Finite Group Databasebeta

← #29695733

Subgroups — #29695733

Order 15616.

1602 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758201C211yesmaximal, characteristic
2C2×#16839860C212yesmaximal, complemented
3C22⋊C4×D244C211yesmaximal, characteristic
4#21446703C212yesmaximal, complemented
5#21446174C212yesmaximal, complemented
6#21446156C212yesmaximal, complemented
7#21445846C211yesmaximal, characteristic, complemented
8#21445754C211yesmaximal, characteristic, complemented
9#21445749C211yesmaximal, characteristic
10#21445597C211yesmaximal, characteristic, complemented
11Q8×C22⋊C4×C61C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758201—28no—
14C22×#11758201C418yescomplemented
15C22×#11758201C2211yescharacteristic
16C22×#11758201C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C23×#92418C2211yescharacteristic
20C2×C4×D244C2212yes—
21C2×C22⋊C4×D61—21no—
22#16839860—28no—
23C2×#14519710C2212yescomplemented
24C2×#14519701C2212yescomplemented
25C2×#14519701C2212yescomplemented
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28C2×C22⋊#92418C2211yescharacteristic
29C2×#14519688C2212yescomplemented
30#16839664C2212yes—
31#16839640—22no—
32#16839622C2211yescharacteristic
33#16839617—21no—
34#16839524C2212yescomplemented
35#16839517C2212yescomplemented
36C23×Q8×C61C2211yescharacteristic
37C2×C4×Q8×C61C2212yescomplemented
38C61×#391—22no—
39C61×#391C2212yescomplemented
40C61×#380—21no—
41C61×#380C2211yescharacteristic
42C4×C22⋊C4×C61—21no—
43C4×C22⋊C4×C61C2211yescharacteristic
44C24×D61—21no—
45C22×D244—24no—
46C22×D244—21no—
47C22×D244—22no—
48C22×D244—42no—
49C22×D244C2×C414yes—
50C22×D244C2311yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C4×D61—22no—
53C2×#11758201—24no—
54C2×#11758201—24no—
55C2×#11758201—232no—
56C2×#11758201—24no—
57C2×#11758201C2×C414yes—
58C2×#11758201C2×C414yes—
59C2×#11758201C2×C414yescomplemented
60C2×#11758201C2×C414yescomplemented
61C2×#11758201D418yes—
62C22×#92418—24no—
63C22×#92418C2×C414yescomplemented
64C22×#92418C2312yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C4×D244—24no—
68C4×D244—42no—
69C2×C4⋊Dic61C2312yes—
70C2×C22⋊Dic61—21no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C22⋊C4×D61—44no—
75#14519710—24no—
76#14519701—24no—
77#14519701—42no—
78#14519697—24no—
79#14519697—42no—
80C22⋊#92418—22no—
81#14519688—24no—
82#14519684—21no—
83C22×Q8×C61—21no—
84C22×Q8×C61—22no—
85C22×Q8×C61—24no—
86C22×Q8×C61—42no—
87C22×Q8×C61C2×C414yescomplemented
88C22×Q8×C61C2311yescharacteristic
89C22×Q8×C61C2311yescharacteristic
90C23×C4×C61—21no—
91C23×C4×C61C2311yescharacteristic
92C4×Q8×C61—24no—
93C4×Q8×C61—42no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61C2312yes—
96C2×C22⋊C4×C61—21no—
97C2×C22⋊C4×C61D411yescharacteristic
98C2×C22⋊C4×C61D411yescharacteristic
99C2×C42×C61—22no—
100C2×C42×C61C2312yes—
101#95×C61—22no—
102#95×C61—22no—
103#95×C61—21no—
104#95×C61D412yes—
105C24×C61C2×D411yescharacteristic
106C23×D61—21no—
107C23×D61—21no—
108C23×D61—24no—
109C23×D61—44no—
110C23×D61—42no—
111C23×D61—42no—
112C23×D61—82no—
113C2×Q8×C61—24no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—22no—
117C2×Q8×C61—28no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—44no—
121C2×Q8×C61—44no—
122C2×Q8×C61—48no—
123C2×Q8×C61C22×C412yes—
124C2×Q8×C61C22×C412yes—
125C2×Q8×C61C2×D414yes—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—24no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—44no—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61#4112yes—
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—22no—
140C2×D244—21no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—28no—
145C2×D244—24no—
146C2×D244—21no—
147C2×D244—44no—
148C2×D244—48no—
149C2×D244—44no—
150C2×D244C22×C412yes—
151C2×D244C22×C412yes—
152C2×D244C2×D414yes—
153C2×C4×D61—44no—
154C2×C4×D61—44no—
155C2×C4×D61—82no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—44no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—42no—
162C42×C61—22no—
163C42×C61—22no—
164C42×C61—44no—
165C22⋊Dic61—42no—
166#11758201—28no—
167#11758201—28no—
168#11758201—28no—
169#11758201C22⋊C4116yescomplemented
170#11758198—44no—
171#11758198—44no—
172#11758198—82no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C2×#92418—22no—
176C2×#92418—28no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418C22×C412yes—
181C2×#92418C22×C412yes—
182C2×#92418C2×D414yes—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61C4×D412yes—
186C23×C61AGL(1,ℤ/8)12yes—
187C23×C61#13712yes—
188C23×C61C22×D411yescharacteristic
189C22×D61—22no—
190C22×D61—24no—
191C22×D61—22no—
192C22×D61—48no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—42no—
200C22×D61—48no—
201C22×D61—84no—
202C22×D61—88no—
203C22×D61—88no—
204Q8×C61—22no—
205Q8×C61—24no—
206Q8×C61—22no—
207Q8×C61—22no—
208Q8×C61—48no—
209Q8×C61—42no—
210Q8×C61C2×C22⋊C414yescomplemented
211C2×C4×C61—22no—
212C2×C4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—48no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61C4×D412yes—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C22×D412yes—
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61H(5,𝔽2)12yes—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237D244—22no—
238D244—22no—
239D244—22no—
240D244—24no—
241D244—48no—
242D244—42no—
243D244C2×C22⋊C414yes—
244C4×D61—88no—
245#92418—22no—
246#92418—42no—
247#92418C2×C22⋊C414yescomplemented
248#22437—611nomaximal, Sylow
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—24no—
252C22×C61—24no—
253C22×C61#44014yescomplemented
254C22×C61C2×C4×D411yescharacteristic
255C22×C61D4212yes—
256C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C61C2×#13711yescharacteristic
258C22×C61#57712yes—
259C2×D61—24no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—44no—
263C2×D61—44no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—816no—
268C2×D61—82no—
269C2×D61—82no—
270C4×C61—22no—
271C4×C61—24no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61—42no—
275C4×C61—82no—
276C4×C61C22×C22⋊C411yescharacteristic
277C4×C61#51811yescharacteristic
278C4×C61#54612yes—
279Dic61—82no—
280C23×SD16—611no—
281C2×#440—612no—
282Q8×C22⋊C4D6111yescharacteristic, complemented
283D4×C22⋊C4—611no—
284#1973—612no—
285#1880—612no—
286#1877—612no—
287#1808—611no—
288#1787—611no—
289#1786—611no—
290#1752—611no—
291C2×C61—24no—
292C2×C61D4×C22⋊C411yescharacteristic
293C2×C61C2×#44011yescharacteristic
294C2×C61#294911yescharacteristic
295C2×C61#326512yes—
296C2×C61#326612yes—
297D61—44no—
298D61—82no—
299C23×Q8C2×D6111yescharacteristic
300C23×D4—611no—
301C22×SD16—611no—
302C22×SD16—618no—
303C22×SD16—611no—
304C22×SD16—611no—
305C22×SD16—611no—
306C22×SD16—1228no—
307C23×C8—611no—
308C2×C4×Q8C2×D6112yescomplemented
309C2×C4×D4—612no—
310C22×C22⋊C4—1221no—
311#440—1228no—
312C2×#107—612no—
313C2×#103—612no—
314C2×#103—612no—
315C2×#102—612no—
316C2×#102—612no—
317C2×C22⋊C8—611no—
318C2×#97—612no—
319#391—22no—
320#391C2×D6112yescomplemented
321#390—612no—
322#386—1222no—
323#380—21no—
324#380—611no—
325#380C2×D6111yescharacteristic
326#379—1221no—
327C4×C22⋊C4—21no—
328C4×C22⋊C4C2×D6111yescharacteristic
329#340—612no—
330#336—612no—
331C61#2243711yescharacteristic, complemented, Sylow
332C25—1221no—
333C22×Q8—22no—
334C22×Q8—21no—
335C22×Q8—24no—
336C22×Q8—42no—
337C22×Q8C4×D6114yescomplemented
338C22×Q8C22×D6111yescharacteristic
339C22×Q8C22×D6111yescharacteristic
340C22×D4—614no—
341C22×D4—611no—
342C22×D4—611no—
343C22×D4—1224no—
344C22×D4—1222no—
345C22×D4—1221no—
346C22×D4—2442no—
347C23×C4—21no—
348C23×C4—1222no—
349C23×C4C22×D6111yescharacteristic
350C2×SD16—614no—
351C2×SD16—618no—
352C2×SD16—614no—
353C2×SD16—614no—
354C2×SD16—614no—
355C2×SD16—1224no—
356C2×SD16—12232no—
357C2×SD16—1224no—
358C2×SD16—1224no—
359C22×C8—611no—
360C22×C8—614no—
361C22×C8—611no—
362C22×C8—612no—
363C22×C8—1224no—
364C4×Q8—24no—
365C4×Q8—42no—
366C4×D4—1224no—
367C4×D4—2442no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4C22×D6112yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—2444no—
377C2×C22⋊C4C22⋊D6111yescharacteristic
378C2×C22⋊C4C22⋊D6111yescharacteristic
379C2×C42—22no—
380C2×C42C22×D6112yes—
381#107—1224no—
382#103—1224no—
383#103—2442no—
384#102—1224no—
385#102—2442no—
386C22⋊C8—1222no—
387#97—1224no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—1221no—
392#95C22⋊D6112yes—
393C24—1221no—
394C24—1224no—
395C24—1221no—
396C24—4882no—
397C24—2444no—
398C24—2442no—
399C24—2442no—
400C24C2×C22⋊D6111yescharacteristic
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—28no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—48no—
411C2×Q8C2×C4×D6112yes—
412C2×Q8C2×C4×D6112yes—
413C2×Q8D4×D6114yes—
414C2×D4—612no—
415C2×D4—612no—
416C2×D4—614no—
417C2×D4—1228no—
418C2×D4—1222no—
419C2×D4—1221no—
420C2×D4—1221no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1224no—
424C2×D4—2444no—
425C2×D4—2448no—
426C2×D4—2444no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—1222no—
435C22×C4—4882no—
436C22×C4—2444no—
437C22×C4—2444no—
438C22×C4C2×C4×D6112yes—
439C22×C4#1175822212yes—
440C22×C4C2×C22⋊D6112yes—
441C22×C4C2×C22⋊D6112yes—
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C23×D6111yescharacteristic
444SD16—6116no—
445SD16—1228no—
446SD16—1228no—
447SD16—1228no—
448C2×C8—612no—
449C2×C8—614no—
450C2×C8—612no—
451C2×C8—1228no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—1222no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—44no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—42no—
464C22⋊C4—4882no—
465C22⋊C4—2444no—
466C22⋊C4—2444no—
467C22⋊C4—2442no—
468C42—22no—
469C42—22no—
470C42—44no—
471C23—22no—
472C23—22no—
473C23—1222no—
474C23—1224no—
475C23—1222no—
476C23—4888no—
477C23—4888no—
478C23—4884no—
479C23—2448no—
480C23—2441no—
481C23—2441no—
482C23—2442no—
483C23—2448no—
484C23—2442no—
485C23—2441no—
486C23—2441no—
487C23—2442no—
488C23C4×C22⋊D6112yes—
489C23#1451980212yes—
490C23#1451980812yes—
491C23C22×C22⋊D6111yescharacteristic
492Q8—22no—
493Q8—24no—
494Q8—22no—
495Q8—22no—
496Q8—48no—
497Q8—42no—
498Q8C22⋊C4×D6114yescomplemented
499D4—614no—
500D4—1222no—
501D4—1222no—
502D4—1222no—
503D4—1224no—
504D4—2448no—
505D4—2442no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—4888no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4C4×C22⋊D6112yes—
529C2×C4C22×C4×D6111yescharacteristic
530C2×C4C2×D4×D6112yes—
531C2×C4C2×#1175822211yescharacteristic
532C2×C4#1451982212yes—
533C8—614no—
534C8—1222no—
535C8—2442no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—22no—
540C22—1224no—
541C22—48816no—
542C22—4882no—
543C22—4882no—
544C22—2442no—
545C22—2442no—
546C22—2442no—
547C22—2442no—
548C22—2442no—
549C22—2444no—
550C22—2444no—
551C22#1683986014yescomplemented
552C22C2×C4×C22⋊D6111yescharacteristic
553C22D4×C22⋊D6112yes—
554C22C2×#1451980211yescharacteristic
555C22C2×#1451980811yescharacteristic
556C22#1684057012yes—
557C4—22no—
558C4—24no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—82no—
563C4—4882no—
564C4C2×C22⋊C4×D6111yescharacteristic
565C4#1684023511yescharacteristic
566C4#1684040212yes—
567C2—24no—
568C2—4882no—
569C2—2444no—
570C2C22⋊C4×C22⋊D6111yescharacteristic
571C2C2×#1683986011yescharacteristic
572C2#2145467811yescharacteristic
573C2#2145731512yes—
574C2#2145733012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingQ8×C22⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC23×D244C22
cosocleC22×C4×C61C24
hypercenterQ8×C22⋊C4D61
hypocenterC61#22437
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16839860C2semidirect
4. #21446703C2semidirect
5. #21446174C2semidirect
6. #21446156C2semidirect
7. #21445846C2semidirect
8. #21445754C2semidirect
10. #21445597C2semidirect
11. Q8×C22⋊C4×C61C2semidirect
14. C22×#11758201C4semidirect
23. C2×#14519710C22semidirect
24. C2×#14519701C22semidirect
25. C2×#14519701C22semidirect
29. C2×#14519688C22semidirect
34. #16839524C22semidirect
35. #16839517C22semidirect
37. C2×C4×Q8×C61C22semidirect
39. C61×#391C22semidirect
59. C2×#11758201C2×C4semidirect
60. C2×#11758201C2×C4semidirect
63. C22×#92418C2×C4semidirect
87. C22×Q8×C61C2×C4semidirect
169. #11758201C22⋊C4semidirect
210. Q8×C61C2×C22⋊C4semidirect
247. #92418C2×C22⋊C4semidirect
253. C22×C61#440semidirect
282. Q8×C22⋊C4D61semidirect
308. C2×C4×Q8C2×D61semidirect
320. #391C2×D61semidirect
331. C61#22437semidirect
337. C22×Q8C4×D61semidirect
498. Q8C22⋊C4×D61semidirect
551. C22#16839860semidirect

Maximal subgroups

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

subgroupindex
C23×#117582012
C2×#168398602
C22⋊C4×D2442
#214467032
#214461742
#214461562
#214458462
#214457542
#214457492
#214455972
Q8×C22⋊C4×C612
#2243761