Finite Group Databasebeta

← #29828839

Subgroups — #29828839

Order 15616.

1436 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic61C211yesmaximal, characteristic
2C2×#16839970C212yesmaximal, complemented
3C2×#16839918C211yesmaximal, characteristic
4#21454167C211yesmaximal, characteristic
5#21446162C212yesmaximal, complemented
6#21446158C212yesmaximal, complemented
7#21446127C211yesmaximal, characteristic
8#21446124C211yesmaximal, characteristic
9#21445822C212yesmaximal, complemented
10#21445797C211yesmaximal, characteristic
11C2×C61×#451C211yesmaximal, characteristic
12C22×#11758202C2212yes—
13C2×#14519775C2211yescharacteristic
14C2×#14519756C2211yescharacteristic
15C2×Q8×Dic61—28no—
16C2×Q8×Dic61—21no—
17C2×Q8×Dic61—21no—
18C2×Q8×Dic61C2212yes—
19C2×Q8×Dic61C2212yes—
20C22×C4⋊Dic61—21no—
21C22×C4⋊Dic61C2211yescharacteristic
22C22×C4×Dic61—21no—
23C22×C4×Dic61C2211yescharacteristic
24#16839970—28no—
25#16839918—24no—
26C2×#14519706C2212yes—
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519701C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839667C2211yescharacteristic
33#16839657—21no—
34#16839657—21no—
35#16839631C2211yescharacteristic
36#16839620—21no—
37#16839620—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C23×Q8×C61C2211yescharacteristic
42C22×Q16×C61C2212yes—
43C2×C22⋊Q8×C61C2211yescharacteristic
44C61×#451—24no—
45C2×#103×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C23×Dic61—21no—
49C23×Dic61D411yescharacteristic
50C23×Dic61D411yescharacteristic
51C2×#11758202—24no—
52C2×#11758202—24no—
53C2×#11758202—24no—
54C22×#92418C2312yes—
55#14519775—24no—
56#14519756—22no—
57#14519756—22no—
58Q8×Dic61—28no—
59Q8×Dic61—28no—
60Q8×Dic61—24no—
61Q8×Dic61—24no—
62Q8×Dic61—24no—
63Q8×Dic61D418yes—
64C2×C4⋊Dic61—28no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C4×Dic61—28no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61—22no—
80C2×C4×Dic61—24no—
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85#14519706—24no—
86#14519702—24no—
87#14519701—24no—
88#14519701—24no—
89C22⋊#92418—22no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94C22×Q8×C61—24no—
95C22×Q8×C61—24no—
96C22×Q8×C61—21no—
97C22×Q8×C61—22no—
98C22×Q8×C61D414yes—
99C22×Q8×C61C2311yescharacteristic
100C22×Q8×C61C2312yes—
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103C2×Q16×C61—24no—
104C2×Q16×C61—24no—
105C2×Q16×C61—24no—
106C22×C8×C61C2312yes—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—22no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112#103×C61—24no—
113C22⋊C8×C61—22no—
114C24×C61C2×D411yescharacteristic
115C2×Q8×C61—216no—
116C2×Q8×C61—24no—
117C2×Q8×C61—24no—
118C2×Q8×C61—216no—
119C2×Q8×C61—24no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61—21no—
126C2×Q8×C61—22no—
127C2×Q8×C61C2×D414yes—
128C2×Q8×C61C2×D414yes—
129C2×Q8×C61#4114yes—
130C22×C4×C61—24no—
131C22×C4×C61—24no—
132C22×C4×C61—24no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D412yes—
140C22×C4×C61#4112yes—
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—24no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—41no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61Q1614yescomplemented
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C2×Dic122—22no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—22no—
162Q16×C61—48no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—42no—
169C22⋊C4×C61—42no—
170C22⋊Dic61—42no—
171#11758202—48no—
172C4⋊Dic61—28no—
173C4⋊Dic61—24no—
174C4⋊Dic61—22no—
175C4⋊Dic61—28no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61C2×D414yes—
179#11758196—42no—
180#11758196—42no—
181#11758196—42no—
182#11758196—42no—
183#11758196—42no—
184C4×Dic61—28no—
185C4×Dic61—22no—
186C4×Dic61—28no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189C4×Dic61C2×D414yes—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61#12112yes—
198C23×C61C2×Q1612yes—
199C23×C61#13712yes—
200C23×C61C22×D411yescharacteristic
201Q8×C61—216no—
202Q8×C61—24no—
203Q8×C61—24no—
204Q8×C61—24no—
205Q8×C61—42no—
206Q8×C61—42no—
207Q8×C61#12118yes—
208C2×C4×C61—216no—
209C2×C4×C61—24no—
210C2×C4×C61—21no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—21no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61#12112yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D412yes—
225C2×C4×C61C2×#4112yes—
226C2×C4×C61C2×#4111yescharacteristic
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—28no—
232C2×Dic61—21no—
233C2×Dic61—24no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—44no—
237C2×Dic61—48no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61#12012yes—
247C2×Dic61#12012yes—
248C2×Dic61C2×Q1612yes—
249C2×Dic61C2×Q1612yes—
250C2×Dic61#13712yes—
251Dic122—42no—
252Dic122—42no—
253C8×C61—42no—
254#92418—42no—
255#32906—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61#46214yescomplemented
263C22×C61C2×#12011yescharacteristic
264C22×C61C2×#12111yescharacteristic
265C22×C61#52511yescharacteristic
266C22×C61C22×Q1611yescharacteristic
267C22×C61C2×#13711yescharacteristic
268C22×C61C2×#13711yescharacteristic
269C22×C61#57511yescharacteristic
270C4×C61—28no—
271C4×C61—22no—
272C4×C61—81no—
273C4×C61—81no—
274C4×C61C2×#12112yes—
275C4×C61#52512yes—
276Dic61—22no—
277Dic61—22no—
278Dic61—41no—
279Dic61—44no—
280Dic61—81no—
281Dic61—81no—
282Dic61#45114yescomplemented
283C22×C4×Q8—611no—
284C2×#462—612no—
285C2×#451—611no—
286C2×#451D6111yescharacteristic
287#2857—611no—
288#1878—612no—
289#1878—612no—
290#1872—611no—
291#1872—611no—
292#1803—612no—
293#1797—611no—
294C2×C61—24no—
295C2×C61#284711yescharacteristic
296C2×C61C2×#45111yescharacteristic
297C2×C61#300711yescharacteristic
298C2×C61C2×#46211yescharacteristic
299C2×C61#303711yescharacteristic
300C2×C61#310411yescharacteristic
301C2×C61#311311yescharacteristic
302C23×Q8C2×D6111yescharacteristic
303C22×Q16—612no—
304C22×Q16C2×D6112yes—
305C2×C4⋊Q8—611no—
306C2×C22⋊Q8—611no—
307C2×C22⋊Q8C2×D6111yescharacteristic
308C2×C4×Q8—612no—
309C2×C4×Q8—612no—
310C2×C4×Q8—1221no—
311C2×C4×Q8—1221no—
312C2×C4×Q8—1228no—
313C22×C4⋊C4—611no—
314C22×C4⋊C4—1221no—
315C22×C42—611no—
316C22×C42—1221no—
317#462—1228no—
318#451—24no—
319#451—1224no—
320C2×C4⋊C8—612no—
321C2×#103—612no—
322C2×#103—612no—
323C2×#103—612no—
324C2×#103C2×D6112yes—
325C2×C22⋊C8—611no—
326C2×C22⋊C8C2×D6111yescharacteristic
327#391—611no—
328#391—611no—
329#389—1221no—
330#389—1221no—
331#384—611no—
332#380—1221no—
333#380—1221no—
334#340—612no—
335#340—612no—
336#336—612no—
337C61#3290611yescharacteristic, complemented, Sylow
338C22×Q8—24no—
339C22×Q8—24no—
340C22×Q8—21no—
341C22×Q8—22no—
342C22×Q8—611no—
343C22×Q8C22⋊D6114yes—
344C22×Q8C22×D6111yescharacteristic
345C22×Q8C22×D6112yes—
346C23×C4—21no—
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—1221no—
350C23×C4C22×D6111yescharacteristic
351C2×Q16—24no—
352C2×Q16—24no—
353C2×Q16—24no—
354C2×Q16—1224no—
355C2×Q16—1224no—
356C2×Q16—1224no—
357C22×C8—612no—
358C22×C8C22×D6112yes—
359C4⋊Q8—1224no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—1222no—
363C22⋊Q8—1222no—
364C4×Q8—618no—
365C4×Q8—1224no—
366C4×Q8—1224no—
367C4×Q8—1228no—
368C4×Q8—1228no—
369C4×Q8—1224no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1224no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1228no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—1221no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—1224no—
391C2×C42—1222no—
392C2×C42—1222no—
393C2×C42—1228no—
394C4⋊C8—1224no—
395#103—24no—
396#103—1224no—
397#103—1224no—
398#103—1224no—
399C22⋊C8—22no—
400C22⋊C8—1222no—
401#95—1221no—
402#95—1221no—
403#95—1221no—
404#95—1221no—
405C24D4×D6111yescharacteristic
406C2×Q8—24no—
407C2×Q8—216no—
408C2×Q8—24no—
409C2×Q8—216no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—1222no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—1222no—
422C2×Q8D4×D6114yes—
423C2×Q8#1175822414yes—
424C2×Q8C2×C22⋊D6114yes—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—614no—
437C22×C4—611no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1224no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—2441no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4D4×D6111yescharacteristic
450C22×C4#1175822412yes—
451C22×C4C2×C22⋊D6112yes—
452C22×C4C23×D6111yescharacteristic
453Q16—48no—
454Q16—2448no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—1222no—
459C2×C8—1222no—
460C2×C8—1222no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—614no—
465C4⋊C4—1228no—
466C4⋊C4—1228no—
467C4⋊C4—1224no—
468C4⋊C4—1222no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—2442no—
472C4⋊C4—2442no—
473C4⋊C4—2442no—
474C4⋊C4—2442no—
475C4⋊C4—2442no—
476C22⋊C4—42no—
477C22⋊C4—2442no—
478C42—614no—
479C42—1222no—
480C42—1222no—
481C42—1228no—
482C42—1228no—
483C42—1222no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1451974912yes—
489C23Q16×D6112yes—
490C23#1451980912yes—
491C23C2×D4×D6111yescharacteristic
492Q8—216no—
493Q8—24no—
494Q8—24no—
495Q8—24no—
496Q8—42no—
497Q8—42no—
498Q8—2442no—
499Q8—2442no—
500Q8#1451974918yes—
501C2×C4—216no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—1228no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1224no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—2444no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2448no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4#1451974912yes—
540C2×C4C2×D4×D6112yes—
541C2×C4C22×C22⋊D6111yescharacteristic
542C2×C4C2×#1175822412yes—
543C2×C4C2×#1175822411yescharacteristic
544C8—42no—
545C8—2442no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22#1683997014yescomplemented
553C22#120×D6111yescharacteristic
554C22C2×#1451974911yescharacteristic
555C22#1684026811yescharacteristic
556C22C2×Q16×D6111yescharacteristic
557C22#137×D6111yescharacteristic
558C22C2×#1451980911yescharacteristic
559C22#1684056111yescharacteristic
560C4—28no—
561C4—22no—
562C4—81no—
563C4—81no—
564C4—614no—
565C4—1222no—
566C4—1222no—
567C4—4881no—
568C4—4881no—
569C4—2441no—
570C4—2444no—
571C4C2×#1451974912yes—
572C4#1684026812yes—
573C2—24no—
574C2#2145409911yescharacteristic
575C2#451×D6111yescharacteristic
576C2#2145507811yescharacteristic
577C2C2×#1683997011yescharacteristic
578C2#2145528811yescharacteristic
579C2#2145585311yescharacteristic
580C2#2145594411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#451C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC2×#451D61
hypocenterC61#32906
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
2. C2×#16839970C2semidirect
5. #21446162C2semidirect
6. #21446158C2semidirect
9. #21445822C2semidirect
153. C22×Dic61Q16semidirect
262. C22×C61#462semidirect
282. Dic61#451semidirect
337. C61#32906semidirect
552. C22#16839970semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic612
C2×#168399702
C2×#168399182
#214541672
#214461622
#214461582
#214461272
#214461242
#214458222
#214457972
C2×C61×#4512
#3290661