Finite Group Databasebeta

← #29601323

Subgroups — #29601323

Order 15616.

1862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic122C211yesmaximal, characteristic
2C2×#16839666C212yesmaximal
3C2×#16839613C211yesmaximal, characteristic
4C2×#16839607C211yesmaximal, characteristic
5#21444053C212yesmaximal
6#21444050C212yesmaximal
7#21444047C212yesmaximal
8#21444046C211yesmaximal, characteristic
9#21444043C211yesmaximal, characteristic
10#21444043C211yesmaximal, characteristic
11C61×#1434C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C2×C4×Dic122—24no—
14C2×C4×Dic122—24no—
15C2×C4×Dic122C418yescomplemented
16C22×C4⋊Dic61C2211yescharacteristic
17C22×#11758196C2211yescharacteristic
18C22×#11758196C2211yescharacteristic
19C22×#11758196C2212yes—
20C22×C4×Dic61C2211yescharacteristic
21C22×C4×Dic61C2212yes—
22C22×C4×Dic61C2212yes—
23C22×C4×Dic61C2211yescharacteristic
24#16839666C4116yescomplemented
25#16839613—24no—
26#16839607—24no—
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2211yescharacteristic
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C2×#14519684C2212yes—
36C22×C42×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2211yescharacteristic
39C2×#95×C61C2212yes—
40C2×#95×C61C2212yes—
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—24no—
44C22×Dic122C2×C414yes—
45C22×Dic122C2×C418yes—
46C23×Dic61C2311yescharacteristic
47C23×Dic61C2312yes—
48C23×Dic61C2312yes—
49C23×Dic61C2311yescharacteristic
50C23×Dic61C2311yescharacteristic
51C23×Dic61C2311yescharacteristic
52C4×Dic122—232no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61C2×C414yes—
56C2×#11758196—22no—
57C2×#11758196—22no—
58C2×#11758196—22no—
59C2×#11758196—22no—
60C2×#11758196—24no—
61C2×#11758196—24no—
62C2×#11758196C2×C414yes—
63C2×#11758196C2×C414yes—
64C2×#11758196C2×C418yes—
65C2×C4×Dic61—22no—
66C2×C4×Dic61—22no—
67C2×C4×Dic61—24no—
68C2×C4×Dic61—24no—
69C2×C4×Dic61—24no—
70C2×C4×Dic61—42no—
71C2×C4×Dic61—42no—
72C2×C4×Dic61C2×C414yes—
73C2×C4×Dic61C2×C418yes—
74C2×C4×Dic61C2×C414yes—
75C2×C4×Dic61D414yes—
76C2×C4×Dic61D414yes—
77#14519684—22no—
78#14519684—22no—
79#14519684—24no—
80#14519684—42no—
81#14519684—42no—
82#14519684—42no—
83#14519684C2×C418yes—
84#14519684C2×C418yes—
85#14519684C2×C418yes—
86C23×C4×C61C2312yes—
87C23×C4×C61C2312yes—
88C23×C4×C61C2311yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C23×C4×C61C2311yescharacteristic
91C2×C42×C61—22no—
92C2×C42×C61—22no—
93C2×C42×C61C2×C414yes—
94#95×C61—22no—
95#95×C61—22no—
96#95×C61—42no—
97#95×C61C2×C418yes—
98C24×C61C2411yescharacteristic
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
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—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61C22×C414yes—
123C22×C4×C61C22×C414yes—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C414yes—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61C22×C414yes—
156C22×Dic61C22×C414yes—
157C22×Dic61C22×C412yes—
158C22×Dic61C22×C414yes—
159C22×Dic61C22×C412yes—
160C22×Dic61C22×C412yes—
161C22×Dic61C22×C412yes—
162C22×Dic61C22×C414yes—
163C22×Dic61C2×D412yes—
164C22×Dic61C2×D412yes—
165C22×Dic61C2×D412yes—
166C22×Dic61C2×D412yes—
167C22×Dic61#4114yes—
168C2×Dic122—28no—
169C2×Dic122—28no—
170C2×Dic122—216no—
171C2×Dic122—216no—
172C2×Dic122C42116yescomplemented
173C42×C61—28no—
174C4⋊Dic61—28no—
175#11758196—28no—
176#11758196—28no—
177#11758196—216no—
178C4×Dic61—28no—
179C4×Dic61—216no—
180C4×Dic61—48no—
181C4×Dic61C22⋊C4116yes—
182C23×C61C23×C411yescharacteristic
183C23×C61C23×C412yes—
184C23×C61C22×D411yescharacteristic
185C23×C61C22×D411yescharacteristic
186C23×C61C2×#4112yes—
187C23×C61H(5,𝔽2)11yescharacteristic
188C23×C61H(5,𝔽2)11yescharacteristic
189C23×C612−1+412yes—
190C23×C612−1+412yes—
191C23×C612−1+411yescharacteristic
192C23×C612−1+411yescharacteristic
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—22no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—24no—
210C2×C4×C61—42no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61C2×C4214yes—
223C2×C4×C61C2×C22⋊C414yes—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—24no—
227C2×Dic61—24no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—24no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—42no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61C2×C4214yes—
256C2×Dic61C2×C4214yes—
257C2×Dic61C2×C22⋊C414yes—
258C2×Dic61C2×C22⋊C414yes—
259C2×Dic61#11714yes—
260C2×Dic61C4×D418yes—
261C2×Dic61C4×D418yes—
262Dic122—232no—
263#13786—611nomaximal, Sylow
264C22×C61C22×C4211yescharacteristic
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61C2×#11711yescharacteristic
267C22×C61C2×C4×D412yes—
268C22×C61C2×C4×D412yes—
269C22×C61#51912yes—
270C22×C61#51911yescharacteristic
271C22×C61#51911yescharacteristic
272C22×C61#51912yes—
273C22×C61#52012yes—
274C22×C61#52011yescharacteristic
275C22×C61#52012yes—
276C22×C61#52011yescharacteristic
277C22×C61#53511yescharacteristic
278C22×C61#53612yes—
279C22×C61#53612yes—
280C22×C61#53611yescharacteristic
281C22×C61#53711yescharacteristic
282C22×C61#53711yescharacteristic
283C22×C61#54112yes—
284C22×C61#54112yes—
285C22×C61#54212yes—
286C22×C61#54212yes—
287C4×C61—24no—
288C4×C61—24no—
289C4×C61—42no—
290C4×C61—44no—
291C4×C61—44no—
292Dic61—24no—
293Dic61—24no—
294Dic61—42no—
295Dic61—44no—
296Dic61—44no—
297Dic61C4×C22⋊C418yes—
298C22×C4×Q8—611no—
299C2×#391—612no—
300C2×C4×C4⋊C4—611no—
301C2×#376—611no—
302#1436—612no—
303#1436—612no—
304C4⋊#95—612no—
305#1434—611no—
306#1434—611no—
307#1434—611no—
308#1434D6111yescharacteristic
309C2×C61C2×C4×C22⋊C411yescharacteristic
310C2×C61#227611yescharacteristic
311C2×C61#227611yescharacteristic
312C2×C61#231011yescharacteristic
313C2×C61#231111yescharacteristic
314C2×C61#232011yescharacteristic
315C2×C61#232011yescharacteristic
316C2×C61#232312yes—
317C2×C61#232312yes—
318C2×C61#232612yes—
319C2×C61#232712yes—
320C23×Q8—611no—
321C2×C4×Q8—618no—
322C2×C4×Q8—1224no—
323C2×C4×Q8—1224no—
324C22×C4⋊C4—611no—
325C22×C4⋊C4—611no—
326C22×C4⋊C4—612no—
327C22×C4⋊C4—611no—
328C22×C42—611no—
329C22×C42—611no—
330C22×C42—612no—
331C22×C42—612no—
332C22×C42C2×D6111yescharacteristic
333#391—6116no—
334C4×C4⋊C4—1224no—
335#376—1224no—
336C2×#95—612no—
337C2×#95—612no—
338C2×#95—612no—
339C2×#95—611no—
340C2×#95—611no—
341C2×#95—612no—
342C2×#95—612no—
343C2×#95—612no—
344C2×#95—612no—
345C2×#95C2×D6111yescharacteristic
346C2×#95C2×D6111yescharacteristic
347C2×#95C2×D6112yes—
348C2×#95C2×D6112yes—
349C61#1378611yescharacteristic, complemented, Sylow
350C22×Q8—614no—
351C22×Q8—618no—
352C22×Q8—1222no—
353C22×Q8—1222no—
354C22×Q8—1224no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—612no—
360C23×C4—612no—
361C23×C4C22×D6111yescharacteristic
362C23×C4C22×D6112yes—
363C23×C4C22×D6112yes—
364C23×C4C22×D6111yescharacteristic
365C23×C4C22×D6111yescharacteristic
366C4×Q8—12232no—
367C2×C4⋊C4—614no—
368C2×C4⋊C4—614no—
369C2×C4⋊C4—614no—
370C2×C4⋊C4—618no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1224no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—614no—
382C2×C42—614no—
383C2×C42—614no—
384C2×C42—614no—
385C2×C42—618no—
386C2×C42—1224no—
387C2×C42—1224no—
388C2×C42—1222no—
389C2×C42—1222no—
390C2×C42—1224no—
391C2×C42—2442no—
392C2×C42—2442no—
393C2×C42C4×D6114yes—
394#95—22no—
395#95—22no—
396#95—42no—
397#95—618no—
398#95—618no—
399#95—618no—
400#95—1224no—
401#95—1222no—
402#95—1222no—
403#95—2442no—
404#95—2442no—
405#95—2442no—
406#95C4×D6118yes—
407C24C23×D6111yescharacteristic
408C2×Q8—6116no—
409C2×Q8—1228no—
410C2×Q8—1228no—
411C2×Q8—12216no—
412C2×Q8—12216no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—612no—
437C22×C4—614no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—614no—
441C22×C4—614no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—614no—
447C22×C4—614no—
448C22×C4—612no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1222no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—1221no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—1221no—
474C22×C4C2×C4×D6114yes—
475C22×C4C2×C4×D6114yes—
476C22×C4C2×C4×D6112yes—
477C22×C4C2×C4×D6114yes—
478C22×C4C2×C4×D6112yes—
479C22×C4C2×C4×D6112yes—
480C22×C4D4×D6112yes—
481C22×C4D4×D6112yes—
482C4⋊C4—12216no—
483C4⋊C4—1228no—
484C4⋊C4—1228no—
485C4⋊C4—1228no—
486C42—28no—
487C42—6116no—
488C42—12216no—
489C42—1228no—
490C42—2448no—
491C23C22×C4×D6111yescharacteristic
492C23C22×C4×D6112yes—
493C23C2×D4×D6111yescharacteristic
494C23C2×D4×D6111yescharacteristic
495C23#41×D6112yes—
496C23#1451982211yescharacteristic
497C23#1451982311yescharacteristic
498C23#1451982412yes—
499C23#1451982412yes—
500C23#1451982411yescharacteristic
501C23#1451982511yescharacteristic
502Q8—12232no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—614no—
533C2×C4—614no—
534C2×C4—614no—
535C2×C4—618no—
536C2×C4—618no—
537C2×C4—614no—
538C2×C4—614no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1224no—
542C2×C4—1224no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—1224no—
555C2×C4—1224no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—2442no—
559C2×C4—2441no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2442no—
565C2×C4—2441no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4C42×D6114yes—
571C2×C4C22⋊C4×D6114yes—
572C22C2×C42×D6111yescharacteristic
573C22C2×C22⋊C4×D6111yescharacteristic
574C22#117×D6111yescharacteristic
575C22C4×D4×D6112yes—
576C22C4×D4×D6112yes—
577C22#1684023812yes—
578C22#1684023911yescharacteristic
579C22#1684023911yescharacteristic
580C22#1684023912yes—
581C22#1684024112yes—
582C22#1684024211yescharacteristic
583C22#1684024212yes—
584C22#1684024311yescharacteristic
585C22#1684032811yescharacteristic
586C22#1684033412yes—
587C22#1684033412yes—
588C22#1684033511yescharacteristic
589C22#1684034111yescharacteristic
590C22#1684034211yescharacteristic
591C22#1684036912yes—
592C22#1684037112yes—
593C22#1684037612yes—
594C22#1684037812yes—
595C4—24no—
596C4—24no—
597C4—44no—
598C4—44no—
599C4—42no—
600C4—618no—
601C4—1224no—
602C4—1224no—
603C4—2444no—
604C4—2444no—
605C4—2442no—
606C2C4×C22⋊C4×D6111yescharacteristic
607C2#2144818211yescharacteristic
608C2#2144818311yescharacteristic
609C2#2144840411yescharacteristic
610C2#2144840811yescharacteristic
611C2#2144846711yescharacteristic
612C2#2144846911yescharacteristic
613C2#2144849812yes—
614C2#2144850012yes—
615C2#2144852812yes—
616C2#2144853412yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC22×C61C22×C42
FittingC61×#1434C2
socle, prime radical, cosocleC24×C61C24
hypercenter#1434D61
hypocenterC61#13786
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
15. C2×C4×Dic122C4semidirect
24. #16839666C4semidirect
172. C2×Dic122C42semidirect
349. C61#13786semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1222
C2×#168396662
C2×#168396132
C2×#168396072
#214440532
#214440502
#214440472
#214440462
#214440432
C61×#14342
#1378661