Finite Group Databasebeta

← #29733328

Subgroups — #29733328

Order 15616.

1162 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840124C211yesmaximal, characteristic, complemented
2C2×#16839984C211yesmaximal, characteristic, complemented
3#21450758C211yesmaximal, characteristic
4#21446827C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446768C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446141C212yesmaximal, complemented
9#21446131C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#556C211yesmaximal, characteristic, complemented
12C22×#11758202C2211yescharacteristic
13C22×#11758201C2211yescharacteristic
14C2×#14519765C2211yescharacteristic
15C2×#14519756C2212yes—
16C2×#14519751C2212yes—
17#16840124—24no—
18#16839984—24no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519701C2212yes—
21C2×#14519700C2212yes—
22C2×#14519700C2211yescharacteristic
23C2×#14519697C2212yes—
24C2×#14519697C2211yescharacteristic
25C2×C22⋊#92418C2212yes—
26C2×C22⋊#92418C2212yes—
27C2×#14519688C2211yescharacteristic
28#16839697—21no—
29#16839670—21no—
30#16839662—22no—
31C4×C4⋊Dic61C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C61×#556—24no—
36C61×#556—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#126×C61—22no—
39C2×#124×C61—21no—
40C2×C22⋊Q8×C61C2212yes—
41C2×C4×Q8×C61—21no—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×#117×C61C2212yes—
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—22no—
48C2×#11758202—22no—
49C2×#11758202—42no—
50C2×#11758201—22no—
51C2×#11758201—42no—
52C22×#92418C2312yes—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519765—22no—
56#14519765—22no—
57#14519756—24no—
58#14519756—42no—
59#14519751—24no—
60#14519751—42no—
61C2×C4⋊Dic61C2312yes—
62C2×#11758196—22no—
63C2×C22⋊Dic61—22no—
64C2×#11758198—22no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×C4×Dic61—21no—
68C4⋊#92418—22no—
69#14519701—24no—
70#14519700—22no—
71#14519700—24no—
72#14519697—22no—
73#14519697—24no—
74C22⋊#92418—42no—
75C22⋊#92418—42no—
76#14519688—22no—
77#14519684—21no—
78C22×Q8×C61—21no—
79C22×Q8×C61C2311yescharacteristic
80C22×D4×C61D411yescharacteristic
81C23×C4×C61C2312yes—
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61D414yescomplemented
84#126×C61—216no—
85#124×C61—28no—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—24no—
88C4×Q8×C61—28no—
89C4×Q8×C61—22no—
90C4×Q8×C61D414yescomplemented
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93#117×C61—24no—
94#117×C61—24no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61D412yes—
104C2×C42×C61—22no—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C24×C61C2×D412yes—
108C23×D61—21no—
109C2×Q8×C61—24no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—44no—
114C2×Q8×C61C2×D412yes—
115C2×D4×C61—22no—
116C2×D4×C61—42no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—82no—
133C22×Dic61—22no—
134C22×Dic61—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—22no—
138C2×D244—22no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C4⋊C4×C61—28no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C2×D412yes—
150C4⋊C4×C61C2×D414yes—
151C22⋊C4×C61—28no—
152C22⋊C4×C61—28no—
153C22⋊C4×C61—24no—
154C42×C61—28no—
155C42×C61—24no—
156C42×C61C2×D412yes—
157C42×C61C2×D412yes—
158C22⋊Dic61—44no—
159#11758202—44no—
160#11758201—44no—
161#11758198—44no—
162#11758198—42no—
163#11758198—42no—
164C4⋊Dic61—22no—
165C4⋊Dic61#4114yes—
166#11758196—44no—
167C4×Dic61—81no—
168C2×#92418—22no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61C22×D411yescharacteristic
183C22×D61—41no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—41no—
187C22×D61—41no—
188Q8×C61—22no—
189Q8×C61—22no—
190Q8×C61—48no—
191Q8×C61—42no—
192D4×C61—44no—
193C2×C4×C61—22no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—42no—
203C2×C4×C61—44no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61#12012yes—
211C2×C4×C61#12012yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4112yes—
215C2×C4×C61H(5,𝔽2)12yes—
216C22⋊D61—88no—
217C2×Dic61—24no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222D244—41no—
223D244—41no—
224D244—42no—
225Dic122—41no—
226Dic122—41no—
227Dic122—42no—
228#92418—41no—
229#92418—41no—
230#92418—42no—
231#25580—611nomaximal, Sylow
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61C2×#12011yescharacteristic
240C22×C61#54612yes—
241C22×C61#57512yes—
242C22×C61#57711yescharacteristic
243C22×C61#57811yescharacteristic
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—82no—
248C2×D61—82no—
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—44no—
254C4×C61—42no—
255C4×C61#53211yescharacteristic
256C4×C61#53511yescharacteristic
257C4×C61#55012yes—
258Dic61—44no—
259Dic61—81no—
260C2×#556D6111yescharacteristic, complemented
261C2×#489—611no—
262C2×#464—611no—
263#2540—611no—
264#2001—612no—
265#2000—612no—
266#1987—611no—
267#1955—611no—
268#1875—612no—
269#1873—612no—
270#1763—611no—
271C2×C61—82no—
272C2×C61#243111yescharacteristic
273C2×C61#300511yescharacteristic
274C2×C61#301311yescharacteristic
275C2×C61#331112yes—
276C2×C61#331312yes—
277D61—161no—
278C22×Q16—611no—
279C22×SD16—611no—
280#556—24no—
281#556—24no—
282C2×C4⋊Q8C2×D6111yescharacteristic
283C2×#126—22no—
284C2×#124—21no—
285C2×#124—611no—
286C2×C22⋊Q8—612no—
287C2×C22⋊Q8C2×D6112yes—
288C2×#121—612no—
289C2×C4×Q8—21no—
290C2×C4×Q8C2×D6111yescharacteristic
291C2×C4×D4C2×D6111yescharacteristic
292C2×#117C2×D6112yes—
293#489—1224no—
294#464—1224no—
295C2×C4⋊C8—611no—
296C2×#103—612no—
297C2×#103—612no—
298C2×#103—611no—
299C2×#102—612no—
300C2×#102—611no—
301C2×C22⋊C8—612no—
302C2×C22⋊C8—612no—
303C2×#97—611no—
304#399—1221no—
305#392—1221no—
306#390—1222no—
307C4×C4⋊C4—611no—
308#340—612no—
309#340—612no—
310#340—612no—
311C61#2558011yescharacteristic, complemented, Sylow
312C22×Q8—21no—
313C22×Q8—611no—
314C22×Q8C22×D6111yescharacteristic
315C22×D4—611no—
316C22×D4—1222no—
317C22×D4C22⋊D6111yescharacteristic
318C23×C4C22×D6112yes—
319C2×Q16—1222no—
320C2×Q16—2442no—
321C2×SD16—1222no—
322C2×SD16—2442no—
323C22×C8—611no—
324C22×C8—611no—
325C22×C8—612no—
326C4⋊Q8—22no—
327C4⋊Q8C22⋊D6114yescomplemented
328#126—216no—
329#124—28no—
330#124—1222no—
331#124—1222no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—1224no—
335C22⋊Q8—2442no—
336#121—1224no—
337#121—2442no—
338C4×Q8—28no—
339C4×Q8—22no—
340C4×Q8C22⋊D6114yescomplemented
341C4×D4—22no—
342C4×D4—22no—
343#117—24no—
344#117—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4C22⋊D6111yescharacteristic
351C2×C4⋊C4C22×D6111yescharacteristic
352C2×C4⋊C4C22×D6112yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—1222no—
356C2×C22⋊C4—1222no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4C22⋊D6112yes—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—1221no—
363C2×C42C22×D6111yescharacteristic
364C4⋊C8—1222no—
365#103—1224no—
366#103—1222no—
367#103—1224no—
368#102—1222no—
369#102—1224no—
370C22⋊C8—2442no—
371C22⋊C8—2442no—
372#97—1222no—
373#95—1221no—
374C24—1221no—
375C24C2×C22⋊D6112yes—
376C2×Q8—21no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—44no—
381C2×Q8—1221no—
382C2×Q8—1221no—
383C2×Q8—1222no—
384C2×Q8—1222no—
385C2×Q8C2×C22⋊D6112yes—
386C2×D4—22no—
387C2×D4—42no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—1222no—
391C2×D4—1222no—
392C2×D4—4882no—
393C2×D4—2444no—
394C2×D4—2444no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—1222no—
405C22×C4—1221no—
406C22×C4C2×C22⋊D6112yes—
407C22×C4C2×C22⋊D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410Q16—2444no—
411SD16—2444no—
412C2×C8—1222no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8—1222no—
416C2×C8—1222no—
417C2×C8—1222no—
418C2×C8—1222no—
419C4⋊C4—28no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—614no—
426C4⋊C4—1222no—
427C4⋊C4—2444no—
428C4⋊C4C2×C22⋊D6112yes—
429C4⋊C4C2×C22⋊D6114yes—
430C22⋊C4—28no—
431C22⋊C4—28no—
432C22⋊C4—24no—
433C22⋊C4—2444no—
434C22⋊C4—2444no—
435C22⋊C4—2442no—
436C22⋊C4—2442no—
437C42—28no—
438C42—24no—
439C42—4881no—
440C42C2×C22⋊D6112yes—
441C42C2×C22⋊D6112yes—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—2441no—
450C23—2442no—
451C23—2441no—
452C23—2441no—
453C23—2442no—
454C23C22×C22⋊D6111yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—48no—
458Q8—42no—
459Q8—2441no—
460Q8—2441no—
461Q8—2442no—
462D4—44no—
463D4—4888no—
464D4—2441no—
465D4—2441no—
466D4—2442no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—1224no—
485C2×C4—2441no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4#1451974612yes—
490C2×C4#1451974612yes—
491C2×C4C22×C22⋊D6111yescharacteristic
492C2×C4C22×C22⋊D6111yescharacteristic
493C2×C4#41×D6112yes—
494C2×C4#1451982312yes—
495C8—2441no—
496C8—2441no—
497C8—2442no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—4881no—
506C22—4882no—
507C22—4881no—
508C22—4882no—
509C22—4881no—
510C22C2×#1451974611yescharacteristic
511C22#1684040012yes—
512C22#1684056012yes—
513C22#1684057011yescharacteristic
514C22#1684057711yescharacteristic
515C4—22no—
516C4—22no—
517C4—42no—
518C4—44no—
519C4—42no—
520C4—42no—
521C4—4881no—
522C4—2444no—
523C4#1684030811yescharacteristic
524C4#1684032611yescharacteristic
525C4#1684043112yes—
526C2—82no—
527C2—9761no—
528C2#2144943411yescharacteristic
529C2#2145506111yescharacteristic
530C2#2145513411yescharacteristic
531C2#2145784912yes—
532C2#2145787912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#556C2
socleC23×C61C22×D4
prime radical#21450758C2
hypercenterC2×#556D61
hypocenterC61#25580
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#16840124C2semidirect
2. C2×#16839984C2semidirect
4. #21446827C2semidirect
5. #21446820C2semidirect
6. #21446768C2semidirect
7. #21446618C2semidirect
8. #21446141C2semidirect
9. #21446131C2semidirect
10. #21445649C2semidirect
11. C2×C61×#556C2semidirect
83. C4⋊Q8×C61D4semidirect
90. C4×Q8×C61D4semidirect
260. C2×#556D61semidirect
311. C61#25580semidirect
327. C4⋊Q8C22⋊D61semidirect
340. C4×Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401242
C2×#168399842
#214507582
#214468272
#214468202
#214467682
#214466182
#214461412
#214461312
#214456492
C2×C61×#5562
#2558061