Finite Group Databasebeta

← #29608520

Subgroups — #29608520

Order 15616.

1810 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519756C211yesmaximal, characteristic
2C2×#16839666C212yesmaximal, complemented
3#21444086C212yesmaximal, complemented
4#21444070C212yesmaximal, complemented
5#21444062C212yesmaximal
6#21444056C212yesmaximal
7#21444050C212yesmaximal, complemented
8#21444042C211yesmaximal, characteristic
9C61×C22⋊#95C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519756—28no—
12C2×#14519756C418yescomplemented
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196C2212yes—
15C22×#11758196C2212yes—
16C22×C22⋊Dic61C2212yes—
17C22×C22⋊Dic61C2211yescharacteristic
18C22×C22⋊Dic61C2211yescharacteristic
19C22×C4×Dic61C2212yes—
20#16839666—28no—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C24×C4×C61C2211yescharacteristic
30C22×C22⋊C4×C61C2212yes—
31C2×#95×C61C2212yes—
32C2×#95×C61C2211yescharacteristic
33C2×#95×C61C2211yescharacteristic
34C22×Dic122—41no—
35C22×Dic122—42no—
36C22×Dic122—41no—
37C22×Dic122C2×C414yes—
38C22×Dic122D418yes—
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43#14519756—232no—
44C2×C4⋊Dic61—24no—
45C2×C4⋊Dic61C2×C414yes—
46C2×#11758196—24no—
47C2×#11758196—24no—
48C2×#11758196—24no—
49C2×#11758196—42no—
50C2×#11758196—42no—
51C2×#11758196C2×C418yes—
52C2×C22⋊Dic61—24no—
53C2×C22⋊Dic61—24no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—41no—
56C2×C22⋊Dic61—41no—
57C2×C22⋊Dic61—41no—
58C2×C22⋊Dic61—41no—
59C2×C22⋊Dic61C2×C418yes—
60C2×C22⋊Dic61Q814yescomplemented
61C2×C4×Dic61—24no—
62C2×C4×Dic61—42no—
63C2×C4×Dic61—42no—
64#14519684—24no—
65#14519684—24no—
66#14519684—24no—
67#14519684—24no—
68#14519684—24no—
69#14519684—42no—
70#14519684—42no—
71#14519684—42no—
72C25×C61C2311yescharacteristic
73C23×C4×C61—24no—
74C23×C4×C61—41no—
75C23×C4×C61—41no—
76C23×C4×C61—42no—
77C23×C4×C61C2×C414yes—
78C23×C4×C61C2312yes—
79C23×C4×C61C2311yescharacteristic
80C23×C4×C61C2312yes—
81C23×C4×C61C2311yescharacteristic
82C2×C22⋊C4×C61—24no—
83C2×C22⋊C4×C61—24no—
84C2×C22⋊C4×C61—42no—
85#95×C61—24no—
86#95×C61—41no—
87#95×C61—41no—
88C24×C61—22no—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—21no—
92C24×C61—21no—
93C24×C61—21no—
94C24×C61—22no—
95C24×C61—21no—
96C24×C61C22×C412yes—
97C24×C61C2×Q812yes—
98C24×C61#4112yes—
99C24×C61C2411yescharacteristic
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—28no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—44no—
122C22×C4×C61—44no—
123C22×C4×C61—48no—
124C22×C4×C61—48no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D414yes—
128C22×C4×C61#4114yes—
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—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61C22×C414yes—
153C22×Dic61C22×C414yes—
154C22×Dic61C2×D414yes—
155C22×Dic61C2×D414yes—
156C22×Dic61C2×Q814yes—
157C2×Dic122—44no—
158C2×Dic122—44no—
159C2×Dic122—48no—
160C2×Dic122—48no—
161C2×Dic122C22⋊C4116yescomplemented
162C22⋊C4×C61—48no—
163C22⋊Dic61—216no—
164C22⋊Dic61—44no—
165C22⋊Dic61—44no—
166C4⋊Dic61—28no—
167#11758196—216no—
168#11758196—48no—
169C4×Dic61—48no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—24no—
181C23×C61—24no—
182C23×C61—24no—
183C23×C61—24no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61C4×Q814yes—
193C23×C61C23×C411yescharacteristic
194C23×C61C22×D412yes—
195C23×C61C22×Q811yescharacteristic
196C23×C61C2×#4111yescharacteristic
197C23×C61C2×#4112yes—
198C23×C61H(5,𝔽2)11yescharacteristic
199C23×C61H(5,𝔽2)11yescharacteristic
200C23×C61H(5,𝔽2)12yes—
201C23×C61H(5,𝔽2)11yescharacteristic
202C23×C612−1+412yes—
203C23×C612−1+411yescharacteristic
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—416no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61C2×C22⋊C414yes—
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—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61C2×C22⋊C418yes—
257Dic122—416no—
258#14449—611nomaximal, Sylow
259C22×C61—24no—
260C22×C61—24no—
261C22×C61—24no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—44no—
267C22×C61—44no—
268C22×C61—44no—
269C22×C61—44no—
270C22×C61C22×C22⋊C411yescharacteristic
271C22×C61C2×C4×Q811yescharacteristic
272C22×C61C4×#4111yescharacteristic
273C22×C61#51811yescharacteristic
274C22×C61#51811yescharacteristic
275C22×C61#51911yescharacteristic
276C22×C61#52011yescharacteristic
277C22×C61#53412yes—
278C22×C61#53412yes—
279C22×C61#53811yescharacteristic
280C22×C61#53912yes—
281C22×C61#53911yescharacteristic
282C22×C61#54212yes—
283C22×C61#54311yescharacteristic
284C22×C61#54411yescharacteristic
285C22×C61#54812yes—
286C22×C61#54812yes—
287C22×C61D4×Q812yes—
288C22×C61#55012yes—
289C22×C61#55112yes—
290C22×C61#55412yes—
291C22×C61#56212yes—
292C22×C61#56312yes—
293C4×C61—24no—
294C4×C61—42no—
295C4×C61—44no—
296C4×C61—82no—
297Dic61—28no—
298Dic61—44no—
299Dic61—44no—
300Dic61—82no—
301C22×C22⋊Q8—611no—
302C2×#391—612no—
303#1445—612no—
304#1441—612no—
305#1439—612no—
306#1437—612no—
307#1436—612no—
308C22⋊#95—611no—
309C22⋊#95D6111yescharacteristic
310C2×C61—44no—
311C2×C61#230811yescharacteristic
312C2×C61#232911yescharacteristic
313C2×C61#233211yescharacteristic
314C2×C61Q8×C22⋊C411yescharacteristic
315C2×C61#234111yescharacteristic
316C2×C61#236811yescharacteristic
317C2×C61#237811yescharacteristic
318C2×C61#272812yes—
319C2×C61#272912yes—
320C2×C61#273012yes—
321C2×C61#273212yes—
322C23×Q8—611no—
323C24×C4C2×D6111yescharacteristic
324C2×C22⋊Q8—618no—
325C2×C22⋊Q8—1228no—
326C22×C4⋊C4—612no—
327C22×C4⋊C4—611no—
328C22×C4⋊C4—612no—
329C22×C22⋊C4—612no—
330C22×C22⋊C4—611no—
331C22×C22⋊C4—611no—
332C22×C22⋊C4C2×D6112yes—
333C22×C42—612no—
334#391—1228no—
335C2×#95—612no—
336C2×#95—612no—
337C2×#95—612no—
338C2×#95—612no—
339C2×#95—612no—
340C2×#95—612no—
341C2×#95—612no—
342C2×#95—612no—
343C2×#95C2×D6112yes—
344C2×#95C2×D6111yescharacteristic
345C2×#95C2×D6111yescharacteristic
346C61#1444911yescharacteristic, complemented, Sylow
347C25C22×D6111yescharacteristic
348C22×Q8—614no—
349C22×Q8—618no—
350C22×Q8—2441no—
351C22×Q8—2442no—
352C22×Q8—2441no—
353C23×C4—24no—
354C23×C4—41no—
355C23×C4—41no—
356C23×C4—42no—
357C23×C4—612no—
358C23×C4—612no—
359C23×C4—612no—
360C23×C4—612no—
361C23×C4C4×D6114yes—
362C23×C4C22×D6112yes—
363C23×C4C22×D6111yescharacteristic
364C23×C4C22×D6112yes—
365C23×C4C22×D6111yescharacteristic
366C22⋊Q8—12232no—
367C2×C4⋊C4—618no—
368C2×C4⋊C4—614no—
369C2×C4⋊C4—1224no—
370C2×C4⋊C4—1224no—
371C2×C4⋊C4—1224no—
372C2×C4⋊C4—1224no—
373C2×C4⋊C4—2442no—
374C2×C4⋊C4—2442no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—618no—
379C2×C22⋊C4—614no—
380C2×C22⋊C4—1224no—
381C2×C22⋊C4—1224no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—2441no—
384C2×C22⋊C4—2441no—
385C2×C22⋊C4—2441no—
386C2×C22⋊C4—2441no—
387C2×C42—1224no—
388C2×C42—2442no—
389C2×C42—2442no—
390#95—24no—
391#95—41no—
392#95—41no—
393#95—1224no—
394#95—1224no—
395#95—1224no—
396#95—1224no—
397#95—1224no—
398#95—2442no—
399#95—2442no—
400#95—2442no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—21no—
405C24—21no—
406C24—21no—
407C24—22no—
408C24—21no—
409C24C2×C4×D6112yes—
410C24#1175822412yes—
411C24Q8×D6112yes—
412C24C23×D6111yescharacteristic
413C2×Q8—6116no—
414C2×Q8—2444no—
415C2×Q8—2444no—
416C2×Q8—2448no—
417C2×Q8—2448no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—614no—
444C22×C4—614no—
445C22×C4—614no—
446C22×C4—614no—
447C22×C4—614no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4C2×C4×D6112yes—
472C22×C4C2×C4×D6112yes—
473C22×C4#1175822214yes—
474C22×C4D4×D6114yes—
475C4⋊C4—12216no—
476C4⋊C4—1228no—
477C4⋊C4—2448no—
478C22⋊C4—48no—
479C22⋊C4—12216no—
480C22⋊C4—2444no—
481C22⋊C4—2444no—
482C42—2448no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23#1451974214yes—
506C23C22×C4×D6111yescharacteristic
507C23C2×D4×D6112yes—
508C23C2×Q8×D6111yescharacteristic
509C23C2×#1175822411yescharacteristic
510C23C2×#1175822212yes—
511C23#1451982212yes—
512C23#1451982211yescharacteristic
513C23#1451982211yescharacteristic
514C23#1451982311yescharacteristic
515C23#1451982412yes—
516C23#1451982411yescharacteristic
517Q8—24416no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—416no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—618no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—1224no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4C22⋊C4×D6114yes—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—44no—
576C22—42no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—44no—
582C22C2×C22⋊C4×D6111yescharacteristic
583C22C2×#1451974211yescharacteristic
584C22C4×#1175822211yescharacteristic
585C22#1684023511yescharacteristic
586C22#1684023511yescharacteristic
587C22#1684023911yescharacteristic
588C22#1684024211yescharacteristic
589C22#1684032212yes—
590C22#1684032212yes—
591C22#1684034911yescharacteristic
592C22#1684035412yes—
593C22#1684035711yescharacteristic
594C22#1684037712yes—
595C22#1684038711yescharacteristic
596C22#1684039111yescharacteristic
597C22#1684041812yes—
598C22#1684042112yes—
599C22#1684042512yes—
600C22#1684043312yes—
601C22#1684043812yes—
602C22#1684046712yes—
603C22#1684050912yes—
604C22#1684051512yes—
605C4—24no—
606C4—42no—
607C4—44no—
608C4—82no—
609C4—1228no—
610C4—4882no—
611C4—2444no—
612C4—2444no—
613C2—44no—
614C2#2144839511yescharacteristic
615C2#2144854911yescharacteristic
616C2#2144857511yescharacteristic
617C2#2144865111yescharacteristic
618C2#2144865911yescharacteristic
619C2#2144889911yescharacteristic
620C2#2144898711yescharacteristic
621C2#2145279012yes—
622C2#2145280112yes—
623C2#2145281712yes—
624C2#2145283812yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×C22⋊#95C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC22⋊#95D61
hypocenterC61#14449
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#16839666C2semidirect
3. #21444086C2semidirect
4. #21444070C2semidirect
7. #21444050C2semidirect
12. C2×#14519756C4semidirect
60. C2×C22⋊Dic61Q8semidirect
161. C2×Dic122C22⋊C4semidirect
346. C61#14449semidirect

Maximal subgroups

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

subgroupindex
C22×#145197562
C2×#168396662
#214440862
#214440702
#214440622
#214440562
#214440502
#214440422
C61×C22⋊#952
#1444961