Finite Group Databasebeta

← #29679816

Subgroups — #29679816

Order 15616.

1470 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519735C211yesmaximal, characteristic
2C2×#16839725C211yesmaximal, characteristic
3#21448198C211yesmaximal, characteristic
4#21445913C214yesmaximal, complemented
5#21445912C214yesmaximal, complemented
6#21445540C211yesmaximal, characteristic, complemented
7#21445539C211yesmaximal, characteristic, complemented
8#21445538C211yesmaximal, characteristic, complemented
9C61×#1736C211yesmaximal, characteristic, complemented
10C23×C4×D61C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13C2×#14519735—21no—
14C2×#14519735—21no—
15C2×#14519735—24no—
16C2×#14519735C418yescomplemented
17C2×#14519735C2212yes—
18C2×#14519735C2212yes—
19C22×C4⋊Dic61C2211yescharacteristic
20C22×#11758198—21no—
21C22×C4×Dic61—21no—
22#16839725—24no—
23C2×#14519689C2212yes—
24C2×#14519689C2212yes—
25#16839623—21no—
26#16839619C2212yes—
27#16839619C2212yes—
28#16839617—21no—
29C4×#11758198—21no—
30C4×#11758198—21no—
31#16839525C2214yescomplemented
32#16839524C2214yescomplemented
33#16839524C2214yescomplemented
34C22×M16×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C61×#340C2214yescomplemented
38C24×D61D411yescharacteristic
39C23×Dic61—21no—
40C23×Dic61D411yescharacteristic
41C22×C4×D61—22no—
42C22×C4×D61—22no—
43C22×C4×D61—22no—
44C22×C4×D61—22no—
45C22×C4×D61C2×C414yes—
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61D414yes—
49C22×C4×D61C2311yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758194—22no—
52C2×#11758194—22no—
53C2×#11758194—22no—
54C22×#92418C2312yes—
55#14519736—22no—
56#14519736—22no—
57#14519735—28no—
58#14519735—28no—
59#14519735—24no—
60#14519735—24no—
61#14519735C2×C418yes—
62#14519735C2×C418yes—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61C2×C414yes—
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2312yes—
68C2×C22⋊Dic61—21no—
69C2×#11758198—28no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×C4×Dic61—28no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61—22no—
77C2×C4×Dic61—21no—
78#14519689—24no—
79#14519689—24no—
80#14519684—22no—
81#14519684—22no—
82C23×C4×C61—21no—
83C23×C4×C61C2311yescharacteristic
84C2×M16×C61—22no—
85C2×M16×C61—22no—
86C2×M16×C61—22no—
87C22×C8×C61C2312yes—
88#117×C61—22no—
89#117×C61—22no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61C2×C414yes—
93C2×C4⋊C4×C61C2312yes—
94C2×C4⋊C4×C61C2312yes—
95C2×C22⋊C4×C61—21no—
96C2×C42×C61—21no—
97C24×C61C2×D411yescharacteristic
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—42no—
101C23×D61—42no—
102C23×D61C22⋊C414yes—
103C23×D61C2×D411yescharacteristic
104C23×D61C2×D411yescharacteristic
105C22×C4×C61—24no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—44no—
109C22×C4×C61C22×C412yes—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D412yes—
112C22×C4×C61#4112yes—
113C22×C4×C61#4112yes—
114C22×C4×C61#4112yes—
115C22×C4×C61#4112yes—
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—24no—
120C22×Dic61—22no—
121C22×Dic61—22no—
122C22×Dic61—44no—
123C22×Dic61C22⋊C414yes—
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126C2×C4×D61—28no—
127C2×C4×D61—28no—
128C2×C4×D61—28no—
129C2×C4×D61—28no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61C22⋊C414yes—
143C2×C4×D61C22⋊C414yes—
144C2×C4×D61C22×C412yes—
145C2×C4×D61C22×C412yes—
146C2×C4×D61C2×D412yes—
147C2×C4×D61C2×D412yes—
148M16×C61—44no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61C22×C414yes—
157C22⋊C4×C61—42no—
158C42×C61—42no—
159C22⋊Dic61—42no—
160#11758198—28no—
161#11758198—28no—
162#11758198—44no—
163#11758198—44no—
164C4⋊Dic61—24no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61C22×C414yes—
169C4×Dic61—28no—
170C4×Dic61—28no—
171C4×Dic61—42no—
172#11758194—44no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61C2×C22⋊C412yes—
179C23×C61#12312yes—
180C23×C61#12312yes—
181C23×C61C22×D411yescharacteristic
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—21no—
185C22×D61—21no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—28no—
191C22×D61—44no—
192C22×D61—44no—
193C22×D61—48no—
194C22×D61—48no—
195C22×D61C2×C22⋊C412yes—
196C22×D61C2×C22⋊C412yes—
197C2×C4×C61—24no—
198C2×C4×C61—48no—
199C2×C4×C61—44no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61#11714yes—
205C2×C4×C61#11714yes—
206C2×C4×C61#12314yes—
207C2×C4×C61#12314yes—
208C2×C4×C61#12312yes—
209C2×C4×C61#12312yes—
210C2×C4×C61C23×C411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—28no—
221C2×Dic61—24no—
222C2×Dic61—48no—
223C2×Dic61—44no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61C2×C22⋊C412yes—
228C2×Dic61C2×C22⋊C412yes—
229C4×D61—22no—
230C4×D61—22no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—22no—
234C4×D61—22no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—28no—
238C4×D61—28no—
239C8×C61—42no—
240#92418—42no—
241#20701—611nomaximal, Sylow
242C22×C61—24no—
243C22×C61—24no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61#38114yes—
247C22×C61C22×C22⋊C411yescharacteristic
248C22×C61C2×#12311yescharacteristic
249C22×C61C2×#12311yescharacteristic
250C22×C61#57512yes—
251C22×C61#57512yes—
252C2×D61—22no—
253C2×D61—22no—
254C2×D61—22no—
255C2×D61—22no—
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—416no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C4×C61—44no—
264C4×C61—82no—
265C4×C61#38114yes—
266C4×C61C2×#11711yescharacteristic
267C4×C61C2×#11711yescharacteristic
268C4×C61C2×#12311yescharacteristic
269C4×C61C2×#12311yescharacteristic
270Dic61—22no—
271Dic61—22no—
272Dic61—44no—
273Dic61—82no—
274C22×#117—611no—
275C2×#407—611no—
276#2280—611no—
277#1825—614no—
278#1825—614no—
279#1736—611no—
280#1736—611no—
281#1736—611no—
282#1736D6111yescharacteristic, complemented
283C2×C61—24no—
284C2×C61C2×#38111yescharacteristic
285C2×C61#289611yescharacteristic
286C2×C61#289611yescharacteristic
287C2×C61#309312yes—
288C2×C61#309312yes—
289D61—42no—
290D61—42no—
291C24×C4—611no—
292C22×M16—611no—
293C22×M16C2×D6111yescharacteristic
294C2×#117—611no—
295C2×#117—618no—
296C2×#117—612no—
297C2×#117—612no—
298C2×#117—1221no—
299C2×#117—1221no—
300C2×#117—1224no—
301C2×#117C2×D6111yescharacteristic
302C22×C4⋊C4—611no—
303C22×C4⋊C4C2×D6111yescharacteristic
304C22×C22⋊C4—1221no—
305C22×C42—1221no—
306#407—1224no—
307C2×C22⋊C8—612no—
308C2×C22⋊C8—612no—
309#381—1221no—
310#380—612no—
311#380—612no—
312#379—1221no—
313C4×C22⋊C4—1221no—
314C4×C22⋊C4—1221no—
315#340—614no—
316#340—614no—
317#340—614no—
318#340C2×D6114yescomplemented
319C61#2070111yescharacteristic, complemented, Sylow
320C25—611no—
321C23×C4—21no—
322C23×C4—611no—
323C23×C4—614no—
324C23×C4—611no—
325C23×C4—611no—
326C23×C4—614no—
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—1222no—
330C23×C4—1222no—
331C23×C4—1222no—
332C23×C4—1222no—
333C23×C4—1221no—
334C23×C4C22×D6111yescharacteristic
335C2×M16—22no—
336C2×M16—22no—
337C2×M16—22no—
338C2×M16—1222no—
339C2×M16—1222no—
340C2×M16—1222no—
341C22×C8—612no—
342C22×C8C22×D6112yes—
343#117—22no—
344#117—22no—
345#117—618no—
346#117—618no—
347#117—1224no—
348#117—1224no—
349#117—1228no—
350#117—1228no—
351#117—1222no—
352#117—1222no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—614no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4C4×D6114yes—
361C2×C4⋊C4C22×D6112yes—
362C2×C4⋊C4C22×D6112yes—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1228no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1221no—
370C2×C42—21no—
371C2×C42—1222no—
372C2×C42—1222no—
373C2×C42—1228no—
374C2×C42—1221no—
375C22⋊C8—1224no—
376C22⋊C8—1224no—
377#95—1222no—
378#95—1222no—
379C24—611no—
380C24—611no—
381C24—614no—
382C24—1222no—
383C24—1222no—
384C24—2442no—
385C24—2442no—
386C24D4×D6111yescharacteristic
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—44no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—614no—
394C22×C4—612no—
395C22×C4—612no—
396C22×C4—612no—
397C22×C4—612no—
398C22×C4—614no—
399C22×C4—614no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4—1228no—
411C22×C4—1228no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1228no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1228no—
418C22×C4—1222no—
419C22×C4—1224no—
420C22×C4—1222no—
421C22×C4—2444no—
422C22×C4C2×C4×D6112yes—
423C22×C4#1175822212yes—
424C22×C4#1175822212yes—
425C22×C4D4×D6112yes—
426C22×C4D4×D6111yescharacteristic
427C22×C4#1175822412yes—
428C22×C4#1175822612yes—
429C22×C4C23×D6111yescharacteristic
430M16—44no—
431M16—2444no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—614no—
443C4⋊C4—1222no—
444C4⋊C4—1224no—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C4⋊C4C2×C4×D6114yes—
448C22⋊C4—42no—
449C22⋊C4—1228no—
450C22⋊C4—1228no—
451C22⋊C4—2444no—
452C22⋊C4—2444no—
453C22⋊C4—2442no—
454C42—42no—
455C42—1228no—
456C42—1228no—
457C42—2442no—
458C23—22no—
459C23—22no—
460C23—612no—
461C23—612no—
462C23—1221no—
463C23—1221no—
464C23—1221no—
465C23—1221no—
466C23—1222no—
467C23—1222no—
468C23—1222no—
469C23—1222no—
470C23—1228no—
471C23—2444no—
472C23—2444no—
473C23—2448no—
474C23—2448no—
475C23C22⋊C4×D6112yes—
476C23#1451975912yes—
477C23#1451976212yes—
478C23C2×D4×D6111yescharacteristic
479C2×C4—24no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—612no—
486C2×C4—612no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—1221no—
491C2×C4—1228no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1228no—
501C2×C4—1228no—
502C2×C4—1224no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2448no—
507C2×C4—2444no—
508C2×C4C22⋊C4×D6112yes—
509C2×C4#1451973414yes—
510C2×C4#1451973514yes—
511C2×C4#1451975914yes—
512C2×C4#1451975912yes—
513C2×C4#1451976214yes—
514C2×C4#1451976212yes—
515C2×C4C22×C4×D6111yescharacteristic
516C2×C4C2×D4×D6111yescharacteristic
517C2×C4C2×#1175822411yescharacteristic
518C2×C4C2×#1175822211yescharacteristic
519C2×C4C2×#1175822211yescharacteristic
520C2×C4C2×#1175822611yescharacteristic
521C8—42no—
522C8—2442no—
523C22—22no—
524C22—24no—
525C22—24no—
526C22—22no—
527C22—1222no—
528C22—1222no—
529C22—1222no—
530C22—1222no—
531C22—1222no—
532C22—1222no—
533C22—24416no—
534C22—2442no—
535C22—2442no—
536C22—2442no—
537C22—2442no—
538C22#1683962414yes—
539C22C2×C22⋊C4×D6111yescharacteristic
540C22C2×#1451975911yescharacteristic
541C22C2×#1451976211yescharacteristic
542C22#1684055812yes—
543C22#1684056112yes—
544C4—44no—
545C4—82no—
546C4—1222no—
547C4—1222no—
548C4—4882no—
549C4—2444no—
550C4#1683962414yes—
551C4C2×#1451973411yescharacteristic
552C4C2×#1451973511yescharacteristic
553C4C2×#1451975911yescharacteristic
554C4C2×#1451976211yescharacteristic
555C2—24no—
556C2—2442no—
557C2—2442no—
558C2C2×#1683962411yescharacteristic
559C2#2145433311yescharacteristic
560C2#2145433711yescharacteristic
561C2#2145574912yes—
562C2#2145575312yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1736D61
hypocenterC61#20701
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #21445913C2semidirect
5. #21445912C2semidirect
6. #21445540C2semidirect
7. #21445539C2semidirect
8. #21445538C2semidirect
9. C61×#1736C2semidirect
16. C2×#14519735C4semidirect
31. #16839525C22semidirect
32. #16839524C22semidirect
33. #16839524C22semidirect
37. C61×#340C22semidirect
282. #1736D61semidirect
318. #340C2×D61semidirect
319. C61#20701semidirect

Maximal subgroups

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

subgroupindex
C22×#145197352
C2×#168397252
#214481982
#214459132
#214459122
#214455402
#214455392
#214455382
C61×#17362
#2070161