Finite Group Databasebeta

← #30609974

Subgroups — #30609974

Order 15616.

1738 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840732C212yesmaximal, complemented
2#21461075C214yesmaximal, complemented
3#21461067C212yesmaximal, complemented
4#21461064C212yesmaximal, complemented
5#21461055C214yesmaximal, complemented
6C22×D4×Dic61C211yesmaximal, characteristic
7C22×#11758217—22no—
8#16840732—22no—
9#16840732—22no—
10#16840732C2214yes—
11#16840732C2214yescomplemented
12C2×C22⋊#92420—22no—
13C2×C22⋊#92420C2214yescomplemented
14#16840607C2218yescomplemented
15#16840605C2218yescomplemented
16C24×Dic61C411yescharacteristic
17C24×Dic61C2211yescharacteristic
18C2×D4×Dic61—22no—
19C2×D4×Dic61—22no—
20C2×D4×Dic61—22no—
21C2×D4×Dic61—22no—
22C2×D4×Dic61C412yes—
23C2×D4×Dic61C412yes—
24C2×D4×Dic61C2212yes—
25C2×D4×Dic61C2212yes—
26C22×C4⋊Dic61C2211yescharacteristic
27C22×C22⋊Dic61C411yescharacteristic
28C22×C22⋊Dic61C2211yescharacteristic
29C22×C4×Dic61C411yescharacteristic
30C23×D4×C61C411yescharacteristic
31C2×#11758217—24no—
32C2×#11758217—24no—
33C2×#11758217—48no—
34C22×#92420—24no—
35C22⋊#92420—24no—
36C22⋊#92420—24no—
37C22⋊#92420—24no—
38C22⋊#92420C2318yes—
39C23×Dic61—21no—
40C23×Dic61—21no—
41C23×Dic61—21no—
42C23×Dic61—21no—
43C23×Dic61—44no—
44C23×Dic61—44no—
45C23×Dic61C2×C411yescharacteristic
46C23×Dic61C2×C411yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C23×Dic61C2×C412yes—
49C23×Dic61C2×C412yes—
50C23×Dic61C2×C411yescharacteristic
51C23×Dic61C2×C411yescharacteristic
52C23×Dic61D412yes—
53C23×Dic61D412yes—
54C23×Dic61D412yes—
55C23×Dic61D412yes—
56C23×Dic61C2311yescharacteristic
57C23×Dic61C2311yescharacteristic
58C23×Dic61C2311yescharacteristic
59D4×Dic61—24no—
60D4×Dic61—24no—
61D4×Dic61—24no—
62D4×Dic61—24no—
63D4×Dic61—24no—
64D4×Dic61—24no—
65D4×Dic61—24no—
66D4×Dic61—24no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61C2×C412yes—
70C2×C4⋊Dic61C2312yes—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61C2×C412yes—
76C2×C22⋊Dic61C2×C412yes—
77C2×C22⋊Dic61C2×C412yes—
78C2×C22⋊Dic61C2312yes—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2×C412yes—
84C2×C4×Dic61C2×C412yes—
85C25×C61C2×C411yescharacteristic
86C25×C61C2×C411yescharacteristic
87C22×D4×C61—22no—
88C22×D4×C61—22no—
89C22×D4×C61—28no—
90C22×D4×C61C2×C412yes—
91C22×D4×C61C2×C412yes—
92C23×C4×C61C2×C411yescharacteristic
93C24×C61—21no—
94C24×C61—21no—
95C24×C61—24no—
96C24×C61—24no—
97C24×C61—42no—
98C24×C61—42no—
99C24×C61—42no—
100C24×C61—42no—
101C24×C61C22⋊C412yes—
102C24×C61C22⋊C412yes—
103C24×C61C22×C411yescharacteristic
104C24×C61C22×C411yescharacteristic
105C24×C61C22×C412yes—
106C24×C61C22×C411yescharacteristic
107C2×D4×C61—216no—
108C2×D4×C61—216no—
109C2×D4×C61—24no—
110C2×D4×C61—24no—
111C2×D4×C61—24no—
112C2×D4×C61—24no—
113C2×D4×C61—24no—
114C2×D4×C61M1618yescomplemented
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—24no—
118C22×C4×C61C22×C412yes—
119C22×Dic61—24no—
120C22×Dic61—28no—
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—42no—
130C22×Dic61—42no—
131C22×Dic61—44no—
132C22×Dic61—44no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61—44no—
136C22×Dic61—42no—
137C22×Dic61—44no—
138C22×Dic61—44no—
139C22×Dic61—44no—
140C22×Dic61—44no—
141C22×Dic61—44no—
142C22×Dic61—44no—
143C22×Dic61—42no—
144C22×Dic61—42no—
145C22×Dic61C22⋊C411yescharacteristic
146C22×Dic61C22⋊C411yescharacteristic
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C22⋊C412yes—
149C22×Dic61C22⋊C412yes—
150C22×Dic61C22⋊C412yes—
151C22×Dic61C22⋊C412yes—
152C22×Dic61C22⋊C411yescharacteristic
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C411yescharacteristic
155C22×Dic61C22×C412yes—
156C22×Dic61C22×C412yes—
157C22×Dic61C22×C411yescharacteristic
158C22×Dic61C2×D412yes—
159C22×Dic61C2×D412yes—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162C22×Dic61C2×D412yes—
163C22×Dic61C2×D412yes—
164C22×Dic61C2411yescharacteristic
165#11758217—48no—
166#11758217—48no—
167C2×#92420—28no—
168C2×#92420—48no—
169C22⋊Dic61—24no—
170C22⋊Dic61—24no—
171C22⋊Dic61—24no—
172C22⋊Dic61—24no—
173C4⋊Dic61—24no—
174C4⋊Dic61—24no—
175C4×Dic61—22no—
176C4×Dic61—22no—
177C4×Dic61—22no—
178C4×Dic61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—24no—
183C23×C61—24no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—24no—
188C23×C61—24no—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61—42no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—42no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—42no—
198C23×C61—416no—
199C23×C61—416no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61C2×C22⋊C412yes—
206C23×C61C2×C22⋊C412yes—
207C23×C61C2×C22⋊C411yescharacteristic
208C23×C61C2×C22⋊C411yescharacteristic
209C23×C61C2×M1614yes—
210C23×C61C2×M1614yescomplemented
211C23×C61C23×C411yescharacteristic
212D4×C61—216no—
213D4×C61—216no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61C2×M1614yescomplemented
219C2×Dic61—24no—
220C2×Dic61—22no—
221C2×Dic61—24no—
222C2×Dic61—24no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—24no—
228C2×Dic61—24no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—41no—
238C2×Dic61—44no—
239C2×Dic61—44no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—44no—
245C2×Dic61—44no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61#9914yes—
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C22×D412yes—
255#92420—48no—
256#17362—611nomaximal, Sylow
257C22×C61—28no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—22no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—24no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—44no—
271C22×C61—48no—
272C22×C61—48no—
273C22×C61—44no—
274C22×C61—44no—
275C22×C61—44no—
276C22×C61—48no—
277C22×C61—48no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61#40714yes—
281C22×C61C2×#9912yes—
282C22×C61#41312yes—
283C22×C61C22×C22⋊C411yescharacteristic
284C22×C61C22×M1612yes—
285C4×C61—24no—
286Dic61—22no—
287Dic61—21no—
288Dic61—21no—
289Dic61—22no—
290Dic61—42no—
291Dic61—42no—
292Dic61C2×#9912yes—
293Dic61C2×#9912yes—
294C22×C4×D4—611no—
295C2×#407—612no—
296#1465—614no—
297#1463—612no—
298#1462—612no—
299#1459—614no—
300C2×C61—22no—
301C2×C61—22no—
302C2×C61—22no—
303C2×C61—24no—
304C2×C61—42no—
305C2×C61—44no—
306C2×C61C2×#40711yescharacteristic
307C2×C61C22×#9911yescharacteristic
308C2×C61C2×#41311yescharacteristic
309C23×D4#692311yescharacteristic
310C24×C4—611no—
311C24×C4—611no—
312C22×M16—1222no—
313C2×C4×D4—612no—
314C2×C4×D4—612no—
315C2×C4×D4—612no—
316C2×C4×D4—612no—
317C2×C4×D4—1222no—
318C2×C4×D4—1222no—
319C2×C4×D4—1222no—
320C2×C4×D4—1222no—
321C22×C4⋊C4—611no—
322C22×C22⋊C4—611no—
323C22×C22⋊C4—611no—
324C22×C42—611no—
325#407—614no—
326#407—614no—
327#407—1222no—
328#407—1222no—
329C2×C22⋊C8—614no—
330C2×C22⋊C8—1222no—
331#324—618no—
332#323—618no—
333C61#1736211yescharacteristic, complemented, Sylow
334C25C2×#692311yescharacteristic
335C25C2×#692311yescharacteristic
336C22×D4—22no—
337C22×D4—22no—
338C22×D4—28no—
339C22×D4C2×#692312yes—
340C22×D4C2×#692312yes—
341C23×C4—612no—
342C23×C4—611no—
343C23×C4—612no—
344C23×C4—612no—
345C23×C4—612no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—612no—
354C23×C4—612no—
355C23×C4—1221no—
356C23×C4—1221no—
357C23×C4—1221no—
358C23×C4—1221no—
359C23×C4—2444no—
360C23×C4—2444no—
361C23×C4C2×#692311yescharacteristic
362C2×M16—1224no—
363C2×M16—1224no—
364C2×M16—2448no—
365C22×C8—1224no—
366C4×D4—1224no—
367C4×D4—1224no—
368C4×D4—1224no—
369C4×D4—1224no—
370C4×D4—1224no—
371C4×D4—1224no—
372C4×D4—1224no—
373C4×D4—1224no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C22⋊C4—612no—
379C2×C22⋊C4—612no—
380C2×C22⋊C4—612no—
381C2×C22⋊C4—612no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C22⋊C4—1222no—
386C2×C42—612no—
387C2×C42—612no—
388C2×C42—1221no—
389C2×C42—1221no—
390C2×C42—1221no—
391C2×C42—1221no—
392C22⋊C8—618no—
393C22⋊C8—1224no—
394C22⋊C8—1224no—
395C22⋊C8—1224no—
396C24—21no—
397C24—21no—
398C24—24no—
399C24—24no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24C22⋊#692312yes—
405C24C22⋊#692312yes—
406C24C22×#692311yescharacteristic
407C24C22×#692311yescharacteristic
408C24C22×#692312yes—
409C24C22×#692311yescharacteristic
410C2×D4—216no—
411C2×D4—216no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4#1175821718yescomplemented
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—612no—
426C22×C4—612no—
427C22×C4—612no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—1228no—
442C22×C4—1224no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—2444no—
452C22×C4—2442no—
453C22×C4—2444no—
454C22×C4—2442no—
455C22×C4—2444no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4—2444no—
460C22×C4—2444no—
461C22×C4—2444no—
462C22×C4—2444no—
463C22×C4—2444no—
464C22×C4—2444no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4C22×#692312yes—
468M16—2448no—
469M16—2448no—
470C2×C8—1228no—
471C2×C8—2448no—
472C4⋊C4—1224no—
473C4⋊C4—1224no—
474C22⋊C4—1224no—
475C22⋊C4—1224no—
476C22⋊C4—1224no—
477C22⋊C4—1224no—
478C42—1222no—
479C42—1222no—
480C42—1222no—
481C42—1222no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—416no—
504C23—416no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23C2×C22⋊#692312yes—
509C23C2×C22⋊#692312yes—
510C23C2×C22⋊#692311yescharacteristic
511C23C2×C22⋊#692311yescharacteristic
512C23C2×#1175821714yes—
513C23C2×#1175821714yescomplemented
514C23C23×#692311yescharacteristic
515D4—216no—
516D4—216no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—614no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—1224no—
529C2×C4—1224no—
530C2×C4—1224no—
531C2×C4—1224no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1224no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—2444no—
541C2×C4—2444no—
542C2×C4—2441no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2441no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2444no—
550C2×C4—2444no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4C2×#1175821714yescomplemented
558C8—2448no—
559C22—28no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—48no—
574C22—48no—
575C22—44no—
576C22—48no—
577C22—48no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22#1684073214yes—
583C22D61⋊#9912yes—
584C22#1684074912yes—
585C22C22×C22⋊#692311yescharacteristic
586C22C22×#1175821712yes—
587C4—24no—
588C4—612no—
589C4—612no—
590C4—1222no—
591C4—1222no—
592C4—1221no—
593C4—1221no—
594C4—2442no—
595C4—2442no—
596C2—22no—
597C2—22no—
598C2—22no—
599C2—24no—
600C2—42no—
601C2—44no—
602C2C2×#1684073211yescharacteristic
603C2C2×D61⋊#9911yescharacteristic
604C2C2×#1684074911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
Fitting, prime radicalC23×D4×C61C4
socleC22×C61C22×C22⋊C4
cosocleC22×Dic61C24
hypercenterC23×D4#6923
hypocenterC61#17362
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16840732C2semidirect
2. #21461075C2semidirect
3. #21461067C2semidirect
4. #21461064C2semidirect
5. #21461055C2semidirect
11. #16840732C22semidirect
13. C2×C22⋊#92420C22semidirect
14. #16840607C22semidirect
15. #16840605C22semidirect
114. C2×D4×C61M16semidirect
210. C23×C61C2×M16semidirect
218. C2×C4×C61C2×M16semidirect
333. C61#17362semidirect
417. C2×D4#11758217semidirect
513. C23C2×#11758217semidirect
557. C2×C4C2×#11758217semidirect

Maximal subgroups

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

subgroupindex
C2×#168407322
#214610752
#214610672
#214610642
#214610552
C22×D4×Dic612
#1736261