×C61C2×D411yescharacteristic
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×D4×C61—28no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—48no—
115C2×D4×C61—44no—
116C2×D4×C61—44no—
117C2×D4×C61—42no—
118C2×D4×C61#4114yes—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D412yes—
127C22×C4×C61#4112yes—
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—24no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—42no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—44no—
141C22⋊C4×C61—44no—
142C22⋊C4×C61—42no—
143C22⋊C4×C61—82no—
144C22⋊C4×C61—82no—
145C42×C61—22no—
146C42×C61—22no—
147C22⋊Dic61—42no—
148C4⋊Dic61—24no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C4⋊Dic61C2×D412yes—
154C4⋊Dic61C2×D412yes—
155#11758196—44no—
156C4×Dic61—44no—
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—24no—
168C23×C61—48no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—88no—
176C23×C61—88no—
177C23×C61—84no—
178C23×C61—84no—
179C23×C61#12112yes—
180C23×C61#13512yes—
181C23×C61AGL(1,ℤ/8)12yes—
182C23×C61C22×D411yescharacteristic
183Q8×C61—41no—
184Q8×C61—41no—
185Q8×C61—42no—
186D4×C61—24no—
187D4×C61—24no—
188D4×C61—48no—
189D4×C61—42no—
190D4×C61—42no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—24no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61#12112yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4112yes—
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61H(5,𝔽2)12yes—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—48no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—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—44no—
229C22×C61—44no—
230C22×C61—44no—
231C22×C61—816no—
232C22×C61—82no—
233C22×C61—82no—
234C22×C61—82no—
235C22×C61—82no—
236C22×C61#46314yes—
237C22×C61C2×#12111yescharacteristic
238C22×C61#54612yes—
239C22×C61C2×#13511yescharacteristic
240C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
241C22×C61#57712yes—
242C4×C61—22no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61—81no—
246C4×C61—81no—
247C4×C61#52511yescharacteristic
248C4×C61#53511yescharacteristic
249C4×C61#55112yes—
250Dic61—42no—
251Dic61—42no—
252Dic61—82no—
253C2×#483—611no—
254C22×#102—611no—
255#2435D6111yescharacteristic
256#2344—611no—
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—84no—
266C2×C61#241111yescharacteristic
267C2×C61C2×#46311yescharacteristic
268C2×C61#304311yescharacteristic
269C2×C61#329612yes—
270C2×C61#330612yes—
271C23×D4C2×D6111yescharacteristic
272C23×C8—611no—
273C2×#124C2×D6112yes—
274C2×C22⋊Q8C2×D6111yescharacteristic
275C2×#117—1221no—
276C22×C4⋊C4—611no—
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—22no—
291#390C2×D6112yes—
292#389—612no—
293#387—1222no—
294#381—1221no—
295#380—611no—
296#379—21no—
297#379—21no—
298C4×C4⋊C4—612no—
299C4×C22⋊C4C2×D6111yescharacteristic
300#340—612no—
301#336—612no—
302C61#2447711yescharacteristic, complemented, Sylow
303C25—21no—
304C22×Q8C22×D6111yescharacteristic
305C22×D4—24no—
306C22×D4—22no—
307C22×D4—22no—
308C22×D4—21no—
309C22×D4—42no—
310C22×D4C22×D6112yes—
311C23×C4—1221no—
312C23×C4C22×D6111yescharacteristic
313C22×C8—612no—
314C22×C8—611no—
315C22×C8—611no—
316C22×C8—1222no—
317C22×C8—2442no—
318#124—24no—
319#124—42no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322#117—2444no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—614no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—1222no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4C22×D6111yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4C22⋊D6111yescharacteristic
339C2×C22⋊C4C22⋊D6111yescharacteristic
340C2×C42—1222no—
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—1222no—
354#95—1221no—
355#95—1222no—
356#95C22⋊D6112yes—
357C24—22no—
358C24—44no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—82no—
363C24C2×C22⋊D6111yescharacteristic
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×D4—28no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—48no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4#1175822414yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—1224no—
386C22×C4—1222no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4D4×D6112yes—
390C22×C4#1175822412yes—
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—42no—
406C4⋊C4—612no—
407C4⋊C4—612no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4—1224no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—2444no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—42no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C22⋊C4—2442no—
422C42—22no—
423C42—22no—
424C42—2444no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—48no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—88no—
436C23—88no—
437C23—84no—
438C23—84no—
439C23#1451974812yes—
440C23#1451979712yes—
441C23#1451980212yes—
442C23C22×C22⋊D6111yescharacteristic
443Q8—41no—
444Q8—41no—
445Q8—42no—
446D4—24no—
447D4—24no—
448D4—48no—
449D4—42no—
450D4—42no—
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—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2448no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4#1451974812yes—
476C2×C4C2×D4×D6111yescharacteristic
477C2×C4C2×#1175822412yes—
478C2×C4C2×#1175822411yescharacteristic
479C2×C4#1451982212yes—
480C8—2441no—
481C8—2441no—
482C8—2442no—
483C22—24no—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—816no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22#1683997414yes—
496C22C2×#1451974811yescharacteristic
497C22#1684040612yes—
498C22C2×#1451979711yescharacteristic
499C22C2×#1451980211yescharacteristic
500C22#1684057312yes—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—81no—
505C4—81no—
506C4—4882no—
507C4—2442no—
508C4—2442no—
509C4#1684026811yescharacteristic
510C4#1684032711yescharacteristic
511C4#1684044312yes—
512C2—24no—
513C2—84no—
514C2#2144920011yescharacteristic
515C2C2×#1683997411yescharacteristic
516C2#2145533611yescharacteristic
517C2#2145765412yes—
518C2#2145778012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2435C2
socleC23×C61C22×D4
prime radicalC23×D4×C61C22
hypercenter#2435D61
hypocenterC61#24477
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
1. C2×#16840090C2semidirect
3. #21448686C2semidirect
4. #21447193C2semidirect
5. #21447055C2semidirect
6. #21446752C2semidirect
7. #21446443C2semidirect
8. #21446243C2semidirect
9. #21445999C2semidirect
16. C2×#14519708C22semidirect
17. C2×#14519708C22semidirect
18. C2×C4⋊#92418C22semidirect
19. C2×#14519701C22semidirect
20. C2×#14519701C22semidirect
21. C2×#14519701C22semidirect
28. #16839660C22semidirect
32. C4×C4⋊Dic61C22semidirect
33. #16839524C22semidirect
34. #16839517C22semidirect
302. C61#24477semidirect

Maximal subgroups

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

subgroupindex
C2×#168400902
C22×#145196982
#214486862
#214471932
#214470552
#214467522
#214464432
#214462432
#214459992
#214457502
C61×#24352
#2447761