Finite Group Databasebeta

← #29711110

Subgroups — #29711110

Order 15616.

1734 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
1C23×#11758202C211yesmaximal, characteristic, complemented
2C2×#16840425C211yesmaximal, characteristic
3C2×#16839916C212yesmaximal, complemented
4#21449178C211yesmaximal, characteristic
5#21446917C212yesmaximal, complemented
6#21446828C212yesmaximal, complemented
7#21446243C211yesmaximal, characteristic, complemented
8#21446157C212yesmaximal, complemented
9#21446109C211yesmaximal, characteristic
10#21445755C211yesmaximal, characteristic, complemented
11C2×C61×#451C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C22×Q8×D61C2211yescharacteristic
14C22×#11758202—28no—
15C22×#11758202—24no—
16C22×#11758202C2212yes—
17C22×#11758202C2212yes—
18C23×#92418C2211yescharacteristic
19#16840425—24no—
20#16840425—24no—
21C2×#14519776—21no—
22C2×#14519775C2211yescharacteristic
23C2×#14519757C2211yescharacteristic
24C2×#14519755—21no—
25C2×#14519754—21no—
26C2×#14519753C2212yes—
27C2×#14519742C2211yescharacteristic
28C2×C4×C22⋊D61C2211yescharacteristic
29C2×#14519737—21no—
30#16839916—28no—
31C2×#14519702C2212yes—
32C2×#14519702C2212yes—
33C2×#14519701C2212yes—
34C2×#14519689C2212yes—
35#16839681—21no—
36#16839673—21no—
37#16839669C2212yes—
38#16839641—21no—
39#16839635—21no—
40#16839622C2211yescharacteristic
41#16839524C2212yes—
42#16839516C2212yes—
43C23×Q8×C61C2211yescharacteristic
44C22×Q16×C61C2212yes—
45C2×C22⋊Q8×C61C2211yescharacteristic
46C61×#451—24no—
47C2×#103×C61C2212yes—
48C2×C22⋊C8×C61C2211yescharacteristic
49C22×Dic122—24no—
50C22×Dic122—21no—
51C22×Dic122—22no—
52C22×Dic122—21no—
53C22×Dic122—42no—
54C22×Dic122D414yes—
55C22×Dic122C2311yescharacteristic
56C22×Dic122C2311yescharacteristic
57C2×Q8×D61—22no—
58C2×Q8×D61—22no—
59C2×Q8×D61—24no—
60C2×Q8×D61—42no—
61C22×C22⋊D61D411yescharacteristic
62C23×Dic61—21no—
63C22×C4×D61—21no—
64C22×C4×D61C2311yescharacteristic
65C2×#11758202—232no—
66C2×#11758202—24no—
67C2×#11758202—24no—
68C2×#11758202—24no—
69C2×#11758202—24no—
70C2×#11758202—24no—
71C2×#11758202D418yescomplemented
72C22×#92418—22no—
73C22×#92418—24no—
74C22×#92418C2312yes—
75#14519776—28no—
76#14519775—22no—
77#14519775—22no—
78#14519757—22no—
79#14519757—22no—
80#14519755—28no—
81#14519754—28no—
82#14519753—24no—
83#14519753—42no—
84#14519742—22no—
85#14519742—22no—
86C4×C22⋊D61—22no—
87C4×C22⋊D61—22no—
88#14519737—28no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61C2312yes—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—22no—
95C2×#11758196D411yescharacteristic
96C2×C22⋊Dic61—21no—
97C2×C22⋊Dic61D411yescharacteristic
98C2×#11758198—21no—
99C2×#11758198—22no—
100C2×#11758198D411yescharacteristic
101C2×C4×Dic61—21no—
102C2×C4×Dic61C2311yescharacteristic
103#14519702—24no—
104#14519702—42no—
105#14519701—24no—
106#14519689—24no—
107#14519684—21no—
108#14519684—22no—
109#14519684—22no—
110C22×Q8×C61—21no—
111C22×Q8×C61—24no—
112C22×Q8×C61—22no—
113C22×Q8×C61—22no—
114C22×Q8×C61—42no—
115C22×Q8×C61C2311yescharacteristic
116C22×Q8×C61C2312yes—
117C23×C4×C61—21no—
118C23×C4×C61C2311yescharacteristic
119C2×Q16×C61—24no—
120C2×Q16×C61—24no—
121C2×Q16×C61—42no—
122C22×C8×C61C2312yes—
123C22⋊Q8×C61—22no—
124C22⋊Q8×C61D414yes—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C22⋊C4×C61—21no—
128#103×C61—24no—
129C22⋊C8×C61—22no—
130C24×C61C2×D411yescharacteristic
131C23×D61C2×D411yescharacteristic
132C2×Q8×C61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61—28no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—22no—
138C2×Q8×C61—22no—
139C2×Q8×C61—22no—
140C2×Q8×C61—22no—
141C2×Q8×C61—44no—
142C2×Q8×C61—48no—
143C2×Q8×C61—44no—
144C2×Q8×C61—42no—
145C2×Q8×C61C2×D414yes—
146C2×Q8×C61C2×D412yes—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—22no—
152C22×C4×C61—44no—
153C22×C4×C61—42no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61Q1614yescomplemented
161C22×Dic61—21no—
162C22×Dic61—24no—
163C22×Dic61—21no—
164C22×Dic61—22no—
165C22×Dic61—44no—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169Q8×D61—48no—
170Q8×D61—44no—
171Q8×D61—44no—
172C2×C4×D61—24no—
173C2×C4×D61—22no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—22no—
177C2×C4×D61—44no—
178C2×Dic122—21no—
179C2×Dic122—22no—
180C2×Dic122—22no—
181C2×Dic122—22no—
182C2×Dic122—28no—
183C2×Dic122—24no—
184C2×Dic122—24no—
185C2×Dic122—21no—
186C2×Dic122—44no—
187C2×Dic122—48no—
188C2×Dic122—44no—
189C2×Dic122—44no—
190C2×Dic122C2×D412yes—
191C2×Dic122C2×D414yes—
192C2×Dic122C2×D412yes—
193Q16×C61—48no—
194C2×C8×C61—22no—
195C2×C8×C61—22no—
196C2×C8×C61—22no—
197C4⋊C4×C61—24no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61C2×D412yes—
200C22⋊C4×C61—24no—
201C22⋊Dic61—22no—
202C22⋊Dic61—42no—
203#11758202—216no—
204#11758202—216no—
205#11758198—24no—
206#11758198—22no—
207#11758198—44no—
208C4⋊Dic61—24no—
209C4⋊Dic61—22no—
210C4⋊Dic61—22no—
211#11758196—24no—
212#11758196—24no—
213#11758196—24no—
214#11758196—22no—
215#11758196—44no—
216C4×Dic61—24no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C2×#92418—28no—
220C2×#92418—22no—
221C2×#92418—22no—
222C2×#92418C2×D414yes—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61#12012yes—
226C23×C61C2×Q1612yes—
227C23×C61#13712yes—
228C23×C61C22×D411yescharacteristic
229C22×D61—21no—
230C22×D61—22no—
231C22×D61—21no—
232C22×D61—22no—
233C22×D61C2×Q1612yescomplemented
234Q8×C61—24no—
235Q8×C61—24no—
236Q8×C61—24no—
237Q8×C61—48no—
238Q8×C61—41no—
239Q8×C61—41no—
240Q8×C61—42no—
241Q8×C61—42no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—24no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—24no—
249C2×C4×C61—48no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61#12012yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D412yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61H(5,𝔽2)12yes—
259C22⋊D61—28no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—24no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—48no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61C2×Q1612yescomplemented
275C4×D61—48no—
276C4×D61—41no—
277C4×D61—41no—
278C4×D61—42no—
279Dic122—22no—
280Dic122—24no—
281Dic122—22no—
282Dic122—22no—
283Dic122—48no—
284Dic122—48no—
285Dic122—42no—
286Dic122#12014yes—
287C8×C61—42no—
288#92418—24no—
289#23856—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—24no—
293C22×C61—24no—
294C22×C61#45114yescomplemented
295C22×C61C2×#12011yescharacteristic
296C22×C61D4212yes—
297C22×C61C22×Q1611yescharacteristic
298C22×C61C2×#13711yescharacteristic
299C22×C61#57812yes—
300C2×D61—22no—
301C2×D61—24no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—42no—
305C4×C61—22no—
306C4×C61—42no—
307C4×C61—44no—
308C4×C61—42no—
309C4×C61C2×#12011yescharacteristic
310C4×C61#53411yescharacteristic
311C4×C61D4212yes—
312Dic61—24no—
313Dic61—22no—
314Dic61—42no—
315Dic61—41no—
316Dic61—41no—
317Dic61—82no—
318C23×Q16—611no—
319C2×D4×Q8—611no—
320C2×#451—612no—
321C2×#451D6111yescharacteristic, complemented
322#2409—611no—
323#2016—612no—
324#2001—612no—
325#1892—611no—
326#1878—612no—
327#1869—611no—
328#1787—611no—
329C2×C61—24no—
330C2×C61#240311yescharacteristic
331C2×C61C2×#45111yescharacteristic
332C2×C61#301111yescharacteristic
333C2×C61#327012yes—
334C2×C61D4×Q1612yes—
335D61—42no—
336C23×Q8—611no—
337C23×Q8—611no—
338C23×Q8C2×D6111yescharacteristic
339C22×Q16—612no—
340C22×Q16—612no—
341C22×Q16—1228no—
342C22×Q16—1224no—
343C22×Q16C2×D6112yes—
344C23×C8—611no—
345D4×Q8—1224no—
346D4×Q8—1224no—
347C2×C4⋊Q8—611no—
348C2×C4⋊Q8—1221no—
349C2×C22⋊Q8—612no—
350C2×C22⋊Q8—611no—
351C2×C22⋊Q8—1221no—
352C2×C22⋊Q8—1221no—
353C2×C22⋊Q8C2×D6111yescharacteristic
354C2×C4×Q8—611no—
355C2×C4×D4—611no—
356C2×C4×D4—1221no—
357#451—24no—
358#451—1228no—
359C2×#103—612no—
360C2×#103—612no—
361C2×#103—612no—
362C2×#103C2×D6112yes—
363C2×C22⋊C8—612no—
364C2×C22⋊C8C2×D6111yescharacteristic
365#395—1221no—
366#393—1221no—
367#391—612no—
368#386—1221no—
369#385—1221no—
370#380—611no—
371#340—612no—
372#336—612no—
373C61#2385611yescharacteristic, complemented, Sylow
374C22×Q8—21no—
375C22×Q8—24no—
376C22×Q8—22no—
377C22×Q8—22no—
378C22×Q8—42no—
379C22×Q8—614no—
380C22×Q8—611no—
381C22×Q8—611no—
382C22×Q8—1221no—
383C22×Q8—1224no—
384C22×Q8—1222no—
385C22×Q8—1222no—
386C22×Q8—1221no—
387C22×Q8—1222no—
388C22×Q8—1224no—
389C22×Q8—2442no—
390C22×Q8—2442no—
391C22×Q8C22×D6112yes—
392C22×Q8C22×D6111yescharacteristic
393C22×D4—611no—
394C23×C4—21no—
395C23×C4—611no—
396C23×C4—1221no—
397C23×C4—1221no—
398C23×C4C22×D6111yescharacteristic
399C2×Q16—24no—
400C2×Q16—24no—
401C2×Q16—42no—
402C2×Q16—618no—
403C2×Q16—1224no—
404C2×Q16—1224no—
405C2×Q16—1224no—
406C2×Q16—1224no—
407C2×Q16—12232no—
408C2×Q16—1224no—
409C22×C8—612no—
410C22×C8—1224no—
411C22×C8—1222no—
412C22×C8C22×D6112yes—
413C4⋊Q8—1228no—
414C4⋊Q8—1222no—
415C4⋊Q8—1222no—
416C22⋊Q8—22no—
417C22⋊Q8—1228no—
418C22⋊Q8—1222no—
419C22⋊Q8—1228no—
420C22⋊Q8—1222no—
421C22⋊Q8—1224no—
422C22⋊Q8—2442no—
423C22⋊Q8C22⋊D6114yes—
424C4×Q8—1222no—
425C4×Q8—1222no—
426C4×D4—1228no—
427C4×D4—1222no—
428C4×D4—1222no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1222no—
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—611no—
440C2×C22⋊C4—611no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1222no—
443C2×C22⋊C4—1221no—
444C2×C42—611no—
445C2×C42—1221no—
446#103—24no—
447#103—1224no—
448#103—1224no—
449#103—2442no—
450C22⋊C8—22no—
451C22⋊C8—1224no—
452#95—1222no—
453#95—1222no—
454#95—1221no—
455C24—611no—
456C24D4×D6111yescharacteristic
457C2×Q8—21no—
458C2×Q8—22no—
459C2×Q8—28no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—44no—
467C2×Q8—48no—
468C2×Q8—44no—
469C2×Q8—42no—
470C2×Q8—614no—
471C2×Q8—612no—
472C2×Q8—612no—
473C2×Q8—1224no—
474C2×Q8—1228no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1222no—
478C2×Q8—1224no—
479C2×Q8—1222no—
480C2×Q8—1222no—
481C2×Q8—2444no—
482C2×Q8—2444no—
483C2×Q8—2448no—
484C2×Q8—2444no—
485C2×Q8—2448no—
486C2×Q8—2444no—
487C2×Q8—2444no—
488C2×Q8D4×D6114yes—
489C2×Q8C2×C22⋊D6112yes—
490C2×D4—614no—
491C2×D4—1222no—
492C2×D4—1222no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—44no—
499C22×C4—42no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—1224no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1224no—
507C22×C4—1222no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—2444no—
513C22×C4—2444no—
514C22×C4D4×D6112yes—
515C22×C4D4×D6111yescharacteristic
516C22×C4C2×C22⋊D6112yes—
517C22×C4C23×D6111yescharacteristic
518Q16—48no—
519Q16—12216no—
520Q16—12216no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—614no—
525C2×C8—1228no—
526C2×C8—1222no—
527C2×C8—1222no—
528C4⋊C4—21no—
529C4⋊C4—24no—
530C4⋊C4—1224no—
531C4⋊C4—1224no—
532C4⋊C4—1224no—
533C4⋊C4—1224no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C4⋊C4—2444no—
538C4⋊C4C2×C22⋊D6112yes—
539C22⋊C4—24no—
540C22⋊C4—1224no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—2444no—
544C22⋊C4—2442no—
545C42—1224no—
546C42—1221no—
547C42—1221no—
548C23—22no—
549C23—22no—
550C23—612no—
551C23—1221no—
552C23—1221no—
553C23—1222no—
554C23—1222no—
555C23#1451974512yes—
556C23Q16×D6112yes—
557C23#1451980912yes—
558C23C2×D4×D6111yescharacteristic
559Q8—24no—
560Q8—24no—
561Q8—24no—
562Q8—48no—
563Q8—41no—
564Q8—42no—
565Q8—41no—
566Q8—42no—
567Q8—614no—
568Q8—1222no—
569Q8—1224no—
570Q8—1222no—
571Q8—1222no—
572Q8—2448no—
573Q8—2448no—
574Q8—2442no—
575D4—1228no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—612no—
589C2×C4—1221no—
590C2×C4—1222no—
591C2×C4—1221no—
592C2×C4—1224no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—2448no—
599C2×C4—2448no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2442no—
607C2×C4#1451974512yes—
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4C2×D4×D6112yes—
610C2×C4C22×C22⋊D6111yescharacteristic
611C2×C4#1451982212yes—
612C8—42no—
613C8—1224no—
614C22—22no—
615C22—24no—
616C22—22no—
617C22—24no—
618C22—1224no—
619C22—1222no—
620C22—2441no—
621C22—2441no—
622C22—2442no—
623C22#1683991614yescomplemented
624C22C2×#1451974511yescharacteristic
625C22#1684039412yes—
626C22C2×Q16×D6111yescharacteristic
627C22C2×#1451980911yescharacteristic
628C22#1684057812yes—
629C4—22no—
630C4—42no—
631C4—44no—
632C4—42no—
633C4—1222no—
634C4—1224no—
635C4—4882no—
636C4—2441no—
637C4—2441no—
638C4—2442no—
639C4#120×D6111yescharacteristic
640C4#1684032211yescharacteristic
641C4D4×C22⋊D6112yes—
642C2—24no—
643C2—2442no—
644C2#2144913011yescharacteristic
645C2C2×#1683991611yescharacteristic
646C2#2145511311yescharacteristic
647C2#2145739312yes—
648C2#2145756412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#451C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
hypercenterC2×#451D61
hypocenterC61#23856
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
1. C23×#11758202C2semidirect
3. C2×#16839916C2semidirect
5. #21446917C2semidirect
6. #21446828C2semidirect
7. #21446243C2semidirect
8. #21446157C2semidirect
10. #21445755C2semidirect
11. C2×C61×#451C2semidirect
71. C2×#11758202D4semidirect
160. C2×C22⋊D61Q16semidirect
233. C22×D61C2×Q16semidirect
274. C2×Dic61C2×Q16semidirect
294. C22×C61#451semidirect
321. C2×#451D61semidirect
373. C61#23856semidirect
623. C22#16839916semidirect

Maximal subgroups

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

subgroupindex
C23×#117582022
C2×#168404252
C2×#168399162
#214491782
#214469172
#214468282
#214462432
#214461572
#214461092
#214457552
C2×C61×#4512
#2385661