Finite Group Databasebeta

← #29682708

Subgroups — #29682708

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×#16839764C211yesmaximal, characteristic
2C2×#16839759C211yesmaximal, characteristic
3#21448435C211yesmaximal, characteristic, complemented
4#21448407C211yesmaximal, characteristic
5#21446516C212yesmaximal, complemented
6#21446509C212yesmaximal, complemented
7#21446502C212yesmaximal, complemented
8#21446495C212yesmaximal, complemented
9#21445539C211yesmaximal, characteristic, complemented
10C2×#16839525C211yesmaximal, characteristic, complemented
11C2×C61×#405C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16839764C418yescomplemented
17#16839759—24no—
18C2×#14519702C2211yescharacteristic
19C2×#14519702C2211yescharacteristic
20C2×#14519702C2212yes—
21C2×#14519699C2211yescharacteristic
22C2×#14519699C2211yescharacteristic
23C2×#14519699C2212yes—
24#16839668C2212yes—
25#16839664C2212yes—
26#16839660C2212yes—
27#16839660C2212yes—
28#16839658—21no—
29#16839658—21no—
30#16839639—22no—
31#16839620—22no—
32#16839619—21no—
33#16839619—21no—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839525C2212yes—
38C22×M16×C61C2211yescharacteristic
39C23×C8×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C61×#405—24no—
42C2×#97×C61C2212yes—
43C2×C4×C8×C61C2212yes—
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—
53C23×Dic61—21no—
54C23×Dic61—21no—
55C22×C4×D61—21no—
56#14519736—22no—
57#14519736—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61—22no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2×C414yes—
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C2×C22⋊Dic61—21no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×C4×Dic61—21no—
74#14519702—22no—
75#14519702—22no—
76#14519702C2×C418yescomplemented
77#14519699—22no—
78#14519699—22no—
79#14519699C2×C418yescomplemented
80#14519684—22no—
81#14519684—22no—
82#14519684—22no—
83#14519684—22no—
84C23×C4×C61C2311yescharacteristic
85C2×M16×C61—22no—
86C2×M16×C61—22no—
87C2×M16×C61C2×C414yes—
88C22×C8×C61—22no—
89C22×C8×C61—22no—
90C22×C8×C61D414yes—
91C22×C8×C61C2312yes—
92C22×C8×C61C2311yescharacteristic
93C22×C8×C61C2311yescharacteristic
94#117×C61—22no—
95#117×C61—22no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C22⋊C4×C61D412yes—
99C2×C42×C61C2312yes—
100#97×C61—24no—
101C4×C8×C61—24no—
102C24×C61C2×D411yescharacteristic
103C23×D61—21no—
104C22×C4×C61C22×C412yes—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C22×C4×C61#4112yes—
110C22×C4×C61#4112yes—
111C22×C4×C61#4112yes—
112C22×C4×C61#4112yes—
113C22×C4×C61C2411yescharacteristic
114C2×C22⋊D61—24no—
115C2×C22⋊D61—42no—
116C2×C22⋊D61—42no—
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61—24no—
120C22×Dic61—22no—
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—42no—
124C22×Dic61—81no—
125C22×Dic61—81no—
126C22×Dic61—81no—
127C22×Dic61—81no—
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—
147M16×C61—28no—
148C2×C8×C61—28no—
149C2×C8×C61—22no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—22no—
155C2×C8×C61C22×C414yes—
156C2×C8×C61C2×D412yes—
157C2×C8×C61C2×D412yes—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C22⋊C4×C61—24no—
161C42×C61—22no—
162C42×C61—22no—
163C22⋊Dic61—42no—
164#11758198—82no—
165#11758198—82no—
166C4⋊Dic61—24no—
167C4⋊Dic61—22no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—42no—
175C4⋊Dic61C22×C414yes—
176#11758196—82no—
177#11758196—82no—
178C4×Dic61—42no—
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—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61#11714yes—
201C2×C4×C61C4×D414yes—
202C2×C4×C61C4×D412yes—
203C2×C4×C61#12112yes—
204C2×C4×C61#12314yes—
205C2×C4×C61#12412yes—
206C2×C4×C61#12614yes—
207C2×C4×C61C23×C411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C22⋊D61—48no—
214C2×Dic61—24no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—41no—
224C2×Dic61—42no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—84no—
230C2×Dic61—84no—
231D244—22no—
232D244—22no—
233D244—41no—
234D244—41no—
235C4×D61—48no—
236Dic122—22no—
237Dic122—22no—
238Dic122—41no—
239Dic122—41no—
240C8×C61—24no—
241C8×C61—22no—
242C8×C61—22no—
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—42no—
266C4×C61—42no—
267C4×C61#38814yes—
268C4×C61C2×#11711yescharacteristic
269C4×C61C2×C4×D411yescharacteristic
270C4×C61C2×#12311yescharacteristic
271C4×C61C2×#12611yescharacteristic
272Dic61—44no—
273Dic61—42no—
274Dic61—82no—
275Dic61—81no—
276Dic61—81no—
277C2×#417—611no—
278C2×#416—611no—
279C2×#405D6111yescharacteristic, complemented
280#2316—611no—
281#2311—611no—
282#1936—612no—
283#1935—612no—
284#1934—612no—
285#1933—612no—
286#1736—611no—
287C2×#340—611no—
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×M16C2×D6111yescharacteristic
299C23×C8C2×D6111yescharacteristic
300C2×#117—611no—
301C2×#117C2×D6111yescharacteristic
302C22×C4⋊C4—611no—
303C22×C4⋊C4—611no—
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—24no—
313C2×#97C2×D6112yes—
314C2×C4×C8C2×D6112yes—
315#391—612no—
316#390—612no—
317#389—612no—
318#389—612no—
319#389—1221no—
320#389—1221no—
321#386—1222no—
322#380—1221no—
323#380—1221no—
324#380—1222no—
325#340—612no—
326#340—612no—
327#340—612no—
328#340—612no—
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—1221no—
340C23×C4—1221no—
341C23×C4—1221no—
342C23×C4C22×D6111yescharacteristic
343C2×M16—22no—
344C2×M16—22no—
345C2×M16C4×D6114yes—
346C22×C8—22no—
347C22×C8—22no—
348C22×C8C22⋊D6114yes—
349C22×C8C22×D6112yes—
350C22×C8C22×D6111yescharacteristic
351C22×C8C22×D6111yescharacteristic
352#117—22no—
353#117—22no—
354#117—1222no—
355#117—1222no—
356C2×C4⋊C4—614no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4D24411yescharacteristic
369C2×C4⋊C4D24411yescharacteristic
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4D24412yes—
374C2×C42—1221no—
375C2×C42C22×D6112yes—
376#103—618no—
377#103—1222no—
378#103—1222no—
379#102—618no—
380#102—1222no—
381#102—1222no—
382#97—24no—
383C4×C8—24no—
384#95—1222no—
385#95—1222no—
386#95—1222no—
387#95—1222no—
388C24—1221no—
389C24C2×D24411yescharacteristic
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—1224no—
410C22×C4—1221no—
411C22×C4—1222no—
412C22×C4—1224no—
413C22×C4—1221no—
414C22×C4—1221no—
415C22×C4—1222no—
416C22×C4—4881no—
417C22×C4—4881no—
418C22×C4—4881no—
419C22×C4—4881no—
420C22×C4—2442no—
421C22×C4—2442no—
422C22×C4—2442no—
423C22×C4C2×C4×D6112yes—
424C22×C4C2×D24412yes—
425C22×C4C2×D24412yes—
426C22×C4C2×D24411yescharacteristic
427C22×C4#1175822212yes—
428C22×C4#1175822212yes—
429C22×C4#1175822212yes—
430C22×C4#1175822212yes—
431C22×C4C2×C22⋊D6112yes—
432C22×C4C23×D6111yescharacteristic
433M16—28no—
434C2×C8—22no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—22no—
440C2×C8—28no—
441C2×C8C2×C4×D6114yes—
442C2×C8C2×C22⋊D6112yes—
443C2×C8C2×C22⋊D6112yes—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—614no—
447C4⋊C4—1222no—
448C4⋊C4—1224no—
449C4⋊C4—1221no—
450C4⋊C4—1221no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—4882no—
456C4⋊C4—4882no—
457C4⋊C4—2442no—
458C22⋊C4—24no—
459C22⋊C4—4882no—
460C22⋊C4—4882no—
461C22⋊C4—2442no—
462C42—22no—
463C42—22no—
464C42—2442no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—1222no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23C4×D24412yes—
477C23#1451975112yes—
478C23#1451976512yes—
479C23C22×D24411yescharacteristic
480Q8—1222no—
481Q8—1222no—
482Q8—2441no—
483Q8—2441no—
484D4—1222no—
485D4—1222no—
486D4—2448no—
487D4—2441no—
488D4—2441no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—1224no—
496C2×C4—1222no—
497C2×C4—4884no—
498C2×C4—4884no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2441no—
504C2×C4—2442no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2448no—
513C2×C4#1451973414yes—
514C2×C4C4×C22⋊D6114yes—
515C2×C4C4×D24412yes—
516C2×C4#1451975112yes—
517C2×C4#1451976014yes—
518C2×C4#1451976512yes—
519C2×C4#1451977114yes—
520C2×C4C22×C4×D6111yescharacteristic
521C2×C4C22×C22⋊D6111yescharacteristic
522C2×C4C2×#1175822211yescharacteristic
523C2×C4C2×#1175822211yescharacteristic
524C2×C4C2×#1175822211yescharacteristic
525C2×C4C2×#1175822211yescharacteristic
526C8—24no—
527C8—22no—
528C8—22no—
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#1683965314yes—
543C22C2×C4×D24411yescharacteristic
544C22C2×#1451975111yescharacteristic
545C22C2×#1451976511yescharacteristic
546C22#1684056711yescharacteristic
547C22#1684057411yescharacteristic
548C22#1684057411yescharacteristic
549C22#1684057911yescharacteristic
550C4—42no—
551C4—42no—
552C4—4882no—
553C4—4881no—
554C4—4881no—
555C4—2442no—
556C4—2444no—
557C4#1683965314yes—
558C4C2×#1451973411yescharacteristic
559C4C2×C4×C22⋊D6111yescharacteristic
560C4C2×#1451976011yescharacteristic
561C4C2×#1451977111yescharacteristic
562C2—24no—
563C2—4882no—
564C2C2×#1683965311yescharacteristic
565C2#2145465311yescharacteristic
566C2#2145466411yescharacteristic
567C2#2145548111yescharacteristic
568C2#2145548911yescharacteristic
569C2#2145617511yescharacteristic
570C2#2145618611yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#405C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#405D61
hypocenterC61#21092
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
3. #21448435C2semidirect
5. #21446516C2semidirect
6. #21446509C2semidirect
7. #21446502C2semidirect
8. #21446495C2semidirect
9. #21445539C2semidirect
10. C2×#16839525C2semidirect
11. C2×C61×#405C2semidirect
16. #16839764C4semidirect
76. #14519702C2×C4semidirect
79. #14519699C2×C4semidirect
279. C2×#405D61semidirect
329. C61#21092semidirect

Maximal subgroups

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

subgroupindex
C2×#168397642
C2×#168397592
#214484352
#214484072
#214465162
#214465092
#214465022
#214464952
#214455392
C2×#168395252
C2×C61×#4052
#2109261