Finite Group Databasebeta

← #29696178

Subgroups — #29696178

Order 15616.

1068 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840453C211yesmaximal, characteristic
2#21454554C212yesmaximal
3#21446811C212yesmaximal, complemented
4#21446284C212yesmaximal, complemented
5#21446284C212yesmaximal, complemented
6#21445901C211yesmaximal, characteristic, complemented
7#21445898C211yesmaximal, characteristic, complemented
8C4×#14519696C211yesmaximal, characteristic, complemented
9#21445556C211yesmaximal, characteristic, complemented
10#343×D61C211yesmaximal, characteristic, complemented
11C61×#1801C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13C2×M16×D61C2212yes—
14#16840453—22no—
15#16840453—22no—
16#16840453—22no—
17#16840453—22no—
18C2×#14519771C412yes—
19C2×#14519769C2211yescharacteristic
20C2×#14519767C412yes—
21C2×#14519762C412yes—
22C2×#14519753C2211yescharacteristic
23C2×#14519753C2211yescharacteristic
24C2×C4×D244C2211yescharacteristic
25C2×#14519735C2211yescharacteristic
26C2×#14519734C412yes—
27C2×C4⋊C4×D61C2211yescharacteristic
28#16839839—22no—
29#16839832—22no—
30#16839832—22no—
31#16839827—22no—
32C2×#14519696C2212yes—
33#16839725C2212yes—
34#16839725C2212yes—
35C4×#11758194C2212yes—
36#16839531C2211yescharacteristic
37#16839531C2211yescharacteristic
38#16839530C2212yes—
39#16839530C2212yes—
40#16839530C2212yes—
41#16839530C2212yes—
42C2×C4×Q8×C61C2211yescharacteristic
43C61×#423C2212yes—
44C2×#101×C61C2212yes—
45C61×#343C2211yescharacteristic
46C61×#343C2211yescharacteristic
47C22×D244C2×C411yescharacteristic
48C22×C4×D61C2×C412yes—
49C22×C4×D61C2311yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758194C2312yes—
52C2×#11758194C2312yes—
53C2×#11758189—22no—
54C2×#11758189—22no—
55M16×D61—24no—
56M16×D61—24no—
57M16×D61—42no—
58M16×D61—42no—
59C2×C8×D61—22no—
60C2×C8×D61—22no—
61#14519771—24no—
62#14519771—24no—
63#14519769—22no—
64#14519769—22no—
65#14519767—24no—
66#14519767—24no—
67#14519762—24no—
68#14519762—24no—
69#14519753—22no—
70#14519753—22no—
71#14519753—22no—
72#14519753—22no—
73C4×D244D412yes—
74C4×D244D412yes—
75C4×D244D412yes—
76C4×D244D412yes—
77#14519735—22no—
78#14519735—22no—
79#14519734—22no—
80#14519734—22no—
81#14519734—22no—
82#14519734—22no—
83C2×C4⋊Dic61C2×C411yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196C2×C412yes—
87C2×#11758196C2×C412yes—
88C2×#11758196C2×C412yes—
89C2×#11758196C2×C412yes—
90C2×#11758196C2×C412yes—
91C4⋊C4×D61—22no—
92C4⋊C4×D61—22no—
93C2×#11758198C2×C412yes—
94C2×#11758198C2×C412yes—
95C2×#11758198C2×C412yes—
96C2×#11758198C2×C411yescharacteristic
97C2×#11758198C2×C411yescharacteristic
98C2×C4×Dic61C2×C412yes—
99#14519696—24no—
100#14519689—22no—
101#14519689—22no—
102#14519689—22no—
103#14519689—22no—
104#14519688—22no—
105C4×#92418—22no—
106C22×Q8×C61C2311yescharacteristic
107C2×M16×C61C2312yes—
108C2×M16×C61C2312yes—
109C4×Q8×C61—22no—
110C4×Q8×C61—22no—
111C2×C4⋊C4×C61C2×C412yes—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C42×C61C2×C412yes—
114C2×C42×C61C2311yescharacteristic
115C4⋊C8×C61—22no—
116C4⋊C8×C61—22no—
117#101×C61—42no—
118C23×D61C22×C411yescharacteristic
119C23×D61C22×C411yescharacteristic
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C22×Dic61C22×C411yescharacteristic
130C22×Dic61C22×C411yescharacteristic
131C22×Dic61C22×C412yes—
132C22×Dic61C22×C412yes—
133C2×D244—42no—
134C2×D244C22⋊C412yes—
135C2×D244C22⋊C412yes—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—44no—
145C2×C4×D61C2×D412yes—
146C2×C4×D61C2×D412yes—
147C2×C4×D61C2×D412yes—
148C2×C4×D61C2×D412yes—
149M16×C61—22no—
150M16×C61—22no—
151M16×C61—22no—
152M16×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61C2×D412yes—
160C42×C61C2×D412yes—
161#11758198—22no—
162#11758198—22no—
163#11758198—24no—
164#11758198—24no—
165#11758198C22⋊C411yescharacteristic
166#11758198C22⋊C411yescharacteristic
167#11758198C22⋊C411yescharacteristic
168#11758198C22⋊C411yescharacteristic
169#11758198C22⋊C412yes—
170#11758198C22⋊C412yes—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61C22⋊C412yes—
174C4⋊Dic61C22⋊C412yes—
175#11758196—22no—
176#11758196—22no—
177#11758196—24no—
178#11758196—24no—
179#11758196—22no—
180#11758196—22no—
181#11758196—24no—
182C4×Dic61—22no—
183C4×Dic61—22no—
184#11758194—22no—
185#11758194—22no—
186#11758194—22no—
187#11758194#4114yes—
188C2×#92418—22no—
189C2×#92418—22no—
190#11758189—44no—
191#11758189—44no—
192C8×D61—44no—
193C8×D61—44no—
194C23×C61C23×C411yescharacteristic
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—22no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61C2×C22⋊C412yes—
204C22×D61C2×C22⋊C411yescharacteristic
205C22×D61C2×C22⋊C411yescharacteristic
206Q8×C61—42no—
207Q8×C61—42no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61C2×C22⋊C412yes—
216C2×C4×C61C2×C22⋊C411yescharacteristic
217C2×C4×C61C2×C22⋊C411yescharacteristic
218C2×C4×C61C22×D412yes—
219C2×C4×C61C2×#4112yes—
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61C2×C22⋊C411yescharacteristic
231C2×Dic61C2×C22⋊C411yescharacteristic
232C2×Dic61C2×C22⋊C412yes—
233D244—24no—
234D244—42no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—44no—
244C4×D61—44no—
245C8×C61—42no—
246C8×C61—42no—
247#92418—24no—
248#92418—42no—
249#22480—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61C22×C22⋊C411yescharacteristic
253C22×C61C4×#4111yescharacteristic
254C22×C61#52011yescharacteristic
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—22no—
259C2×D61—21no—
260C2×D61—22no—
261C2×D61—21no—
262C2×D61—22no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—81no—
266C2×D61—81no—
267C2×D61#41312yes—
268C4×C61—22no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61#54612yes—
274C4×C61#54712yes—
275Dic61—21no—
276Dic61—22no—
277Dic61—21no—
278Dic61—42no—
279Dic61—42no—
280C2×#552—611no—
281D4×M16—612no—
282#1998—612no—
283#1901—612no—
284#1901—612no—
285#1822—611no—
286#1821—611no—
287#1801D6111yescharacteristic, complemented
288C4×#101—611no—
289#1742—611no—
290C2×#343—611no—
291C2×C61—21no—
292C2×C61—21no—
293C2×C61#233911yescharacteristic
294C2×C61C2×#41311yescharacteristic
295C2×C61#288911yescharacteristic
296D61—22no—
297D61—41no—
298D61—42no—
299C22×M16—612no—
300C22×M16—612no—
301#552—1222no—
302#552—1222no—
303#552—1222no—
304#552—1222no—
305C2×#126—612no—
306C2×#125—612no—
307C2×#125—611no—
308C2×#123—612no—
309C2×C22⋊Q8—611no—
310C2×C22⋊Q8—611no—
311C2×C4×Q8C2×D6111yescharacteristic
312C2×C4×D4—611no—
313C2×#117—611no—
314C2×#117—612no—
315C22×C4⋊C4—611no—
316#436—1222no—
317#435—1222no—
318#435—1222no—
319C8×D4—1222no—
320#423C2×D6112yes—
321C2×#101—612no—
322C2×#101C2×D6112yes—
323#407—612no—
324#407—612no—
325C4×M16—612no—
326#343—612no—
327#343—612no—
328#343—612no—
329#343—612no—
330#343—611no—
331#343—611no—
332#343C2×D6111yescharacteristic
333#343C2×D6111yescharacteristic
334C61#2248011yescharacteristic, complemented, Sylow
335C22×Q8C22×D6111yescharacteristic
336C22×D4—611no—
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—612no—
340C2×M16—612no—
341C2×M16—612no—
342C2×M16—1222no—
343C2×M16—1222no—
344C2×M16—1224no—
345C2×M16—1224no—
346C2×M16—2442no—
347C2×M16—2442no—
348C2×M16C22×D6112yes—
349C2×M16C22×D6112yes—
350C22×C8—1222no—
351C22×C8—1222no—
352#126—1224no—
353#126—1224no—
354#125—1224no—
355#125—1224no—
356#125—1222no—
357#125—1222no—
358#123—1224no—
359#123—1224no—
360C22⋊Q8—1222no—
361C22⋊Q8—1222no—
362C22⋊Q8—1222no—
363C22⋊Q8—1222no—
364C4×Q8—22no—
365C4×Q8—22no—
366C4×D4—612no—
367C4×D4—612no—
368C4×D4—612no—
369C4×D4—612no—
370#117—1222no—
371#117—1222no—
372#117—1222no—
373#117—1222no—
374#117—1222no—
375#117—1222no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4C4×D6112yes—
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—611no—
389C2×C22⋊C4—611no—
390C2×C22⋊C4—612no—
391C2×C22⋊C4—612no—
392C2×C22⋊C4—612no—
393C2×C42—612no—
394C2×C42C4×D6112yes—
395C2×C42C22×D6111yescharacteristic
396C4⋊C8—22no—
397C4⋊C8—22no—
398#101—42no—
399#101—1224no—
400C22⋊C8—1222no—
401C22⋊C8—1222no—
402C22⋊C8—1222no—
403C22⋊C8—1222no—
404#97—1222no—
405C4×C8—1222no—
406C24—611no—
407C24—611no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×D4—612no—
413C2×D4—612no—
414C2×D4—2442no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—612no—
418C22×C4—612no—
419C22×C4—612no—
420C22×C4—612no—
421C22×C4—612no—
422C22×C4—612no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—2444no—
432C22×C4C2×C4×D6112yes—
433C22×C4C2×C4×D6112yes—
434C22×C4C2×C4×D6111yescharacteristic
435C22×C4C2×C4×D6111yescharacteristic
436C22×C4C23×D6111yescharacteristic
437M16—22no—
438M16—22no—
439M16—22no—
440M16—22no—
441M16—614no—
442M16—1222no—
443M16—1222no—
444M16—1222no—
445M16—2444no—
446M16—2444no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—1222no—
450C2×C8—1222no—
451C2×C8—2444no—
452C2×C8—2444no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—612no—
456C4⋊C4—612no—
457C4⋊C4—1224no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1224no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—1224no—
466C22⋊C4—611no—
467C22⋊C4—611no—
468C22⋊C4—611no—
469C22⋊C4—611no—
470C22⋊C4—612no—
471C22⋊C4—612no—
472C22⋊C4—1222no—
473C22⋊C4—1222no—
474C22⋊C4—1224no—
475C22⋊C4—1224no—
476C42—22no—
477C42—22no—
478C42—1222no—
479C42—1222no—
480C42D4×D6112yes—
481C42D4×D6112yes—
482C23—611no—
483C23—612no—
484C23—611no—
485C23—1222no—
486C23—1221no—
487C23—1221no—
488C23—1221no—
489C23—1221no—
490C23—2441no—
491C23—2441no—
492C23—2441no—
493C23C22×C4×D6111yescharacteristic
494Q8—42no—
495Q8—42no—
496D4—1224no—
497D4—2442no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—612no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—2444no—
523C2×C4—2444no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4C22⋊C4×D6111yescharacteristic
527C2×C4C22⋊C4×D6111yescharacteristic
528C2×C4C22⋊C4×D6112yes—
529C2×C4C2×D4×D6112yes—
530C2×C4#41×D6112yes—
531C2×C4#1451982211yescharacteristic
532C2×C4#1451982511yescharacteristic
533C8—42no—
534C8—42no—
535C8—1224no—
536C8—2442no—
537C22—21no—
538C22—21no—
539C22—612no—
540C22—1221no—
541C22—1221no—
542C22—1222no—
543C22—1221no—
544C22—1221no—
545C22—1222no—
546C22—1222no—
547C22—1221no—
548C22—4881no—
549C22—4881no—
550C22—2442no—
551C22—2442no—
552C22C2×C22⋊C4×D6111yescharacteristic
553C22#1684023011yescharacteristic
554C22#1684024311yescharacteristic
555C4—22no—
556C4—21no—
557C4—21no—
558C4—42no—
559C4—42no—
560C4—1221no—
561C4—1221no—
562C4—1222no—
563C4—2442no—
564C4—2442no—
565C4#1684040912yes—
566C4#1684041412yes—
567C2—21no—
568C2—21no—
569C2—1222no—
570C2—2442no—
571C2—2441no—
572C2#2144864711yescharacteristic
573C2#413×D6111yescharacteristic
574C2#2145429911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1801C2
socleC22×C61C22×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC22×C4×C61C24
hypercenter#1801D61
hypocenterC61#22480
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
3. #21446811C2semidirect
4. #21446284C2semidirect
5. #21446284C2semidirect
6. #21445901C2semidirect
7. #21445898C2semidirect
8. C4×#14519696C2semidirect
9. #21445556C2semidirect
10. #343×D61C2semidirect
11. C61×#1801C2semidirect
287. #1801D61semidirect
334. C61#22480semidirect

Maximal subgroups

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

subgroupindex
C2×#168404532
#214545542
#214468112
#214462842
#214459012
#214458982
C4×#145196962
#214455562
#343×D612
C61×#18012
#2248061