Finite Group Databasebeta

← #29733335

Subgroups — #29733335

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×#16840485C211yesmaximal, characteristic, complemented
2C2×#16840121C211yesmaximal, characteristic, complemented
3#21450759C211yesmaximal, characteristic
4#21446826C212yesmaximal, complemented
5#21446819C212yesmaximal, complemented
6#21446766C211yesmaximal, characteristic, complemented
7#21446616C211yesmaximal, characteristic, complemented
8#21446138C212yesmaximal, complemented
9#21446128C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#464C211yesmaximal, characteristic, complemented
12#16840485—24no—
13#16840485—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519772—22no—
16C2×#14519765—21no—
17C2×#14519753C2212yes—
18C2×#14519753C2212yes—
19C2×#14519748C2212yes—
20C2×C4×Dic122—21no—
21C2×C4×D244C2211yescharacteristic
22C2×#14519735C2212yes—
23#16840121—24no—
24C2×#14519701C2212yes—
25C2×#14519701C2212yes—
26C2×#14519698C2212yes—
27C2×#14519689C2212yes—
28C2×#14519689C2212yes—
29C2×#14519688C2211yescharacteristic
30#16839698—21no—
31#16839671—21no—
32#16839662—22no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37C22×Q16×C61C2211yescharacteristic
38C22×SD16×C61C2211yescharacteristic
39C2×#124×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#464—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2211yescharacteristic
44C2×#102×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C4×D61C2312yes—
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777D414yescomplemented
52#14519772—216no—
53#14519765—28no—
54#14519753—24no—
55#14519753—24no—
56#14519753—24no—
57#14519753—42no—
58#14519748—24no—
59#14519748—42no—
60C4×Dic122—28no—
61C4×D244—22no—
62C4×D244—22no—
63#14519735—24no—
64#14519735—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2312yes—
69C2×#11758196—22no—
70C2×#11758196—22no—
71C2×C22⋊Dic61—22no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198D412yes—
76C2×C4×Dic61—22no—
77C2×C4×Dic61—21no—
78#14519701—24no—
79#14519701—24no—
80#14519698—24no—
81#14519689—42no—
82#14519689—42no—
83#14519688—22no—
84#14519684—21no—
85C22×Q8×C61C2311yescharacteristic
86C22×Q8×C61C2311yescharacteristic
87C22×D4×C61C2311yescharacteristic
88C2×Q16×C61—22no—
89C2×Q16×C61—42no—
90C2×SD16×C61—22no—
91C2×SD16×C61—42no—
92C22×C8×C61C2311yescharacteristic
93C22×C8×C61C2311yescharacteristic
94#124×C61—22no—
95#124×C61—22no—
96C4×Q8×C61—22no—
97C4×Q8×C61D414yescomplemented
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—21no—
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—22no—
105#103×C61—22no—
106#102×C61—22no—
107C24×C61—21no—
108C23×D61C2×D412yes—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C2×D412yes—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—82no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61C2×D412yes—
133C2×D244—22no—
134C2×D244—42no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×Dic122—24no—
142C2×Dic122—44no—
143Q16×C61—44no—
144SD16×C61—44no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—42no—
153C22⋊C4×C61—42no—
154C42×C61—24no—
155C42×C61C2×D412yes—
156C42×C61C2×D412yes—
157C22⋊Dic61—44no—
158#11758198—28no—
159#11758198—28no—
160#11758198—24no—
161#11758198—44no—
162C4⋊Dic61—24no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61C2×D414yes—
167C4⋊Dic61#4114yes—
168#11758196—28no—
169#11758196—44no—
170C4×Dic61—28no—
171C4×Dic61—81no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—41no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61C22×D411yescharacteristic
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188Q8×C61—22no—
189Q8×C61—22no—
190Q8×C61—42no—
191Q8×C61—41no—
192Q8×C61—41no—
193Q8×C61—42no—
194D4×C61—41no—
195D4×C61—41no—
196D4×C61—42no—
197C2×C4×C61—22no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—42no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61#12012yes—
208C2×C4×C61#12012yes—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4112yes—
212C2×C4×C61H(5,𝔽2)12yes—
213C22⋊D61—88no—
214C2×Dic61—22no—
215C2×Dic61—24no—
216C2×Dic61—22no—
217C2×Dic61—24no—
218C2×Dic61—22no—
219C2×Dic61—44no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222D244—44no—
223C4×D61—42no—
224C4×D61—42no—
225C4×D61—42no—
226C4×D61—42no—
227Dic122—48no—
228C8×C61—41no—
229C8×C61—41no—
230#92418—42no—
231#25580—611nomaximal, Sylow
232C22×C61—81no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61—82no—
236C22×C61—82no—
237C22×C61C2×#12011yescharacteristic
238C22×C61#54612yes—
239C22×C61#57512yes—
240C22×C61#57711yescharacteristic
241C22×C61#57811yescharacteristic
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—42no—
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—81no—
254C4×C61#53211yescharacteristic
255C4×C61#53511yescharacteristic
256C4×C61#55012yes—
257Dic61—42no—
258Dic61—44no—
259Dic61—44no—
260C2×#556—611no—
261C2×#489—611no—
262C2×#464D6111yescharacteristic, complemented
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—161no—
272C2×C61#243111yescharacteristic
273C2×C61#300511yescharacteristic
274C2×C61#301311yescharacteristic
275C2×C61#331112yes—
276C2×C61#331312yes—
277D61—82no—
278C22×Q16C2×D6111yescharacteristic
279C22×SD16C2×D6111yescharacteristic
280#556—1224no—
281#556—1224no—
282C2×C4⋊Q8—611no—
283C2×#126—1222no—
284C2×#124—1221no—
285C2×#124C2×D6111yescharacteristic
286C2×C22⋊Q8—612no—
287C2×C22⋊Q8—612no—
288C2×#121—612no—
289C2×C4×Q8—1221no—
290C2×C4×Q8C2×D6111yescharacteristic
291C2×C4×D4—611no—
292C2×#117—612no—
293#489—1224no—
294#464—24no—
295C2×C4⋊C8C2×D6111yescharacteristic
296C2×#103—612no—
297C2×#103—612no—
298C2×#103C2×D6111yescharacteristic
299C2×#102—612no—
300C2×#102C2×D6111yescharacteristic
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—1221no—
313C22×Q8C22×D6111yescharacteristic
314C22×Q8C22×D6111yescharacteristic
315C22×D4—611no—
316C22×D4—1222no—
317C22×D4C22×D6111yescharacteristic
318C23×C4—612no—
319C2×Q16—22no—
320C2×Q16—42no—
321C2×SD16—22no—
322C2×SD16—42no—
323C22×C8—612no—
324C22×C8C22×D6111yescharacteristic
325C22×C8C22×D6111yescharacteristic
326C4⋊Q8—614no—
327C4⋊Q8—1222no—
328#126—12216no—
329#124—22no—
330#124—22no—
331#124—1228no—
332C22⋊Q8—1224no—
333C22⋊Q8—1224no—
334C22⋊Q8—1224no—
335C22⋊Q8—2442no—
336#121—1224no—
337#121—2442no—
338C4×Q8—22no—
339C4×Q8—1228no—
340C4×Q8C22⋊D6114yescomplemented
341C4×D4—1222no—
342C4×D4—1222no—
343#117—1224no—
344#117—1224no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—612no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4C22×D6111yescharacteristic
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—612no—
356C2×C22⋊C4—1222no—
357C2×C22⋊C4—1222no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1222no—
360C2×C42—21no—
361C2×C42—1222no—
362C2×C42—1221no—
363C2×C42C22×D6111yescharacteristic
364C4⋊C8—22no—
365#103—22no—
366#103—1224no—
367#103—1224no—
368#102—22no—
369#102—1224no—
370C22⋊C8—2442no—
371C22⋊C8—2442no—
372#97—1222no—
373#95—1221no—
374C24—21no—
375C24—612no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—1224no—
384C2×Q8—2444no—
385C2×Q8C2×C22⋊D6112yes—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—1222no—
391C2×D4—4882no—
392C2×D4—2442no—
393C2×D4—2444no—
394C2×D4—2444no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—612no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4D4×D6111yescharacteristic
408C22×C4D4×D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410Q16—44no—
411SD16—44no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—1222no—
417C2×C8—1222no—
418C2×C8—1222no—
419C4⋊C4—21no—
420C4⋊C4—24no—
421C4⋊C4—614no—
422C4⋊C4—614no—
423C4⋊C4—1228no—
424C4⋊C4—1222no—
425C4⋊C4—1222no—
426C4⋊C4—1222no—
427C4⋊C4—1224no—
428C4⋊C4—2444no—
429C4⋊C4C2×C22⋊D6112yes—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—1228no—
433C22⋊C4—1228no—
434C22⋊C4—1224no—
435C22⋊C4—2444no—
436C22⋊C4—2444no—
437C42—24no—
438C42—1228no—
439C42—4881no—
440C42D4×D6112yes—
441C42C2×C22⋊D6112yes—
442C23—41no—
443C23—42no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—1222no—
448C23—1222no—
449C23—1222no—
450C23—1222no—
451C23—1222no—
452C23—1222no—
453C23—1222no—
454C23C2×D4×D6111yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—42no—
458Q8—41no—
459Q8—41no—
460Q8—42no—
461Q8—2448no—
462D4—41no—
463D4—41no—
464D4—42no—
465D4—4888no—
466D4—2444no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—1222no—
478C2×C4—1224no—
479C2×C4—1224no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—2442no—
483C2×C4—2444no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4#1451974512yes—
490C2×C4#1451974512yes—
491C2×C4C2×D4×D6111yescharacteristic
492C2×C4C22×C22⋊D6111yescharacteristic
493C2×C4#41×D6112yes—
494C2×C4#1451982212yes—
495C8—41no—
496C8—41no—
497C8—2442no—
498C22—81no—
499C22—82no—
500C22—81no—
501C22—81no—
502C22—82no—
503C22—2442no—
504C22—2442no—
505C22—2442no—
506C22—2442no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22C2×#1451974511yescharacteristic
511C22#1684040912yes—
512C22#1684056112yes—
513C22#1684056911yescharacteristic
514C22#1684057611yescharacteristic
515C4—22no—
516C4—22no—
517C4—42no—
518C4—42no—
519C4—81no—
520C4—2444no—
521C4—2444no—
522C4—2442no—
523C4#1684030811yescharacteristic
524C4#1684032911yescharacteristic
525C4#1684043012yes—
526C2—161no—
527C2—4882no—
528C2#2144943911yescharacteristic
529C2#2145506011yescharacteristic
530C2#2145513011yescharacteristic
531C2#2145786212yes—
532C2#2145789212yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#464C2
socleC23×C61C22×D4
prime radical#21450759C2
hypercenterC2×#464D61
hypocenterC61#25580
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
1. C2×#16840485C2semidirect
2. C2×#16840121C2semidirect
4. #21446826C2semidirect
5. #21446819C2semidirect
6. #21446766C2semidirect
7. #21446616C2semidirect
8. #21446138C2semidirect
9. #21446128C2semidirect
10. #21445649C2semidirect
11. C2×C61×#464C2semidirect
51. #14519777D4semidirect
97. C4×Q8×C61D4semidirect
262. C2×#464D61semidirect
311. C61#25580semidirect
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×#168404852
C2×#168401212
#214507592
#214468262
#214468192
#214467662
#214466162
#214461382
#214461282
#214456492
C2×C61×#4642
#2558061