Finite Group Databasebeta

← #29708689

Subgroups — #29708689

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839790C211yesmaximal, characteristic, complemented
2C2×#16839764C211yesmaximal, characteristic
3C2×#16839756C211yesmaximal, characteristic
4#21448442C211yesmaximal, characteristic, complemented
5#21448412C211yesmaximal, characteristic
6#21447209C212yesmaximal, complemented
7#21447202C212yesmaximal, complemented
8#21447195C212yesmaximal, complemented
9#21447188C212yesmaximal, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C2×C61×#428C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16839790—24no—
17C2×#14519706C2212yes—
18C2×#14519706C2212yes—
19#16839764—24no—
20#16839756—24no—
21C2×#14519702C2212yes—
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519699C2212yes—
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27#16839667C2212yes—
28#16839663C2212yes—
29#16839661C2212yes—
30#16839661C2212yes—
31#16839641—22no—
32#16839637—22no—
33#16839632—21no—
34#16839628—21no—
35#16839619—21no—
36#16839619—21no—
37#16839524C2212yes—
38#16839524C2212yes—
39C22×M16×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C61×#428—24no—
43C2×#107×C61C2212yes—
44C2×#106×C61C2212yes—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222Q814yes—
51C22×Dic122C2311yescharacteristic
52C22×D244C2311yescharacteristic
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C22×C4×D61—21no—
56C22×#92418—22no—
57C22×#92418—22no—
58C22×#92418—22no—
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61#14519736—22no—
62#14519736—22no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×C22⋊Dic61D412yes—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×C4×Dic61C2312yes—
75#14519706—24no—
76#14519706—24no—
77#14519702—24no—
78#14519700—22no—
79#14519700—22no—
80#14519699—24no—
81#14519697—22no—
82#14519697—22no—
83#14519684—22no—
84#14519684—22no—
85#14519684—22no—
86#14519684—22no—
87C23×C4×C61—21no—
88C23×C4×C61C2311yescharacteristic
89C2×M16×C61—22no—
90C2×M16×C61—22no—
91C2×M16×C61D414yes—
92C22×C8×C61C2312yes—
93#117×C61—22no—
94#117×C61D414yes—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61—21no—
103#107×C61—24no—
104#106×C61—24no—
105C24×C61C2×D411yescharacteristic
106C23×D61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—42no—
111C22×C4×C61—81no—
112C22×C4×C61—81no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×Q812yes—
117C22×C4×C61#4112yes—
118C22×C4×C61#4112yes—
119C22×C4×C61#4112yes—
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—24no—
122C2×C22⋊D61—42no—
123C2×C22⋊D61—42no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61—42no—
127C22×Dic61—81no—
128C22×Dic61—81no—
129C22×Dic61C2×D412yes—
130C22×Dic61C2×D412yes—
131#11758222—28no—
132#11758222—28no—
133#11758222—44no—
134#11758222—44no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244C2×Q812yes—
141C2×C4×D61—24no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122C2×Q812yes—
150M16×C61—28no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61C2×D414yes—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61C2×D412yes—
160C4⋊C4×C61C2×D412yes—
161C22⋊C4×C61—24no—
162C42×C61—24no—
163C22⋊Dic61—24no—
164#11758198—82no—
165#11758198—82no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—42no—
171C4⋊Dic61—82no—
172C4⋊Dic61—82no—
173C4×Dic61—22no—
174C4×Dic61—22no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—44no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61#12112yes—
187C23×C61#12112yes—
188C23×C61C22⋊Q812yes—
189C23×C61C22×D411yescharacteristic
190C22×D61—22no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—84no—
207C2×C4×C61#12114yes—
208C2×C4×C61#12112yes—
209C2×C4×C61#12112yes—
210C2×C4×C61C22⋊Q814yes—
211C2×C4×C61C22⋊Q814yes—
212C2×C4×C61C22⋊Q812yes—
213C2×C4×C61#12614yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×Q811yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C22⋊D61—48no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—84no—
234D244—22no—
235D244—22no—
236D244—41no—
237D244—41no—
238C4×D61—48no—
239Dic122—22no—
240Dic122—22no—
241Dic122—41no—
242Dic122—41no—
243C8×C61—24no—
244#92418—41no—
245#92418—41no—
246#23616—611nomaximal, Sylow
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61#39614yes—
254C22×C61C2×#12111yescharacteristic
255C22×C61C2×#12111yescharacteristic
256C22×C61C2×C22⋊Q811yescharacteristic
257C22×C61#57611yescharacteristic
258C22×C61#57711yescharacteristic
259C22×C61#57711yescharacteristic
260C22×C61#57811yescharacteristic
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61—82no—
271C4×C61#39614yes—
272C4×C61C2×#12111yescharacteristic
273C4×C61C2×C22⋊Q811yescharacteristic
274C4×C61C2×C22⋊Q811yescharacteristic
275C4×C61C2×#12611yescharacteristic
276Dic61—42no—
277Dic61—42no—
278Dic61—42no—
279Dic61—82no—
280C2×#428D6111yescharacteristic, complemented
281C2×#424—611no—
282C2×#417—611no—
283C2×#416—611no—
284#2317—611no—
285#2311—611no—
286#2068—612no—
287#2067—612no—
288#2066—612no—
289#2065—612no—
290C2×#340—611no—
291C2×C61—24no—
292C2×C61C2×#39611yescharacteristic
293C2×C61#304511yescharacteristic
294C2×C61#304611yescharacteristic
295C2×C61#306211yescharacteristic
296C2×C61#306311yescharacteristic
297C2×C61#308311yescharacteristic
298C2×C61#308411yescharacteristic
299D61—82no—
300C22×#41—611no—
301C22×M16C2×D6111yescharacteristic
302C23×C8—611no—
303C2×#117—611no—
304C2×#117C2×D6111yescharacteristic
305C22×C4⋊C4—611no—
306C22×C4⋊C4C2×D6111yescharacteristic
307#428—24no—
308C2×#107C2×D6112yes—
309C2×#106C2×D6112yes—
310#424—1224no—
311C2×C4⋊C8—612no—
312C2×C4⋊C8—612no—
313#417—1224no—
314#416—1224no—
315C2×#103—612no—
316C2×#103—611no—
317C2×#103—611no—
318C2×#102—612no—
319C2×#102—611no—
320C2×#102—611no—
321#391—612no—
322#390—612no—
323#389—612no—
324#389—612no—
325#386—1222no—
326#385—1222no—
327#384—1221no—
328#382—1221no—
329#380—1221no—
330#380—1221no—
331#340—612no—
332#340—612no—
333C61#2361611yescharacteristic, complemented, Sylow
334C2×#41—614no—
335C2×#41—1222no—
336C2×#41—1222no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C22×Q8—611no—
341C22×D4—611no—
342C22×D4—1221no—
343C23×C4—21no—
344C23×C4—1221no—
345C23×C4—1221no—
346C23×C4C22×D6111yescharacteristic
347C2×M16—22no—
348C2×M16—22no—
349C2×M16D24414yes—
350C22×C8—611no—
351C22×C8—611no—
352C22×C8—1222no—
353C22×C8—1222no—
354C22×C8—1222no—
355C22×C8C22×D6112yes—
356#117—22no—
357#117—1222no—
358#117—1222no—
359#117C22⋊D6114yes—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22×D6112yes—
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—612no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C42—21no—
379C2×C42—612no—
380#107—24no—
381#106—24no—
382C4⋊C8—1224no—
383C4⋊C8—1224no—
384#103—1224no—
385#103—1222no—
386#103—1222no—
387#102—1224no—
388#102—1222no—
389#102—1222no—
390#95—1222no—
391#95—1222no—
392#95—1222no—
393#95—1222no—
394C24—1221no—
395C24D4×D6111yescharacteristic
396#41—1228no—
397#41—1228no—
398#41—2444no—
399#41—2444no—
400C2×Q8—612no—
401C2×Q8—1221no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8—1221no—
406C2×D4—612no—
407C2×D4—1224no—
408C2×D4—1221no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—2442no—
414C2×D4—2442no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—612no—
422C22×C4—612no—
423C22×C4—1221no—
424C22×C4—1224no—
425C22×C4—1222no—
426C22×C4—4881no—
427C22×C4—4881no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4—2442no—
431C22×C4C2×D24412yes—
432C22×C4#1175822212yes—
433C22×C4D4×D6111yescharacteristic
434C22×C4#1175822412yes—
435C22×C4Q8×D6112yes—
436C22×C4#1175822612yes—
437C22×C4C2×C22⋊D6112yes—
438C22×C4C23×D6111yescharacteristic
439M16—28no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—2444no—
449C2×C8C2×D24414yes—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—4882no—
460C4⋊C4—4882no—
461C4⋊C4—2442no—
462C4⋊C4C2×C22⋊D6112yes—
463C4⋊C4C2×C22⋊D6112yes—
464C22⋊C4—24no—
465C22⋊C4—1224no—
466C22⋊C4—4882no—
467C22⋊C4—4882no—
468C42—24no—
469C42—1222no—
470C42—1222no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—1222no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23—2441no—
482C23#1451974912yes—
483C23#1451975212yes—
484C23#1451975512yes—
485C23C2×D4×D6111yescharacteristic
486Q8—1222no—
487Q8—1222no—
488Q8—2441no—
489Q8—2441no—
490D4—1222no—
491D4—1222no—
492D4—2448no—
493D4—2441no—
494D4—2441no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—84no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—4884no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2448no—
519C2×C4#1451974912yes—
520C2×C4#1451975114yes—
521C2×C4#1451975212yes—
522C2×C4#1451975314yes—
523C2×C4#1451975512yes—
524C2×C4#1451975714yes—
525C2×C4#1451977214yes—
526C2×C4C22×C22⋊D6111yescharacteristic
527C2×C4C22×D24411yescharacteristic
528C2×C4C2×Q8×D6111yescharacteristic
529C2×C4C2×#1175822411yescharacteristic
530C2×C4C2×#1175822211yescharacteristic
531C2×C4C2×#1175822611yescharacteristic
532C8—24no—
533C8—2441no—
534C8—2441no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—4881no—
542C22—4881no—
543C22—4881no—
544C22—4881no—
545C22—2442no—
546C22—2442no—
547C22—2442no—
548C22#1683968414yes—
549C22C2×#1451974911yescharacteristic
550C22C2×#1451975211yescharacteristic
551C22C2×#1451975511yescharacteristic
552C22#1684056611yescharacteristic
553C22#1684057111yescharacteristic
554C22#1684057211yescharacteristic
555C22#1684057811yescharacteristic
556C4—42no—
557C4—42no—
558C4—82no—
559C4—4882no—
560C4—2442no—
561C4—2442no—
562C4—2442no—
563C4#1683968414yes—
564C4C2×#1451975111yescharacteristic
565C4C2×#1451975311yescharacteristic
566C4C2×#1451975711yescharacteristic
567C4C2×#1451977211yescharacteristic
568C2—24no—
569C2—4882no—
570C2C2×#1683968411yescharacteristic
571C2#2145536211yescharacteristic
572C2#2145537311yescharacteristic
573C2#2145550011yescharacteristic
574C2#2145551111yescharacteristic
575C2#2145566911yescharacteristic
576C2#2145568011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC2×#428D61
hypocenterC61#23616
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#16839790C2semidirect
4. #21448442C2semidirect
6. #21447209C2semidirect
7. #21447202C2semidirect
8. #21447195C2semidirect
9. #21447188C2semidirect
10. C2×#16839524C2semidirect
11. C2×C61×#428C2semidirect
280. C2×#428D61semidirect
333. C61#23616semidirect

Maximal subgroups

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

subgroupindex
C2×#168397902
C2×#168397642
C2×#168397562
#214484422
#214484122
#214472092
#214472022
#214471952
#214471882
C2×#168395242
C2×C61×#4282
#2361661