Finite Group Databasebeta

← #29685239

Subgroups — #29685239

Order 15616.

1154 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×#16840316C211yesmaximal, characteristic
2C2×#97×D61C211yesmaximal, characteristic
3#21448215C211yesmaximal, characteristic
4#21446790C212yesmaximal, complemented
5#21446637C212yesmaximal, complemented
6#21446241C212yesmaximal
7#21446239C212yesmaximal
8#21445652C211yesmaximal, characteristic, complemented
9#21445650C211yesmaximal, characteristic, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C61×#1763C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13#16840316—24no—
14#16840316C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519775C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519755—21no—
20C2×#14519753C2211yescharacteristic
21C2×C4×Dic122—21no—
22C2×C4×Dic122C2211yescharacteristic
23C2×#14519734—21no—
24C2×C42×D61C2211yescharacteristic
25C2×#14519702C2212yes—
26C2×#14519701C2212yes—
27C2×#14519700C2212yes—
28C2×#14519688C2211yescharacteristic
29C2×#14519687C2212yes—
30#97×D61—24no—
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×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52C2×C8×D61—24no—
53C2×C8×D61—24no—
54C2×C8×D61—24no—
55#14519776—22no—
56#14519776C2×C414yes—
57#14519775—22no—
58#14519775C2×C414yes—
59#14519767—28no—
60#14519757—22no—
61#14519757C2×C414yes—
62#14519755—28no—
63#14519753—22no—
64#14519753C2×C414yes—
65C4×Dic122—22no—
66C4×Dic122—28no—
67C4×Dic122C2×C414yes—
68#14519734—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61C2×C414yes—
86C42×D61D414yes—
87#14519702—24no—
88#14519701—24no—
89#14519700—24no—
90#14519688—22no—
91#14519687—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×D61C2×D411yescharacteristic
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61C22×C412yes—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61—22no—
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126C2×C4×D61—24no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—41no—
130C2×C4×D61—41no—
131C2×C4×D61C22⋊C414yes—
132C2×C4×D61C22⋊C414yes—
133C2×C4×D61C22×C412yes—
134C2×C4×D61C2×D412yes—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—21no—
138C2×Dic122—24no—
139C2×Dic122—42no—
140C2×Dic122—42no—
141C2×Dic122C22×C412yes—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61#4114yes—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C22×C412yes—
150C42×C61—24no—
151C42×C61—42no—
152C42×C61C22×C412yes—
153C42×C61C2×D412yes—
154#11758198—24no—
155#11758198—24no—
156#11758198—44no—
157C4⋊Dic61—21no—
158C4⋊Dic61—24no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61C22×C412yes—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165#11758196—24no—
166#11758196—24no—
167C4×Dic61—24no—
168C4×Dic61—42no—
169C4×Dic61C22⋊C414yes—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22×C412yes—
172C4×Dic61C2×D412yes—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418#4114yes—
176C8×D61—216no—
177C23×C61C22×D411yescharacteristic
178C22×D61—22no—
179C22×D61—21no—
180C22×D61—21no—
181C22×D61C2×C22⋊C412yes—
182C22×D61#13714yes—
183Q8×C61—22no—
184Q8×C61—22no—
185Q8×C61—42no—
186C2×C4×C61—22no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61C2×C22⋊C412yes—
198C2×C4×C61C2×C22⋊C412yes—
199C2×C4×C61C23×C411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4112yes—
202C2×C4×C61C2×#4112yes—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61C2×C22⋊C412yes—
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218C4×D61—28no—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—24no—
222C4×D61—44no—
223Dic122—22no—
224Dic122—22no—
225Dic122—48no—
226Dic122—42no—
227C8×C61—24no—
228#92418—24no—
229#21407—611nomaximal, Sylow
230C22×C61C22×C22⋊C411yescharacteristic
231C22×C61#53212yes—
232C22×C61C2×#13712yes—
233C22×C61#57511yescharacteristic
234C22×C61#57511yescharacteristic
235C2×D61—22no—
236C2×D61—24no—
237C2×D61—24no—
238C2×D61—22no—
239C2×D61#41914yes—
240C4×C61—22no—
241C4×C61—22no—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61—82no—
245C4×C61C4×#4111yescharacteristic
246C4×C61C4×#4111yescharacteristic
247C4×C61#52512yes—
248Dic61—22no—
249Dic61—22no—
250Dic61—22no—
251Dic61—42no—
252Dic61—42no—
253Dic61—82no—
254C2×#533—611no—
255C22×#97—611no—
256#2283—611no—
257#1993—612no—
258#1960—612no—
259#1892—612no—
260#1892—612no—
261#1763—611no—
262#1763—611no—
263#1763—611no—
264#1763D6111yescharacteristic, complemented
265C2×C61#229711yescharacteristic
266C2×C61C2×#41911yescharacteristic
267C2×C61#289611yescharacteristic
268C2×C61#318112yes—
269C2×C61#318112yes—
270D61—24no—
271C23×C8—612no—
272#533—618no—
273#533—1224no—
274C2×C4⋊Q8—611no—
275C2×C4⋊Q8—611no—
276C2×#125—1221no—
277C2×C22⋊Q8—611no—
278C2×C22⋊Q8—611no—
279C2×C22⋊Q8—1221no—
280C2×C4×Q8—611no—
281C2×C4×Q8—1221no—
282C2×C4×Q8C2×D6111yescharacteristic
283C2×#117—1221no—
284C22×C42—611no—
285C2×#103—612no—
286C2×#103—612no—
287C2×#103—612no—
288C2×#103C2×D6112yes—
289C2×#97—612no—
290C2×#97—611no—
291C2×#97—1224no—
292C2×#97C2×D6111yescharacteristic
293#383—1221no—
294#382—612no—
295#381—1221no—
296#380—612no—
297C4×C4⋊C4—611no—
298C4×C4⋊C4C2×D6111yescharacteristic
299C4×C22⋊C4—1221no—
300#376—1221no—
301#340—612no—
302#340—612no—
303#340—612no—
304#340C2×D6112yes—
305C61#2140711yescharacteristic, complemented, Sylow
306C22×Q8—611no—
307C22×Q8—1221no—
308C22×Q8C22×D6111yescharacteristic
309C23×C4—611no—
310C23×C4—611no—
311C23×C4—611no—
312C22×C8—612no—
313C22×C8—1224no—
314C22×C8—1224no—
315C22×C8—1224no—
316C22×C8C22×D6112yes—
317C4⋊Q8—614no—
318C4⋊Q8—614no—
319C4⋊Q8—1222no—
320C4⋊Q8—1222no—
321#125—1228no—
322C22⋊Q8—614no—
323C22⋊Q8—614no—
324C22⋊Q8—1222no—
325C22⋊Q8—1228no—
326C22⋊Q8—1222no—
327C4×Q8—22no—
328C4×Q8—614no—
329C4×Q8—1228no—
330C4×Q8—1222no—
331C4×Q8C4×D6114yes—
332#117—1228no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—612no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4C22×D6112yes—
343C2×C4⋊C4C22×D6111yescharacteristic
344C2×C22⋊C4—1221no—
345C2×C22⋊C4—1221no—
346C2×C22⋊C4—1222no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—611no—
352C2×C42—614no—
353C2×C42—614no—
354C2×C42—1221no—
355C2×C42—1221no—
356C2×C42C22×D6111yescharacteristic
357#103—24no—
358#103—1224no—
359#103—1224no—
360#103—1224no—
361#97—22no—
362#97—1224no—
363#97—1222no—
364#95—21no—
365#95—1222no—
366#95—1222no—
367#95—1222no—
368#95—1221no—
369C24—611no—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—612no—
374C2×Q8—1224no—
375C2×Q8—1221no—
376C2×Q8—1222no—
377C2×Q8—1222no—
378C2×Q8—2442no—
379C2×Q8—2442no—
380C2×Q8C2×C4×D6112yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—612no—
388C22×C4—612no—
389C22×C4—614no—
390C22×C4—614no—
391C22×C4—1224no—
392C22×C4—1222no—
393C22×C4—1222no—
394C22×C4—1221no—
395C22×C4—1221no—
396C22×C4—1222no—
397C22×C4—2441no—
398C22×C4—2441no—
399C22×C4D4×D6111yescharacteristic
400C22×C4D4×D6111yescharacteristic
401C22×C4C23×D6111yescharacteristic
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—614no—
405C2×C8—12216no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8#1175822214yes—
409C4⋊C4—21no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—612no—
414C4⋊C4—1224no—
415C4⋊C4—1224no—
416C4⋊C4—1224no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—1221no—
420C4⋊C4—1224no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4C2×C4×D6112yes—
424C22⋊C4—1224no—
425C22⋊C4—1224no—
426C22⋊C4—2444no—
427C42—24no—
428C42—42no—
429C42—612no—
430C42—612no—
431C42—614no—
432C42—614no—
433C42—1224no—
434C42—2442no—
435C42C2×C4×D6112yes—
436C42D4×D6112yes—
437C23—612no—
438C23—614no—
439C23—1221no—
440C23—1221no—
441C23—1222no—
442C23C2×D4×D6111yescharacteristic
443Q8—22no—
444Q8—22no—
445Q8—42no—
446Q8—1222no—
447Q8—1222no—
448Q8—2448no—
449Q8—2442no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—612no—
462C2×C4—612no—
463C2×C4—612no—
464C2×C4—1228no—
465C2×C4—1222no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1222no—
471C2×C4—1224no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—2444no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2442no—
481C2×C4C22⋊C4×D6112yes—
482C2×C4C22⋊C4×D6112yes—
483C2×C4C22×C4×D6111yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4#41×D6112yes—
486C2×C4C2×#1175822212yes—
487C8—24no—
488C8—1224no—
489C22—614no—
490C22—1222no—
491C22—1224no—
492C22—1224no—
493C22—1222no—
494C22C2×C22⋊C4×D6111yescharacteristic
495C22#1684030912yes—
496C22#137×D6112yes—
497C22#1684055811yescharacteristic
498C22#1684056111yescharacteristic
499C4—22no—
500C4—22no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—1222no—
505C4—1222no—
506C4—1222no—
507C4—4882no—
508C4—2442no—
509C4—2442no—
510C4#1684023011yescharacteristic
511C4C4×#1175822211yescharacteristic
512C4#1684027112yes—
513C2—1224no—
514C2#2144830511yescharacteristic
515C2#419×D6111yescharacteristic
516C2#2145433511yescharacteristic
517C2#2145652412yes—
518C2#2145652612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1763C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1763D61
hypocenterC61#21407
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
4. #21446790C2semidirect
5. #21446637C2semidirect
8. #21445652C2semidirect
9. #21445650C2semidirect
10. #21445649C2semidirect
11. C61×#1763C2semidirect
14. #16840316C4semidirect
264. #1763D61semidirect
305. C61#21407semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×#97×D612
#214482152
#214467902
#214466372
#214462412
#214462392
#214456522
#214456502
#214456492
C61×#17632
#2140761