Finite Group Databasebeta

← #29685058

Subgroups — #29685058

Order 15616.

1258 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840314C211yesmaximal, characteristic
2C2×#16839812C211yesmaximal, characteristic
3C2×C4⋊C4×Dic61C211yesmaximal, characteristic
4#21446765C212yesmaximal, complemented
5#21446740C212yesmaximal, complemented
6#21446127C211yesmaximal, characteristic
7#21446126C211yesmaximal, characteristic
8#21446124C211yesmaximal, characteristic
9#103×Dic61C212yesmaximal, complemented
10#21445638C212yesmaximal, complemented
11C61×#1872C211yesmaximal, characteristic
12#16840314—24no—
13#16840314C418yescomplemented
14C2×#14519776C2211yescharacteristic
15C2×#14519775C2211yescharacteristic
16C2×#14519767—21no—
17C2×#14519756C2211yescharacteristic
18C2×#14519754—21no—
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519742—21no—
21C2×#14519742C2211yescharacteristic
22C2×#14519733—21no—
23C22×C4⋊Dic61C2211yescharacteristic
24C22×C4×Dic61—21no—
25C22×C4×Dic61C2211yescharacteristic
26#16839812—24no—
27C2×#14519706C2212yes—
28C2×#14519702C2212yes—
29C2×#14519701C2212yes—
30C2×#14519700C2212yes—
31C2×C22⋊#92418C2211yescharacteristic
32C2×C8×Dic61C2212yes—
33C4⋊C4×Dic61—21no—
34C4⋊C4×Dic61—21no—
35C4⋊C4×Dic61C2212yes—
36C4⋊C4×Dic61C2212yes—
37C2×#14519684—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C23×Dic61—21no—
49C23×Dic61D411yescharacteristic
50C23×Dic61D411yescharacteristic
51C22×#92418C2312yes—
52#14519776—22no—
53#14519776C2×C414yescomplemented
54#14519775—22no—
55#14519775C2×C414yescomplemented
56#14519767—28no—
57#14519756—22no—
58#14519756C2×C414yes—
59#14519754—28no—
60Q8×Dic61—22no—
61Q8×Dic61C2×C414yescomplemented
62#14519742—28no—
63#14519742—22no—
64#14519742C2×C414yescomplemented
65#14519733—28no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—22no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61—44no—
82C2×C4×Dic61C2×C414yes—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D414yes—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519706—24no—
89#14519702—24no—
90#14519701—24no—
91#14519700—24no—
92C22⋊#92418—22no—
93C8×Dic61—24no—
94#14519684—22no—
95#14519684—22no—
96C22×Q8×C61C2311yescharacteristic
97C23×C4×C61—21no—
98C23×C4×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61C2×C414yes—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—24no—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2312yes—
107C2×C22⋊C4×C61—21no—
108#103×C61—24no—
109C22⋊C8×C61—22no—
110C24×C61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—24no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—24no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—41no—
132C22×Dic61—41no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61C22⋊C414yes—
136C22×Dic61C22⋊C414yes—
137C22×Dic61SD1614yes—
138C22×Dic61Q1614yes—
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—21no—
146C2×Dic122—24no—
147C2×Dic122—42no—
148C2×Dic122—42no—
149C2×Dic122C22×C412yes—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61C22×C412yes—
158C4⋊C4×C61#4114yes—
159C22⋊C4×C61—24no—
160C22⋊Dic61—24no—
161C4⋊Dic61—24no—
162C4⋊Dic61—21no—
163C4⋊Dic61—22no—
164C4⋊Dic61—24no—
165C4⋊Dic61C22×C412yes—
166C4⋊Dic61#4114yes—
167#11758196—24no—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—24no—
172C4×Dic61—24no—
173C4×Dic61—24no—
174C4×Dic61—44no—
175C4×Dic61—44no—
176C4×Dic61C22⋊C414yes—
177C4×Dic61C22⋊C414yes—
178C4×Dic61C22×C412yes—
179C4×Dic61C22×C412yes—
180C4×Dic61C2×D412yes—
181C4×Dic61C2×D412yes—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61C2×C22⋊C412yes—
190C23×C61C2×SD1612yes—
191C23×C61C2×Q1612yes—
192C23×C61C22×D411yescharacteristic
193Q8×C61—24no—
194Q8×C61—42no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—48no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61C2×C22⋊C412yes—
210C2×C4×C61C23×C411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C2×#4112yes—
213C2×C4×C61C2×#4112yes—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—24no—
221C2×Dic61—44no—
222C2×Dic61—48no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—42no—
229C2×Dic61#103116yescomplemented
230C2×Dic61C2×C22⋊C412yes—
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C2×C22⋊C412yes—
233C2×Dic61C2×C22⋊C412yes—
234C2×Dic61C2×SD1612yes—
235C2×Dic61C2×SD1612yes—
236C2×Dic61C2×Q1612yes—
237C2×Dic61C2×Q1612yes—
238Dic122—24no—
239Dic122—48no—
240Dic122—42no—
241C8×C61—42no—
242#92418—42no—
243#21385—611nomaximal, Sylow
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61C2×#10314yes—
251C22×C61C22×C22⋊C411yescharacteristic
252C22×C61#52511yescharacteristic
253C22×C61#52511yescharacteristic
254C22×C61C22×SD1611yescharacteristic
255C22×C61C22×Q1611yescharacteristic
256C22×C61#57511yescharacteristic
257C22×C61#57511yescharacteristic
258C4×C61—22no—
259C4×C61—44no—
260C4×C61—42no—
261C4×C61—42no—
262C4×C61C4×#4112yes—
263C4×C61#53212yes—
264Dic61—22no—
265Dic61—22no—
266Dic61—44no—
267Dic61—42no—
268Dic61—42no—
269Dic61C2×#10314yes—
270Dic61C2×#10314yes—
271C2×#533—611no—
272C2×#431—611no—
273C2×C4×C4⋊C4—611no—
274#1987—612no—
275#1981—612no—
276#1872—611no—
277#1872—611no—
278#1872—611no—
279#1872D6111yescharacteristic
280C4×#103—612no—
281C4×#103—612no—
282C2×C61—24no—
283C2×C61#229711yescharacteristic
284C2×C61C22×#10311yescharacteristic
285C2×C61#289611yescharacteristic
286C2×C61#310111yescharacteristic
287C2×C61#310111yescharacteristic
288C2×C61#310411yescharacteristic
289C2×C61#310411yescharacteristic
290#533—618no—
291#533—1224no—
292C2×C4⋊Q8—611no—
293C2×C4⋊Q8—611no—
294C2×#125—1221no—
295C2×C22⋊Q8—611no—
296C2×C22⋊Q8—1221no—
297C2×C22⋊Q8C2×D6111yescharacteristic
298C2×C4×Q8—611no—
299C2×C4×Q8—611no—
300C2×C4×Q8—1221no—
301C2×#117—1221no—
302C22×C4⋊C4—611no—
303C22×C4⋊C4C2×D6111yescharacteristic
304C22×C42—611no—
305C22×C42—1221no—
306#431—1224no—
307C2×C4⋊C8—612no—
308C2×#103—612no—
309C2×#103—612no—
310C2×#103—612no—
311C2×#103C2×D6112yes—
312C2×C22⋊C8—611no—
313C2×C22⋊C8C2×D6111yescharacteristic
314C2×C4×C8—612no—
315C4×C4⋊C4—612no—
316C4×C4⋊C4—612no—
317C4×C4⋊C4—1221no—
318C4×C4⋊C4—1221no—
319C2×#95—1221no—
320#340—612no—
321#340—612no—
322#340—612no—
323#340C2×D6112yes—
324C61#2138511yescharacteristic, complemented, Sylow
325C22×Q8—611no—
326C22×Q8—1221no—
327C22×Q8C22×D6111yescharacteristic
328C23×C4—21no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—1221no—
332C23×C4C22×D6111yescharacteristic
333C22×C8—612no—
334C22×C8C22×D6112yes—
335C4⋊Q8—614no—
336C4⋊Q8—614no—
337C4⋊Q8—1222no—
338C4⋊Q8—1222no—
339#125—1228no—
340C22⋊Q8—22no—
341C22⋊Q8—614no—
342C22⋊Q8—1228no—
343C22⋊Q8—1222no—
344C22⋊Q8C4×D6114yes—
345C4×Q8—614no—
346C4×Q8—614no—
347C4×Q8—1222no—
348C4×Q8—1228no—
349C4×Q8—1222no—
350#117—1228no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—1224no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C4⋊C4C22×D6112yes—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1221no—
368C2×C42—614no—
369C2×C42—611no—
370C2×C42—611no—
371C2×C42—614no—
372C2×C42—611no—
373C2×C42—611no—
374C2×C42—1221no—
375C2×C42—1221no—
376C2×C42—1222no—
377C2×C42—1222no—
378C2×C42—2444no—
379C4⋊C8—1224no—
380#103—24no—
381#103—1224no—
382#103—1224no—
383#103—1224no—
384C22⋊C8—22no—
385C22⋊C8—1222no—
386C4×C8—1224no—
387#95—1222no—
388#95—1222no—
389C24D4×D6111yescharacteristic
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—612no—
394C2×Q8—1224no—
395C2×Q8—1221no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×Q8—2442no—
399C2×Q8—2442no—
400C2×Q8C2×C4×D6112yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—614no—
412C22×C4—614no—
413C22×C4—614no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—614no—
417C22×C4—1224no—
418C22×C4—1222no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—2441no—
422C22×C4—2441no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4C2×C4×D6112yes—
426C22×C4D4×D6112yes—
427C22×C4D4×D6111yescharacteristic
428C22×C4C23×D6111yescharacteristic
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—1222no—
433C2×C8—1222no—
434C2×C8—1222no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—612no—
440C4⋊C4—614no—
441C4⋊C4—1224no—
442C4⋊C4—1224no—
443C4⋊C4—1224no—
444C4⋊C4—1224no—
445C4⋊C4—1224no—
446C4⋊C4—1224no—
447C4⋊C4—1221no—
448C4⋊C4—1222no—
449C4⋊C4—1224no—
450C4⋊C4C2×C4×D6112yes—
451C4⋊C4#1175822414yes—
452C22⋊C4—24no—
453C22⋊C4—1224no—
454C42—612no—
455C42—612no—
456C42—612no—
457C42—612no—
458C42—614no—
459C42—614no—
460C42—1224no—
461C42—1224no—
462C42—2444no—
463C42—2444no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23C22⋊C4×D6112yes—
469C23SD16×D6112yes—
470C23Q16×D6112yes—
471C23C2×D4×D6111yescharacteristic
472Q8—24no—
473Q8—42no—
474Q8—1224no—
475Q8—2448no—
476Q8—2442no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—48no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—612no—
492C2×C4—612no—
493C2×C4—6116no—
494C2×C4—612no—
495C2×C4—612no—
496C2×C4—612no—
497C2×C4—612no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1224no—
507C2×C4—2444no—
508C2×C4—2448no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4—2441no—
515C2×C4C22⋊C4×D6112yes—
516C2×C4C22×C4×D6111yescharacteristic
517C2×C4C2×D4×D6111yescharacteristic
518C2×C4#41×D6112yes—
519C2×C4C2×#1175822412yes—
520C8—42no—
521C8—2442no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22#103×D6114yes—
529C22C2×C22⋊C4×D6111yescharacteristic
530C22#1684026811yescharacteristic
531C22#1684027011yescharacteristic
532C22C2×SD16×D6111yescharacteristic
533C22C2×Q16×D6111yescharacteristic
534C22#1684055811yescharacteristic
535C22#1684056111yescharacteristic
536C4—22no—
537C4—42no—
538C4—44no—
539C4—42no—
540C4—614no—
541C4—614no—
542C4—1222no—
543C4—1222no—
544C4—2442no—
545C4—2444no—
546C4—2442no—
547C4C4×#1175822412yes—
548C4#1684030712yes—
549C2—24no—
550C2#2144830411yescharacteristic
551C2C2×#103×D6111yescharacteristic
552C2#2145433511yescharacteristic
553C2#2145582211yescharacteristic
554C2#2145582411yescharacteristic
555C2#2145585311yescharacteristic
556C2#2145585511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1872D61
hypocenterC61#21385
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #21446765C2semidirect
5. #21446740C2semidirect
9. #103×Dic61C2semidirect
10. #21445638C2semidirect
13. #16840314C4semidirect
53. #14519776C2×C4semidirect
55. #14519775C2×C4semidirect
61. Q8×Dic61C2×C4semidirect
64. #14519742C2×C4semidirect
229. C2×Dic61#103semidirect
324. C61#21385semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
C2×#168398122
C2×C4⋊C4×Dic612
#214467652
#214467402
#214461272
#214461262
#214461242
#103×Dic612
#214456382
C61×#18722
#2138561