Finite Group Databasebeta

← #29852914

Subgroups — #29852914

Order 15616.

2322 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840357C212yesmaximal, complemented
2#21454145C212yesmaximal, complemented
3#21454118C212yesmaximal, complemented
4#21451933C211yesmaximal, characteristic, complemented
5#21451563C212yesmaximal, complemented
6#21450904C214yesmaximal, complemented
7#21450880C214yesmaximal, complemented
8#21450487C214yesmaximal, complemented
9#21449781C214yesmaximal, complemented
10#21448524C212yesmaximal, complemented
11#121×Dic61C212yesmaximal, complemented
12C2×#16839613C211yesmaximal, characteristic, complemented
13C61×#2656C211yesmaximal, characteristic
14#16840357—216no—
15C2×#14519770C2214yes—
16C2×#14519767C2214yes—
17C2×#14519764C2212yes—
18C2×#14519758C2214yes—
19C2×#14519758C2214yes—
20C2×#14519756C2212yes—
21C2×#14519754C2214yes—
22C2×D4×Dic61C2214yes—
23C2×D4×Dic61C2212yes—
24C22×#11758196C2211yescharacteristic
25C22×#11758196C2211yescharacteristic
26C22×C4×Dic61C2211yescharacteristic
27C22×C4×Dic61C2211yescharacteristic
28C22×C4×Dic61C2211yescharacteristic
29#16839694C2214yes—
30#16839693C2214yes—
31#16839685C2214yes—
32#16839685C2214yes—
33#16839683C2214yes—
34#16839676C2214yes—
35#16839667C2212yes—
36#16839665C2212yes—
37#16839665C2212yes—
38#16839665C2212yes—
39#16839665C2212yes—
40#16839661C2212yes—
41#16839658C2212yes—
42#16839658C2212yes—
43#16839656C2214yes—
44#16839656C2214yes—
45#16839655C2214yes—
46#16839655C2214yes—
47#16839650C2214yes—
48#16839650C2214yes—
49#16839648C2214yes—
50#16839642C2214yes—
51#16839642C2214yes—
52#16839642C2214yes—
53#16839637C2214yes—
54#16839627C2212yes—
55#16839622C2212yes—
56#16839622C2212yes—
57C4⋊C4×Dic61C2212yes—
58#16839613C2214yes—
59#16839613C2214yes—
60C22⋊C4×Dic61C2214yes—
61C2×#14519684C2211yescharacteristic
62C2×#121×C61C2212yes—
63C2×#121×C61C2212yes—
64C2×#121×C61C2212yes—
65C61×#399C2214yes—
66C61×#392C2214yes—
67C2×#95×C61C2211yescharacteristic
68C22×Dic122C2312yes—
69C23×Dic61C2311yescharacteristic
70C23×Dic61C2311yescharacteristic
71C23×Dic61C2311yescharacteristic
72C23×Dic61C2314yes—
73C23×Dic61C2311yescharacteristic
74#14519770—216no—
75#14519767—216no—
76#14519764—44no—
77#14519758—216no—
78#14519758—216no—
79#14519756—28no—
80#14519754—216no—
81D4×Dic61—216no—
82D4×Dic61—44no—
83C2×C4⋊Dic61C2312yes—
84C2×C4⋊Dic61C2314yes—
85C2×C4⋊Dic61C2312yes—
86C2×C4⋊Dic61C2312yes—
87C2×C4⋊Dic61C2312yes—
88C2×C4⋊Dic61C2312yes—
89C2×#11758196—24no—
90C2×#11758196D418yescomplemented
91C2×#11758196C2314yes—
92C2×#11758196C2314yes—
93C2×#11758196C2314yes—
94C2×#11758196C2314yes—
95C2×#11758196C2314yes—
96C2×C22⋊Dic61C2314yes—
97C2×C22⋊Dic61C2314yes—
98C2×C22⋊Dic61C2314yes—
99C2×C22⋊Dic61C2314yes—
100C2×C22⋊Dic61C2314yes—
101C2×C22⋊Dic61C2314yes—
102C2×C22⋊Dic61C2314yes—
103C2×C22⋊Dic61C2314yes—
104C2×C4×Dic61—24no—
105C2×C4×Dic61—42no—
106C2×C4×Dic61—42no—
107C2×C4×Dic61C2312yes—
108C2×C4×Dic61C2312yes—
109C2×C4×Dic61C2312yes—
110C2×C4×Dic61C2312yes—
111C2×C4×Dic61C2314yes—
112C2×C4×Dic61C2312yes—
113C2×C4×Dic61C2312yes—
114#14519684C2314yes—
115#14519684C2314yes—
116#14519684C2314yes—
117#14519684C2314yes—
118#14519684C2314yes—
119#14519684C2314yes—
120#14519684C2314yes—
121C22×D4×C61C2314yes—
122C22×D4×C61C2312yes—
123C22×D4×C61C2312yes—
124C22×D4×C61C2312yes—
125C23×C4×C61C2311yescharacteristic
126C23×C4×C61C2311yescharacteristic
127C23×C4×C61C2311yescharacteristic
128#121×C61—28no—
129#121×C61—28no—
130#121×C61—28no—
131C2×C4⋊C4×C61C2312yes—
132C2×C4⋊C4×C61C2312yes—
133C2×C4⋊C4×C61C2312yes—
134C2×C22⋊C4×C61C2314yes—
135C2×C22⋊C4×C61C2314yes—
136C2×C22⋊C4×C61C2314yes—
137#95×C61C2314yes—
138C24×C61C2414yes—
139C24×C61C2411yescharacteristic
140C2×D4×C61—28no—
141C2×D4×C61—28no—
142C2×D4×C61—28no—
143C2×D4×C61—24no—
144C2×D4×C61—24no—
145C2×D4×C61—24no—
146C2×D4×C61—42no—
147C2×D4×C61—42no—
148C2×D4×C61—42no—
149C2×D4×C61—42no—
150C2×D4×C61—42no—
151C2×D4×C61—42no—
152C22×C4×C61—41no—
153C22×C4×C61—41no—
154C22×C4×C61—41no—
155C22×C4×C61—41no—
156C22×C4×C61—41no—
157C22×C4×C61—41no—
158C22×C4×C61C2×D414yes—
159C22×C4×C61#4114yes—
160C22×C4×C61#4114yes—
161C22×C4×C61C2414yes—
162C22×C4×C61C2412yes—
163C22×C4×C61C2412yes—
164C22×C4×C61C2412yes—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—28no—
172C22×Dic61—41no—
173C22×Dic61—41no—
174C22×Dic61—41no—
175C22×Dic61—41no—
176C22×Dic61—44no—
177C22×Dic61—44no—
178C22×Dic61C2×D414yes—
179C22×Dic61C2×D414yes—
180C22×Dic61C2414yes—
181C22×Dic61C2412yes—
182C22×Dic61C2414yes—
183C22×Dic61C2412yes—
184C22×Dic61C2412yes—
185C22×Dic61C2411yescharacteristic
186C22×Dic61C2411yescharacteristic
187C2×Dic122—24no—
188C2×Dic122—42no—
189C2×Dic122—42no—
190C4⋊C4×C61—24no—
191C4⋊C4×C61—24no—
192C4⋊C4×C61—24no—
193C22⋊C4×C61—28no—
194C22⋊C4×C61—28no—
195C22⋊C4×C61—28no—
196C22⋊Dic61—28no—
197C22⋊Dic61—28no—
198C22⋊Dic61—28no—
199C22⋊Dic61—28no—
200C22⋊Dic61—28no—
201C22⋊Dic61—28no—
202C22⋊Dic61—44no—
203C22⋊Dic61—44no—
204C4⋊Dic61—24no—
205C4⋊Dic61—28no—
206C4⋊Dic61—42no—
207C4⋊Dic61—42no—
208C4⋊Dic61—42no—
209C4⋊Dic61—42no—
210#11758196—28no—
211#11758196—28no—
212#11758196—28no—
213#11758196—28no—
214#11758196C2×D4116yes—
215C4×Dic61—24no—
216C4×Dic61—24no—
217C4×Dic61—28no—
218C4×Dic61—24no—
219C4×Dic61—24no—
220C4×Dic61—42no—
221C4×Dic61—42no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—24no—
227C23×C61—24no—
228C23×C61—24no—
229C23×C61—24no—
230C23×C61—24no—
231C23×C61—24no—
232C23×C61—24no—
233C23×C61C22×D412yes—
234C23×C61C2×#4112yes—
235C23×C61C2×#4112yes—
236C23×C61C2511yescharacteristic
237D4×C61—48no—
238D4×C61—416no—
239D4×C61—84no—
240D4×C61—84no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—24no—
248C2×C4×C61—24no—
249C2×C4×C61—24no—
250C2×C4×C61—44no—
251C2×C4×C61—44no—
252C2×C4×C61—44no—
253C2×C4×C61C22×D414yes—
254C2×C4×C61C2×#4114yes—
255C2×C4×C61C2×#4114yes—
256C2×Dic61—28no—
257C2×Dic61—28no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—24no—
267C2×Dic61—24no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—24no—
271C2×Dic61—24no—
272C2×Dic61—24no—
273C2×Dic61—24no—
274C2×Dic61—22no—
275C2×Dic61—44no—
276C2×Dic61—44no—
277C2×Dic61—88no—
278C2×Dic61C22×D414yes—
279C2×Dic61C22×D414yes—
280C2×Dic61H(5,𝔽2)12yes—
281C2×Dic612−1+412yes—
282Dic122—84no—
283#35004—611nomaximal, Sylow
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—44no—
291C22×C61—44no—
292C22×C61—44no—
293C22×C61—44no—
294C22×C61—44no—
295C22×C61—44no—
296C22×C61—44no—
297C22×C61#53214yes—
298C22×C61C23×D411yescharacteristic
299C22×C61C22×#4111yescharacteristic
300C22×C61C22×#4111yescharacteristic
301C22×C61C2×H(5,𝔽2)11yescharacteristic
302C22×C61C2×2−1+411yescharacteristic
303C22×C61#58511yescharacteristic
304C22×C61#58511yescharacteristic
305C4×C61—24no—
306C4×C61—24no—
307C4×C61—24no—
308C4×C61—44no—
309Dic61—24no—
310Dic61—22no—
311Dic61—22no—
312Dic61—24no—
313Dic61—42no—
314Dic61—44no—
315Dic61—44no—
316C2×#539—612no—
317#2854—612no—
318#2850—612no—
319#2659—611no—
320#2656D6111yescharacteristic
321#2618—612no—
322#2556—614no—
323#2554—614no—
324#2519—614no—
325#2461—614no—
326#2325—612no—
327C4×#121—612no—
328C2×C4×C4⋊C4—611no—
329C2×C61—24no—
330C2×C61—84no—
331C2×C61C2×#53211yescharacteristic
332C2×C61#345911yescharacteristic
333C2×C61#346111yescharacteristic
334C2×C61#349111yescharacteristic
335C2×C61#349211yescharacteristic
336C2×C61#349611yescharacteristic
337C2×C61#349711yescharacteristic
338#539—12216no—
339C2×#126—614no—
340C2×#125—614no—
341C2×#124—612no—
342C2×#123—614no—
343C2×#123—614no—
344C2×C22⋊Q8—612no—
345C2×C22⋊Q8—614no—
346C2×#121C2×D6112yes—
347C2×#121C2×D6112yes—
348C2×#121C2×D6112yes—
349C2×C4×D4—614no—
350C2×C4×D4—612no—
351C22×C4⋊C4—611no—
352C22×C4⋊C4—611no—
353C22×C42—611no—
354C22×C42—611no—
355C22×C42—611no—
356#399C2×D6114yes—
357#398—614no—
358#398—614no—
359#396—614no—
360#396—614no—
361#395—614no—
362#394—614no—
363#392C2×D6114yes—
364#391—612no—
365#390—612no—
366#390—612no—
367#390—612no—
368#390—612no—
369#389—612no—
370#389—612no—
371#389—612no—
372#388—614no—
373#388—614no—
374#388—614no—
375#388—614no—
376#388—614no—
377#388—614no—
378#387—614no—
379#386—614no—
380#386—614no—
381#386—614no—
382#385—614no—
383#382—612no—
384#380—612no—
385#380—612no—
386C4×C4⋊C4—612no—
387C4×C4⋊C4—614no—
388C4×C4⋊C4—614no—
389C4×C22⋊C4—614no—
390C2×#95—611no—
391C2×#95C2×D6111yescharacteristic
392C61#3500411yescharacteristic, complemented, Sylow
393C22×Q8—612no—
394C22×D4C22×D6114yes—
395C22×D4C22×D6112yes—
396C22×D4C22×D6112yes—
397C22×D4C22×D6112yes—
398C23×C4—611no—
399C23×C4—611no—
400C23×C4—611no—
401C23×C4—611no—
402C23×C4—614no—
403C23×C4C22×D6111yescharacteristic
404C23×C4C22×D6111yescharacteristic
405C23×C4C22×D6111yescharacteristic
406#126—12216no—
407#125—12216no—
408#124—2444no—
409#123—12216no—
410#123—12216no—
411C22⋊Q8—12216no—
412C22⋊Q8—1228no—
413#121—28no—
414#121—28no—
415#121—28no—
416C4×D4—12216no—
417C4×D4—2444no—
418C2×C4⋊C4—618no—
419C2×C4⋊C4—614no—
420C2×C4⋊C4—614no—
421C2×C4⋊C4—614no—
422C2×C4⋊C4—614no—
423C2×C4⋊C4—614no—
424C2×C4⋊C4—612no—
425C2×C4⋊C4—614no—
426C2×C4⋊C4—612no—
427C2×C4⋊C4—612no—
428C2×C4⋊C4—612no—
429C2×C4⋊C4—612no—
430C2×C4⋊C4—1224no—
431C2×C4⋊C4C22×D6112yes—
432C2×C4⋊C4C22×D6112yes—
433C2×C4⋊C4C22×D6112yes—
434C2×C22⋊C4—614no—
435C2×C22⋊C4—614no—
436C2×C22⋊C4—614no—
437C2×C22⋊C4—614no—
438C2×C22⋊C4—614no—
439C2×C22⋊C4—614no—
440C2×C22⋊C4—614no—
441C2×C22⋊C4—614no—
442C2×C22⋊C4C22×D6114yes—
443C2×C22⋊C4C22×D6114yes—
444C2×C22⋊C4C22×D6114yes—
445C2×C42—614no—
446C2×C42—612no—
447C2×C42—612no—
448C2×C42—612no—
449C2×C42—612no—
450C2×C42—612no—
451C2×C42—612no—
452C2×C42—1224no—
453C2×C42—2442no—
454C2×C42—2442no—
455#95—614no—
456#95—614no—
457#95—614no—
458#95—614no—
459#95—614no—
460#95—614no—
461#95—614no—
462#95C22×D6114yes—
463C24C23×D6114yes—
464C24C23×D6111yescharacteristic
465C2×Q8—1224no—
466C2×Q8—2442no—
467C2×Q8—2442no—
468C2×D4—28no—
469C2×D4—28no—
470C2×D4—28no—
471C2×D4—24no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—614no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—614no—
492C22×C4—612no—
493C22×C4—614no—
494C22×C4—614no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1228no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—2444no—
503C22×C4—2444no—
504C22×C4—2441no—
505C22×C4—2441no—
506C22×C4—2441no—
507C22×C4—2441no—
508C22×C4D4×D6114yes—
509C22×C4#1175822414yes—
510C22×C4#1175822414yes—
511C22×C4C23×D6114yes—
512C22×C4C23×D6112yes—
513C22×C4C23×D6112yes—
514C22×C4C23×D6112yes—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—6116no—
519C4⋊C4—1228no—
520C4⋊C4—1228no—
521C4⋊C4—1228no—
522C4⋊C4—1228no—
523C4⋊C4—1224no—
524C4⋊C4—1228no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4—2442no—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—1228no—
533C22⋊C4—1228no—
534C22⋊C4—1228no—
535C22⋊C4—1228no—
536C22⋊C4—1228no—
537C22⋊C4—1228no—
538C22⋊C4—2444no—
539C22⋊C4—2444no—
540C42—1224no—
541C42—1224no—
542C42—1228no—
543C42—1224no—
544C42—1224no—
545C42—2442no—
546C42—2442no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23C2×D4×D6112yes—
559C23C2×#1175822412yes—
560C23C2×#1175822412yes—
561C23C24×D6111yescharacteristic
562Q8—4884no—
563D4—48no—
564D4—416no—
565D4—84no—
566D4—84no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—614no—
580C2×C4—614no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—1228no—
584C2×C4—1228no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1224no—
594C2×C4—1224no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1224no—
598C2×C4—1224no—
599C2×C4—1222no—
600C2×C4—1224no—
601C2×C4—1224no—
602C2×C4—4888no—
603C2×C4—2444no—
604C2×C4—2444no—
605C2×C4C2×D4×D6114yes—
606C2×C4C2×#1175822414yes—
607C2×C4C2×#1175822414yes—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22#1684031014yes—
622C22C22×D4×D6111yescharacteristic
623C22C22×#1175822411yescharacteristic
624C22C22×#1175822411yescharacteristic
625C22H(5,𝔽2)×D6111yescharacteristic
626C222−1+4×D6111yescharacteristic
627C22#1684059711yescharacteristic
628C22#1684059711yescharacteristic
629C4—24no—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—1222no—
634C4—1222no—
635C4—1224no—
636C4—1224no—
637C4—2442no—
638C4—2444no—
639C4—2444no—
640C2—24no—
641C2—84no—
642C2C2×#1684031011yescharacteristic
643C2#2145922511yescharacteristic
644C2#2145924711yescharacteristic
645C2#2145951611yescharacteristic
646C2#2145952611yescharacteristic
647C2#2145958011yescharacteristic
648C2#2145958811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
Fitting, prime radicalC61×#2656C2
hypercenter#2656D61
hypocenterC61#35004
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2656

Complements

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

NH ≅ G/Ntype
1. C2×#16840357C2semidirect
2. #21454145C2semidirect
3. #21454118C2semidirect
4. #21451933C2semidirect
5. #21451563C2semidirect
6. #21450904C2semidirect
7. #21450880C2semidirect
8. #21450487C2semidirect
9. #21449781C2semidirect
10. #21448524C2semidirect
11. #121×Dic61C2semidirect
12. C2×#16839613C2semidirect
90. C2×#11758196D4semidirect
392. C61#35004semidirect

Maximal subgroups

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

subgroupindex
C2×#168403572
#214541452
#214541182
#214519332
#214515632
#214509042
#214508802
#214504872
#214497812
#214485242
#121×Dic612
C2×#168396132
C61×#26562
#3500461