Finite Group Databasebeta

← #29681604

Subgroups — #29681604

Order 15616.

1694 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840332C211yesmaximal, characteristic
2#519×D61C211yesmaximal, characteristic, complemented
3#21454333C212yesmaximal
4C2×#101×D61C211yesmaximal, characteristic, complemented
5#21446223C211yesmaximal, characteristic, complemented
6#21446219C212yesmaximal, complemented
7#21446212C211yesmaximal, characteristic, complemented
8#21446211C212yesmaximal, complemented
9#21446210C211yesmaximal, characteristic, complemented
10#21445527C212yesmaximal, complemented
11C61×#1888C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C22×Q8×D61C2211yescharacteristic
14C2×M16×D61C2212yes—
15#16840332—24no—
16#16840332C418yescomplemented
17C2×#14519777C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19C2×#14519764C2211yescharacteristic
20C2×#14519759—22no—
21C2×#14519754—22no—
22#16840240C2211yescharacteristic
23#16840239—22no—
24#16840239C2212yes—
25#16840238—24no—
26C4×Q8×D61—24no—
27C2×#14519734C2211yescharacteristic
28#117×D61—22no—
29#117×D61C2212yes—
30#16839772C2212yes—
31#16839770C2212yes—
32#16839767C2212yes—
33#16839760—24no—
34#16839755—24no—
35C2×#14519696C2211yescharacteristic
36C2×#14519695C2212yes—
37#101×D61—24no—
38#16839523C2212yes—
39#16839522C2212yes—
40#16839521C2212yes—
41C2×#124×C61C2211yescharacteristic
42C61×#519C2211yescharacteristic
43C61×#418C2212yes—
44C2×#101×C61C2211yescharacteristic
45C61×#339C2212yes—
46C2×#11758224—22no—
47C2×#11758224—22no—
48C2×#11758224—22no—
49C2×#11758224—24no—
50C2×#11758224C2×C414yes—
51C22×Dic122D412yes—
52C22×Dic122C2311yescharacteristic
53C22×Dic122C2311yescharacteristic
54C2×Q8×D61—22no—
55C2×Q8×D61—22no—
56C2×Q8×D61—24no—
57C2×Q8×D61C2×C414yes—
58C2×Q8×D61D414yes—
59C22×C22⋊D61—21no—
60C23×Dic61—21no—
61C22×C4×D61D412yes—
62C22×C4×D61C2311yescharacteristic
63C2×#11758194C2312yes—
64C2×#11758189—22no—
65M16×D61—24no—
66M16×D61—24no—
67C2×C8×D61—22no—
68#14519777—22no—
69#14519777C2×C414yescomplemented
70#14519775—22no—
71#14519775C2×C414yescomplemented
72#14519764—22no—
73#14519764C2×C414yescomplemented
74#14519759—216no—
75#14519754—216no—
76Q8×Dic61—24no—
77C4×Dic122—28no—
78C4×Dic122—24no—
79#14519742—28no—
80#14519742—24no—
81#14519736—22no—
82#14519736C2312yes—
83#14519735—24no—
84#14519735D414yes—
85#14519734—24no—
86#14519734—22no—
87#14519734—22no—
88#14519734C2×C414yes—
89#14519733—24no—
90#14519733—22no—
91C2×C4⋊Dic61—22no—
92C2×#11758196—22no—
93C2×#11758196—21no—
94C4⋊C4×D61—24no—
95C4⋊C4×D61D414yes—
96C2×C22⋊Dic61—22no—
97C2×#11758198—21no—
98C22⋊C4×D61—22no—
99C2×C4×Dic61C2311yescharacteristic
100C42×D61—24no—
101C42×D61—22no—
102#14519702—24no—
103#14519700—24no—
104#14519698—24no—
105#14519696—22no—
106#14519695—24no—
107C22×Q8×C61C2311yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C2×M16×C61C2312yes—
110#124×C61—22no—
111#124×C61C2×C414yescomplemented
112C4×Q8×C61—24no—
113#117×C61—22no—
114#117×C61C2312yes—
115C2×C22⋊C4×C61—22no—
116C2×C42×C61C2311yescharacteristic
117#102×C61—24no—
118#101×C61—22no—
119C24×C61—21no—
120C23×D61C2×D411yescharacteristic
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61C22×C412yes—
124C2×Q8×C61C2×D412yes—
125C2×D4×C61—22no—
126C2×D4×C61—21no—
127C2×D4×C61—22no—
128C2×D4×C61C22×C412yes—
129C22×C4×C61—21no—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—24no—
133C2×C22⋊D61—41no—
134C2×C22⋊D61—41no—
135C2×C22⋊D61—42no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—24no—
139C22×Dic61—42no—
140C22×Dic61—42no—
141C22×Dic61C2×D412yes—
142C22×Dic61C2×D411yescharacteristic
143Q8×D61—28no—
144Q8×D61—28no—
145Q8×D61—28no—
146Q8×D61—28no—
147#11758224—28no—
148#11758224—28no—
149#11758224—44no—
150#11758224—44no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—24no—
154C2×C4×D61—24no—
155C2×C4×D61C22⋊C418yes—
156C2×C4×D61C22×C412yes—
157C2×C4×D61C2×D412yes—
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×D412yes—
160C2×C4×D61#4114yes—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—24no—
164C2×Dic122—24no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—21no—
168C2×Dic122C22⋊C418yes—
169C2×Dic122C22×C412yes—
170C2×Dic122C22×C412yes—
171C2×Dic122C2×D412yes—
172M16×C61—22no—
173M16×C61—22no—
174C2×C8×C61—22no—
175C4⋊C4×C61—24no—
176C4⋊C4×C61C2×D414yes—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—28no—
179C42×C61—24no—
180C42×C61—21no—
181C42×C61—22no—
182C42×C61C22×C412yes—
183C22⋊Dic61—22no—
184C22⋊Dic61—28no—
185#11758198—24no—
186#11758198—24no—
187C4⋊Dic61—24no—
188C4⋊Dic61—28no—
189C4⋊Dic61C2×D414yes—
190#11758196—28no—
191#11758196—28no—
192#11758196—24no—
193#11758196—24no—
194C4×Dic61—28no—
195C4×Dic61—24no—
196C4×Dic61—22no—
197C4×Dic61—21no—
198C4×Dic61—24no—
199C4×Dic61C22×C412yes—
200#11758194—22no—
201#11758194—22no—
202C2×#92418—22no—
203#11758189—44no—
204C8×D61—44no—
205C23×C61—22no—
206C23×C61—44no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61C22×D411yescharacteristic
210C22×D61—42no—
211C22×D61C2×C22⋊C412yes—
212C22×D61#12011yescharacteristic
213C22×D61#12011yescharacteristic
214C22×D61#12312yes—
215Q8×C61—24no—
216Q8×C61—24no—
217D4×C61—24no—
218D4×C61—42no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—42no—
224C2×C4×C61C2×C22⋊C414yes—
225C2×C4×C61C23×C411yescharacteristic
226C2×C4×C61C22×D412yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4112yes—
229C22⋊D61—44no—
230C22⋊D61—44no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—24no—
236C2×Dic61—42no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—48no—
240C2×Dic61C2×C22⋊C414yes—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61#12012yes—
243C2×Dic61#12312yes—
244C4×D61—28no—
245C4×D61—28no—
246C4×D61—48no—
247C4×D61C4×D414yes—
248C4×D61C4×D414yes—
249C4×D61#12114yes—
250C4×D61#12114yes—
251Dic122—24no—
252Dic122—24no—
253Dic122—216no—
254Dic122—24no—
255Dic122—42no—
256C8×C61—42no—
257#92418—42no—
258#20954—611nomaximal, Sylow
259C22×C61—22no—
260C22×C61—44no—
261C22×C61—42no—
262C22×C61—82no—
263C22×C61—82no—
264C22×C61C22×C22⋊C411yescharacteristic
265C22×C61C2×#12011yescharacteristic
266C22×C61C2×#12311yescharacteristic
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61#38612yes—
272C2×D61#38612yes—
273C4×C61—24no—
274C4×C61—42no—
275C4×C61—44no—
276C4×C61C2×C4×D412yes—
277C4×C61C2×#12112yes—
278Dic61—24no—
279Dic61—42no—
280Dic61—44no—
281Dic61—42no—
282Dic61#38614yes—
283C2×#536—611no—
284C2×#519—611no—
285#2896—612no—
286C22×#101—611no—
287#1888—612no—
288#1888—611no—
289#1888D6111yescharacteristic, complemented
290#1886—611no—
291#1886—612no—
292#1886—611no—
293#1733—612no—
294C2×C61—21no—
295C2×C61—21no—
296C2×C61—82no—
297C2×C61C2×#38611yescharacteristic
298C2×C61#302012yes—
299D61—22no—
300D61—22no—
301C22×#41—611no—
302C23×Q8—611no—
303C22×M16—612no—
304#536—618no—
305#536—1224no—
306C2×C4⋊Q8—611no—
307C2×C4⋊Q8—611no—
308C2×#124—611no—
309C2×#124C2×D6111yescharacteristic
310C2×#123—1222no—
311C2×C22⋊Q8—1222no—
312#519—611no—
313#519—612no—
314#519—1222no—
315#519—1224no—
316#519C2×D6111yescharacteristic
317C2×C4×Q8—1224no—
318C2×#117—611no—
319C2×#117—612no—
320C2×#117—1222no—
321#419—612no—
322#419—612no—
323#418—612no—
324#418C2×D6112yes—
325#417—1224no—
326#416—1224no—
327C2×#101—611no—
328C2×#101—612no—
329C2×#101—1224no—
330C2×#101C2×D6111yescharacteristic
331#339—612no—
332#339—612no—
333#339—612no—
334#339C2×D6112yes—
335C61#2095411yescharacteristic, complemented, Sylow
336C2×#41—614no—
337C2×#41—1222no—
338C2×#41—1224no—
339C2×#41—1222no—
340C2×#41—1222no—
341C22×Q8—614no—
342C22×Q8—614no—
343C22×Q8—612no—
344C22×Q8—611no—
345C22×Q8—611no—
346C22×Q8—1222no—
347C22×Q8—1222no—
348C22×Q8—1224no—
349C22×Q8C22×D6111yescharacteristic
350C22×D4—1221no—
351C22×D4C22×D6111yescharacteristic
352C23×C4—611no—
353C23×C4—612no—
354C23×C4—1221no—
355C2×M16—612no—
356C2×M16—1222no—
357C2×M16—1224no—
358C2×M16—1224no—
359C2×M16C22×D6112yes—
360C22×C8—1222no—
361C4⋊Q8—614no—
362C4⋊Q8—614no—
363C4⋊Q8—1222no—
364C4⋊Q8—1222no—
365#124—22no—
366#124—614no—
367#124—1222no—
368#124C4×D6114yescomplemented
369#123—12216no—
370C22⋊Q8—12216no—
371C4×Q8—24no—
372C4×Q8—1224no—
373C4×Q8—1228no—
374C4×Q8—1228no—
375C4×Q8—1224no—
376C4×Q8—1224no—
377#117—22no—
378#117—614no—
379#117—612no—
380#117—614no—
381#117—1222no—
382#117—1222no—
383#117—1224no—
384#117—1222no—
385#117—1224no—
386#117—1224no—
387#117—1222no—
388#117C22×D6112yes—
389C2×C4⋊C4—614no—
390C2×C4⋊C4—1224no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1222no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1222no—
398C2×C42—611no—
399C2×C42—1222no—
400C2×C42—1224no—
401C2×C42C22×D6111yescharacteristic
402#103—1224no—
403#103—1224no—
404#102—24no—
405#102—1224no—
406#101—22no—
407#101—1224no—
408#101—1222no—
409C24—21no—
410C24—611no—
411#41—1228no—
412#41—1228no—
413#41—2444no—
414#41—2444no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—612no—
418C2×Q8—612no—
419C2×Q8—618no—
420C2×Q8—612no—
421C2×Q8—1224no—
422C2×Q8—1224no—
423C2×Q8—1228no—
424C2×Q8—1228no—
425C2×Q8—1228no—
426C2×Q8—1228no—
427C2×Q8—1222no—
428C2×Q8—1221no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×Q8—1222no—
432C2×Q8C2×C4×D6112yes—
433C2×Q8D4×D6112yes—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—1224no—
438C2×D4—2441no—
439C2×D4—2441no—
440C2×D4—2442no—
441C2×D4C2×C4×D6112yes—
442C22×C4—21no—
443C22×C4—611no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—614no—
449C22×C4—612no—
450C22×C4—618no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1224no—
454C22×C4—1224no—
455C22×C4—1224no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4D4×D6112yes—
461C22×C4C23×D6111yescharacteristic
462M16—22no—
463M16—22no—
464M16—1222no—
465M16—1222no—
466M16—2444no—
467C2×C8—22no—
468C2×C8—1222no—
469C2×C8—2444no—
470C4⋊C4—24no—
471C4⋊C4—614no—
472C4⋊C4—1224no—
473C4⋊C4—1224no—
474C4⋊C4—1228no—
475C4⋊C4—1228no—
476C4⋊C4—1224no—
477C4⋊C4—1228no—
478C4⋊C4D4×D6114yes—
479C22⋊C4—22no—
480C22⋊C4—28no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C22⋊C4—1228no—
484C22⋊C4—1222no—
485C42—24no—
486C42—22no—
487C42—21no—
488C42—612no—
489C42—1224no—
490C42—1224no—
491C42—1228no—
492C42—1221no—
493C42—1222no—
494C42C2×C4×D6112yes—
495C23—22no—
496C23—44no—
497C23—41no—
498C23—41no—
499C23—611no—
500C23—611no—
501C23—612no—
502C23—612no—
503C23—2442no—
504C23C2×D4×D6111yescharacteristic
505Q8—24no—
506Q8—24no—
507Q8—12216no—
508Q8—1224no—
509Q8—1224no—
510Q8—1224no—
511Q8—2442no—
512D4—24no—
513D4—42no—
514D4—2444no—
515D4—2444no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—612no—
524C2×C4—614no—
525C2×C4—614no—
526C2×C4—614no—
527C2×C4—614no—
528C2×C4—614no—
529C2×C4—1228no—
530C2×C4—1228no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1224no—
536C2×C4—2448no—
537C2×C4—2448no—
538C2×C4—2442no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4C22⋊C4×D6114yes—
542C2×C4C22×C4×D6111yescharacteristic
543C2×C4C2×D4×D6112yes—
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4#41×D6112yes—
546C8—42no—
547C8—2442no—
548C22—22no—
549C22—44no—
550C22—42no—
551C22—82no—
552C22—82no—
553C22—612no—
554C22—612no—
555C22—1222no—
556C22—1222no—
557C22—2442no—
558C22—2442no—
559C22C2×C22⋊C4×D6111yescharacteristic
560C22#120×D6111yescharacteristic
561C22#123×D6111yescharacteristic
562C4—24no—
563C4—42no—
564C4—44no—
565C4—614no—
566C4—1224no—
567C4—2442no—
568C4—2442no—
569C4—2444no—
570C4C4×D4×D6112yes—
571C4#121×D6112yes—
572C2—21no—
573C2—21no—
574C2—82no—
575C2—1222no—
576C2—1222no—
577C2#386×D6111yescharacteristic
578C2#2145518012yes—
subgroupG/H
centerC22#120×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1888C2
socleC22×C61C2×#120
prime radicalC22×#11758224C22
cosocleC22×C4×C61C24
hypercenter#1888D61
hypocenterC61#20954
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1888

Complements

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

NH ≅ G/Ntype
2. #519×D61C2semidirect
4. C2×#101×D61C2semidirect
5. #21446223C2semidirect
6. #21446219C2semidirect
7. #21446212C2semidirect
8. #21446211C2semidirect
9. #21446210C2semidirect
10. #21445527C2semidirect
11. C61×#1888C2semidirect
16. #16840332C4semidirect
69. #14519777C2×C4semidirect
71. #14519775C2×C4semidirect
73. #14519764C2×C4semidirect
111. #124×C61C2×C4semidirect
289. #1888D61semidirect
335. C61#20954semidirect
368. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403322
#519×D612
#214543332
C2×#101×D612
#214462232
#214462192
#214462122
#214462112
#214462102
#214455272
C61×#18882
#2095461