Finite Group Databasebeta

← #29681707

Subgroups — #29681707

Order 15616.

1914 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×#11758189C211yesmaximal, characteristic, complemented
2C2×#16840274C211yesmaximal, characteristic
3C2×#16839759C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5#21448198C211yesmaximal, characteristic, complemented
6#21446256C214yesmaximal, complemented
7#21446251C214yesmaximal, complemented
8#21445535C211yesmaximal, characteristic, complemented
9C61×#1735C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×C4×D61C2211yescharacteristic
12C22×#11758189—24no—
13C22×#11758189—24no—
14C22×#11758189—24no—
15C22×#11758189C2214yes—
16C23×#92418C2211yescharacteristic
17#16840274—24no—
18#16840274C418yescomplemented
19C2×#14519761—21no—
20C2×#14519757C2212yes—
21C2×#14519752C2212yes—
22C2×#14519747—21no—
23C2×C4×D244—22no—
24C2×#14519736C2211yescharacteristic
25#16839759—24no—
26#16839756—24no—
27C2×#14519702C2212yescomplemented
28C2×#14519700C2212yescomplemented
29C2×#14519699C2212yescomplemented
30C2×#14519697C2212yescomplemented
31#16839623—21no—
32#16839619C2214yes—
33#16839617—21no—
34C4×#11758198—22no—
35#16839524C2214yescomplemented
36C23×C8×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C61×#340C2214yescomplemented
40C24×D61D411yescharacteristic
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222C2×C414yes—
47C22×Dic122C2311yescharacteristic
48C22×D244—22no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C23×Dic61D411yescharacteristic
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61—22no—
55C22×C4×D61—22no—
56C22×C4×D61—21no—
57C22×C4×D61C2×C414yes—
58C22×C4×D61D412yes—
59C22×C4×D61D414yes—
60C22×C4×D61C2312yes—
61C2×#11758189—232no—
62C2×#11758189—28no—
63C2×#11758189—28no—
64C2×#11758189—28no—
65C22×#92418—22no—
66C22×#92418—22no—
67C22×#92418D414yescomplemented
68C22×#92418C2312yes—
69#14519761—28no—
70#14519757—24no—
71#14519757C2×C418yes—
72#14519752—24no—
73#14519752C2×C418yes—
74#14519747—28no—
75C4×D244—216no—
76#14519736—22no—
77#14519736—22no—
78C2×C4⋊Dic61—22no—
79C2×C4⋊Dic61C2312yes—
80C2×C22⋊Dic61—21no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×#11758198—24no—
84C2×C4×Dic61—21no—
85#14519702—24no—
86#14519700—24no—
87#14519699—24no—
88#14519697—24no—
89#14519684—24no—
90C23×C4×C61C2311yescharacteristic
91C22×C8×C61—22no—
92C22×C8×C61—22no—
93C22×C8×C61D414yescomplemented
94C22×C8×C61C2312yes—
95#117×C61—22no—
96#117×C61—22no—
97#117×C61—22no—
98#117×C61C2×C414yes—
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61—21no—
104C2×C42×C61—21no—
105C24×C61C2×D411yescharacteristic
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—21no—
109C23×D61—44no—
110C23×D61C22⋊C414yes—
111C23×D61C2×D412yes—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61C22×C412yes—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D412yes—
119C22×C4×C61#4112yes—
120C22×C4×C61#4112yes—
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—24no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61C22⋊C414yes—
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—28no—
140C2×D244—21no—
141C2×D244—48no—
142C2×D244C22×C412yes—
143C2×C4×D61—28no—
144C2×C4×D61—28no—
145C2×C4×D61—28no—
146C2×C4×D61—24no—
147C2×C4×D61—24no—
148C2×C4×D61—28no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—24no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61C22⋊C418yes—
157C2×C4×D61C22×C414yes—
158C2×C4×D61C2×D414yes—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122C22×C412yes—
165C2×C8×C61—28no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61C2×D414yes—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61C22×C414yes—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—42no—
175C42×C61—24no—
176C42×C61—42no—
177C22⋊Dic61—42no—
178#11758198—28no—
179#11758198—28no—
180#11758198—48no—
181C4⋊Dic61—28no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4×Dic61—42no—
185C2×#92418—28no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418C2×D414yes—
189#11758189—232no—
190C23×C61—24no—
191C23×C61C2×C22⋊C412yes—
192C23×C61#12112yes—
193C23×C61#12112yes—
194C23×C61C22×D411yescharacteristic
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—24no—
198C22×D61—24no—
199C22×D61—28no—
200C22×D61—22no—
201C22×D61—48no—
202C22×D61—48no—
203C22×D61—48no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—44no—
207C22×D61C2×C22⋊C414yes—
208C2×C4×C61—24no—
209C2×C4×C61—44no—
210C2×C4×C61—42no—
211C2×C4×C61—44no—
212C2×C4×C61C2×C22⋊C412yes—
213C2×C4×C61C4×D414yes—
214C2×C4×C61C4×D414yes—
215C2×C4×C61#12014yes—
216C2×C4×C61#12112yes—
217C2×C4×C61#12112yes—
218C2×C4×C61#12314yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C22⋊D61—48no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—28no—
229C2×Dic61—22no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—44no—
234C2×Dic61C2×C22⋊C414yes—
235D244—22no—
236D244—22no—
237D244—416no—
238D244—41no—
239D244—41no—
240C4×D61—24no—
241C4×D61—24no—
242C4×D61—24no—
243C4×D61—24no—
244C4×D61—216no—
245C4×D61—48no—
246Dic122—22no—
247Dic122—22no—
248Dic122—41no—
249Dic122—41no—
250C8×C61—24no—
251#92418—24no—
252#20970—611nomaximal, Sylow
253C22×C61—24no—
254C22×C61—24no—
255C22×C61—24no—
256C22×C61#38614yes—
257C22×C61C22×C22⋊C411yescharacteristic
258C22×C61C2×#12111yescharacteristic
259C22×C61C2×#12111yescharacteristic
260C22×C61#57514yes—
261C2×D61—24no—
262C2×D61—24no—
263C2×D61—24no—
264C2×D61—416no—
265C2×D61—44no—
266C2×D61—44no—
267C2×D61—44no—
268C2×D61—42no—
269C2×D61—82no—
270C2×D61—82no—
271C4×C61—44no—
272C4×C61—82no—
273C4×C61#38614yes—
274C4×C61C2×C4×D411yescharacteristic
275C4×C61C2×C4×D411yescharacteristic
276C4×C61C2×#12011yescharacteristic
277C4×C61C2×#12311yescharacteristic
278Dic61—24no—
279Dic61—42no—
280Dic61—82no—
281C23×M16—611no—
282C2×#525—611no—
283C2×#416—611no—
284C2×#416—611no—
285#2280—611no—
286#1895—614no—
287#1894—614no—
288#1735—611no—
289#1735D6111yescharacteristic, complemented
290C2×C61—24no—
291C2×C61C2×#38611yescharacteristic
292C2×C61#289612yes—
293C2×C61#305412yes—
294C2×C61#305412yes—
295D61—44no—
296D61—82no—
297C22×#41—611no—
298C24×C4—611no—
299C22×M16—614no—
300C22×M16—1224no—
301C22×M16—1224no—
302C22×M16—1224no—
303C23×C8—611no—
304C23×C8C2×D6111yescharacteristic
305#525—618no—
306#525—1224no—
307C2×#123—1221no—
308C2×C22⋊Q8—612no—
309C2×#121—612no—
310C2×#120—1221no—
311C2×C4×D4—1222no—
312C2×#117—611no—
313C2×#117C2×D6111yescharacteristic
314C2×#117C2×D6111yescharacteristic
315#416—1224no—
316#416—1224no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#102—612no—
320C2×#102—612no—
321#381—1221no—
322#380—614no—
323#379—1221no—
324C4×C22⋊C4—1222no—
325#340—614no—
326#340C2×D6114yescomplemented
327C61#2097011yescharacteristic, complemented, Sylow
328C25—611no—
329C2×#41—614no—
330C2×#41—1222no—
331C2×#41—1222no—
332C2×#41—1222no—
333C2×#41—1222no—
334C2×#41—1222no—
335C22×Q8—611no—
336C22×D4—611no—
337C22×D4—1222no—
338C22×D4—1221no—
339C23×C4—611no—
340C23×C4—614no—
341C23×C4—614no—
342C23×C4—612no—
343C23×C4—612no—
344C23×C4—1222no—
345C23×C4—1222no—
346C23×C4—1222no—
347C23×C4—1222no—
348C23×C4—1221no—
349C23×C4C22×D6111yescharacteristic
350C2×M16—1228no—
351C2×M16—1228no—
352C2×M16—12232no—
353C2×M16—1228no—
354C22×C8—22no—
355C22×C8—22no—
356C22×C8—614no—
357C22×C8—612no—
358C22×C8—1222no—
359C22×C8—1222no—
360C22×C8D24414yescomplemented
361C22×C8C22×D6112yes—
362#123—1228no—
363C22⋊Q8—618no—
364C22⋊Q8—1224no—
365#121—618no—
366#121—1224no—
367#120—1228no—
368C4×D4—12216no—
369#117—22no—
370#117—22no—
371#117—22no—
372#117—1222no—
373#117—1222no—
374#117C4×D6114yes—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4C22×D6112yes—
378C2×C4⋊C4C22×D6112yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1224no—
384C2×C22⋊C4—1221no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—1221no—
388#103—1224no—
389#103—1224no—
390#102—1224no—
391#102—1224no—
392#95—1224no—
393C24—612no—
394C24—614no—
395C24—1222no—
396C24—1222no—
397C24—1221no—
398C24—2444no—
399C24D4×D6111yescharacteristic
400#41—1228no—
401#41—1228no—
402#41—2444no—
403#41—2444no—
404C2×Q8—612no—
405C2×Q8—1221no—
406C2×Q8—1221no—
407C2×Q8—1221no—
408C2×Q8—1221no—
409C2×Q8—1221no—
410C2×D4—612no—
411C2×D4—1228no—
412C2×D4—1224no—
413C2×D4—1221no—
414C2×D4—1221no—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4—1221no—
418C2×D4—2448no—
419C2×D4—2442no—
420C2×D4—2442no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—612no—
424C22×C4—614no—
425C22×C4—614no—
426C22×C4—614no—
427C22×C4—618no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1228no—
435C22×C4—1228no—
436C22×C4—1228no—
437C22×C4—1224no—
438C22×C4—1224no—
439C22×C4—1228no—
440C22×C4—1221no—
441C22×C4—1224no—
442C22×C4—1222no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4C2×C4×D6112yes—
446C22×C4C2×D24412yes—
447C22×C4#1175822212yes—
448C22×C4D4×D6112yes—
449C22×C4D4×D6112yes—
450C22×C4D4×D6111yescharacteristic
451C22×C4#1175822612yes—
452C22×C4C23×D6111yescharacteristic
453M16—12232no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—28no—
457C2×C8—614no—
458C2×C8—1222no—
459C2×C8—1222no—
460C2×C8—1228no—
461C2×C8C2×D24414yes—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—1228no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4C2×C4×D6114yes—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—1228no—
472C22⋊C4—1228no—
473C22⋊C4—2448no—
474C22⋊C4—2442no—
475C42—24no—
476C42—42no—
477C42—2442no—
478C23—24no—
479C23—614no—
480C23—1222no—
481C23—1222no—
482C23—1224no—
483C23—1224no—
484C23—1228no—
485C23—1222no—
486C23—2448no—
487C23—2448no—
488C23—2448no—
489C23—2441no—
490C23—2441no—
491C23—2444no—
492C23C22⋊C4×D6112yes—
493C23#1451975012yes—
494C23#1451975212yes—
495C23C2×D4×D6111yescharacteristic
496Q8—1222no—
497Q8—1222no—
498Q8—2441no—
499Q8—2441no—
500D4—1222no—
501D4—1222no—
502D4—24416no—
503D4—2448no—
504D4—2441no—
505D4—2441no—
506C2×C4—24no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—614no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1228no—
514C2×C4—1224no—
515C2×C4—1224no—
516C2×C4—1224no—
517C2×C4—1224no—
518C2×C4—12216no—
519C2×C4—1222no—
520C2×C4—2442no—
521C2×C4—2444no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2448no—
525C2×C4C22⋊C4×D6112yes—
526C2×C4#1451973914yes—
527C2×C4C4×D24414yes—
528C2×C4#1451974714yes—
529C2×C4#1451975012yes—
530C2×C4#1451975212yes—
531C2×C4#1451976214yes—
532C2×C4C22×C4×D6111yescharacteristic
533C2×C4C2×D4×D6111yescharacteristic
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C22×D24411yescharacteristic
536C2×C4C2×#1175822211yescharacteristic
537C2×C4C2×#1175822611yescharacteristic
538C8—24no—
539C8—1224no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—1224no—
544C22—1224no—
545C22—1224no—
546C22—4882no—
547C22—4882no—
548C22—24416no—
549C22—2442no—
550C22—2444no—
551C22—2444no—
552C22—2444no—
553C22#1683964014yes—
554C22C2×C22⋊C4×D6111yescharacteristic
555C22C2×#1451975011yescharacteristic
556C22C2×#1451975211yescharacteristic
557C22#1684055914yes—
558C4—44no—
559C4—82no—
560C4—1224no—
561C4—4882no—
562C4—2442no—
563C4#1683964014yes—
564C4C2×#1451973911yescharacteristic
565C4C2×C4×D24411yescharacteristic
566C4C2×#1451974711yescharacteristic
567C4C2×#1451976211yescharacteristic
568C2—24no—
569C2—4882no—
570C2—2444no—
571C2C2×#1683964011yescharacteristic
572C2#2145433412yes—
573C2#2145542712yes—
574C2#2145542912yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1735C2
socleC23×C61C22×D4
prime radicalC2×#16840274C2
cosocleC22×C4×C61C24
hypercenter#1735D61
hypocenterC61#20970
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
1. C23×#11758189C2semidirect
3. C2×#16839759C2semidirect
4. C2×#16839756C2semidirect
5. #21448198C2semidirect
6. #21446256C2semidirect
7. #21446251C2semidirect
8. #21445535C2semidirect
9. C61×#1735C2semidirect
18. #16840274C4semidirect
27. C2×#14519702C22semidirect
28. C2×#14519700C22semidirect
29. C2×#14519699C22semidirect
30. C2×#14519697C22semidirect
35. #16839524C22semidirect
39. C61×#340C22semidirect
67. C22×#92418D4semidirect
93. C22×C8×C61D4semidirect
289. #1735D61semidirect
326. #340C2×D61semidirect
327. C61#20970semidirect
360. C22×C8D244semidirect

Maximal subgroups

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

subgroupindex
C23×#117581892
C2×#168402742
C2×#168397592
C2×#168397562
#214481982
#214462562
#214462512
#214455352
C61×#17352
#2097061