Finite Group Databasebeta

← #29689531

Subgroups — #29689531

Order 15616.

1054 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840318C211yesmaximal, characteristic
2C2×#16839817C211yesmaximal, characteristic
3#21448224C211yesmaximal, characteristic
4#21446789C211yesmaximal, characteristic, complemented
5#21446788C211yesmaximal, characteristic, complemented
6#21446787C211yesmaximal, characteristic, complemented
7#21446769C212yesmaximal, complemented
8#21446765C212yesmaximal, complemented
9#21445913C212yesmaximal
10#21445912C212yesmaximal
11C61×#1992C211yesmaximal, characteristic, complemented
12#16840318—24no—
13#16840318C418yescomplemented
14C2×#14519776C2211yescharacteristic
15C2×#14519776C2211yescharacteristic
16C2×#14519769C2211yescharacteristic
17C2×#14519755—21no—
18C2×#14519755—21no—
19C2×#14519742—21no—
20C2×#14519742—21no—
21C2×#14519735—21no—
22C2×#14519735C2211yescharacteristic
23C2×#14519735C2211yescharacteristic
24C2×C42×D61C2211yescharacteristic
25#16839817—24no—
26C2×#14519706C2212yes—
27C2×#14519689C2212yes—
28C2×#14519688C2211yescharacteristic
29#16839632C2211yescharacteristic
30#16839627—21no—
31#16839619—21no—
32#16839619C2212yes—
33C4⋊C4×Dic61—21no—
34C4⋊C4×Dic61C2212yes—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#125×C61C2211yescharacteristic
42C2×#97×C61C2211yescharacteristic
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519776—22no—
53#14519776—22no—
54#14519776C2×C414yes—
55#14519776C2×C414yes—
56#14519769—22no—
57#14519769C2×C414yes—
58#14519755—28no—
59#14519755—28no—
60#14519742—28no—
61#14519742—28no—
62#14519735—22no—
63#14519735—22no—
64#14519735—44no—
65#14519735C2×C414yes—
66#14519735C2×C414yes—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—22no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61C2×C414yes—
86C42×D61D414yes—
87#14519706—24no—
88#14519689—24no—
89#14519688—22no—
90#14519684—22no—
91#14519684—22no—
92C22×C8×C61C2312yes—
93#125×C61—22no—
94#125×C61C2×C414yes—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C42×C61C2311yescharacteristic
103#97×C61—22no—
104C23×D61C2×D411yescharacteristic
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—22no—
114C22×Dic61C2×D411yescharacteristic
115C22×Dic61C2×D411yescharacteristic
116C22×Dic61C2×D411yescharacteristic
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—41no—
122C2×C4×D61—41no—
123C2×C4×D61C22⋊C414yes—
124C2×C4×D61C22⋊C414yes—
125C2×C4×D61C22×C412yes—
126C2×C4×D61C2×D412yes—
127C2×Dic122—24no—
128C2×Dic122—24no—
129C2×Dic122—42no—
130C2×Dic122—42no—
131C2×Dic122—42no—
132C2×Dic122—42no—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—42no—
142C4⋊C4×C61—42no—
143C4⋊C4×C61C22×C412yes—
144C4⋊C4×C61C22×C412yes—
145C42×C61C22×C412yes—
146C42×C61C2×D412yes—
147#11758198—24no—
148#11758198—24no—
149#11758198—44no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—24no—
153C4⋊Dic61—42no—
154C4⋊Dic61—42no—
155C4⋊Dic61C22×C412yes—
156C4⋊Dic61C22×C412yes—
157#11758196—24no—
158#11758196—24no—
159#11758196—24no—
160#11758196—24no—
161C4×Dic61—24no—
162C4×Dic61—24no—
163C4×Dic61—44no—
164C4×Dic61C22⋊C414yes—
165C4×Dic61C22⋊C414yes—
166C4×Dic61C22×C412yes—
167C4×Dic61C2×D412yes—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C23×C61C22×D411yescharacteristic
172C22×D61—21no—
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—21no—
177C22×D61—21no—
178C22×D61C2×C22⋊C412yes—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61C2×C22⋊C412yes—
193C2×C4×C61C2×C22⋊C412yes—
194C2×C4×C61C23×C411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C612−1+412yes—
197C2×C4×C612−1+412yes—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61C2×C22⋊C412yes—
212C2×Dic61C2×C22⋊C412yes—
213C2×Dic61C2×C22⋊C412yes—
214C2×Dic61AGL(1,ℤ/8)12yes—
215C2×Dic61#13712yes—
216C4×D61—24no—
217C4×D61—24no—
218C4×D61—28no—
219C4×D61—44no—
220Dic122—48no—
221Dic122—48no—
222C8×C61—42no—
223#92418—42no—
224#21839—611nomaximal, Sylow
225C22×C61C22×C22⋊C411yescharacteristic
226C22×C61#53611yescharacteristic
227C22×C61#53611yescharacteristic
228C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
229C22×C61C2×#13711yescharacteristic
230C22×C61#57511yescharacteristic
231C22×C61#57511yescharacteristic
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—22no—
235C2×D61—41no—
236C2×D61—41no—
237C2×D61—41no—
238C2×D61—41no—
239C4×C61—22no—
240C4×C61—22no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—82no—
244C4×C61#51912yes—
245C4×C61#53612yes—
246Dic61—22no—
247Dic61—22no—
248Dic61—42no—
249Dic61—42no—
250Dic61—82no—
251Dic61#41714yes—
252C2×#533—611no—
253C2×#432—611no—
254#2285—611no—
255#1992—611no—
256#1992—611no—
257#1992—611no—
258#1992D6111yescharacteristic, complemented
259#1988—612no—
260#1987—612no—
261#1825—612no—
262#1825—612no—
263C2×C61#231011yescharacteristic
264C2×C61C2×#41711yescharacteristic
265C2×C61#289611yescharacteristic
266C2×C61#323511yescharacteristic
267C2×C61#323511yescharacteristic
268C2×C61#323711yescharacteristic
269C2×C61#323711yescharacteristic
270D61—42no—
271#533—618no—
272#533—1224no—
273C2×C4⋊Q8—611no—
274C2×C4⋊Q8—611no—
275C2×#125—611no—
276C2×#125C2×D6111yescharacteristic
277C2×C22⋊Q8—1221no—
278C2×C22⋊Q8—1221no—
279C2×C4×Q8—1221no—
280C2×C4×Q8—1221no—
281C2×#117—611no—
282C2×#117—611no—
283C2×#117—1221no—
284C22×C42—611no—
285#432—1224no—
286C2×C4⋊C8—612no—
287C2×C22⋊C8—612no—
288C2×#97—611no—
289C2×#97C2×D6111yescharacteristic
290#384—611no—
291#384C2×D6111yescharacteristic
292#382—1221no—
293#380—612no—
294#380—1221no—
295C4×C4⋊C4—612no—
296C4×C4⋊C4—1221no—
297#340—612no—
298#340—612no—
299#340—612no—
300#340—612no—
301#340—612no—
302#340—612no—
303#340C2×D6112yes—
304#340C2×D6112yes—
305C61#2183911yescharacteristic, complemented, Sylow
306C22×Q8—1221no—
307C22×Q8—1221no—
308C23×C4—611no—
309C23×C4—611no—
310C23×C4—611no—
311C22×C8—612no—
312C22×C8C22×D6112yes—
313C4⋊Q8—614no—
314C4⋊Q8—614no—
315C4⋊Q8—1222no—
316C4⋊Q8—1222no—
317#125—22no—
318#125—614no—
319#125—1222no—
320#125C4×D6114yes—
321C22⋊Q8—1228no—
322C22⋊Q8—1228no—
323C4×Q8—1228no—
324C4×Q8—1228no—
325#117—614no—
326#117—614no—
327#117—1222no—
328#117—1222no—
329#117—2444no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4C22×D6112yes—
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1222no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—611no—
352C2×C42—614no—
353C2×C42—614no—
354C2×C42—1221no—
355C2×C42—1221no—
356C2×C42—1222no—
357C2×C42C22×D6111yescharacteristic
358C4⋊C8—1224no—
359C22⋊C8—1224no—
360#97—22no—
361#97—1222no—
362#95—1222no—
363#95—1222no—
364C24—611no—
365C2×Q8—1224no—
366C2×Q8—1224no—
367C2×Q8—2442no—
368C2×Q8—2442no—
369C2×Q8—2442no—
370C2×Q8—2442no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—611no—
375C22×C4—611no—
376C22×C4—611no—
377C22×C4—612no—
378C22×C4—612no—
379C22×C4—614no—
380C22×C4—614no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1222no—
388C22×C4—2441no—
389C22×C4—2441no—
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4C23×D6111yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—1222no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—612no—
407C4⋊C4—612no—
408C4⋊C4—1224no—
409C4⋊C4—1224no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—1224no—
413C4⋊C4—1224no—
414C4⋊C4—1224no—
415C4⋊C4—2442no—
416C4⋊C4—2442no—
417C4⋊C4C2×C4×D6112yes—
418C4⋊C4C2×C4×D6112yes—
419C22⋊C4—1224no—
420C22⋊C4—1224no—
421C22⋊C4—2444no—
422C42—612no—
423C42—612no—
424C42—614no—
425C42—614no—
426C42—1224no—
427C42—1224no—
428C42—2444no—
429C42C2×C4×D6112yes—
430C42D4×D6112yes—
431C23—612no—
432C23—1221no—
433C23—1221no—
434C23—1221no—
435C23—1221no—
436C23—1221no—
437C23—1221no—
438C23C2×D4×D6111yescharacteristic
439Q8—2448no—
440Q8—2448no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—612no—
455C2×C4—612no—
456C2×C4—612no—
457C2×C4—612no—
458C2×C4—612no—
459C2×C4—1228no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—2444no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4C22⋊C4×D6112yes—
477C2×C4C22⋊C4×D6112yes—
478C2×C4C22×C4×D6111yescharacteristic
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4#1451982412yes—
481C2×C4#1451982512yes—
482C8—42no—
483C8—2442no—
484C22—1222no—
485C22—1222no—
486C22—1222no—
487C22—2441no—
488C22—2441no—
489C22—2441no—
490C22—2441no—
491C22C2×C22⋊C4×D6111yescharacteristic
492C22#1684033211yescharacteristic
493C22#1684033711yescharacteristic
494C22AGL(1,ℤ/8)×D6111yescharacteristic
495C22#137×D6111yescharacteristic
496C22#1684055811yescharacteristic
497C22#1684056111yescharacteristic
498C4—22no—
499C4—22no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—614no—
504C4—1222no—
505C4—1222no—
506C4—4882no—
507C4—2442no—
508C4—2442no—
509C4#1684023812yes—
510C4#1684033512yes—
511C2—2442no—
512C2#2144840211yescharacteristic
513C2#417×D6111yescharacteristic
514C2#2145433511yescharacteristic
515C2#2145701211yescharacteristic
516C2#2145701811yescharacteristic
517C2#2145702811yescharacteristic
518C2#2145703411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1992C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1992D61
hypocenterC61#21839
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
4. #21446789C2semidirect
5. #21446788C2semidirect
6. #21446787C2semidirect
7. #21446769C2semidirect
8. #21446765C2semidirect
11. C61×#1992C2semidirect
13. #16840318C4semidirect
258. #1992D61semidirect
305. C61#21839semidirect

Maximal subgroups

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

subgroupindex
C2×#168403182
C2×#168398172
#214482242
#214467892
#214467882
#214467872
#214467692
#214467652
#214459132
#214459122
C61×#19922
#2183961