Finite Group Databasebeta

← #29601821

Subgroups — #29601821

Order 15616.

1942 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
1C22×#14519742C211yesmaximal, characteristic
2C2×#16839682C211yesmaximal, characteristic
3C2×C4⋊C4×Dic61C211yesmaximal, characteristic
4C2×#16839613C212yesmaximal
5#21444089C211yesmaximal, characteristic
6#21444083C212yesmaximal
7#21444082C211yesmaximal, characteristic
8#21444070C212yesmaximal
9#95×Dic61C212yesmaximal
10#95×Dic61C211yesmaximal, characteristic
11C61×#1441C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C2×#14519742—24no—
14C2×#14519742—28no—
15C22×C4⋊Dic61C2211yescharacteristic
16C22×#11758196C2212yes—
17C22×#11758196C2212yes—
18C22×#11758196C2212yes—
19C22×C4×Dic61C2211yescharacteristic
20C22×C4×Dic61C2212yes—
21C22×C4×Dic61C2211yescharacteristic
22C22×C4×Dic61C2212yes—
23C22×C4×Dic61C2211yescharacteristic
24#16839682C418yescomplemented
25C4⋊C4×Dic61—24no—
26#16839613—28no—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2211yescharacteristic
32C2×#14519684C2211yescharacteristic
33C2×#14519684C2211yescharacteristic
34C2×#14519684C2212yes—
35C22×C4⋊C4×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C2×#95×C61C2212yes—
38C2×#95×C61C2211yescharacteristic
39C2×#95×C61C2212yes—
40C22×Dic122—22no—
41C22×Dic122—22no—
42C22×Dic122—24no—
43C22×Dic122—24no—
44C22×Dic122C2×C414yes—
45C23×Dic61C2311yescharacteristic
46C23×Dic61C2311yescharacteristic
47C23×Dic61C2311yescharacteristic
48C23×Dic61C2312yes—
49C23×Dic61C2312yes—
50C23×Dic61C2311yescharacteristic
51#14519742—232no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—22no—
55C2×#11758196—24no—
56C2×#11758196—24no—
57C2×#11758196—24no—
58C2×#11758196—24no—
59C2×#11758196—24no—
60C2×#11758196—24no—
61C2×#11758196—24no—
62C2×#11758196—24no—
63C2×#11758196C2×C418yes—
64C2×C4×Dic61—24no—
65C2×C4×Dic61—24no—
66C2×C4×Dic61—24no—
67C2×C4×Dic61—24no—
68C2×C4×Dic61—22no—
69C2×C4×Dic61—24no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—24no—
72C2×C4×Dic61—22no—
73C2×C4×Dic61D418yes—
74C2×C4×Dic61D414yes—
75C2×C4×Dic61Q814yescomplemented
76C2×C4×Dic61Q818yescomplemented
77#14519684—22no—
78#14519684—24no—
79#14519684—22no—
80#14519684—24no—
81#14519684—24no—
82#14519684—24no—
83#14519684C2×C418yes—
84#14519684C2×C414yes—
85C23×C4×C61C2311yescharacteristic
86C23×C4×C61C2312yes—
87C23×C4×C61C2312yes—
88C23×C4×C61C2311yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61—24no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61C2×C414yes—
95#95×C61—24no—
96#95×C61—22no—
97#95×C61—24no—
98C24×C61C2411yescharacteristic
99C22×C4×C61—22no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
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—21no—
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×C61C22×C412yes—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C2×D414yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×Q814yes—
127C22×C4×C61C2×Q812yes—
128C22×C4×C61#4112yes—
129C22×C4×C61#4112yes—
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—21no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—21no—
146C22×Dic61—22no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C22×C414yes—
151C22×Dic61C22×C414yes—
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D414yes—
154C22×Dic61C2×D414yes—
155C22×Dic61C2×D412yes—
156C22×Dic61C2×Q812yes—
157C22×Dic61C2×Q814yes—
158C22×Dic61C2×Q814yes—
159C22×Dic61C2×Q812yes—
160C22×Dic61#4112yes—
161C22×Dic61#4114yes—
162C22×Dic61#4114yes—
163C22×Dic61#4112yes—
164C2×Dic122—216no—
165C2×Dic122—28no—
166C2×Dic122—28no—
167C2×Dic122—28no—
168C2×Dic122—216no—
169C4⋊C4×C61—28no—
170C4⋊C4×C61—28no—
171C4⋊Dic61—28no—
172#11758196—216no—
173#11758196—216no—
174#11758196—216no—
175C4×Dic61—216no—
176C4×Dic61—28no—
177C4×Dic61—216no—
178C4×Dic61—28no—
179C4×Dic61—44no—
180C23×C61C23×C411yescharacteristic
181C23×C61C22×D412yes—
182C23×C61C22×D411yescharacteristic
183C23×C61C22×Q812yes—
184C23×C61C22×Q811yescharacteristic
185C23×C61C2×#4112yes—
186C23×C61C2×#4111yescharacteristic
187C23×C61C2×#4112yes—
188C23×C61C2×#4111yescharacteristic
189C23×C61C2×#4111yescharacteristic
190C23×C61C2×#4111yescharacteristic
191C2×C4×C61—22no—
192C2×C4×C61—24no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—22no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61C4×D414yes—
221C2×C4×C61C4×Q814yescomplemented
222C2×Dic61—24no—
223C2×Dic61—24no—
224C2×Dic61—24no—
225C2×Dic61—24no—
226C2×Dic61—24no—
227C2×Dic61—24no—
228C2×Dic61—24no—
229C2×Dic61—24no—
230C2×Dic61—24no—
231C2×Dic61—24no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—24no—
241C2×Dic61—41no—
242C2×Dic61—42no—
243C2×Dic61—41no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61C4×D418yes—
250C2×Dic61C4×Q818yescomplemented
251C2×Dic61#12014yes—
252C2×Dic61C22⋊Q814yes—
253C2×Dic61C22⋊Q818yes—
254C2×Dic61C4⋊Q814yes—
255C2×Dic61C4⋊Q818yes—
256Dic122—232no—
257C4×#395—611nomaximal, Sylow
258C22×C61C2×C4×D412yes—
259C22×C61C2×C4×D411yescharacteristic
260C22×C61C2×C4×Q812yes—
261C22×C61C2×C4×Q811yescharacteristic
262C22×C61C4×#4111yescharacteristic
263C22×C61C2×#12011yescharacteristic
264C22×C61C2×C22⋊Q811yescharacteristic
265C22×C61C2×C22⋊Q812yes—
266C22×C61#52512yes—
267C22×C61#52511yescharacteristic
268C22×C61#52511yescharacteristic
269C22×C61#52512yes—
270C22×C61#52911yescharacteristic
271C22×C61#52912yes—
272C22×C61C2×C4⋊Q811yescharacteristic
273C22×C61C2×C4⋊Q812yes—
274C22×C61#53212yes—
275C22×C61#53211yescharacteristic
276C22×C61#53312yes—
277C22×C61#53312yes—
278C22×C61#53311yescharacteristic
279C22×C61#53312yes—
280C22×C61#53312yes—
281C4×C61—24no—
282C4×C61—24no—
283C4×C61—44no—
284C4×C61—44no—
285C4×C61—42no—
286Dic61—28no—
287Dic61—28no—
288Dic61—42no—
289Dic61—42no—
290Dic61—42no—
291Dic61#39518yescomplemented
292C22×C4×Q8—611no—
293C2×#395—611no—
294C2×C4×C4⋊C4—612no—
295C2×C4×C4⋊C4—611no—
296#1446—611no—
297#1444—612no—
298#1444—611no—
299#1441—612no—
300#1441D6111yescharacteristic
301C4×#95—611no—
302C4×#95—612no—
303C2×C61C4×#12011yescharacteristic
304C2×C61C4×C22⋊Q811yescharacteristic
305C2×C61C4×C22⋊Q812yes—
306C2×C61C4×C4⋊Q811yescharacteristic
307C2×C61C4×C4⋊Q812yes—
308C2×C61C2×#39511yescharacteristic
309C2×C61#238811yescharacteristic
310C2×C61#239611yescharacteristic
311C2×C61#239612yes—
312C2×C61#239912yes—
313C2×C61#239911yescharacteristic
314C23×Q8—611no—
315C2×C4×Q8—1224no—
316C2×C4×Q8—1228no—
317C22×C4⋊C4—612no—
318C22×C4⋊C4—612no—
319C22×C4⋊C4—612no—
320C22×C4⋊C4—611no—
321C22×C4⋊C4C2×D6111yescharacteristic
322C22×C4⋊C4C2×D6111yescharacteristic
323C22×C42—612no—
324C22×C42—611no—
325C22×C42—612no—
326C22×C42—611no—
327C22×C42—611no—
328#395—618no—
329C4×C4⋊C4—1228no—
330C4×C4⋊C4—1224no—
331C2×#95—612no—
332C2×#95—611no—
333C2×#95—612no—
334C2×#95—612no—
335C2×#95—612no—
336C2×#95—611no—
337C2×#95—612no—
338C2×#95—611no—
339C2×#95C2×D6112yes—
340C2×#95C2×D6111yescharacteristic
341C2×#95C2×D6112yes—
342C61C4×#39511yescharacteristic, complemented, Sylow
343C22×Q8—614no—
344C22×Q8—1224no—
345C22×Q8—1224no—
346C22×Q8—1222no—
347C22×Q8—1222no—
348C23×C4—611no—
349C23×C4—612no—
350C23×C4—612no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4C22×D6111yescharacteristic
355C23×C4C22×D6112yes—
356C23×C4C22×D6112yes—
357C23×C4C22×D6111yescharacteristic
358C23×C4C22×D6111yescharacteristic
359C4×Q8—12232no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—618no—
365C2×C4⋊C4—1224no—
366C2×C4⋊C4—1224no—
367C2×C4⋊C4—1224no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1224no—
370C2×C4⋊C4—1224no—
371C2×C4⋊C4—1224no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1224no—
374C2×C4⋊C4—1224no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4C4×D6114yes—
377C2×C42—618no—
378C2×C42—614no—
379C2×C42—618no—
380C2×C42—614no—
381C2×C42—1224no—
382C2×C42—1224no—
383C2×C42—1224no—
384C2×C42—1222no—
385C2×C42—1224no—
386C2×C42—1222no—
387C2×C42—1224no—
388C2×C42—1224no—
389C2×C42—1222no—
390#95—24no—
391#95—22no—
392#95—24no—
393#95—614no—
394#95—618no—
395#95—1224no—
396#95—1222no—
397#95—1224no—
398#95—1224no—
399#95—1224no—
400#95—1222no—
401C24C23×D6111yescharacteristic
402C2×Q8—12216no—
403C2×Q8—1228no—
404C2×Q8—1228no—
405C2×Q8—12216no—
406C2×Q8—1228no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—614no—
430C22×C4—612no—
431C22×C4—614no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—614no—
435C22×C4—614no—
436C22×C4—614no—
437C22×C4—614no—
438C22×C4—614no—
439C22×C4—614no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1221no—
452C22×C4—1222no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1221no—
459C22×C4—1222no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4C2×C4×D6112yes—
464C22×C4C2×C4×D6112yes—
465C22×C4C2×C4×D6112yes—
466C22×C4D4×D6114yes—
467C22×C4D4×D6112yes—
468C22×C4#1175822412yes—
469C22×C4#1175822412yes—
470C22×C4Q8×D6114yes—
471C22×C4Q8×D6112yes—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—12216no—
475C4⋊C4—12216no—
476C4⋊C4—12216no—
477C4⋊C4—1228no—
478C42—1228no—
479C42—12216no—
480C42—1228no—
481C42—12216no—
482C42—2444no—
483C23C22×C4×D6111yescharacteristic
484C23C2×D4×D6112yes—
485C23C2×D4×D6111yescharacteristic
486C23C2×Q8×D6112yes—
487C23C2×Q8×D6111yescharacteristic
488C23#41×D6112yes—
489C23#41×D6112yes—
490C23#41×D6111yescharacteristic
491C23#41×D6111yescharacteristic
492C23C2×#1175822411yescharacteristic
493C23C2×#1175822411yescharacteristic
494Q8—12232no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—614no—
525C2×C4—614no—
526C2×C4—618no—
527C2×C4—614no—
528C2×C4—618no—
529C2×C4—618no—
530C2×C4—618no—
531C2×C4—1224no—
532C2×C4—1224no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1224no—
541C2×C4—1224no—
542C2×C4—1224no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1224no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1224no—
550C2×C4—2441no—
551C2×C4—2442no—
552C2×C4—2441no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4#1451973714yes—
559C2×C4#1451974214yescomplemented
560C22C4×D4×D6112yes—
561C22C2×#1451973711yescharacteristic
562C22C4×Q8×D6112yes—
563C22C2×#1451974211yescharacteristic
564C22#1684023011yescharacteristic
565C22#120×D6111yescharacteristic
566C22C22⋊Q8×D6111yescharacteristic
567C22C22⋊Q8×D6112yes—
568C22#1684027012yes—
569C22#1684027011yescharacteristic
570C22#1684027211yescharacteristic
571C22#1684027212yes—
572C22#1684029311yescharacteristic
573C22#1684029512yes—
574C22C4⋊Q8×D6111yescharacteristic
575C22C4⋊Q8×D6112yes—
576C22#1684030712yes—
577C22#1684031111yescharacteristic
578C22#1684031412yes—
579C22#1684031412yes—
580C22#1684031411yescharacteristic
581C22#1684031812yes—
582C22#1684031812yes—
583C4—24no—
584C4—24no—
585C4—44no—
586C4—42no—
587C4—44no—
588C4—618no—
589C4—1228no—
590C4—1228no—
591C4—2442no—
592C4—2442no—
593C4—2442no—
594C2#2144832411yescharacteristic
595C2#2144834711yescharacteristic
596C2#2144835312yes—
597C2#2144838811yescharacteristic
598C2#2144839012yes—
599C2#395×D6111yescharacteristic
600C2#2144903311yescharacteristic
601C2#2144909211yescharacteristic
602C2#2144909412yes—
603C2#2144911112yes—
604C2#2144911111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1441C2
socle, prime radical, cosocleC24×C61C24
hypercenter#1441D61
hypocenterC61C4×#395
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
24. #16839682C4semidirect
75. C2×C4×Dic61Q8semidirect
76. C2×C4×Dic61Q8semidirect
221. C2×C4×C61C4×Q8semidirect
250. C2×Dic61C4×Q8semidirect
291. Dic61#395semidirect
342. C61C4×#395semidirect
559. C2×C4#14519742semidirect

Maximal subgroups

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

subgroupindex
C22×#145197422
C2×#168396822
C2×C4⋊C4×Dic612
C2×#168396132
#214440892
#214440832
#214440822
#214440702
#95×Dic612
C61×#14412
C4×#39561