Finite Group Databasebeta

← #29689888

Subgroups — #29689888

Order 15616.

1442 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840307C211yesmaximal, characteristic
2C22×#14519706C211yesmaximal, characteristic, complemented
3#21448221C211yesmaximal, characteristic, complemented
4#21446769C212yesmaximal, complemented
5#21446744C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446247C211yesmaximal, characteristic, complemented
8#21446245C211yesmaximal, characteristic, complemented
9#21445818C212yesmaximal, complemented
10#21445817C212yesmaximal, complemented
11C61×#1893C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C23×#92418C2211yescharacteristic
14#16840307—24no—
15#16840307C418yescomplemented
16C2×#14519776C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519751C2211yescharacteristic
20C2×#14519749—21no—
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23C2×#14519737—21no—
24C2×#14519733—21no—
25C22×C4×Dic61C2211yescharacteristic
26C2×#14519706—24no—
27C2×#14519706C2212yes—
28C2×#14519706C2212yes—
29C2×#14519699C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yes—
32#16839631C2212yes—
33#16839626—21no—
34#16839622C2211yescharacteristic
35#16839620—21no—
36C4⋊C4×Dic61C2212yes—
37#16839613—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C23×C8×C61C2211yescharacteristic
42C2×#121×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C61×#380C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758224—28no—
47C2×#11758224—44no—
48C2×#11758224—44no—
49C22×Dic122—21no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C23×Dic61D411yescharacteristic
56C23×Dic61D411yescharacteristic
57C22×C4×D61—21no—
58C22×#92418—22no—
59C22×#92418—42no—
60C22×#92418C2312yes—
61#14519776—22no—
62#14519776C2×C414yes—
63#14519773—22no—
64#14519773C2×C414yes—
65#14519763—28no—
66#14519751—22no—
67#14519751C2×C414yes—
68#14519749—28no—
69D4×Dic61—22no—
70D4×Dic61C2×C414yes—
71#14519739—22no—
72#14519739C2×C414yes—
73#14519737—28no—
74#14519733—44no—
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—22no—
79C2×#11758196—22no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61C2×C414yes—
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D414yes—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519706—24no—
92#14519706—24no—
93#14519699—42no—
94#14519698—42no—
95#14519697—42no—
96#14519684—21no—
97#14519684—22no—
98C22×D4×C61—21no—
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61C2311yescharacteristic
101C22×C8×C61—22no—
102C22×C8×C61—42no—
103C22×C8×C61C2312yes—
104#121×C61—22no—
105#121×C61C2×C414yes—
106C2×C4⋊C4×C61C2312yes—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110#102×C61—42no—
111#95×C61—21no—
112C24×C61—21no—
113C24×C61C2×D411yescharacteristic
114C23×D61—21no—
115C2×D4×C61—24no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—42no—
120C2×D4×C61—42no—
121C2×D4×C61C22×C412yes—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—24no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61—24no—
133C2×C22⋊D61—44no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61—42no—
138C22×Dic61—21no—
139C22×Dic61—24no—
140C22×Dic61—22no—
141C22×Dic61—42no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61C22⋊C414yes—
145C22×Dic61C22⋊C414yes—
146C22×Dic61D814yes—
147C22×Dic61SD1614yes—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152#11758224—432no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—21no—
156C2×D244C22×C412yes—
157C2×C4×D61—24no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×Dic122—24no—
161C2×Dic122—42no—
162C2×Dic122—42no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—44no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61C22×C412yes—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—42no—
173C22⋊Dic61—24no—
174C22⋊Dic61—42no—
175#11758198—24no—
176#11758198—24no—
177C4⋊Dic61—21no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61C22×C412yes—
181#11758196—24no—
182#11758196—44no—
183#11758196—44no—
184C4×Dic61—44no—
185C4×Dic61C22⋊C414yes—
186C4×Dic61C22⋊C414yes—
187C4×Dic61C22×C412yes—
188C4×Dic61C22×C412yes—
189C4×Dic61C2×D412yes—
190C4×Dic61C2×D412yes—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—44no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—42no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—42no—
202C23×C61C2×C22⋊C412yes—
203C23×C61C2×D812yes—
204C23×C61C2×SD1612yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—22no—
207C22×D61—42no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—42no—
211D4×C61—24no—
212D4×C61—48no—
213D4×C61—41no—
214D4×C61—41no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61C2×C22⋊C412yes—
226C2×C4×C61C23×C411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61H(5,𝔽2)12yes—
229C2×C4×C612−1+412yes—
230C22⋊D61—48no—
231C22⋊D61—48no—
232C22⋊D61—48no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—44no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—48no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61#102116yescomplemented
244C2×Dic61C2×C22⋊C412yes—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×D812yes—
249C2×Dic61C2×D812yes—
250C2×Dic61C2×SD1612yes—
251C2×Dic61C2×SD1612yes—
252D244—24no—
253D244—41no—
254D244—41no—
255C4×D61—48no—
256Dic122—48no—
257C8×C61—42no—
258#92418—42no—
259#21884—611nomaximal, Sylow
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—24no—
263C22×C61—24no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—82no—
268C22×C61—82no—
269C22×C61C2×#10214yes—
270C22×C61C22×C22⋊C411yescharacteristic
271C22×C61#53712yes—
272C22×C61C22×D811yescharacteristic
273C22×C61C22×SD1611yescharacteristic
274C22×C61#57512yes—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—82no—
279C2×D61—82no—
280C4×C61—22no—
281C4×C61—42no—
282C4×C61—82no—
283C4×C61#52012yes—
284C4×C61#53511yescharacteristic
285C4×C61#53611yescharacteristic
286Dic61—22no—
287Dic61—22no—
288Dic61—42no—
289Dic61—82no—
290Dic61C2×#10214yes—
291Dic61C2×#10214yes—
292C2×#532—611no—
293C22×C4⋊C8—611no—
294#2285—611no—
295#1988—612no—
296#1982—612no—
297#1893—611no—
298#1893—611no—
299#1893—611no—
300#1893D6111yescharacteristic, complemented
301#1802—612no—
302#1802—612no—
303C2×C61—24no—
304C2×C61—82no—
305C2×C61#231111yescharacteristic
306C2×C61C22×#10211yescharacteristic
307C2×C61#289611yescharacteristic
308C2×C61#324512yes—
309C2×C61#324612yes—
310D61—82no—
311C22×#41—1221no—
312C23×C8—611no—
313C23×C8C2×D6111yescharacteristic
314#532—618no—
315#532—1224no—
316C2×C4⋊Q8—611no—
317C2×#127—611no—
318C2×#124—1221no—
319C2×#121—611no—
320C2×#121—1221no—
321C2×#121C2×D6111yescharacteristic
322C2×C4×D4—611no—
323C2×C4×D4—611no—
324C2×C4×D4—1221no—
325C2×#117—1221no—
326C22×C42—611no—
327C2×C4⋊C8—612no—
328C2×C4⋊C8—612no—
329C2×C4⋊C8—1224no—
330C2×#102—612no—
331C2×#102—612no—
332C2×#102—612no—
333C2×#102C2×D6112yes—
334#384—612no—
335#382—1221no—
336#380—611no—
337#380—1221no—
338#380C2×D6111yescharacteristic
339C4×C4⋊C4—612no—
340C4×C4⋊C4—1221no—
341#340—612no—
342#340—612no—
343#340—612no—
344#340C2×D6112yes—
345C61#2188411yescharacteristic, complemented, Sylow
346C2×#41—1228no—
347C2×#41—2444no—
348C2×#41—2444no—
349C22×Q8—1221no—
350C22×D4—21no—
351C22×D4—611no—
352C22×D4—1221no—
353C22×D4—1221no—
354C22×D4—1221no—
355C22×D4C22×D6111yescharacteristic
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—1221no—
359C23×C4—1221no—
360C23×C4C22×D6111yescharacteristic
361C22×C8—22no—
362C22×C8—42no—
363C22×C8—612no—
364C22×C8—1222no—
365C22×C8—2442no—
366C22×C8C22×D6112yes—
367C4⋊Q8—614no—
368C4⋊Q8—1222no—
369#127—614no—
370#127—1222no—
371#124—1228no—
372#121—22no—
373#121—614no—
374#121—1228no—
375#121—1222no—
376#121C4×D6114yes—
377C4×D4—614no—
378C4×D4—614no—
379C4×D4—1228no—
380C4×D4—1222no—
381C4×D4—1222no—
382#117—2444no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4C22×D6112yes—
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C42—614no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—614no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—1222no—
403C4⋊C8—1224no—
404C4⋊C8—1224no—
405#102—42no—
406#102—2442no—
407#102—2442no—
408#102—2442no—
409#95—21no—
410#95—1222no—
411#95—1221no—
412C24—21no—
413C24—1221no—
414C24D4×D6111yescharacteristic
415#41—24432no—
416C2×Q8—1224no—
417C2×Q8—2442no—
418C2×Q8—2442no—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—612no—
426C2×D4—1224no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1221no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—2442no—
433C2×D4—2442no—
434C2×D4—2444no—
435C2×D4—2442no—
436C2×D4—2442no—
437C2×D4C2×C4×D6112yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—614no—
445C22×C4—614no—
446C22×C4—614no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—614no—
450C22×C4—1224no—
451C22×C4—1224no—
452C22×C4—1221no—
453C22×C4—1222no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4C2×C4×D6112yes—
460C22×C4D4×D6112yes—
461C22×C4D4×D6111yescharacteristic
462C22×C4C23×D6111yescharacteristic
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—44no—
467C2×C8—1222no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—2444no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—612no—
475C4⋊C4—1224no—
476C4⋊C4—1221no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—2444no—
480C4⋊C4—2444no—
481C4⋊C4C2×C4×D6112yes—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—1224no—
485C22⋊C4—1224no—
486C22⋊C4—1224no—
487C22⋊C4—2442no—
488C42—612no—
489C42—612no—
490C42—612no—
491C42—612no—
492C42—614no—
493C42—614no—
494C42—2444no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—1222no—
503C23—2441no—
504C23—2441no—
505C23—2442no—
506C23—2442no—
507C23C22⋊C4×D6112yes—
508C23D8×D6112yes—
509C23SD16×D6112yes—
510C23C2×D4×D6111yescharacteristic
511Q8—2448no—
512D4—24no—
513D4—48no—
514D4—41no—
515D4—41no—
516D4—1224no—
517D4—2448no—
518D4—2448no—
519D4—2448no—
520D4—2441no—
521D4—2441no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—6116no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—2448no—
545C2×C4—2448no—
546C2×C4—2444no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4C22⋊C4×D6112yes—
553C2×C4C22×C4×D6111yescharacteristic
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4#1451982212yes—
556C2×C4#1451982412yes—
557C8—42no—
558C8—2442no—
559C22—22no—
560C22—24no—
561C22—24no—
562C22—22no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—82no—
567C22—82no—
568C22—4882no—
569C22—4882no—
570C22—2442no—
571C22—2442no—
572C22—2442no—
573C22#102×D6114yes—
574C22C2×C22⋊C4×D6111yescharacteristic
575C22#1684033912yes—
576C22C2×D8×D6111yescharacteristic
577C22C2×SD16×D6111yescharacteristic
578C22#1684055912yes—
579C4—22no—
580C4—42no—
581C4—82no—
582C4—614no—
583C4—614no—
584C4—1222no—
585C4—1222no—
586C4—4882no—
587C4—2442no—
588C4#1684024112yes—
589C4#1684032711yescharacteristic
590C4#1684033411yescharacteristic
591C2—24no—
592C2—82no—
593C2—4882no—
594C2#2144840611yescharacteristic
595C2C2×#102×D6111yescharacteristic
596C2#2145433411yescharacteristic
597C2#2145712512yes—
598C2#2145713012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1893C2
socleC23×C61C22×D4
prime radicalC2×#16840307C2
cosocleC22×C4×C61C24
hypercenter#1893D61
hypocenterC61#21884
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
2. C22×#14519706C2semidirect
3. #21448221C2semidirect
4. #21446769C2semidirect
5. #21446744C2semidirect
6. #21446248C2semidirect
7. #21446247C2semidirect
8. #21446245C2semidirect
9. #21445818C2semidirect
10. #21445817C2semidirect
11. C61×#1893C2semidirect
15. #16840307C4semidirect
243. C2×Dic61#102semidirect
300. #1893D61semidirect
345. C61#21884semidirect

Maximal subgroups

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

subgroupindex
C2×#168403072
C22×#145197062
#214482212
#214467692
#214467442
#214462482
#214462472
#214462452
#214458182
#214458172
C61×#18932
#2188461