Finite Group Databasebeta

← #29685267

Subgroups — #29685267

Order 15616.

1174 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840299C211yesmaximal, characteristic
2C2×#16839714C211yesmaximal, characteristic, complemented
3#21448215C211yesmaximal, characteristic, complemented
4#21446782C212yesmaximal, complemented
5#21446627C212yesmaximal, complemented
6#21446256C212yesmaximal
7#21446249C212yesmaximal, complemented
8#21445649C211yesmaximal, characteristic, complemented
9#21445647C211yesmaximal, characteristic, complemented
10#21445645C211yesmaximal, characteristic, complemented
11C61×#1763C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840299—24no—
14#16840299C418yescomplemented
15C2×#14519772—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×#14519762—21no—
19C2×#14519753C2211yescharacteristic
20C2×#14519752C2211yescharacteristic
21C2×C4×D244—21no—
22C2×C4×D244C2211yescharacteristic
23C2×#14519734—21no—
24C2×C42×D61C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519699C2212yes—
27C2×#14519697C2212yes—
28#16839714—24no—
29C2×#14519688C2211yescharacteristic
30C2×C8×Dic61C2212yes—
31#16839629—21no—
32#16839627C2212yes—
33#16839624—21no—
34#16839619C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36C4×#11758198—21no—
37#16839608—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#97×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×D244—21no—
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C2×#11758189—24no—
53C2×#11758189—24no—
54C2×#11758189—24no—
55C22×#92418C2312yes—
56#14519772—28no—
57#14519768—22no—
58#14519768C2×C414yes—
59#14519766—22no—
60#14519766C2×C414yes—
61#14519762—28no—
62#14519753—22no—
63#14519753C2×C414yes—
64#14519752—22no—
65#14519752C2×C414yes—
66C4×D244—28no—
67C4×D244—22no—
68C4×D244C2×C414yes—
69#14519734—28no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61C2×C414yes—
86C42×D61D414yes—
87#14519701—24no—
88#14519699—24no—
89#14519697—24no—
90#14519688—22no—
91C8×Dic61—24no—
92#14519684—21no—
93#14519684—22no—
94#14519684—22no—
95#14519684—22no—
96C22×Q8×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C4×Q8×C61—22no—
99C4×Q8×C61C2×C414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C42×C61—21no—
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#103×C61—24no—
107#97×C61—22no—
108#95×C61—21no—
109C23×D61—21no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—22no—
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—24no—
129C2×D244—21no—
130C2×D244—42no—
131C2×D244—42no—
132C2×D244C22×C412yes—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—24no—
136C2×C4×D61—41no—
137C2×C4×D61—41no—
138C2×C4×D61—42no—
139C2×C4×D61—44no—
140C2×C4×D61C22⋊C414yes—
141C2×C4×D61C22⋊C414yes—
142C2×C4×D61C22×C412yes—
143C2×C4×D61C2×D412yes—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61#4114yes—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61C22×C412yes—
152C42×C61—24no—
153C42×C61—42no—
154C42×C61C22×C412yes—
155C42×C61C2×D412yes—
156#11758198—24no—
157#11758198—24no—
158#11758198—24no—
159#11758198—24no—
160#11758198—24no—
161#11758198—44no—
162C4⋊Dic61—21no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61C22×C412yes—
167#11758196—24no—
168#11758196—24no—
169C4×Dic61—42no—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22⋊C414yes—
172C4×Dic61C22×C412yes—
173C4×Dic61C2×D412yes—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418#4114yes—
177#11758189—216no—
178C23×C61C22×D411yescharacteristic
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—22no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61C2×C22⋊C412yes—
189Q8×C61—22no—
190Q8×C61—22no—
191Q8×C61—42no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—42no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61C2×C22⋊C412yes—
205C2×C4×C61C23×C411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4112yes—
208C2×C4×C61C2×#4112yes—
209C2×Dic61—22no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61C2×C22⋊C412yes—
219C2×Dic61C2×C22⋊C412yes—
220C2×Dic61C2×C22⋊C412yes—
221D244—22no—
222D244—22no—
223D244—48no—
224D244—42no—
225C4×D61—22no—
226C4×D61—22no—
227C4×D61—24no—
228C4×D61—44no—
229C4×D61—44no—
230C4×D61—48no—
231C8×C61—24no—
232#92418—24no—
233#21411—611nomaximal, Sylow
234C22×C61C22×C22⋊C411yescharacteristic
235C22×C61#53212yes—
236C22×C61#57511yescharacteristic
237C22×C61#57512yes—
238C22×C61#57511yescharacteristic
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—22no—
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—82no—
248C2×D61—82no—
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—82no—
254C4×C61C4×#4111yescharacteristic
255C4×C61C4×#4111yescharacteristic
256C4×C61#52512yes—
257Dic61—22no—
258Dic61—22no—
259Dic61—22no—
260Dic61—42no—
261Dic61—82no—
262C2×#529—611no—
263C2×C4×M16—611no—
264#2283—611no—
265#1991—612no—
266#1958—612no—
267#1895—612no—
268#1894—612no—
269#1763—611no—
270#1763D6111yescharacteristic, complemented
271#1762—611no—
272#1762—611no—
273C2×C61#229711yescharacteristic
274C2×C61#289611yescharacteristic
275C2×C61#289611yescharacteristic
276C2×C61#318312yes—
277C2×C61#318312yes—
278D61—42no—
279D61—82no—
280C22×M16—612no—
281#529—618no—
282#529—1224no—
283C2×#126—1221no—
284C2×#125—611no—
285C2×#124—611no—
286C2×#123—1221no—
287C2×C22⋊Q8—611no—
288C2×#121—611no—
289C2×C4×Q8C2×D6111yescharacteristic
290C2×C4×D4—611no—
291C2×C4×D4—1221no—
292C2×#117—1221no—
293C22×C42—611no—
294C2×#103—612no—
295C2×#103C2×D6112yes—
296C2×#102—612no—
297C2×#102—612no—
298C4×M16—1224no—
299C2×#97—611no—
300C2×#97C2×D6111yescharacteristic
301C2×C4×C8—612no—
302#383—1221no—
303#382—612no—
304#381—1221no—
305#380—612no—
306C4×C4⋊C4—611no—
307C4×C4⋊C4C2×D6111yescharacteristic
308C4×C22⋊C4—1221no—
309#376—1221no—
310#340—612no—
311#340—612no—
312#340—612no—
313#340C2×D6112yes—
314C61#2141111yescharacteristic, complemented, Sylow
315C22×Q8C22×D6111yescharacteristic
316C22×D4—611no—
317C22×D4—1221no—
318C23×C4—611no—
319C23×C4—611no—
320C23×C4—611no—
321C23×C4—1221no—
322C2×M16—1224no—
323C2×M16—1224no—
324C2×M16—1224no—
325C22×C8—612no—
326C22×C8C22×D6112yes—
327#126—1228no—
328#125—614no—
329#125—1222no—
330#124—614no—
331#124—1222no—
332#123—1228no—
333C22⋊Q8—614no—
334C22⋊Q8—1222no—
335#121—614no—
336#121—1222no—
337C4×Q8—22no—
338C4×Q8C4×D6114yes—
339C4×D4—614no—
340C4×D4—1228no—
341C4×D4—1222no—
342#117—1228no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4C22×D6112yes—
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1222no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—611no—
362C2×C42—614no—
363C2×C42—614no—
364C2×C42—1221no—
365C2×C42C22×D6111yescharacteristic
366#103—24no—
367#103—1224no—
368#102—1224no—
369#102—1224no—
370#97—22no—
371#97—1222no—
372C4×C8—1224no—
373#95—21no—
374#95—1222no—
375#95—1222no—
376#95—1222no—
377#95—1221no—
378C24—611no—
379C24—1221no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8C2×C4×D6112yes—
384C2×D4—612no—
385C2×D4—1224no—
386C2×D4—1221no—
387C2×D4—1222no—
388C2×D4—1222no—
389C2×D4—2442no—
390C2×D4—2442no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—612no—
398C22×C4—612no—
399C22×C4—614no—
400C22×C4—614no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1221no—
404C22×C4—1224no—
405C22×C4—1222no—
406C22×C4—2441no—
407C22×C4—2441no—
408C22×C4—2442no—
409C22×C4—2444no—
410C22×C4D4×D6111yescharacteristic
411C22×C4D4×D6111yescharacteristic
412C22×C4C23×D6111yescharacteristic
413M16—12216no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—614no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8#1175822214yes—
420C4⋊C4—21no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—612no—
425C4⋊C4—1224no—
426C4⋊C4—1224no—
427C4⋊C4—1224no—
428C4⋊C4—1221no—
429C4⋊C4—1222no—
430C4⋊C4—1222no—
431C4⋊C4C2×C4×D6112yes—
432C22⋊C4—1224no—
433C22⋊C4—1224no—
434C22⋊C4—1224no—
435C22⋊C4—1224no—
436C22⋊C4—1224no—
437C22⋊C4—2444no—
438C42—24no—
439C42—42no—
440C42—612no—
441C42—612no—
442C42—614no—
443C42—614no—
444C42—2442no—
445C42C2×C4×D6112yes—
446C42D4×D6112yes—
447C23—612no—
448C23—1221no—
449C23—1222no—
450C23—1221no—
451C23—1222no—
452C23—1222no—
453C23—2441no—
454C23—2442no—
455C23—2441no—
456C23—2442no—
457C23C2×D4×D6111yescharacteristic
458Q8—22no—
459Q8—22no—
460Q8—42no—
461D4—1222no—
462D4—1222no—
463D4—2448no—
464D4—2442no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—612no—
477C2×C4—612no—
478C2×C4—612no—
479C2×C4—1221no—
480C2×C4—1222no—
481C2×C4—1221no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—1224no—
486C2×C4—1222no—
487C2×C4—2444no—
488C2×C4—2444no—
489C2×C4—2448no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4C22⋊C4×D6112yes—
495C2×C4C22⋊C4×D6112yes—
496C2×C4C22×C4×D6111yescharacteristic
497C2×C4C2×D4×D6111yescharacteristic
498C2×C4#41×D6112yes—
499C2×C4C2×#1175822212yes—
500C8—24no—
501C8—1224no—
502C22—1222no—
503C22—1222no—
504C22—1222no—
505C22—4882no—
506C22—4882no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22—2442no—
511C22—2442no—
512C22C2×C22⋊C4×D6111yescharacteristic
513C22#1684030912yes—
514C22#1684055811yescharacteristic
515C22#1684056112yes—
516C22#1684056111yescharacteristic
517C4—22no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—82no—
522C4—1222no—
523C4—1222no—
524C4—1222no—
525C4—4882no—
526C4—2442no—
527C4#1684023011yescharacteristic
528C4C4×#1175822211yescharacteristic
529C4#1684027112yes—
530C2—4882no—
531C2—2442no—
532C2#2144830511yescharacteristic
533C2#2145433511yescharacteristic
534C2#2145433711yescharacteristic
535C2#2145654412yes—
536C2#2145655112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1763C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1763D61
hypocenterC61#21411
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. C2×#16839714C2semidirect
3. #21448215C2semidirect
4. #21446782C2semidirect
5. #21446627C2semidirect
7. #21446249C2semidirect
8. #21445649C2semidirect
9. #21445647C2semidirect
10. #21445645C2semidirect
11. C61×#1763C2semidirect
14. #16840299C4semidirect
270. #1763D61semidirect
314. C61#21411semidirect

Maximal subgroups

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

subgroupindex
C2×#168402992
C2×#168397142
#214482152
#214467822
#214466272
#214462562
#214462492
#214456492
#214456472
#214456452
C61×#17632
#2141161