Finite Group Databasebeta

← #29601608

Subgroups — #29601608

Order 15616.

2022 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839650C212yesmaximal
2C2×#16839634C211yesmaximal, characteristic
3C2×#16839613C212yesmaximal, complemented
4C2×C22⋊C4×Dic61C212yesmaximal
5#21444056C211yesmaximal, characteristic
6#21444050C212yesmaximal, complemented
7#21444050C212yesmaximal, complemented
8C4×#14519684C212yesmaximal, complemented
9C61×#1437C211yesmaximal, characteristic
10C24×Dic61C2211yescharacteristic
11C22×#11758196C2212yes—
12C22×#11758196C2212yes—
13C22×C22⋊Dic61C2212yes—
14C22×C22⋊Dic61C2211yescharacteristic
15C22×C4×Dic61C2212yes—
16C22×C4×Dic61C2212yes—
17C22×C4×Dic61C2211yescharacteristic
18C22×C4×Dic61C2212yes—
19#16839650C4116yescomplemented
20#16839634C418yescomplemented
21#16839613—28no—
22C22⋊C4×Dic61—28no—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C22×C22⋊C4×C61C2212yes—
31C22×C22⋊C4×C61C2211yescharacteristic
32C22×C42×C61C2211yescharacteristic
33C2×#95×C61C2212yes—
34C2×#95×C61C2211yescharacteristic
35C23×Dic61—24no—
36C23×Dic61—24no—
37C23×Dic61—41no—
38C23×Dic61—41no—
39C23×Dic61C2×C414yes—
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43C23×Dic61C2312yes—
44C2×#11758196—24no—
45C2×#11758196—24no—
46C2×#11758196—24no—
47C2×#11758196—24no—
48C2×#11758196C2×C418yes—
49C2×#11758196C2×C418yescomplemented
50C2×C22⋊Dic61—24no—
51C2×C22⋊Dic61—24no—
52C2×C22⋊Dic61C2×C418yes—
53C2×C22⋊Dic61C2×C414yes—
54C2×C22⋊Dic61C2×C418yes—
55C2×C4×Dic61—24no—
56C2×C4×Dic61—24no—
57C2×C4×Dic61—24no—
58C2×C4×Dic61—24no—
59C2×C4×Dic61—24no—
60C2×C4×Dic61—24no—
61C2×C4×Dic61—24no—
62C2×C4×Dic61—42no—
63C2×C4×Dic61—41no—
64C2×C4×Dic61—42no—
65C2×C4×Dic61C2×C418yescomplemented
66#14519684—24no—
67#14519684—24no—
68#14519684—41no—
69#14519684C2×C418yes—
70#14519684C2×C418yescomplemented
71#14519684C2×C418yescomplemented
72#14519684D418yes—
73C25×C61C2311yescharacteristic
74C23×C4×C61C2311yescharacteristic
75C23×C4×C61C2312yes—
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2312yes—
78C2×C22⋊C4×C61—24no—
79C2×C22⋊C4×C61—24no—
80C2×C22⋊C4×C61—41no—
81C2×C22⋊C4×C61—41no—
82C2×C22⋊C4×C61C2×C418yes—
83C2×C22⋊C4×C61C2×C414yes—
84C2×C42×C61—24no—
85C2×C42×C61—41no—
86#95×C61—24no—
87#95×C61—41no—
88C24×C61—22no—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—21no—
93C24×C61—21no—
94C24×C61—22no—
95C24×C61C22×C412yes—
96C24×C61C22×C414yes—
97C24×C61C2411yescharacteristic
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61C22×C412yes—
117C22×C4×C61C22×C414yes—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C22×C414yes—
120C22×C4×C61C2×D414yes—
121C22×C4×C61#4114yes—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—216no—
141C22×Dic61—44no—
142C22×Dic61—44no—
143C22×Dic61—48no—
144C22×Dic61—44no—
145C22×Dic61C22×C414yes—
146C22×Dic61C22×C414yes—
147C22×Dic61C22×C414yes—
148C22×Dic61C22×C414yes—
149C22×Dic61C22×C414yes—
150C22×Dic61C22×C414yes—
151C22×Dic61C2×D414yes—
152C22×Dic61C2×D414yes—
153C22×Dic61#4114yes—
154C22×Dic61#4114yes—
155C22⋊C4×C61—216no—
156C22⋊C4×C61—44no—
157C42×C61—44no—
158C22⋊Dic61—216no—
159C22⋊Dic61C42116yescomplemented
160#11758196—216no—
161#11758196—216no—
162C4×Dic61—216no—
163C4×Dic61—44no—
164C4×Dic61—48no—
165C4×Dic61—48no—
166C23×C61—24no—
167C23×C61—24no—
168C23×C61—24no—
169C23×C61—24no—
170C23×C61—24no—
171C23×C61—24no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—24no—
175C23×C61—24no—
176C23×C61—41no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—41no—
186C23×C61C2×C4214yes—
187C23×C61C23×C411yescharacteristic
188C23×C61C23×C412yes—
189C23×C61C22×D412yes—
190C23×C61C2×#4112yes—
191C23×C61C2×#4112yes—
192C23×C61C2×#4112yes—
193C23×C61H(5,𝔽2)11yescharacteristic
194C23×C61H(5,𝔽2)11yescharacteristic
195C23×C612−1+412yes—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61C4×D418yes—
220C2×Dic61—24no—
221C2×Dic61—24no—
222C2×Dic61—24no—
223C2×Dic61—24no—
224C2×Dic61—24no—
225C2×Dic61—24no—
226C2×Dic61—24no—
227C2×Dic61—24no—
228C2×Dic61—24no—
229C2×Dic61—24no—
230C2×Dic61—24no—
231C2×Dic61—24no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—24no—
235C2×Dic61—24no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—416no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61C2×C4218yes—
247C2×Dic61C4×D418yes—
248C2×Dic61C4×D418yes—
249#13823—611nomaximal, Sylow
250C22×C61—28no—
251C22×C61—24no—
252C22×C61—42no—
253C22×C61—44no—
254C22×C61—44no—
255C22×C61—44no—
256C22×C61—44no—
257C22×C61—44no—
258C22×C61—42no—
259C22×C61C22×C4211yescharacteristic
260C22×C61C2×C4×D412yes—
261C22×C61C2×C4×D412yes—
262C22×C61C2×C4×D412yes—
263C22×C61C4×#4112yes—
264C22×C61C4×#4112yes—
265C22×C61C4×#4112yes—
266C22×C61#51811yescharacteristic
267C22×C61#51911yescharacteristic
268C22×C61#52012yes—
269C22×C61#52012yes—
270C22×C61D4211yescharacteristic
271C22×C61#54612yes—
272C22×C61#54712yes—
273C22×C61#54812yes—
274C22×C61#55111yescharacteristic
275C22×C61#55111yescharacteristic
276C22×C61#55212yes—
277C22×C61#55312yes—
278C22×C61#55612yes—
279C22×C61#55911yescharacteristic
280C4×C61—28no—
281C4×C61—42no—
282C4×C61—44no—
283C4×C61—42no—
284Dic61—28no—
285Dic61—28no—
286Dic61—28no—
287Dic61—44no—
288C2×#388—612no—
289C2×#385—611no—
290C2×C4×C4⋊C4—612no—
291C2×C4×C22⋊C4—612no—
292#1437—611no—
293#1437D6111yescharacteristic
294#1436—612no—
295#1436—612no—
296C4×#95—612no—
297C2×C61—44no—
298C2×C61C42×D412yes—
299C2×C61#227411yescharacteristic
300C2×C61#235811yescharacteristic
301C2×C61#235912yes—
302C2×C61#236112yes—
303C2×C61#236212yes—
304C2×C61#236712yes—
305C2×C61#237312yes—
306C2×C61#237411yescharacteristic
307C24×C4—611no—
308C22×C4⋊C4—612no—
309C22×C4⋊C4—612no—
310C22×C22⋊C4—611no—
311C22×C22⋊C4—612no—
312C22×C22⋊C4C2×D6112yes—
313C22×C22⋊C4C2×D6111yescharacteristic
314C22×C42—612no—
315C22×C42—612no—
316C22×C42—612no—
317C22×C42—611no—
318C22×C42C2×D6111yescharacteristic
319#388—6116no—
320#385—618no—
321C4×C4⋊C4—1228no—
322C4×C22⋊C4—1228no—
323C2×#95—612no—
324C2×#95—612no—
325C2×#95—612no—
326C2×#95—612no—
327C2×#95—612no—
328C2×#95—612no—
329C2×#95—611no—
330C2×#95C2×D6112yes—
331C2×#95C2×D6111yescharacteristic
332C61#1382311yescharacteristic, complemented, Sylow
333C25C22×D6111yescharacteristic
334C23×C4—612no—
335C23×C4—612no—
336C23×C4—612no—
337C23×C4—612no—
338C23×C4—614no—
339C23×C4—1224no—
340C23×C4—1224no—
341C23×C4—2441no—
342C23×C4—2441no—
343C23×C4C22×D6111yescharacteristic
344C23×C4C22×D6112yes—
345C23×C4C22×D6111yescharacteristic
346C23×C4C22×D6112yes—
347C2×C4⋊C4—618no—
348C2×C4⋊C4—618no—
349C2×C4⋊C4—1224no—
350C2×C4⋊C4—1224no—
351C2×C4⋊C4—1224no—
352C2×C4⋊C4—1224no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—41no—
356C2×C22⋊C4—41no—
357C2×C22⋊C4—614no—
358C2×C22⋊C4—618no—
359C2×C22⋊C4—618no—
360C2×C22⋊C4—1224no—
361C2×C22⋊C4—1224no—
362C2×C22⋊C4C4×D6118yes—
363C2×C22⋊C4C4×D6114yes—
364C2×C42—24no—
365C2×C42—41no—
366C2×C42—618no—
367C2×C42—1224no—
368C2×C42—1224no—
369C2×C42—1224no—
370C2×C42—1224no—
371C2×C42—1224no—
372C2×C42—1224no—
373C2×C42—1224no—
374C2×C42—2442no—
375C2×C42—2442no—
376C2×C42—2441no—
377#95—24no—
378#95—41no—
379#95—618no—
380#95—618no—
381#95—618no—
382#95—618no—
383#95—1224no—
384#95—1224no—
385#95—2441no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—21no—
391C24—21no—
392C24—22no—
393C24C2×C4×D6114yes—
394C24C2×C4×D6112yes—
395C24C23×D6111yescharacteristic
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—614no—
415C22×C4—614no—
416C22×C4—614no—
417C22×C4—614no—
418C22×C4—614no—
419C22×C4—614no—
420C22×C4—614no—
421C22×C4—614no—
422C22×C4—614no—
423C22×C4—614no—
424C22×C4—12216no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—2444no—
444C22×C4—2444no—
445C22×C4—2448no—
446C22×C4—2444no—
447C22×C4C2×C4×D6112yes—
448C22×C4C2×C4×D6114yes—
449C22×C4C2×C4×D6112yes—
450C22×C4C2×C4×D6114yes—
451C22×C4D4×D6114yes—
452C22×C4#1175822414yes—
453C4⋊C4—12216no—
454C4⋊C4—12216no—
455C22⋊C4—216no—
456C22⋊C4—44no—
457C22⋊C4—6116no—
458C22⋊C4—12216no—
459C42—44no—
460C42—12216no—
461C42—2448no—
462C42—2444no—
463C42—2448no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23C42×D6114yes—
485C23C22×C4×D6111yescharacteristic
486C23C22×C4×D6112yes—
487C23C2×D4×D6112yes—
488C23#41×D6112yes—
489C23#41×D6112yes—
490C23C2×#1175822412yes—
491C23#1451982211yescharacteristic
492C23#1451982211yescharacteristic
493C23#1451982412yes—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—618no—
518C2×C4—618no—
519C2×C4—618no—
520C2×C4—1224no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—1224no—
529C2×C4—1224no—
530C2×C4—1224no—
531C2×C4—1224no—
532C2×C4—1224no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1224no—
541C2×C4—24416no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4#1451973718yes—
547C22—28no—
548C22—24no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22C2×C42×D6111yescharacteristic
557C22C4×D4×D6112yes—
558C22C4×D4×D6112yes—
559C22C2×#1451973712yes—
560C22C4×#1175822412yes—
561C22C4×#1175822412yes—
562C22#1684023012yes—
563C22#1684023511yescharacteristic
564C22#1684023911yescharacteristic
565C22#1684024112yes—
566C22#1684024112yes—
567C22#1684039711yescharacteristic
568C22#1684040112yes—
569C22#1684041212yes—
570C22#1684042012yes—
571C22#1684044211yescharacteristic
572C22#1684044311yescharacteristic
573C22#1684044912yes—
574C22#1684045912yes—
575C22#1684047912yes—
576C22#1684049611yescharacteristic
577C4—28no—
578C4—42no—
579C4—44no—
580C4—42no—
581C4—1228no—
582C4—1228no—
583C4—1228no—
584C4—2444no—
585C2—44no—
586C2C4×#1451973712yes—
587C2#2144817411yescharacteristic
588C2#2144880011yescharacteristic
589C2#2144881712yes—
590C2#2144884012yes—
591C2#2144884312yes—
592C2#2144888512yes—
593C2#2144893612yes—
594C2#2144895211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC22×C61C22×C42
FittingC61×#1437C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenter#1437D61
hypocenterC61#13823
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
3. C2×#16839613C2semidirect
6. #21444050C2semidirect
7. #21444050C2semidirect
8. C4×#14519684C2semidirect
19. #16839650C4semidirect
20. #16839634C4semidirect
49. C2×#11758196C2×C4semidirect
65. C2×C4×Dic61C2×C4semidirect
70. #14519684C2×C4semidirect
71. #14519684C2×C4semidirect
159. C22⋊Dic61C42semidirect
332. C61#13823semidirect

Maximal subgroups

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

subgroupindex
C2×#168396502
C2×#168396342
C2×#168396132
C2×C22⋊C4×Dic612
#214440562
#214440502
C4×#145196842
C61×#14372
#1382361