Finite Group Databasebeta

← #29679970

Subgroups — #29679970

Order 15616.

1694 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
1C2×C4×D4×D61C211yesmaximal, characteristic
2#21454451C212yesmaximal, complemented
3#21445898C212yesmaximal, complemented
4#21445811C211yesmaximal, characteristic, complemented
5#21445808C212yesmaximal, complemented
6#21445738C212yesmaximal
7#100×Dic61C212yesmaximal, complemented
8#21445610C212yesmaximal, complemented
9C61×#1800C211yesmaximal, characteristic, complemented
10C22×D4×D61C2211yescharacteristic
11C23×C4×D61C412yes—
12C2×C4×D244C2211yescharacteristic
13C2×D4×Dic61C2211yescharacteristic
14C2×#14519739C2211yescharacteristic
15C2×C4×C22⋊D61C412yes—
16C2×#14519737C412yes—
17C4×D4×D61—22no—
18C4×D4×D61—22no—
19C4×D4×D61—22no—
20C4×D4×D61—22no—
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C22⋊C4×D61C412yes—
23C2×C42×D61C2211yescharacteristic
24#16839818—22no—
25#16839818—22no—
26#16839814—22no—
27#16839814—22no—
28#16839786C2212yescomplemented
29C2×#14519694C2212yes—
30C2×#14519693C2214yes—
31#16839725C2214yes—
32M16×Dic61C2214yescomplemented
33#16839531C2212yescomplemented
34#16839530C2214yescomplemented
35C2×C4×D4×C61C2211yescharacteristic
36C61×#423C2212yescomplemented
37C2×#100×C61C2212yes—
38C61×#343C2212yescomplemented
39C24×D61C2×C412yes—
40C22×D244C2311yescharacteristic
41C22×C22⋊D61C2×C412yes—
42C2×D4×D61—22no—
43C2×D4×D61—22no—
44C2×D4×D61—24no—
45C2×D4×D61—22no—
46C2×D4×D61—22no—
47C23×Dic61C2×C412yes—
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61—48no—
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C412yes—
55C22×C4×D61C2×C412yes—
56C22×C4×D61C2×C412yes—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C2×#11758194C2314yes—
60C4×D244—22no—
61C4×D244—22no—
62D4×Dic61—22no—
63D4×Dic61—22no—
64#14519739—22no—
65#14519739—22no—
66C4×C22⋊D61—24no—
67C4×C22⋊D61—24no—
68#14519737—24no—
69#14519737—24no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196C2×C412yes—
72C4⋊C4×D61—22no—
73C4⋊C4×D61—22no—
74C2×C22⋊Dic61C2×C412yes—
75C2×#11758198C2×C412yes—
76C2×#11758198C2×C412yes—
77C22⋊C4×D61—24no—
78C22⋊C4×D61—24no—
79C2×C4×Dic61C2×C411yescharacteristic
80C2×C4×Dic61C2×C411yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C42×D61D412yes—
83C42×D61D412yes—
84C42×D61D412yes—
85C42×D61D412yes—
86C4⋊#92418—22no—
87C4⋊#92418—22no—
88#14519694—24no—
89#14519693—28no—
90#14519689—24no—
91#14519689—24no—
92#14519687—24no—
93C8×Dic61—24no—
94C22×D4×C61C2311yescharacteristic
95C23×C4×C61C2×C412yes—
96C2×M16×C61C2314yes—
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61C2×C412yes—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—22no—
103C4⋊C8×C61—22no—
104#100×C61—24no—
105C24×C61C22×C412yes—
106C23×D61—24no—
107C23×D61—22no—
108C23×D61—22no—
109C23×D61—24no—
110C23×D61—44no—
111C23×D61—44no—
112C23×D61C22×C411yescharacteristic
113C23×D61C22×C412yes—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61#4112yes—
117C2×D4×C61#4112yes—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—44no—
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—24no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61—24no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—44no—
131C22×Dic61C22×C411yescharacteristic
132C22×Dic61C22×C411yescharacteristic
133C22×Dic61C22×C412yes—
134C22×Dic61C22×C411yescharacteristic
135D4×D61—28no—
136D4×D61—28no—
137D4×D61—28no—
138D4×D61—28no—
139C2×D244—22no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244#4114yes—
143C2×C4×D61—22no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—24no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—42no—
154C2×C4×D61—48no—
155C2×C4×D61—48no—
156C2×C4×D61—48no—
157C2×C4×D61—48no—
158C2×C4×D61—44no—
159C2×C4×D61—44no—
160C2×C4×D61—44no—
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22⋊C411yescharacteristic
163C2×C4×D61C22⋊C411yescharacteristic
164C2×C4×D61C22⋊C412yes—
165C2×C4×D61C22⋊C412yes—
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C2×D412yes—
168C2×C4×D61C2×D412yes—
169M16×C61—24no—
170M16×C61—24no—
171C2×C8×C61—24no—
172C4⋊C4×C61—22no—
173C22⋊C4×C61—24no—
174C42×C61C2×D412yes—
175C42×C61C2×D412yes—
176C22⋊Dic61—24no—
177#11758198—24no—
178#11758198—24no—
179C4⋊Dic61—22no—
180#11758196—24no—
181C4×Dic61C22⋊C412yes—
182C4×Dic61C22⋊C412yes—
183C4×Dic61C22⋊C412yes—
184C4×Dic61C22⋊C412yes—
185C4×Dic61C2×D412yes—
186C4×Dic61C2×D412yes—
187#11758194—24no—
188#11758194—24no—
189C2×#92418—24no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61#11714yes—
195C23×C61C23×C411yescharacteristic
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—28no—
201C22×D61—28no—
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—44no—
205C22×D61—416no—
206C22×D61—416no—
207C22×D61—44no—
208C22×D61—44no—
209C22×D61—44no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61C2×C22⋊C411yescharacteristic
213C22×D61C2×C22⋊C411yescharacteristic
214C22×D61#11714yes—
215D4×C61—24no—
216D4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—41no—
220C2×C4×C61—44no—
221C2×C4×C61—44no—
222C2×C4×C61—42no—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61C22×D412yes—
227C2×C4×C61C2×#4112yes—
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C22⋊D61—216no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—41no—
236C2×Dic61—44no—
237C2×Dic61—44no—
238C2×Dic61—42no—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C2×C22⋊C411yescharacteristic
243D244—24no—
244D244—24no—
245C4×D61—22no—
246C4×D61—24no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—42no—
254C4×D61—44no—
255C4×D61—44no—
256C4×D61—44no—
257C4×D61—44no—
258C4×D61—44no—
259C4×D61—44no—
260C4×D61#10014yescomplemented
261C8×C61—44no—
262#92418—44no—
263#20726—611nomaximal, Sylow
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—24no—
267C22×C61—44no—
268C22×C61—44no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61C22×C22⋊C411yescharacteristic
272C22×C61C2×#11711yescharacteristic
273C22×C61C2×#11711yescharacteristic
274C2×D61—24no—
275C2×D61—24no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—24no—
281C2×D61—416no—
282C2×D61—416no—
283C2×D61—44no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—44no—
287C2×D61C2×#10012yes—
288C2×D61#41312yes—
289C4×C61—21no—
290C4×C61—22no—
291C4×C61—42no—
292C4×C61C2×#10012yes—
293C4×C61#52512yes—
294C4×C61#52512yes—
295Dic61—21no—
296Dic61—22no—
297Dic61—22no—
298Dic61—42no—
299Dic61C2×#10012yes—
300C22×C4×D4—611no—
301#2920—612no—
302#1821—612no—
303#1800—612no—
304#1800—611no—
305#1800D6111yescharacteristic, complemented
306#1783—612no—
307C4×#100—612no—
308C4×#100—612no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—44no—
312C2×C61#228011yescharacteristic
313C2×C61C22×#10011yescharacteristic
314C2×C61C2×#41311yescharacteristic
315D61—22no—
316D61—22no—
317D61—44no—
318C23×D4—611no—
319C24×C4—612no—
320C2×C4×D4—611no—
321C2×C4×D4—612no—
322C2×C4×D4—611no—
323C2×C4×D4—612no—
324C2×C4×D4—611no—
325C2×C4×D4—1222no—
326C2×C4×D4—1222no—
327C2×C4×D4—1222no—
328C2×C4×D4—1222no—
329C2×C4×D4C2×D6111yescharacteristic
330C22×C4⋊C4—611no—
331C22×C22⋊C4—612no—
332C22×C42—611no—
333#432—1222no—
334#432—1222no—
335#431—1222no—
336#431—1222no—
337#423—612no—
338#423C2×D6112yescomplemented
339C2×#100—612no—
340C2×#100—614no—
341C2×#100C2×D6112yes—
342#407—614no—
343C4×M16—614no—
344#343—614no—
345#343—612no—
346#343C2×D6112yescomplemented
347C61#2072611yescharacteristic, complemented, Sylow
348C25—612no—
349C22×D4—612no—
350C22×D4—611no—
351C22×D4—1222no—
352C22×D4—1222no—
353C22×D4—1224no—
354C22×D4—1222no—
355C22×D4—1222no—
356C22×D4C22×D6111yescharacteristic
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—612no—
361C23×C4—612no—
362C23×C4—612no—
363C23×C4—612no—
364C23×C4—1222no—
365C23×C4—1222no—
366C23×C4—1222no—
367C23×C4—1222no—
368C23×C4—2448no—
369C23×C4C4×D6112yes—
370C2×M16—614no—
371C2×M16C22×D6114yes—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—1224no—
375C4×D4—1224no—
376C4×D4—1222no—
377C4×D4—1222no—
378C4×D4—1224no—
379C4×D4—1224no—
380C4×D4—1222no—
381C4×D4—1222no—
382C4×D4—1222no—
383C4×D4—1222no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—612no—
390C2×C22⋊C4—612no—
391C2×C22⋊C4—612no—
392C2×C22⋊C4—1224no—
393C2×C22⋊C4—1224no—
394C2×C22⋊C4C4×D6112yes—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—611no—
398C2×C42—612no—
399C2×C42—612no—
400C2×C42—612no—
401C2×C42—612no—
402C2×C42C22×D6111yescharacteristic
403C4⋊C8—22no—
404C4⋊C8—22no—
405C4⋊C8—1222no—
406C4⋊C8—1222no—
407#100—24no—
408#100—1228no—
409#100—1224no—
410C22⋊C8—1224no—
411C22⋊C8—1224no—
412#97—1224no—
413C4×C8—1224no—
414C24—611no—
415C24—612no—
416C24—1222no—
417C24—1222no—
418C24—1224no—
419C24—1224no—
420C24—2444no—
421C24—2444no—
422C24C2×C4×D6112yes—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—614no—
426C2×D4—1224no—
427C2×D4—1228no—
428C2×D4—1228no—
429C2×D4—1228no—
430C2×D4—1228no—
431C2×D4—1224no—
432C2×D4—1224no—
433C2×D4—1222no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4#1175822412yes—
437C2×D4#1175822612yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—44no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—1224no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—2444no—
466C22×C4—2448no—
467C22×C4—2442no—
468C22×C4—2448no—
469C22×C4—2448no—
470C22×C4—2448no—
471C22×C4—2444no—
472C22×C4—2444no—
473C22×C4—2444no—
474C22×C4C2×C4×D6111yescharacteristic
475C22×C4C2×C4×D6112yes—
476C22×C4C2×C4×D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478M16—24no—
479M16—24no—
480M16—1224no—
481M16—1224no—
482C2×C8—24no—
483C2×C8—1224no—
484C4⋊C4—22no—
485C4⋊C4—1224no—
486C4⋊C4—1222no—
487C22⋊C4—24no—
488C22⋊C4—1224no—
489C22⋊C4—1224no—
490C22⋊C4—1224no—
491C42—612no—
492C42—612no—
493C42—612no—
494C42—612no—
495C42—612no—
496C42—612no—
497C42D4×D6112yes—
498C42D4×D6112yes—
499C23—22no—
500C23—22no—
501C23—42no—
502C23—42no—
503C23—611no—
504C23—611no—
505C23—614no—
506C23—1222no—
507C23—1222no—
508C23—1228no—
509C23—1228no—
510C23—1221no—
511C23—1221no—
512C23—1222no—
513C23—1222no—
514C23—2442no—
515C23—2442no—
516C23—2444no—
517C23—2444no—
518C23—2444no—
519C23—24416no—
520C23—24416no—
521C23—2444no—
522C23#1451973514yes—
523C23C22×C4×D6111yescharacteristic
524D4—24no—
525D4—24no—
526D4—12216no—
527D4—1224no—
528D4—1224no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—611no—
536C2×C4—614no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—611no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1224no—
543C2×C4—1222no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—2442no—
553C2×C4—2444no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4—2444no—
558C2×C4—2444no—
559C2×C4—2441no—
560C2×C4—2444no—
561C2×C4—2444no—
562C2×C4—2442no—
563C2×C4C22⋊C4×D6112yes—
564C2×C4C22⋊C4×D6111yescharacteristic
565C2×C4C22⋊C4×D6111yescharacteristic
566C2×C4C2×D4×D6112yes—
567C2×C4#41×D6112yes—
568C2×C4C2×#1175822411yescharacteristic
569C2×C4C2×#1175822611yescharacteristic
570C8—44no—
571C8—2444no—
572C22—21no—
573C22—21no—
574C22—24no—
575C22—44no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22—612no—
580C22—612no—
581C22—1224no—
582C22—1224no—
583C22—1221no—
584C22—1221no—
585C22—1221no—
586C22—1221no—
587C22—1224no—
588C22—2444no—
589C22—2444no—
590C22—24416no—
591C22—24416no—
592C22—2442no—
593C22—2442no—
594C22C2×C22⋊C4×D6111yescharacteristic
595C22#117×D6111yescharacteristic
596C22C2×#1451973511yescharacteristic
597C4—21no—
598C4—22no—
599C4—42no—
600C4—612no—
601C4—1222no—
602C4—1221no—
603C4—1222no—
604C4—2442no—
605C4#100×D6112yes—
606C4#1684027212yes—
607C4#1684027512yes—
608C2—21no—
609C2—21no—
610C2—44no—
611C2—1222no—
612C2—1222no—
613C2—2444no—
614C2#2144820111yescharacteristic
615C2C2×#100×D6111yescharacteristic
616C2#413×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1800C2
socleC22×C61C22×C22⋊C4
prime radicalC22×D4×D61C22
cosocleC22×C4×C61C24
hypercenter#1800D61
hypocenterC61#20726
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #21454451C2semidirect
3. #21445898C2semidirect
4. #21445811C2semidirect
5. #21445808C2semidirect
7. #100×Dic61C2semidirect
8. #21445610C2semidirect
9. C61×#1800C2semidirect
28. #16839786C22semidirect
32. M16×Dic61C22semidirect
33. #16839531C22semidirect
34. #16839530C22semidirect
36. C61×#423C22semidirect
38. C61×#343C22semidirect
260. C4×D61#100semidirect
305. #1800D61semidirect
338. #423C2×D61semidirect
346. #343C2×D61semidirect
347. C61#20726semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4×D612
#214544512
#214458982
#214458112
#214458082
#214457382
#100×Dic612
#214456102
C61×#18002
#2072661