Finite Group Databasebeta

← #29696201

Subgroups — #29696201

Order 15616.

1132 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
1C2×#16840456C211yesmaximal, characteristic
2M16×C22⋊D61C212yesmaximal, complemented
3#21446810C212yesmaximal, complemented
4#21446282C212yesmaximal, complemented
5#21446281C212yesmaximal, complemented
6#21445902C211yesmaximal, characteristic, complemented
7#21445898C211yesmaximal, characteristic, complemented
8#21445810C211yesmaximal, characteristic, complemented
9#21445610C211yesmaximal, characteristic, complemented
10#343×D61C211yesmaximal, characteristic, complemented
11C61×#1742C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13#16840456—22no—
14#16840456—22no—
15#16840456—22no—
16#16840456—22no—
17C2×#14519771C412yes—
18C2×#14519768C2211yescharacteristic
19C2×#14519759C412yes—
20C2×#14519752C2211yescharacteristic
21C2×#14519751C2211yescharacteristic
22C2×#14519750C412yes—
23C2×C4×D244C412yes—
24C2×#14519739C2211yescharacteristic
25C2×C4×C22⋊D61C2211yescharacteristic
26C2×C4⋊C4×D61C2211yescharacteristic
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×D244C2×C412yes—
48C22×D244C2311yescharacteristic
49C22×C22⋊D61C2×C412yes—
50C22×C22⋊D61C2×C411yescharacteristic
51C22×C4×D61C2×C412yes—
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#14519771—24no—
61#14519771—24no—
62#14519768—22no—
63#14519768—22no—
64#14519759—24no—
65#14519759—24no—
66#14519752—22no—
67#14519752—22no—
68#14519751—22no—
69#14519751—22no—
70#14519750—24no—
71#14519750—24no—
72C4×D244—22no—
73C4×D244—22no—
74C4×D244—22no—
75C4×D244—22no—
76#14519739—22no—
77#14519739—22no—
78C4×C22⋊D61D412yes—
79C4×C22⋊D61D412yes—
80C4×C22⋊D61D412yes—
81C4×C22⋊D61D412yes—
82C2×C4⋊Dic61C2×C412yes—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196C2×C412yes—
85C2×#11758196C2×C412yes—
86C2×#11758196C2×C411yescharacteristic
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C2×C22⋊Dic61C2×C411yescharacteristic
90C2×#11758198C2×C412yes—
91C2×#11758198C2×C412yes—
92C2×#11758198C2×C412yes—
93C2×#11758198C2×C411yescharacteristic
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×C411yescharacteristic
120C23×D61C22×C412yes—
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×C412yes—
137C22×Dic61C22×C411yescharacteristic
138C22×Dic61C22×C411yescharacteristic
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—44no—
146C2×C4×D61—22no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—82no—
154C2×C4×D61—82no—
155C2×C4×D61C2×D412yes—
156C2×C4×D61C2×D412yes—
157M16×C61—22no—
158M16×C61—22no—
159M16×C61—22no—
160M16×C61—44no—
161M16×C61#4114yes—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—44no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C22⋊C4×C61—24no—
168C42×C61—22no—
169C42×C61—22no—
170C22⋊Dic61C22⋊C412yes—
171C22⋊Dic61C22⋊C412yes—
172#11758198—22no—
173#11758198—22no—
174#11758198—24no—
175#11758198—24no—
176#11758198C22⋊C411yescharacteristic
177#11758198C22⋊C411yescharacteristic
178#11758198C22⋊C411yescharacteristic
179#11758198C22⋊C411yescharacteristic
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183#11758196—24no—
184#11758196—24no—
185#11758196C22⋊C412yes—
186#11758196C22⋊C412yes—
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—22no—
209C22×D61—41no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61C2×C22⋊C411yescharacteristic
213C22×D61C2×C22⋊C411yescharacteristic
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61C22×D412yes—
227C2×C4×C61C2×#4112yes—
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×C4×C612−1+411yescharacteristic
230C22⋊D61—24no—
231C22⋊D61—42no—
232C22⋊D61—84no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—42no—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×C22⋊C411yescharacteristic
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C2×C22⋊C412yes—
243D244—44no—
244D244—44no—
245D244—42no—
246D244—42no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—82no—
252C4×D61—82no—
253C4×D61—82no—
254C4×D61—82no—
255C8×C61—24no—
256C8×C61—42no—
257#92418—42no—
258#92418—42no—
259#22482—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61C4×#4111yescharacteristic
269C22×C61#52011yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—82no—
281C2×D61—82no—
282C2×D61#41312yes—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—42no—
286C4×C61—42no—
287C4×C61#54612yes—
288C4×C61#54712yes—
289Dic61—22no—
290Dic61—22no—
291Dic61—21no—
292Dic61—21no—
293Dic61—42no—
294C2×#553—611no—
295D4×M16—612no—
296#1998—612no—
297#1900—612no—
298#1900—612no—
299#1822—611no—
300#1821—611no—
301#1800—611no—
302C4×#100—611no—
303#1742D6111yescharacteristic, complemented
304C2×#343—611no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61—42no—
308C2×C61#233911yescharacteristic
309C2×C61C2×#41311yescharacteristic
310C2×C61#288811yescharacteristic
311D61—22no—
312D61—41no—
313D61—82no—
314C22×M16—612no—
315C22×M16C2×D6112yes—
316#553—1222no—
317#553—1222no—
318#553—1222no—
319#553—1222no—
320C2×#126—612no—
321C2×#125—611no—
322C2×#123—612no—
323C2×#121—611no—
324C2×#121—611no—
325C2×#121—612no—
326C2×C4×D4—611no—
327C2×C4×D4—611no—
328C2×C4×D4—612no—
329C2×#117C2×D6111yescharacteristic
330C22×C4⋊C4—611no—
331#436—1222no—
332#435—1222no—
333#435—1222no—
334C8×D4—1222no—
335#423—612no—
336C2×#100—612no—
337C2×#100—612no—
338#407—612no—
339#407—612no—
340C4×M16—612no—
341#343—612no—
342#343—612no—
343#343—611no—
344#343—611no—
345#343—612no—
346#343C2×D6111yescharacteristic
347#343C2×D6112yes—
348#343C2×D6111yescharacteristic
349C61#2248211yescharacteristic, complemented, Sylow
350C22×D4—611no—
351C22×D4—612no—
352C22×D4—612no—
353C22×D4—611no—
354C23×C4—611no—
355C23×C4—612no—
356C23×C4—612no—
357C23×C4C22×D6111yescharacteristic
358C2×M16—22no—
359C2×M16—24no—
360C2×M16—42no—
361C2×M16—612no—
362C2×M16—612no—
363C2×M16—1222no—
364C2×M16—1224no—
365C2×M16—2442no—
366C2×M16C22×D6112yes—
367C2×M16C22×D6112yes—
368C22×C8—22no—
369C22×C8—1222no—
370#126—1224no—
371#126—1224no—
372#125—1222no—
373#125—1222no—
374#123—1224no—
375#123—1224no—
376#121—1224no—
377#121—1224no—
378#121—1222no—
379#121—1222no—
380#121—1222no—
381#121—1222no—
382C4×D4—612no—
383C4×D4—612no—
384C4×D4—612no—
385C4×D4—612no—
386C4×D4—1222no—
387C4×D4—1222no—
388C4×D4—1222no—
389C4×D4—1222no—
390C4×D4—1222no—
391C4×D4—1222no—
392#117—22no—
393#117—22no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—612no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C22⋊C4—611no—
404C2×C22⋊C4—611no—
405C2×C22⋊C4—612no—
406C2×C22⋊C4—612no—
407C2×C22⋊C4—612no—
408C2×C22⋊C4C4×D6112yes—
409C2×C42—611no—
410C2×C42C4×D6112yes—
411C4⋊C8—1222no—
412C4⋊C8—1222no—
413#100—1224no—
414#100—2442no—
415C22⋊C8—1222no—
416C22⋊C8—1222no—
417C22⋊C8—1222no—
418C22⋊C8—1222no—
419#97—1222no—
420C4×C8—1222no—
421C24—611no—
422C24—612no—
423C24C2×C4×D6111yescharacteristic
424C2×D4—612no—
425C2×D4—612no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—1224no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—4882no—
434C2×D4—2444no—
435C2×D4—2442no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—4882no—
453C22×C4—4882no—
454C22×C4C2×C4×D6111yescharacteristic
455C22×C4C2×C4×D6112yes—
456C22×C4C2×C4×D6112yes—
457C22×C4D4×D6112yes—
458C22×C4D4×D6112yes—
459C22×C4C23×D6111yescharacteristic
460M16—22no—
461M16—22no—
462M16—22no—
463M16—44no—
464M16—1222no—
465M16—1222no—
466M16—1222no—
467M16—1222no—
468M16—2444no—
469M16#1175822214yes—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—44no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—2444no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—612no—
479C4⋊C4—612no—
480C4⋊C4—1224no—
481C4⋊C4—1224no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C22⋊C4—24no—
486C22⋊C4—612no—
487C22⋊C4—612no—
488C22⋊C4—611no—
489C22⋊C4—611no—
490C22⋊C4—611no—
491C22⋊C4—611no—
492C22⋊C4—1224no—
493C22⋊C4—1224no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C42—22no—
497C42—22no—
498C42—612no—
499C42—612no—
500C23—21no—
501C23—21no—
502C23—41no—
503C23—41no—
504C23—611no—
505C23—611no—
506C23—1222no—
507C23—1221no—
508C23—1221no—
509C23—1222no—
510C23—1222no—
511C23—1222no—
512C23—2441no—
513C23—2442no—
514C23—2442no—
515C23C22⋊C4×D6112yes—
516C23C22×C4×D6111yescharacteristic
517D4—1224no—
518D4—4884no—
519D4—2444no—
520D4—2444no—
521D4—2442no—
522D4—2442no—
523D4—2442no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—4882no—
546C2×C4—4882no—
547C2×C4—4882no—
548C2×C4—4882no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2442no—
552C2×C4C22⋊C4×D6111yescharacteristic
553C2×C4C22⋊C4×D6111yescharacteristic
554C2×C4C2×D4×D6112yes—
555C2×C4C2×#1175822212yes—
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982511yescharacteristic
558C8—24no—
559C8—42no—
560C8—2442no—
561C8—2442no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—81no—
568C22—81no—
569C22—612no—
570C22—1221no—
571C22—1221no—
572C22—1221no—
573C22—1221no—
574C22—1221no—
575C22—4882no—
576C22—4882no—
577C22—2442no—
578C22—2442no—
579C22—2442no—
580C22—2442no—
581C22—2442no—
582C22C2×C22⋊C4×D6111yescharacteristic
583C22C4×#1175822211yescharacteristic
584C22#1684024311yescharacteristic
585C4—21no—
586C4—21no—
587C4—42no—
588C4—42no—
589C4—1222no—
590C4—1222no—
591C4—1221no—
592C4—1221no—
593C4—2442no—
594C4#1684040212yes—
595C4#1684041312yes—
596C2—21no—
597C2—21no—
598C2—42no—
599C2—1222no—
600C2—4882no—
601C2—2441no—
602C2#2144864611yescharacteristic
603C2#413×D6111yescharacteristic
604C2#2145429611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1742C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840456C2
cosocleC22×C4×C61C24
hypercenter#1742D61
hypocenterC61#22482
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. #21446810C2semidirect
4. #21446282C2semidirect
5. #21446281C2semidirect
6. #21445902C2semidirect
7. #21445898C2semidirect
8. #21445810C2semidirect
9. #21445610C2semidirect
10. #343×D61C2semidirect
11. C61×#1742C2semidirect
303. #1742D61semidirect
349. C61#22482semidirect

Maximal subgroups

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

subgroupindex
C2×#168404562
M16×C22⋊D612
#214468102
#214462822
#214462812
#214459022
#214458982
#214458102
#214456102
#343×D612
C61×#17422
#2248261