Finite Group Databasebeta

← #29730746

Subgroups — #29730746

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839994C212yesmaximal, complemented
2C2×#16839964C211yesmaximal, characteristic, complemented
3C2×C8×D244C211yesmaximal, characteristic, complemented
4#21450773C211yesmaximal, characteristic, complemented
5#21450735C211yesmaximal, characteristic
6#21447013C212yesmaximal, complemented
7#21446821C212yesmaximal, complemented
8#21446580C211yesmaximal, characteristic, complemented
9#21446239C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C61×#1956C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C22×C8×D61C2212yes—
14C2×#14519753C2212yes—
15C2×#14519753C2212yes—
16C2×#14519752—21no—
17C2×#14519752C2212yes—
18C2×C4×D244C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20#16839994—28no—
21#16839964—24no—
22C8×D244—24no—
23C2×#14519708C2212yes—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519689C2212yes—
29#16839697—22no—
30#16839679—22no—
31#16839663—21no—
32#16839631—21no—
33#16839619—21no—
34#16839619C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839525C2211yescharacteristic
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×#103×C61C2212yes—
41C2×#103×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C61×#384C2211yescharacteristic
44C22×D244—22no—
45C22×D244D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61C2312yes—
49C2×#11758201—24no—
50C2×#11758201—24no—
51C2×#11758201—42no—
52C22×#92418C2312yes—
53C2×C8×D61—24no—
54C2×C8×D61—24no—
55C2×C8×D61—42no—
56#14519753—24no—
57#14519753—24no—
58#14519753—42no—
59#14519753—42no—
60#14519752—24no—
61#14519752—42no—
62#14519752—44no—
63C4×D244—22no—
64C4×D244—22no—
65C4×D244—22no—
66C4×D244—22no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—22no—
72C2×#11758196—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×#11758198—21no—
77C2×#11758198—22no—
78C2×#11758198D412yes—
79C2×C4×Dic61—21no—
80#14519708—24no—
81C4⋊#92418D414yescomplemented
82#14519701—24no—
83#14519699—22no—
84#14519699D414yescomplemented
85#14519689—24no—
86#14519684—21no—
87#14519684—22no—
88C22×Q8×C61C2312yes—
89C22×C8×C61C2311yescharacteristic
90C22×C8×C61C2311yescharacteristic
91C4⋊Q8×C61—22no—
92C4⋊Q8×C61—22no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2312yes—
98C2×C42×C61C2311yescharacteristic
99#103×C61—24no—
100#103×C61—42no—
101C4×C8×C61—22no—
102C23×D61—21no—
103C23×D61C2×D412yes—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—21no—
118C22×Dic61C2×D412yes—
119C2×D244—22no—
120C2×D244—22no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—44no—
127C2×D244—44no—
128C2×D244—82no—
129C2×D244SD1614yescomplemented
130C2×D244C2×D412yes—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61C2×D412yes—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—42no—
149C4⋊C4×C61—42no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—81no—
152C42×C61C2×D412yes—
153C42×C61#4112yes—
154#11758201—48no—
155#11758198—24no—
156#11758198—44no—
157#11758198—44no—
158#11758198—44no—
159#11758198—44no—
160#11758198—42no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61C2×D412yes—
167C4⋊Dic61#4112yes—
168#11758196—44no—
169#11758196—44no—
170C4×Dic61—42no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418C2×D414yes—
174C8×D61—48no—
175C23×C61C22×D411yescharacteristic
176C22×D61—22no—
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61C2×SD1614yes—
190Q8×C61—42no—
191Q8×C61—42no—
192Q8×C61—42no—
193Q8×C61—42no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61#12112yes—
203C2×C4×C61#12112yes—
204C2×C4×C61C2×SD1612yes—
205C2×C4×C61#13512yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61H(5,𝔽2)11yescharacteristic
211C2×C4×C61H(5,𝔽2)11yescharacteristic
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224D244—22no—
225D244—22no—
226D244—28no—
227D244—41no—
228D244—41no—
229D244—88no—
230C4×D61—42no—
231C4×D61—42no—
232C4×D61—42no—
233C4×D61—42no—
234C4×D61—84no—
235C8×C61—22no—
236C8×C61—41no—
237#92418—24no—
238#25385—611nomaximal, Sylow
239C22×C61C2×#12111yescharacteristic
240C22×C61D4211yescharacteristic
241C22×C61#54611yescharacteristic
242C22×C61C22×SD1611yescharacteristic
243C22×C61C2×#13511yescharacteristic
244C22×C61#57711yescharacteristic
245C22×C61#57811yescharacteristic
246C2×D61—24no—
247C2×D61—24no—
248C2×D61—24no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—81no—
260C4×C61—22no—
261C4×C61—82no—
262C4×C61—82no—
263C4×C61#46514yescomplemented
264C4×C61#53211yescharacteristic
265C4×C61#54011yescharacteristic
266C4×C61D4211yescharacteristic
267C4×C61#55011yescharacteristic
268Dic61—42no—
269Dic61—42no—
270Dic61—42no—
271Dic61—81no—
272C2×#465—612no—
273C2×#461—611no—
274C2×C8×D4—611no—
275#2542—611no—
276#2537—611no—
277#2035—612no—
278#2000—612no—
279#1956D6111yescharacteristic, complemented
280#1948—611no—
281#1892—612no—
282C4×#102—611no—
283C2×C61#242811yescharacteristic
284C2×C61C2×#46511yescharacteristic
285C2×C61#306311yescharacteristic
286C2×C61D4×SD1611yescharacteristic
287C2×C61#328311yescharacteristic
288C2×C61#329811yescharacteristic
289C2×C61#330311yescharacteristic
290D61—44no—
291D61—82no—
292C22×SD16—612no—
293C23×C8—612no—
294C2×C4⋊Q8C2×D6111yescharacteristic
295C2×C22⋊Q8—612no—
296C2×C22⋊Q8—612no—
297C2×#121—612no—
298C2×#121—1221no—
299C2×C4×D4—611no—
300C2×C4×D4—611no—
301#465—1228no—
302#461—1224no—
303C8×D4—1224no—
304C2×#106—612no—
305C2×C4⋊C8—611no—
306C2×#103—612no—
307C2×#103C2×D6112yes—
308C2×#103C2×D6112yes—
309C2×#102—611no—
310C2×#102—611no—
311C2×C22⋊C8—612no—
312C2×C4×C8C2×D6111yescharacteristic
313#399—1222no—
314#394—1222no—
315#390—1221no—
316#384—1221no—
317#384C2×D6111yescharacteristic
318#380—612no—
319#380—1221no—
320C4×C4⋊C4—611no—
321#340—611no—
322#340—611no—
323#340—612no—
324C61#2538511yescharacteristic, complemented, Sylow
325C22×Q8C22×D6112yes—
326C22×D4—611no—
327C22×D4—611no—
328C22×D4—1222no—
329C23×C4—612no—
330C23×C4—1221no—
331C2×SD16—1224no—
332C2×SD16—1224no—
333C2×SD16—2442no—
334C22×C8—612no—
335C22×C8—1224no—
336C22×C8—1224no—
337C22×C8—2442no—
338C22×C8C22×D6111yescharacteristic
339C22×C8C22×D6111yescharacteristic
340C4⋊Q8—22no—
341C4⋊Q8—22no—
342C22⋊Q8—1224no—
343C22⋊Q8—1224no—
344C22⋊Q8—2442no—
345C22⋊Q8—2442no—
346#121—1224no—
347#121—2442no—
348#121—2444no—
349C4×D4—1222no—
350C4×D4—1222no—
351C4×D4—1222no—
352C4×D4—1222no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4C22×D6112yes—
364C2×C22⋊C4—612no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1221no—
370C2×C42—1221no—
371C2×C42C22×D6111yescharacteristic
372#106—1224no—
373C4⋊C8—614no—
374#103—24no—
375#103—42no—
376#103—1224no—
377#102—614no—
378#102—1222no—
379C22⋊C8—1224no—
380C4×C8—22no—
381#95—1222no—
382#95—1221no—
383C24—612no—
384C24—1221no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×D4—612no—
392C2×D4—614no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—4882no—
401C2×D4—2444no—
402C2×D4—2444no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—612no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1221no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4C23×D6111yescharacteristic
421SD16—2448no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—614no—
428C2×C8—1222no—
429C2×C8—1222no—
430C2×C8—2448no—
431C2×C8D4×D6112yes—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—81no—
438C4⋊C4—612no—
439C4⋊C4—612no—
440C4⋊C4—1221no—
441C4⋊C4—1221no—
442C4⋊C4—1222no—
443C4⋊C4—1221no—
444C4⋊C4—1221no—
445C4⋊C4—2444no—
446C4⋊C4—2444no—
447C22⋊C4—1224no—
448C22⋊C4—2444no—
449C22⋊C4—2444no—
450C22⋊C4—2444no—
451C22⋊C4—2444no—
452C22⋊C4—2442no—
453C42—2442no—
454C42#1175822212yes—
455C42D4×D6112yes—
456C23—614no—
457C23—1222no—
458C23—1222no—
459C23—1222no—
460C23—1222no—
461C23—1222no—
462C23—1222no—
463C23—1222no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23—2441no—
470C23C2×D4×D6111yescharacteristic
471Q8—42no—
472Q8—42no—
473Q8—42no—
474Q8—42no—
475D4—1228no—
476D4—1222no—
477D4—1222no—
478D4—4888no—
479D4—2441no—
480D4—2441no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—4884no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4#1451975012yes—
507C2×C4#1451975012yes—
508C2×C4SD16×D6112yes—
509C2×C4#1451980012yes—
510C2×C4C2×D4×D6111yescharacteristic
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4#41×D6111yescharacteristic
513C2×C4C2×#1175822211yescharacteristic
514C2×C4#1451982211yescharacteristic
515C2×C4#1451982311yescharacteristic
516C8—22no—
517C8—41no—
518C8—1224no—
519C22—1224no—
520C22—1224no—
521C22—1224no—
522C22—4881no—
523C22—4881no—
524C22—4881no—
525C22—4881no—
526C22—2442no—
527C22—2442no—
528C22—2442no—
529C22—2442no—
530C22—2442no—
531C22—2442no—
532C22—2442no—
533C22C2×#1451975011yescharacteristic
534C22#1684039711yescharacteristic
535C22#1684040511yescharacteristic
536C22C2×SD16×D6111yescharacteristic
537C22C2×#1451980011yescharacteristic
538C22#1684056811yescharacteristic
539C22#1684057511yescharacteristic
540C4—22no—
541C4—82no—
542C4—82no—
543C4—4881no—
544C4—2442no—
545C4—2442no—
546C4—2442no—
547C4#1683999414yescomplemented
548C4#1684030911yescharacteristic
549C4#1684036311yescharacteristic
550C4#1684039611yescharacteristic
551C4#1684043011yescharacteristic
552C2—4882no—
553C2—2444no—
554C2#2144940811yescharacteristic
555C2C2×#1683999411yescharacteristic
556C2#2145550811yescharacteristic
557C2#2145751411yescharacteristic
558C2#2145753911yescharacteristic
559C2#2145768311yescharacteristic
560C2#2145774611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1956C2
socleC23×C61C22×D4
prime radical#21450735C2
hypercenter#1956D61
hypocenterC61#25385
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. C2×#16839994C2semidirect
2. C2×#16839964C2semidirect
3. C2×C8×D244C2semidirect
4. #21450773C2semidirect
6. #21447013C2semidirect
7. #21446821C2semidirect
8. #21446580C2semidirect
9. #21446239C2semidirect
10. C4×#14519699C2semidirect
11. C61×#1956C2semidirect
81. C4⋊#92418D4semidirect
84. #14519699D4semidirect
129. C2×D244SD16semidirect
263. C4×C61#465semidirect
279. #1956D61semidirect
324. C61#25385semidirect
547. C4#16839994semidirect

Maximal subgroups

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

subgroupindex
C2×#168399942
C2×#168399642
C2×C8×D2442
#214507732
#214507352
#214470132
#214468212
#214465802
#214462392
C4×#145196992
C61×#19562
#2538561