Finite Group Databasebeta

← #29733337

Subgroups — #29733337

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×#16840486C211yesmaximal, characteristic, complemented
2C2×#16839981C211yesmaximal, characteristic, complemented
3#21450759C211yesmaximal, characteristic
4#21446828C212yesmaximal, complemented
5#21446823C212yesmaximal, complemented
6#21446768C211yesmaximal, characteristic, complemented
7#21446617C211yesmaximal, characteristic, complemented
8#21446138C212yesmaximal, complemented
9#21446128C212yesmaximal, complemented
10#21445650C211yesmaximal, characteristic, complemented
11C2×C61×#489C211yesmaximal, characteristic, complemented
12C22×#11758202C2211yescharacteristic
13C22×#11758200C2211yescharacteristic
14#16840486—24no—
15#16840486—24no—
16C2×#14519772—22no—
17C2×#14519766—21no—
18C2×#14519757C2212yes—
19C2×#14519753C2212yes—
20C2×#14519748C2212yes—
21C2×Q8×Dic61—21no—
22C2×C4×Dic122C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24C2×#14519735C2212yes—
25#16839981—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2212yes—
28C2×#14519701C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519689C2212yes—
31C2×#14519689C2212yes—
32#16839698—21no—
33#16839671—21no—
34#16839662—22no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×#124×C61C2211yescharacteristic
41C61×#489—24no—
42C2×#103×C61C2212yes—
43C2×#102×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C22×Dic122C2311yescharacteristic
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C4×D61C2312yes—
49C2×#11758202—22no—
50C2×#11758202—42no—
51C2×#11758200—22no—
52C2×#11758200—42no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519772—216no—
56#14519766—28no—
57#14519757—24no—
58#14519757—24no—
59#14519753—24no—
60#14519753—42no—
61#14519748—24no—
62#14519748—42no—
63Q8×Dic61—28no—
64C4×Dic122—22no—
65C4×Dic122D414yescomplemented
66C4×D244—22no—
67C4×D244—22no—
68#14519735—24no—
69#14519735—24no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—21no—
75C2×#11758196—22no—
76C2×C22⋊Dic61—22no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198D412yes—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—21no—
84C4⋊#92418—22no—
85#14519702—24no—
86#14519701—22no—
87#14519698—22no—
88#14519689—42no—
89#14519689—42no—
90#14519684—21no—
91C22×Q8×C61—21no—
92C22×Q8×C61C2311yescharacteristic
93C22×D4×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C4⋊Q8×C61—22no—
96C4⋊Q8×C61D414yescomplemented
97#124×C61—22no—
98#124×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#103×C61—24no—
105#102×C61—24no—
106#97×C61—22no—
107C24×C61—21no—
108C23×D61C2×D412yes—
109C2×Q8×C61—24no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—44no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—44no—
125C2×C22⋊D61—44no—
126C2×C22⋊D61—82no—
127C22×Dic61—22no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61C2×D412yes—
131C2×D244—22no—
132C2×D244—42no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—21no—
142C2×Dic122C2×D412yes—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61C2×D414yes—
149C22⋊C4×C61—42no—
150C22⋊C4×C61—42no—
151C42×C61C2×D412yes—
152C42×C61C2×D412yes—
153C22⋊Dic61—44no—
154#11758202—44no—
155#11758200—44no—
156#11758198—28no—
157#11758198—28no—
158#11758198—24no—
159#11758198—44no—
160C4⋊Dic61—28no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—22no—
164C4⋊Dic61C2×D412yes—
165C4⋊Dic61#4114yes—
166#11758196—24no—
167#11758196—44no—
168C4×Dic61—28no—
169C4×Dic61—24no—
170C4×Dic61—81no—
171C2×#92418—21no—
172C2×#92418—21no—
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—48no—
189Q8×C61—41no—
190Q8×C61—41no—
191Q8×C61—42no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—42no—
195C2×C4×C61—22no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—24no—
199C2×C4×C61—44no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61#12012yes—
203C2×C4×C61#12012yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4112yes—
207C2×C4×C61H(5,𝔽2)12yes—
208C22⋊D61—88no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—24no—
214C2×Dic61—22no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220D244—44no—
221C4×D61—42no—
222C4×D61—42no—
223C4×D61—42no—
224C4×D61—42no—
225Dic122—22no—
226Dic122—22no—
227Dic122—42no—
228C8×C61—42no—
229#92418—41no—
230#92418—41no—
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—44no—
252C4×C61—81no—
253C4×C61#53211yescharacteristic
254C4×C61#53511yescharacteristic
255C4×C61#55012yes—
256Dic61—42no—
257Dic61—42no—
258Dic61—42no—
259Dic61—44no—
260C2×#556—611no—
261C2×#489D6111yescharacteristic, complemented
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—161no—
272C2×C61#243111yescharacteristic
273C2×C61#300511yescharacteristic
274C2×C61#301311yescharacteristic
275C2×C61#331112yes—
276C2×C61#331312yes—
277D61—82no—
278C22×Q16—611no—
279C22×SD16—611no—
280#556—1224no—
281#556—1224no—
282C2×C4⋊Q8C2×D6111yescharacteristic
283C2×#126—1222no—
284C2×#124—1221no—
285C2×#124C2×D6111yescharacteristic
286C2×C22⋊Q8—612no—
287C2×C22⋊Q8—612no—
288C2×#121—612no—
289C2×C4×Q8—611no—
290C2×C4×Q8—1221no—
291C2×C4×D4—611no—
292C2×#117—612no—
293#489—24no—
294#464—1224no—
295C2×C4⋊C8—611no—
296C2×#103—611no—
297C2×#103—612no—
298C2×#103C2×D6112yes—
299C2×#102—611no—
300C2×#102C2×D6112yes—
301C2×C22⋊C8—612no—
302C2×C22⋊C8—612no—
303C2×#97C2×D6111yescharacteristic
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×C4—612no—
319C2×Q16—1222no—
320C2×Q16—2442no—
321C2×SD16—1222no—
322C2×SD16—2442no—
323C22×C8—611no—
324C22×C8—611no—
325C22×C8C22×D6112yes—
326C4⋊Q8—22no—
327C4⋊Q8C22⋊D6114yescomplemented
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—614no—
339C4×Q8—1228no—
340C4×Q8—1222no—
341C4×D4—1222no—
342C4×D4—1222no—
343#117—1224no—
344#117—1224no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4C22×D6112yes—
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—1222no—
361C2×C42—1221no—
362C2×C42—1221no—
363C2×C42C22×D6111yescharacteristic
364C4⋊C8—1222no—
365#103—24no—
366#103—1224no—
367#103—1222no—
368#102—24no—
369#102—1222no—
370C22⋊C8—2442no—
371C22⋊C8—2442no—
372#97—22no—
373#95—1221no—
374C24—21no—
375C24—612no—
376C2×Q8—24no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—44no—
382C2×Q8—612no—
383C2×Q8—1221no—
384C2×Q8—1222no—
385C2×Q8—1222no—
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—22no—
396C22×C4—21no—
397C22×C4—612no—
398C22×C4—1222no—
399C22×C4—1221no—
400C22×C4—1221no—
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—2444no—
411SD16—2444no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—1221no—
416C2×C8—1221no—
417C2×C8—1222no—
418C2×C8—1222no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—612no—
422C4⋊C4—614no—
423C4⋊C4—1228no—
424C4⋊C4—1221no—
425C4⋊C4—1222no—
426C4⋊C4—1222no—
427C4⋊C4—1224no—
428C4⋊C4—2444no—
429C4⋊C4C2×C22⋊D6114yes—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—1228no—
433C22⋊C4—1228no—
434C22⋊C4—1224no—
435C22⋊C4—2444no—
436C22⋊C4—2444no—
437C42—1228no—
438C42—1224no—
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—48no—
456Q8—41no—
457Q8—41no—
458Q8—42no—
459Q8—1222no—
460Q8—1222no—
461Q8—2442no—
462D4—41no—
463D4—41no—
464D4—42no—
465D4—4888no—
466D4—2444no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—1222no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—1224no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2441no—
483C2×C4—2441no—
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#1451982312yes—
495C8—42no—
496C8—2441no—
497C8—2441no—
498C22—81no—
499C22—82no—
500C22—81no—
501C22—82no—
502C22—81no—
503C22—2442no—
504C22—2442no—
505C22—2442no—
506C22—2442no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22C2×#1451974511yescharacteristic
511C22#1684040512yes—
512C22#1684055812yes—
513C22#1684057111yescharacteristic
514C22#1684057811yescharacteristic
515C4—22no—
516C4—22no—
517C4—44no—
518C4—81no—
519C4—2442no—
520C4—2444no—
521C4—2442no—
522C4—2442no—
523C4#1684031111yescharacteristic
524C4#1684033011yescharacteristic
525C4#1684043112yes—
526C2—161no—
527C2—4882no—
528C2#2144944111yescharacteristic
529C2#2145505811yescharacteristic
530C2#2145513311yescharacteristic
531C2#2145785412yes—
532C2#2145788412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#489C2
socleC23×C61C22×D4
prime radical#21450759C2
hypercenterC2×#489D61
hypocenterC61#25580
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×#16840486C2semidirect
2. C2×#16839981C2semidirect
4. #21446828C2semidirect
5. #21446823C2semidirect
6. #21446768C2semidirect
7. #21446617C2semidirect
8. #21446138C2semidirect
9. #21446128C2semidirect
10. #21445650C2semidirect
11. C2×C61×#489C2semidirect
65. C4×Dic122D4semidirect
96. C4⋊Q8×C61D4semidirect
261. C2×#489D61semidirect
311. C61#25580semidirect
327. C4⋊Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×#168399812
#214507592
#214468282
#214468232
#214467682
#214466172
#214461382
#214461282
#214456502
C2×C61×#4892
#2558061