Finite Group Databasebeta

← #29679969

Subgroups — #29679969

Order 15616.

1694 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×D61C211yesmaximal, characteristic
2#21454450C212yesmaximal, complemented
3#21445898C212yesmaximal, complemented
4#21445810C212yesmaximal, complemented
5#21445809C212yesmaximal, complemented
6#21445738C212yesmaximal
7C4×#14519693C212yesmaximal, complemented
8C4×#14519694C211yesmaximal, characteristic, complemented
9C4×#100×C61C211yesmaximal, 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#16839817—22no—
25#16839817—22no—
26#16839813—22no—
27#16839813—22no—
28#16839785C2214yescomplemented
29C2×#14519694C2212yes—
30C2×#14519693C2214yes—
31#16839725C2214yes—
32C4×#11758194C2212yescomplemented
33#16839531C2212yescomplemented
34#16839530C2214yescomplemented
35C2×C4×D4×C61C2211yescharacteristic
36C2×#100×C61C2212yes—
37C4×M16×C61C2212yescomplemented
38C61×#343C2212yescomplemented
39C24×D61C2×C412yes—
40C22×D244C2311yescharacteristic
41C22×C22⋊D61C2×C412yes—
42C2×D4×D61—24no—
43C2×D4×D61—24no—
44C2×D4×D61—22no—
45C2×D4×D61—22no—
46C23×Dic61C2×C412yes—
47C22×C4×D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61—48no—
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C412yes—
54C22×C4×D61C2×C412yes—
55C22×C4×D61C2×C412yes—
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C2×#11758194C2314yes—
59C4×D244—22no—
60C4×D244—22no—
61D4×Dic61—22no—
62D4×Dic61—22no—
63#14519739—22no—
64#14519739—22no—
65C4×C22⋊D61—24no—
66C4×C22⋊D61—24no—
67#14519737—24no—
68#14519737—24no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196C2×C412yes—
71C4⋊C4×D61—22no—
72C4⋊C4×D61—22no—
73C2×C22⋊Dic61C2×C412yes—
74C2×#11758198C2×C412yes—
75C2×#11758198C2×C412yes—
76C22⋊C4×D61—24no—
77C22⋊C4×D61—24no—
78C2×C4×Dic61C2×C411yescharacteristic
79C2×C4×Dic61C2×C411yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81C42×D61D412yes—
82C42×D61D412yes—
83C42×D61D412yes—
84C42×D61D412yes—
85#14519706—24no—
86#14519706—24no—
87#14519694—24no—
88#14519693—28no—
89#14519689—24no—
90#14519689—24no—
91#14519688—22no—
92C4×#92418—22no—
93C22×D4×C61C2311yescharacteristic
94C23×C4×C61C2×C412yes—
95C2×M16×C61C2314yes—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61C2×C412yes—
100C2×C42×C61C2311yescharacteristic
101#100×C61—24no—
102#97×C61—22no—
103C4×C8×C61—22no—
104C24×C61C22×C412yes—
105C23×D61—24no—
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—24no—
109C23×D61—44no—
110C23×D61—44no—
111C23×D61C22×C411yescharacteristic
112C23×D61C22×C412yes—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61#4114yes—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—44no—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—24no—
124C2×C22⋊D61—24no—
125C2×C22⋊D61—24no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—44no—
129C22×Dic61C22×C411yescharacteristic
130C22×Dic61C22×C411yescharacteristic
131C22×Dic61C22×C412yes—
132C22×Dic61C22×C411yescharacteristic
133D4×D61—216no—
134D4×D61—216no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244#4114yes—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—24no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—42no—
148C2×C4×D61—48no—
149C2×C4×D61—48no—
150C2×C4×D61—48no—
151C2×C4×D61—48no—
152C2×C4×D61—44no—
153C2×C4×D61—44no—
154C2×C4×D61—44no—
155C2×C4×D61C22⋊C411yescharacteristic
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22⋊C411yescharacteristic
158C2×C4×D61C22⋊C412yes—
159C2×C4×D61C22⋊C412yes—
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C2×D412yes—
162C2×C4×D61C2×D412yes—
163M16×C61—24no—
164M16×C61—24no—
165C2×C8×C61—24no—
166C4⋊C4×C61—22no—
167C22⋊C4×C61—24no—
168C42×C61C2×D412yes—
169C42×C61C2×D412yes—
170C22⋊Dic61—24no—
171#11758198—24no—
172#11758198—24no—
173C4⋊Dic61—22no—
174#11758196—24no—
175C4×Dic61C22⋊C412yes—
176C4×Dic61C22⋊C412yes—
177C4×Dic61C22⋊C412yes—
178C4×Dic61C22⋊C412yes—
179C4×Dic61C2×D412yes—
180C4×Dic61C2×D412yes—
181#11758194—24no—
182#11758194—24no—
183C2×#92418—24no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61#11714yes—
189C23×C61C23×C411yescharacteristic
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—28no—
195C22×D61—28no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—44no—
199C22×D61—416no—
200C22×D61—416no—
201C22×D61—44no—
202C22×D61—44no—
203C22×D61—44no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61C2×C22⋊C411yescharacteristic
207C22×D61C2×C22⋊C411yescharacteristic
208C22×D61#11714yes—
209D4×C61—28no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—44no—
214C2×C4×C61—44no—
215C2×C4×C61—42no—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61C2×C22⋊C411yescharacteristic
218C2×C4×C61C2×C22⋊C411yescharacteristic
219C2×C4×C61C22×D412yes—
220C2×C4×C61C2×#4112yes—
221C2×C4×C61C2×#4112yes—
222C22⋊D61—216no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—41no—
228C2×Dic61—44no—
229C2×Dic61—44no—
230C2×Dic61—42no—
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C2×C22⋊C412yes—
233C2×Dic61C2×C22⋊C411yescharacteristic
234C2×Dic61C2×C22⋊C411yescharacteristic
235D244—28no—
236C4×D61—22no—
237C4×D61—24no—
238C4×D61—24no—
239C4×D61—24no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—42no—
243C4×D61—44no—
244C4×D61—44no—
245C4×D61—44no—
246C4×D61—44no—
247C4×D61—44no—
248C4×D61—44no—
249C4×D61#10014yescomplemented
250C8×C61—44no—
251#92418—44no—
252#20726—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—24no—
256C22×C61—44no—
257C22×C61—44no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61C22×C22⋊C411yescharacteristic
261C22×C61C2×#11711yescharacteristic
262C22×C61C2×#11711yescharacteristic
263C2×D61—24no—
264C2×D61—24no—
265C2×D61—21no—
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—24no—
270C2×D61—416no—
271C2×D61—416no—
272C2×D61—44no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—44no—
276C2×D61C2×#10012yes—
277C2×D61#41312yes—
278C4×C61—21no—
279C4×C61—22no—
280C4×C61—42no—
281C4×C61C2×#10012yes—
282C4×C61#52514yes—
283Dic61—21no—
284Dic61—22no—
285Dic61—22no—
286Dic61—42no—
287Dic61C2×#10012yes—
288C22×C4×D4—611no—
289#2920—612no—
290#1821—612no—
291#1800—612no—
292#1800—612no—
293#1783—612no—
294C4×#100—612no—
295C4×#100—611no—
296C4×#100D6111yescharacteristic, complemented
297C2×C61—21no—
298C2×C61—21no—
299C2×C61—44no—
300C2×C61#228011yescharacteristic
301C2×C61C22×#10011yescharacteristic
302C2×C61C2×#41311yescharacteristic
303D61—22no—
304D61—22no—
305D61—44no—
306C23×D4—611no—
307C24×C4—612no—
308C2×C4×D4—611no—
309C2×C4×D4—612no—
310C2×C4×D4—611no—
311C2×C4×D4—612no—
312C2×C4×D4—611no—
313C2×C4×D4—1222no—
314C2×C4×D4—1222no—
315C2×C4×D4—1222no—
316C2×C4×D4—1222no—
317C2×C4×D4C2×D6111yescharacteristic
318C22×C4⋊C4—611no—
319C22×C22⋊C4—612no—
320C22×C42—611no—
321#432—1222no—
322#432—1222no—
323#431—1222no—
324#431—1222no—
325#423—614no—
326C2×#100—612no—
327C2×#100—614no—
328C2×#100C2×D6112yes—
329#407—614no—
330C4×M16—612no—
331C4×M16C2×D6112yescomplemented
332#343—614no—
333#343—612no—
334#343C2×D6112yescomplemented
335C61#2072611yescharacteristic, complemented, Sylow
336C25—612no—
337C22×D4—612no—
338C22×D4—611no—
339C22×D4—1222no—
340C22×D4—1222no—
341C22×D4—1224no—
342C22×D4—1224no—
343C22×D4C22×D6111yescharacteristic
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—612no—
348C23×C4—612no—
349C23×C4—612no—
350C23×C4—612no—
351C23×C4—1222no—
352C23×C4—1222no—
353C23×C4—1222no—
354C23×C4—1222no—
355C23×C4—2448no—
356C23×C4C4×D6112yes—
357C2×M16—614no—
358C2×M16C22×D6114yes—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—1224no—
362C4×D4—1224no—
363C4×D4—1222no—
364C4×D4—1222no—
365C4×D4—1224no—
366C4×D4—1224no—
367C4×D4—1222no—
368C4×D4—1222no—
369C4×D4—1222no—
370C4×D4—1222no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4C22×D6111yescharacteristic
376C2×C22⋊C4—612no—
377C2×C22⋊C4—612no—
378C2×C22⋊C4—612no—
379C2×C22⋊C4—1224no—
380C2×C22⋊C4—1224no—
381C2×C22⋊C4C4×D6112yes—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—612no—
386C2×C42—612no—
387C2×C42—612no—
388C2×C42—612no—
389C2×C42C22×D6111yescharacteristic
390C4⋊C8—1224no—
391C4⋊C8—1224no—
392#100—24no—
393#100—1228no—
394#100—1224no—
395C22⋊C8—1224no—
396C22⋊C8—1224no—
397#97—22no—
398#97—1222no—
399C4×C8—22no—
400C4×C8—1222no—
401C24—611no—
402C24—612no—
403C24—1222no—
404C24—1222no—
405C24—1224no—
406C24—1224no—
407C24—2444no—
408C24—2444no—
409C24C2×C4×D6112yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—614no—
413C2×D4—1224no—
414C2×D4—12216no—
415C2×D4—12216no—
416C2×D4—1224no—
417C2×D4—1224no—
418C2×D4—1222no—
419C2×D4—1222no—
420C2×D4#1175822214yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—44no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—612no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—1224no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—2444no—
448C22×C4—2448no—
449C22×C4—2442no—
450C22×C4—2448no—
451C22×C4—2448no—
452C22×C4—2448no—
453C22×C4—2444no—
454C22×C4—2444no—
455C22×C4—2444no—
456C22×C4C2×C4×D6111yescharacteristic
457C22×C4C2×C4×D6112yes—
458C22×C4C2×C4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460M16—24no—
461M16—24no—
462M16—1224no—
463M16—1224no—
464C2×C8—24no—
465C2×C8—1224no—
466C4⋊C4—22no—
467C4⋊C4—1224no—
468C4⋊C4—1222no—
469C22⋊C4—24no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C42—612no—
474C42—612no—
475C42—612no—
476C42—612no—
477C42—612no—
478C42—612no—
479C42D4×D6112yes—
480C42D4×D6112yes—
481C23—22no—
482C23—22no—
483C23—42no—
484C23—42no—
485C23—611no—
486C23—611no—
487C23—614no—
488C23—1222no—
489C23—1222no—
490C23—1228no—
491C23—1228no—
492C23—1221no—
493C23—1221no—
494C23—1222no—
495C23—1222no—
496C23—2442no—
497C23—2442no—
498C23—2444no—
499C23—2444no—
500C23—2444no—
501C23—24416no—
502C23—24416no—
503C23—2444no—
504C23#1451973414yes—
505C23C22×C4×D6111yescharacteristic
506D4—28no—
507D4—12216no—
508D4—1228no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—611no—
516C2×C4—614no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—611no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1224no—
523C2×C4—1222no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1222no—
527C2×C4—1224no—
528C2×C4—1224no—
529C2×C4—1222no—
530C2×C4—2442no—
531C2×C4—2444no—
532C2×C4—2444no—
533C2×C4—2444no—
534C2×C4—2444no—
535C2×C4—2444no—
536C2×C4—2444no—
537C2×C4—2441no—
538C2×C4—2444no—
539C2×C4—2444no—
540C2×C4—2442no—
541C2×C4C22⋊C4×D6112yes—
542C2×C4C22⋊C4×D6111yescharacteristic
543C2×C4C22⋊C4×D6111yescharacteristic
544C2×C4C2×D4×D6112yes—
545C2×C4#41×D6112yes—
546C2×C4C2×#1175822212yes—
547C8—44no—
548C8—2444no—
549C22—21no—
550C22—21no—
551C22—24no—
552C22—44no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—612no—
557C22—612no—
558C22—1224no—
559C22—1224no—
560C22—1221no—
561C22—1221no—
562C22—1221no—
563C22—1221no—
564C22—1224no—
565C22—2444no—
566C22—2444no—
567C22—24416no—
568C22—24416no—
569C22—2442no—
570C22—2442no—
571C22C2×C22⋊C4×D6111yescharacteristic
572C22#117×D6111yescharacteristic
573C22C2×#1451973411yescharacteristic
574C4—21no—
575C4—22no—
576C4—42no—
577C4—612no—
578C4—1222no—
579C4—1221no—
580C4—1222no—
581C4—2442no—
582C4#100×D6112yes—
583C4#1684027114yes—
584C2—21no—
585C2—21no—
586C2—44no—
587C2—1222no—
588C2—1222no—
589C2—2444no—
590C2#2144820011yescharacteristic
591C2C2×#100×D6111yescharacteristic
592C2#413×D6111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#100×C61C2
socleC22×C61C22×C22⋊C4
prime radicalC22×D4×D61C22
cosocleC22×C4×C61C24
hypercenterC4×#100D61
hypocenterC61#20726
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
2. #21454450C2semidirect
3. #21445898C2semidirect
4. #21445810C2semidirect
5. #21445809C2semidirect
7. C4×#14519693C2semidirect
8. C4×#14519694C2semidirect
9. C4×#100×C61C2semidirect
28. #16839785C22semidirect
32. C4×#11758194C22semidirect
33. #16839531C22semidirect
34. #16839530C22semidirect
37. C4×M16×C61C22semidirect
38. C61×#343C22semidirect
249. C4×D61#100semidirect
296. C4×#100D61semidirect
331. C4×M16C2×D61semidirect
334. #343C2×D61semidirect
335. 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
#214544502
#214458982
#214458102
#214458092
#214457382
C4×#145196932
C4×#145196942
C4×#100×C612
#2072661