Finite Group Databasebeta

← #29696207

Subgroups — #29696207

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840475C211yesmaximal, characteristic
2M16×C22⋊D61C212yesmaximal, complemented
3#21446806C212yesmaximal, complemented
4#21446282C212yesmaximal, complemented
5#21446281C212yesmaximal, complemented
6#21445902C211yesmaximal, characteristic, complemented
7#21445901C211yesmaximal, characteristic, complemented
8#21445810C211yesmaximal, characteristic, complemented
9#21445610C211yesmaximal, characteristic, complemented
10#21445556C211yesmaximal, characteristic, complemented
11C61×#1742C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13#16840475—22no—
14#16840475—22no—
15#16840475—22no—
16#16840475—22no—
17C2×#14519776C2211yescharacteristic
18C2×#14519765C412yes—
19C2×#14519763C412yes—
20C2×#14519752C2211yescharacteristic
21C2×#14519751C2211yescharacteristic
22C2×#14519750C412yes—
23C2×#14519739C2211yescharacteristic
24C2×C4×C22⋊D61C2211yescharacteristic
25C2×#14519735C2211yescharacteristic
26C2×#14519734C412yes—
27#16839838—22no—
28#16839833—22no—
29#16839831—22no—
30C8×C22⋊D61—22no—
31#16839785C2212yes—
32C2×#14519693C2212yes—
33C2×#14519693C2212yes—
34#16839726C2212yes—
35#16839725C2212yes—
36M16×Dic61C2212yes—
37#16839531C2211yescharacteristic
38#16839531C2212yes—
39#16839531C2211yescharacteristic
40#16839530C2212yes—
41#16839530C2212yes—
42C22×M16×C61C2212yes—
43C2×#117×C61C2211yescharacteristic
44C61×#343C2211yescharacteristic
45C61×#343C2212yes—
46C61×#343C2211yescharacteristic
47C22×Dic122C2×C412yes—
48C22×D244C2×C412yes—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61C2×C412yes—
51C22×C22⋊D61C2×C411yescharacteristic
52C22×C4×D61C2×C412yes—
53C22×C4×D61C2311yescharacteristic
54C2×#11758194C2312yes—
55C2×#11758194C2312yes—
56C2×#11758189—22no—
57M16×D61—24no—
58M16×D61—42no—
59C2×C8×D61—22no—
60#14519776—22no—
61#14519776—22no—
62#14519765—24no—
63#14519765—24no—
64#14519763—24no—
65#14519763—24no—
66#14519752—22no—
67#14519752—22no—
68#14519751—22no—
69#14519751—22no—
70#14519750—24no—
71#14519750—24no—
72#14519739—22no—
73#14519739—22no—
74C4×C22⋊D61D412yes—
75C4×C22⋊D61D412yes—
76C4×C22⋊D61D412yes—
77C4×C22⋊D61D412yes—
78#14519735—22no—
79#14519735—22no—
80#14519734—22no—
81#14519734—22no—
82#14519734—22no—
83#14519734—22no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196C2×C412yes—
86C2×#11758196C2×C411yescharacteristic
87C2×C22⋊Dic61C2×C411yescharacteristic
88C2×#11758198C2×C412yes—
89C2×#11758198C2×C412yes—
90C2×#11758198C2×C412yes—
91C2×#11758198C2×C412yes—
92C2×#11758198C2×C411yescharacteristic
93C2×C4×Dic61C2×C412yes—
94C2×C4×Dic61C2311yescharacteristic
95#14519706—22no—
96#14519706—22no—
97#14519693—24no—
98#14519693—42no—
99C22⋊#92418—22no—
100C22⋊#92418—22no—
101#14519689—22no—
102#14519689—22no—
103#14519687—22no—
104C8×Dic61—22no—
105C23×C4×C61C2311yescharacteristic
106C2×M16×C61—22no—
107C2×M16×C61—24no—
108C2×M16×C61—42no—
109C2×M16×C61C2312yes—
110C2×M16×C61C2312yes—
111C22×C8×C61—22no—
112#117×C61—22no—
113#117×C61—22no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61C2×C412yes—
117C2×C42×C61C2×C412yes—
118C24×C61C22×C411yescharacteristic
119C23×D61C22×C412yes—
120C23×D61C22×C411yescharacteristic
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—24no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—82no—
133C2×C22⋊D61C22⋊C412yes—
134C2×C22⋊D61C22⋊C412yes—
135C22×Dic61C22×C411yescharacteristic
136C22×Dic61C22×C411yescharacteristic
137C22×Dic61C22×C412yes—
138C22×Dic61C22×C411yescharacteristic
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—24no—
144C2×D244—82no—
145C2×C4×D61—22no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—44no—
151C2×C4×D61C2×D412yes—
152C2×C4×D61C2×D412yes—
153C2×Dic122—24no—
154C2×Dic122—82no—
155M16×C61—22no—
156M16×C61—22no—
157M16×C61—22no—
158M16×C61—44no—
159M16×C61#4114yes—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—44no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C22⋊C4×C61—24no—
166C42×C61—22no—
167C42×C61—22no—
168C22⋊Dic61C22⋊C412yes—
169C22⋊Dic61C22⋊C412yes—
170#11758198—24no—
171#11758198—22no—
172#11758198—22no—
173#11758198—22no—
174#11758198—22no—
175#11758198—24no—
176#11758198C22⋊C412yes—
177#11758198C22⋊C412yes—
178C4⋊Dic61—22no—
179#11758196—22no—
180#11758196—22no—
181#11758196C22⋊C411yescharacteristic
182#11758196C22⋊C411yescharacteristic
183#11758196C22⋊C411yescharacteristic
184#11758196C22⋊C411yescharacteristic
185C4×Dic61—22no—
186C4×Dic61—22no—
187C4×Dic61C2×D412yes—
188C4×Dic61C2×D412yes—
189#11758194—22no—
190#11758194—22no—
191#11758194—22no—
192#11758194—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195#11758189—44no—
196C8×D61—44no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61C2×C22⋊C412yes—
202C23×C61C23×C411yescharacteristic
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61C2×C22⋊C412yes—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C2×C4×C61C22×D412yes—
226C2×C4×C61C2×#4112yes—
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C22⋊D61—24no—
230C22⋊D61—42no—
231C22⋊D61—84no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—41no—
237C2×Dic61—42no—
238C2×Dic61C2×C22⋊C411yescharacteristic
239C2×Dic61C2×C22⋊C411yescharacteristic
240C2×Dic61C2×C22⋊C411yescharacteristic
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C2×C22⋊C411yescharacteristic
243D244—42no—
244D244—42no—
245D244—84no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—44no—
251C4×D61—44no—
252Dic122—84no—
253C8×C61—24no—
254C8×C61—42no—
255#92418—42no—
256#92418—42no—
257#22483—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61C4×#4111yescharacteristic
267C22×C61#52011yescharacteristic
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—82no—
277C2×D61—82no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—42no—
281C4×C61—42no—
282C4×C61#54612yes—
283C4×C61#54712yes—
284Dic61—21no—
285Dic61—21no—
286Dic61—21no—
287Dic61—22no—
288Dic61—42no—
289Dic61#41312yes—
290C2×#555—611no—
291D4×M16—612no—
292#1997—612no—
293#1900—612no—
294#1900—612no—
295#1822—611no—
296#1822—611no—
297#1800—611no—
298C4×#100—611no—
299#1742—611no—
300#1742D6111yescharacteristic, complemented
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—42no—
304C2×C61#233911yescharacteristic
305C2×C61C2×#41311yescharacteristic
306C2×C61#288811yescharacteristic
307D61—42no—
308D61—82no—
309C22×M16—612no—
310C22×M16C2×D6112yes—
311#555—1222no—
312#555—1222no—
313#555—1222no—
314#555—1222no—
315C2×C4⋊Q8—611no—
316C2×#124—612no—
317C2×#124—612no—
318C2×#121—611no—
319C2×#121—611no—
320C2×#121—612no—
321C2×C4×D4—611no—
322C2×C4×D4—611no—
323C2×#117—611no—
324C2×#117—612no—
325C2×#117C2×D6111yescharacteristic
326#436—1222no—
327#435—1222no—
328#435—1222no—
329C8×D4—1222no—
330#423—612no—
331C2×#100—612no—
332C2×#100—612no—
333#407—612no—
334#407—612no—
335C4×M16—612no—
336#343—612no—
337#343—612no—
338#343—611no—
339#343—611no—
340#343—612no—
341#343C2×D6111yescharacteristic
342#343C2×D6112yes—
343#343C2×D6111yescharacteristic
344C61#2248311yescharacteristic, complemented, Sylow
345C22×Q8—612no—
346C22×D4—611no—
347C22×D4—612no—
348C22×D4—612no—
349C22×D4—611no—
350C23×C4—611no—
351C23×C4—612no—
352C23×C4C22×D6111yescharacteristic
353C2×M16—22no—
354C2×M16—24no—
355C2×M16—42no—
356C2×M16—612no—
357C2×M16—612no—
358C2×M16—1222no—
359C2×M16—1224no—
360C2×M16—2442no—
361C2×M16C22×D6112yes—
362C2×M16C22×D6112yes—
363C22×C8—22no—
364C22×C8—1222no—
365C4⋊Q8—1222no—
366C4⋊Q8—1222no—
367#124—1224no—
368#124—1224no—
369#124—1224no—
370#124—1224no—
371#121—1224no—
372#121—1224no—
373#121—1222no—
374#121—1222no—
375#121—1222no—
376#121—1222no—
377C4×D4—612no—
378C4×D4—612no—
379C4×D4—612no—
380C4×D4—612no—
381C4×D4—1222no—
382C4×D4—1222no—
383#117—22no—
384#117—22no—
385#117—1222no—
386#117—1222no—
387#117—1222no—
388#117—1222no—
389#117—1222no—
390#117—1222no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C22⋊C4—611no—
397C2×C22⋊C4—611no—
398C2×C22⋊C4—612no—
399C2×C22⋊C4—612no—
400C2×C22⋊C4—612no—
401C2×C22⋊C4—612no—
402C2×C22⋊C4C4×D6112yes—
403C2×C42—611no—
404C2×C42—612no—
405C2×C42C4×D6112yes—
406C4⋊C8—1222no—
407C4⋊C8—1222no—
408#100—1224no—
409#100—2442no—
410C22⋊C8—1222no—
411C22⋊C8—1222no—
412C22⋊C8—1222no—
413C22⋊C8—1222no—
414#97—1222no—
415C4×C8—1222no—
416C24—611no—
417C24—612no—
418C24C2×C4×D6111yescharacteristic
419C2×Q8—1224no—
420C2×Q8—4882no—
421C2×D4—612no—
422C2×D4—612no—
423C2×D4—1224no—
424C2×D4—1224no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—4882no—
430C2×D4—4882no—
431C2×D4—2442no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—2444no—
447C22×C4C2×C4×D6111yescharacteristic
448C22×C4C2×C4×D6112yes—
449C22×C4C2×C4×D6112yes—
450C22×C4D4×D6112yes—
451C22×C4D4×D6112yes—
452C22×C4C23×D6111yescharacteristic
453M16—22no—
454M16—22no—
455M16—22no—
456M16—44no—
457M16—1222no—
458M16—1222no—
459M16—1222no—
460M16—1222no—
461M16—2444no—
462M16#1175822214yes—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—44no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8—2444no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—611no—
472C4⋊C4—611no—
473C4⋊C4—611no—
474C4⋊C4—611no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C22⋊C4—24no—
479C22⋊C4—612no—
480C22⋊C4—612no—
481C22⋊C4—612no—
482C22⋊C4—612no—
483C22⋊C4—1222no—
484C22⋊C4—1222no—
485C22⋊C4—1224no—
486C22⋊C4—1224no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C42—22no—
490C42—22no—
491C42—612no—
492C42—612no—
493C42—1222no—
494C42—1222no—
495C23—21no—
496C23—21no—
497C23—41no—
498C23—41no—
499C23—612no—
500C23—1221no—
501C23—1221no—
502C23—1222no—
503C23—1222no—
504C23—1222no—
505C23—2441no—
506C23—2441no—
507C23—2442no—
508C23—2442no—
509C23C22⋊C4×D6112yes—
510C23C22×C4×D6111yescharacteristic
511Q8—4884no—
512D4—1224no—
513D4—4884no—
514D4—4884no—
515D4—2442no—
516D4—2442no—
517D4—2442no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—611no—
529C2×C4—612no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—1222no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—2444no—
542C2×C4—2444no—
543C2×C4—2441no—
544C2×C4—2442no—
545C2×C4C22⋊C4×D6111yescharacteristic
546C2×C4C22⋊C4×D6111yescharacteristic
547C2×C4C2×D4×D6112yes—
548C2×C4C2×#1175822212yes—
549C2×C4#1451982211yescharacteristic
550C2×C4#1451982511yescharacteristic
551C8—24no—
552C8—42no—
553C8—2442no—
554C8—2442no—
555C22—21no—
556C22—21no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—81no—
561C22—81no—
562C22—1222no—
563C22—1222no—
564C22—1222no—
565C22—4882no—
566C22—4882no—
567C22—4881no—
568C22—4881no—
569C22—2442no—
570C22—2442no—
571C22—2442no—
572C22C2×C22⋊C4×D6111yescharacteristic
573C22C4×#1175822211yescharacteristic
574C22#1684024311yescharacteristic
575C4—21no—
576C4—21no—
577C4—42no—
578C4—42no—
579C4—612no—
580C4—1221no—
581C4—1221no—
582C4—1221no—
583C4—1222no—
584C4—2442no—
585C4#1684040212yes—
586C4#1684041312yes—
587C2—21no—
588C2—21no—
589C2—42no—
590C2—4882no—
591C2—2442no—
592C2#2144864611yescharacteristic
593C2#413×D6111yescharacteristic
594C2#2145429611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1742C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840475C2
cosocleC22×C4×C61C24
hypercenter#1742D61
hypocenterC61#22483
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. M16×C22⋊D61C2semidirect
3. #21446806C2semidirect
4. #21446282C2semidirect
5. #21446281C2semidirect
6. #21445902C2semidirect
7. #21445901C2semidirect
8. #21445810C2semidirect
9. #21445610C2semidirect
10. #21445556C2semidirect
11. C61×#1742C2semidirect
300. #1742D61semidirect
344. C61#22483semidirect

Maximal subgroups

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

subgroupindex
C2×#168404752
M16×C22⋊D612
#214468062
#214462822
#214462812
#214459022
#214459012
#214458102
#214456102
#214455562
C61×#17422
#2248361