Finite Group Databasebeta

← #29682710

Subgroups — #29682710

Order 15616.

1122 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839761C211yesmaximal, characteristic
2C2×#16839759C211yesmaximal, characteristic
3C2×#16839717C211yesmaximal, characteristic, complemented
4#21448437C211yesmaximal, characteristic, complemented
5#21448412C211yesmaximal, characteristic
6#21446517C212yesmaximal, complemented
7#21446510C212yesmaximal, complemented
8#21446503C212yesmaximal, complemented
9#21446496C212yesmaximal, complemented
10#21445538C211yesmaximal, characteristic, complemented
11C2×C61×#340C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C2×#14519736C2211yescharacteristic
16#16839761C418yescomplemented
17#16839759—24no—
18C2×#14519702C2211yescharacteristic
19C2×#14519702C2211yescharacteristic
20C2×#14519700C2212yes—
21C2×#14519699C2211yescharacteristic
22C2×#14519699C2211yescharacteristic
23C2×#14519697C2212yes—
24#16839717—24no—
25C2×#14519687C2212yes—
26C2×C8×Dic61C2212yes—
27#16839667C2212yes—
28#16839663C2212yes—
29#16839661—21no—
30#16839661—21no—
31#16839661C2212yes—
32#16839661C2212yes—
33#16839641—22no—
34#16839622—22no—
35#16839619—21no—
36#16839619—21no—
37#16839524C2212yes—
38#16839524C2212yes—
39C23×C8×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222C2×C414yes—
50C22×Dic122C2311yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C4×D61—21no—
54C2×#11758194—22no—
55C2×#11758194—22no—
56C2×#11758194C2×C414yes—
57C22×#92418C2312yes—
58#14519736—22no—
59#14519736—22no—
60#14519736—22no—
61#14519736—22no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C22⋊Dic61—21no—
69C2×C22⋊Dic61D412yes—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×C4×Dic61—21no—
73C2×C4×Dic61C2312yes—
74#14519702—22no—
75#14519702—22no—
76#14519700C2×C418yescomplemented
77#14519699—22no—
78#14519699—22no—
79#14519697C2×C418yescomplemented
80#14519687—24no—
81C8×Dic61—24no—
82#14519684—22no—
83#14519684—22no—
84#14519684—22no—
85#14519684—22no—
86C23×C4×C61—21no—
87C23×C4×C61—21no—
88C23×C4×C61C2311yescharacteristic
89C22×C8×C61—22no—
90C22×C8×C61—22no—
91C22×C8×C61D414yes—
92C22×C8×C61C2311yescharacteristic
93C22×C8×C61C2311yescharacteristic
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2×C414yes—
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2312yes—
102C24×C61C2×D411yescharacteristic
103C23×D61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—24no—
106C22×C4×C61—22no—
107C22×C4×C61—42no—
108C22×C4×C61—81no—
109C22×C4×C61—81no—
110C22×C4×C61—81no—
111C22×C4×C61—81no—
112C22×C4×C61C22×C412yes—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D412yes—
115C22×C4×C61#4112yes—
116C22×C4×C61#4112yes—
117C22×C4×C61#4112yes—
118C22×C4×C61#4112yes—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—24no—
121C2×C22⋊D61—42no—
122C2×C22⋊D61—42no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61C2×D412yes—
127C22×Dic61C2×D412yes—
128#11758222—28no—
129#11758222—28no—
130#11758222—44no—
131#11758222—44no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244C22×C412yes—
138C2×C4×D61—24no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122C22×C412yes—
147C2×C8×C61—28no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61C2×D412yes—
153C2×C8×C61C2×D412yes—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61C22×C414yes—
160C22⋊Dic61—24no—
161C22⋊Dic61—42no—
162#11758198—82no—
163#11758198—82no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—82no—
171C4⋊Dic61—82no—
172C4×Dic61—22no—
173C4×Dic61—22no—
174C4×Dic61—42no—
175#11758194—28no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418C22×C414yes—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61C4×D412yes—
184C23×C61#12112yes—
185C23×C61#12412yes—
186C23×C61C22×D411yescharacteristic
187C22×D61—22no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C2×C4×C61—24no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—84no—
201C2×C4×C61—84no—
202C2×C4×C61#11714yes—
203C2×C4×C61C4×D414yes—
204C2×C4×C61C4×D412yes—
205C2×C4×C61#12112yes—
206C2×C4×C61#12314yes—
207C2×C4×C61#12412yes—
208C2×C4×C61#12614yes—
209C2×C4×C61C23×C411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C22⋊D61—48no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231D244—22no—
232D244—22no—
233D244—41no—
234D244—41no—
235C4×D61—48no—
236Dic122—22no—
237Dic122—22no—
238Dic122—41no—
239Dic122—41no—
240C8×C61—22no—
241C8×C61—22no—
242#92418—24no—
243#21092—611nomaximal, Sylow
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61#38814yes—
251C22×C61C2×C4×D411yescharacteristic
252C22×C61C2×#12111yescharacteristic
253C22×C61C2×#12411yescharacteristic
254C22×C61#57611yescharacteristic
255C22×C61#57711yescharacteristic
256C22×C61#57711yescharacteristic
257C22×C61#57811yescharacteristic
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C4×C61—44no—
266C4×C61—82no—
267C4×C61#38814yes—
268C4×C61C2×#11711yescharacteristic
269C4×C61C2×C4×D411yescharacteristic
270C4×C61C2×#12311yescharacteristic
271C4×C61C2×#12611yescharacteristic
272Dic61—42no—
273Dic61—42no—
274Dic61—42no—
275Dic61—81no—
276Dic61—81no—
277C2×#417—611no—
278C2×#416—611no—
279C2×#405—611no—
280#2316—611no—
281#2311—611no—
282#1936—612no—
283#1935—612no—
284#1934—612no—
285#1933—612no—
286#1736—611no—
287C2×#340D6111yescharacteristic, complemented
288C2×C61—24no—
289C2×C61C2×#38811yescharacteristic
290C2×C61#294611yescharacteristic
291C2×C61#294711yescharacteristic
292C2×C61#306011yescharacteristic
293C2×C61#306111yescharacteristic
294C2×C61#313611yescharacteristic
295C2×C61#313711yescharacteristic
296D61—82no—
297C22×#41—611no—
298C22×M16—611no—
299C23×C8C2×D6111yescharacteristic
300C2×#117—611no—
301C2×#117—611no—
302C22×C4⋊C4C2×D6111yescharacteristic
303C22×C4⋊C4C2×D6111yescharacteristic
304#417—618no—
305#416—1224no—
306C2×#103—612no—
307C2×#103—611no—
308C2×#103—611no—
309C2×#102—612no—
310C2×#102—611no—
311C2×#102—611no—
312#405—1224no—
313C2×#97—612no—
314C2×C4×C8—612no—
315#391—612no—
316#390—612no—
317#389—612no—
318#389—612no—
319#389—1221no—
320#389—1221no—
321#386—1222no—
322#380—1222no—
323#380—1221no—
324#380—1221no—
325#340—612no—
326#340—612no—
327#340C2×D6112yes—
328#340C2×D6112yes—
329C61#2109211yescharacteristic, complemented, Sylow
330C2×#41—614no—
331C2×#41—1222no—
332C2×#41—1222no—
333C2×#41—1222no—
334C2×#41—1222no—
335C2×#41—1222no—
336C22×Q8—611no—
337C22×D4—611no—
338C22×D4—1221no—
339C23×C4—21no—
340C23×C4—21no—
341C23×C4—1221no—
342C23×C4C22×D6111yescharacteristic
343C2×M16—614no—
344C2×M16—1222no—
345C2×M16—1222no—
346C22×C8—22no—
347C22×C8—22no—
348C22×C8—612no—
349C22×C8D24414yes—
350C22×C8C22×D6111yescharacteristic
351C22×C8C22×D6111yescharacteristic
352#117—1222no—
353#117—1222no—
354#117—1222no—
355#117—1222no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4C4×D6114yes—
368C2×C4⋊C4C22×D6112yes—
369C2×C4⋊C4C22×D6112yes—
370C2×C22⋊C4—612no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1221no—
374C2×C42—612no—
375C2×C42—1221no—
376#103—618no—
377#103—1222no—
378#103—1222no—
379#102—618no—
380#102—1222no—
381#102—1222no—
382#97—1224no—
383C4×C8—1224no—
384#95—1222no—
385#95—1222no—
386#95—1222no—
387#95—1222no—
388C24—1221no—
389C24D4×D6111yescharacteristic
390#41—1228no—
391#41—1228no—
392#41—2444no—
393#41—2444no—
394C2×Q8—612no—
395C2×Q8—1221no—
396C2×Q8—1221no—
397C2×Q8—1221no—
398C2×Q8—1221no—
399C2×Q8—1221no—
400C2×D4—612no—
401C2×D4—1224no—
402C2×D4—1221no—
403C2×D4—1221no—
404C2×D4—1221no—
405C2×D4—1221no—
406C2×D4—1221no—
407C2×D4—2442no—
408C2×D4—2442no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—612no—
418C22×C4—612no—
419C22×C4—1221no—
420C22×C4—1224no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4C2×C4×D6112yes—
426C22×C4C2×D24412yes—
427C22×C4#1175822212yes—
428C22×C4D4×D6111yescharacteristic
429C22×C4#1175822412yes—
430C22×C4#1175822412yes—
431C22×C4#1175822612yes—
432C22×C4C23×D6111yescharacteristic
433M16—1228no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—28no—
439C2×C8—614no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8C2×D24412yes—
443C2×C8C2×D24412yes—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—4882no—
456C4⋊C4—4882no—
457C4⋊C4C2×C4×D6114yes—
458C22⋊C4—1224no—
459C22⋊C4—4882no—
460C22⋊C4—4882no—
461C22⋊C4—2442no—
462C42—1222no—
463C42—1222no—
464C42—2442no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—1222no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23#1451973712yes—
477C23#1451975212yes—
478C23#1451976312yes—
479C23C2×D4×D6111yescharacteristic
480Q8—1222no—
481Q8—1222no—
482Q8—2441no—
483Q8—2441no—
484D4—1222no—
485D4—1222no—
486D4—2448no—
487D4—2441no—
488D4—2441no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2448no—
513C2×C4#1451973514yes—
514C2×C4#1451973712yes—
515C2×C4C4×D24414yes—
516C2×C4#1451975212yes—
517C2×C4#1451976114yes—
518C2×C4#1451976312yes—
519C2×C4#1451977214yes—
520C2×C4C22×C4×D6111yescharacteristic
521C2×C4C22×D24411yescharacteristic
522C2×C4C2×#1175822411yescharacteristic
523C2×C4C2×#1175822411yescharacteristic
524C2×C4C2×#1175822211yescharacteristic
525C2×C4C2×#1175822611yescharacteristic
526C8—22no—
527C8—22no—
528C8—1224no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—4881no—
536C22—4881no—
537C22—4881no—
538C22—4881no—
539C22—2442no—
540C22—2442no—
541C22—2442no—
542C22#1683965414yes—
543C22C2×#1451973711yescharacteristic
544C22C2×#1451975211yescharacteristic
545C22C2×#1451976311yescharacteristic
546C22#1684056311yescharacteristic
547C22#1684056811yescharacteristic
548C22#1684056811yescharacteristic
549C22#1684057511yescharacteristic
550C4—44no—
551C4—82no—
552C4—4881no—
553C4—4881no—
554C4—2442no—
555C4—2442no—
556C4—2442no—
557C4#1683965414yes—
558C4C2×#1451973511yescharacteristic
559C4C2×C4×D24411yescharacteristic
560C4C2×#1451976111yescharacteristic
561C4C2×#1451977211yescharacteristic
562C2—24no—
563C2—4882no—
564C2C2×#1683965411yescharacteristic
565C2#2145464511yescharacteristic
566C2#2145465611yescharacteristic
567C2#2145548211yescharacteristic
568C2#2145549111yescharacteristic
569C2#2145617111yescharacteristic
570C2#2145618211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#340C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#340D61
hypocenterC61#21092
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C2×#16839717C2semidirect
4. #21448437C2semidirect
6. #21446517C2semidirect
7. #21446510C2semidirect
8. #21446503C2semidirect
9. #21446496C2semidirect
10. #21445538C2semidirect
11. C2×C61×#340C2semidirect
16. #16839761C4semidirect
76. #14519700C2×C4semidirect
79. #14519697C2×C4semidirect
287. C2×#340D61semidirect
329. C61#21092semidirect

Maximal subgroups

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

subgroupindex
C2×#168397612
C2×#168397592
C2×#168397172
#214484372
#214484122
#214465172
#214465102
#214465032
#214464962
#214455382
C2×C61×#3402
#2109261