Finite Group Databasebeta

← #29607196

Subgroups — #29607196

Order 15616.

1926 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×#14519736C211yesmaximal, characteristic
2C2×#16839631C211yesmaximal, characteristic, complemented
3C2×#16839627C211yesmaximal, characteristic, complemented
4C2×#16839622C211yesmaximal, characteristic
5#21444071C214yesmaximal, complemented
6#21444062C214yesmaximal
7#21444048C211yesmaximal, characteristic, complemented
8#21444048C211yesmaximal, characteristic, complemented
9C2×C61×#380C211yesmaximal, characteristic
10C2×#14519736—24no—
11C2×#14519736—24no—
12C2×#14519736C418yescomplemented
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C4⋊Dic61C2212yes—
16C22×#11758196C2214yes—
17C22×C22⋊Dic61C2212yes—
18C22×C22⋊Dic61C2212yes—
19C22×C4×Dic61C2211yescharacteristic
20C22×C4×Dic61C2211yescharacteristic
21#16839631—24no—
22#16839627—24no—
23#16839622—24no—
24C2×#14519684C2212yes—
25C2×#14519684C2214yes—
26C2×#14519684C2214yes—
27C2×#14519684C2214yes—
28C24×C4×C61C2211yescharacteristic
29C22×C4⋊C4×C61C2211yescharacteristic
30C22×C4⋊C4×C61C2211yescharacteristic
31C22×C22⋊C4×C61C2212yes—
32C61×#380—24no—
33C2×#95×C61C2212yes—
34C23×Dic61C2312yes—
35C23×Dic61C2312yes—
36C23×Dic61C2312yes—
37C23×Dic61C2312yes—
38#14519736—232no—
39C2×C4⋊Dic61—24no—
40C2×C4⋊Dic61—24no—
41C2×C4⋊Dic61—22no—
42C2×C4⋊Dic61—22no—
43C2×C4⋊Dic61—22no—
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—22no—
47C2×C4⋊Dic61C2×C418yes—
48C2×#11758196—28no—
49C2×#11758196—44no—
50C2×#11758196—44no—
51C2×C22⋊Dic61—24no—
52C2×C22⋊Dic61—24no—
53C2×C22⋊Dic61—24no—
54C2×C22⋊Dic61—44no—
55C2×C22⋊Dic61C2×C418yes—
56C2×C4×Dic61—22no—
57C2×C4×Dic61—22no—
58C2×C4×Dic61C2×C414yes—
59C2×C4×Dic61C2×C414yes—
60C2×C4×Dic61D414yes—
61C2×C4×Dic61D414yes—
62#14519684—24no—
63#14519684—28no—
64#14519684—44no—
65#14519684—44no—
66C25×C61C2311yescharacteristic
67C23×C4×C61—22no—
68C23×C4×C61—22no—
69C23×C4×C61—22no—
70C23×C4×C61—22no—
71C23×C4×C61C2×C414yes—
72C23×C4×C61D414yes—
73C23×C4×C61Q814yes—
74C23×C4×C61C2312yes—
75C23×C4×C61C2312yes—
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61—22no—
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61—22no—
84C2×C22⋊C4×C61—24no—
85C2×C22⋊C4×C61—44no—
86#95×C61—24no—
87C24×C61—21no—
88C24×C61—21no—
89C24×C61—21no—
90C24×C61—21no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—22no—
94C24×C61—22no—
95C24×C61C22×C412yes—
96C24×C61C2×D412yes—
97C24×C61C2×Q812yes—
98C24×C61C2411yescharacteristic
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—28no—
104C22×C4×C61—28no—
105C22×C4×C61—28no—
106C22×C4×C61—28no—
107C22×C4×C61—28no—
108C22×C4×C61—28no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—24no—
116C22×C4×C61—24no—
117C22×C4×C61—24no—
118C22×C4×C61—24no—
119C22×C4×C61C4⋊C4116yescomplemented
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×Q812yes—
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—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—24no—
147C22×Dic61—24no—
148C22×Dic61—24no—
149C22×Dic61—24no—
150C22×Dic61C22×C414yes—
151C22×Dic61C22×C414yes—
152C22×Dic61C2×D414yes—
153C22×Dic61C2×D414yes—
154C4⋊C4×C61—44no—
155C4⋊C4×C61—44no—
156C22⋊C4×C61—48no—
157C22⋊Dic61—216no—
158C22⋊Dic61—48no—
159C4⋊Dic61—216no—
160C4⋊Dic61—44no—
161C4⋊Dic61—44no—
162#11758196—416no—
163C4×Dic61—28no—
164C4×Dic61C22⋊C4116yescomplemented
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—24no—
168C23×C61—24no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—24no—
178C23×C61—24no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61C2×C4⋊C414yes—
188C23×C61C23×C411yescharacteristic
189C23×C61C22×D411yescharacteristic
190C23×C61C22×D411yescharacteristic
191C23×C61C22×D411yescharacteristic
192C23×C61C22×Q811yescharacteristic
193C23×C61C2×#4111yescharacteristic
194C23×C61C2×#4111yescharacteristic
195C23×C61H(5,𝔽2)12yes—
196C23×C61H(5,𝔽2)12yes—
197C23×C612−1+412yes—
198C23×C612−1+412yes—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—232no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—44no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—44no—
214C2×C4×C61—44no—
215C2×C4×C61—44no—
216C2×C4×C61C2×C22⋊C414yes—
217C2×C4×C61C2×C4⋊C414yes—
218C2×C4×C61C2×C4⋊C414yes—
219C2×C4×C61#11714yes—
220C2×C4×C61#12114yes—
221C2×C4×C61#12114yes—
222C2×C4×C61C22⋊Q814yes—
223C2×C4×C61C22⋊Q814yes—
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—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—44no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—44no—
245C2×Dic61—44no—
246C2×Dic61—44no—
247C2×Dic61C2×C22⋊C418yes—
248#14340—611nomaximal, Sylow
249C22×C61—24no—
250C22×C61—24no—
251C22×C61—24no—
252C22×C61—44no—
253C22×C61—44no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—44no—
259C22×C61—44no—
260C22×C61C22×C22⋊C411yescharacteristic
261C22×C61C22×C4⋊C411yescharacteristic
262C22×C61C2×#11711yescharacteristic
263C22×C61#52012yes—
264C22×C61#52012yes—
265C22×C61C2×#12111yescharacteristic
266C22×C61C2×#12111yescharacteristic
267C22×C61C2×C22⋊Q811yescharacteristic
268C22×C61C2×C22⋊Q811yescharacteristic
269C22×C61#53511yescharacteristic
270C22×C61#53512yes—
271C22×C61#53511yescharacteristic
272C22×C61#53611yescharacteristic
273C22×C61#53612yes—
274C22×C61#53611yescharacteristic
275C22×C61#53712yes—
276C22×C61#53712yes—
277C22×C61#54012yes—
278C22×C61#54112yes—
279C22×C61#54212yes—
280C22×C61#54212yes—
281C22×C61#54412yes—
282C22×C61#54412yes—
283C4×C61—24no—
284C4×C61—82no—
285C4×C61—82no—
286C4×C61#38018yes—
287Dic61—28no—
288Dic61—44no—
289Dic61—82no—
290Dic61—82no—
291C22×#117—611no—
292C2×#384—611no—
293C2×#382—611no—
294C2×#380—611no—
295C2×#380D6111yescharacteristic
296#1441—614no—
297#1439—614no—
298C4⋊#95—611no—
299C4⋊#95—611no—
300C2×C61—44no—
301C2×C61C2×#38011yescharacteristic
302C2×C61#231112yes—
303C2×C61#231412yes—
304C2×C61#231711yescharacteristic
305C2×C61#231711yescharacteristic
306C2×C61#261911yescharacteristic
307C2×C61#262011yescharacteristic
308C2×C61#262011yescharacteristic
309C2×C61#262111yescharacteristic
310C2×C61#262812yes—
311C2×C61#262912yes—
312C24×C4C2×D6111yescharacteristic
313C2×#117—618no—
314C2×#117—1224no—
315C2×#117—1224no—
316C22×C4⋊C4—612no—
317C22×C4⋊C4—611no—
318C22×C4⋊C4—611no—
319C22×C4⋊C4—614no—
320C22×C4⋊C4C2×D6111yescharacteristic
321C22×C4⋊C4C2×D6111yescharacteristic
322C22×C22⋊C4—612no—
323C22×C22⋊C4—612no—
324C22×C22⋊C4C2×D6112yes—
325C22×C42—611no—
326C22×C42—611no—
327#384—1224no—
328#382—1224no—
329#380—24no—
330#380—1224no—
331C2×#95—614no—
332C2×#95—612no—
333C2×#95—614no—
334C2×#95—614no—
335C2×#95C2×D6112yes—
336C61#1434011yescharacteristic, complemented, Sylow
337C25C22×D6111yescharacteristic
338C23×C4—22no—
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—22no—
342C23×C4—612no—
343C23×C4—612no—
344C23×C4—612no—
345C23×C4—612no—
346C23×C4Dic12214yes—
347C23×C4C4×D6114yes—
348C23×C4C22⋊D6114yes—
349C23×C4C22×D6112yes—
350C23×C4C22×D6112yes—
351C23×C4C22×D6111yescharacteristic
352C23×C4C22×D6111yescharacteristic
353#117—12232no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—618no—
361C2×C4⋊C4—1224no—
362C2×C4⋊C4—1224no—
363C2×C4⋊C4—1228no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—2444no—
371C2×C4⋊C4—2444no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—44no—
374C2×C22⋊C4—618no—
375C2×C22⋊C4—1224no—
376C2×C22⋊C4—1224no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—2444no—
379C2×C42—614no—
380C2×C42—614no—
381C2×C42—614no—
382C2×C42—614no—
383C2×C42—1222no—
384C2×C42—1222no—
385#95—24no—
386#95—1228no—
387#95—1224no—
388#95—2444no—
389#95—2444no—
390C24—21no—
391C24—21no—
392C24—21no—
393C24—21no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24C2×Dic12212yes—
399C24C2×C4×D6112yes—
400C24C2×C22⋊D6112yes—
401C24C23×D6111yescharacteristic
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—614no—
423C22×C4—614no—
424C22×C4—614no—
425C22×C4—614no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4#11758196116yescomplemented
447C22×C4C2×Dic12212yes—
448C22×C4C2×Dic12212yes—
449C22×C4C2×C4×D6112yes—
450C22×C4C2×C4×D6112yes—
451C22×C4D4×D6112yes—
452C22×C4D4×D6112yes—
453C22×C4#1175822412yes—
454C22×C4#1175822412yes—
455C22×C4C2×C22⋊D6112yes—
456C22×C4C2×C22⋊D6112yes—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—12216no—
460C4⋊C4—24416no—
461C4⋊C4—2444no—
462C4⋊C4—2444no—
463C22⋊C4—48no—
464C22⋊C4—12216no—
465C22⋊C4—2448no—
466C42—6116no—
467C42—1228no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23C2×#1175819614yes—
491C23C22×C4×D6111yescharacteristic
492C23C2×D4×D6111yescharacteristic
493C23C2×D4×D6111yescharacteristic
494C23C22×C22⋊D6111yescharacteristic
495C23C22×Dic12211yescharacteristic
496C23C2×#1175822411yescharacteristic
497C23C2×#1175822411yescharacteristic
498C23#1451982212yes—
499C23#1451982312yes—
500C23#1451982412yes—
501C23#1451982512yes—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—232no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—618no—
520C2×C4—1224no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—1224no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2444no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2444no—
537C2×C4—2444no—
538C2×C4—2444no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4C22⋊C4×D6114yes—
544C2×C4C2×#1175819614yes—
545C2×C4C2×#1175819614yes—
546C2×C4#1451973314yes—
547C2×C4#1451974914yes—
548C2×C4#1451974914yes—
549C2×C4#1451975414yes—
550C2×C4#1451975414yes—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22C2×C22⋊C4×D6111yescharacteristic
563C22C22×#1175819611yescharacteristic
564C22C2×#1451973311yescharacteristic
565C22#1684024212yes—
566C22#1684024312yes—
567C22C2×#1451974911yescharacteristic
568C22C2×#1451974911yescharacteristic
569C22C2×#1451975411yescharacteristic
570C22C2×#1451975411yescharacteristic
571C22#1684032611yescharacteristic
572C22#1684032812yes—
573C22#1684032911yescharacteristic
574C22#1684033311yescharacteristic
575C22#1684033512yes—
576C22#1684033611yescharacteristic
577C22#1684034112yes—
578C22#1684034212yes—
579C22#1684036512yes—
580C22#1684037212yes—
581C22#1684037912yes—
582C22#1684038512yes—
583C22#1684039012yes—
584C22#1684039312yes—
585C4—24no—
586C4—82no—
587C4—82no—
588C4—1228no—
589C4—4882no—
590C4—4882no—
591C4—2444no—
592C4#1683962018yes—
593C2—44no—
594C2C2×#1683962011yescharacteristic
595C2#2144840812yes—
596C2#2144842612yes—
597C2#2144843911yescharacteristic
598C2#2144844311yescharacteristic
599C2#2145157311yescharacteristic
600C2#2145158011yescharacteristic
601C2#2145158111yescharacteristic
602C2#2145158911yescharacteristic
603C2#2145164412yes—
604C2#2145165312yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#380C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#380D61
hypocenterC61#14340
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#16839631C2semidirect
3. C2×#16839627C2semidirect
5. #21444071C2semidirect
7. #21444048C2semidirect
8. #21444048C2semidirect
12. C2×#14519736C4semidirect
119. C22×C4×C61C4⋊C4semidirect
164. C4×Dic61C22⋊C4semidirect
336. C61#14340semidirect
446. C22×C4#11758196semidirect

Maximal subgroups

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

subgroupindex
C22×#145197362
C2×#168396312
C2×#168396272
C2×#168396222
#214440712
#214440622
#214440482
C2×C61×#3802
#1434061