Finite Group Databasebeta

← #29736722

Subgroups — #29736722

Order 15616.

1110 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×#16840389C211yesmaximal, characteristic, complemented
2C2×#16840126C211yesmaximal, characteristic, complemented
3#21451570C211yesmaximal, characteristic
4#21446963C212yesmaximal, complemented
5#21446953C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446784C211yesmaximal, characteristic, complemented
8#21446138C212yesmaximal, complemented
9#21446133C212yesmaximal, complemented
10#21445586C211yesmaximal, characteristic, complemented
11C2×C61×#486C211yesmaximal, characteristic, complemented
12#16840389—24no—
13#16840389—24no—
14C2×#14519776C2211yescharacteristic
15C2×#14519775—21no—
16C2×#14519769C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519767—21no—
19C2×#14519755—21no—
20C2×#14519753—21no—
21C2×#14519748C2212yes—
22C2×#14519735C2211yescharacteristic
23C2×#14519734C2211yescharacteristic
24C2×C4⋊C4×D61C2211yescharacteristic
25#16840126—24no—
26C2×#14519698C2212yes—
27C2×#14519698C2212yes—
28C2×#14519689C2212yes—
29C2×#14519688C2211yescharacteristic
30#16839676—21no—
31#16839676—21no—
32#16839670—21no—
33#16839670—21no—
34#16839665C2212yes—
35#16839628C2211yescharacteristic
36#16839525C2212yes—
37#16839525C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#127×C61C2211yescharacteristic
43C61×#486—24no—
44C2×#102×C61C2212yes—
45C2×#102×C61C2212yes—
46C2×C4×C8×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×C22⋊D61—22no—
49C22×C4×D61—21no—
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519776—22no—
53#14519776—22no—
54#14519775—28no—
55#14519769—22no—
56#14519769D414yescomplemented
57#14519768—22no—
58#14519768—22no—
59#14519767—28no—
60#14519755—28no—
61#14519753—28no—
62#14519748—28no—
63#14519735—22no—
64#14519735—22no—
65#14519734—22no—
66#14519734—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196D411yescharacteristic
77C2×#11758196D411yescharacteristic
78C4⋊C4×D61—22no—
79C4⋊C4×D61—22no—
80C2×C22⋊Dic61—22no—
81C2×C22⋊Dic61—22no—
82C2×C22⋊Dic61—22no—
83C2×#11758198—21no—
84C2×#11758198D411yescharacteristic
85C2×#11758198D411yescharacteristic
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88#14519698—24no—
89#14519698—24no—
90#14519689—24no—
91#14519688—22no—
92#14519684—21no—
93#14519684—21no—
94C22×Q8×C61—21no—
95C22×D4×C61—21no—
96C22×D4×C61—21no—
97C22×D4×C61C2312yes—
98C22×C8×C61C2312yes—
99C4⋊Q8×C61—22no—
100C4⋊Q8×C61D414yescomplemented
101#127×C61—22no—
102#127×C61—22no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C42×C61C2311yescharacteristic
107#102×C61—24no—
108#102×C61—24no—
109C4×C8×C61—22no—
110C24×C61—22no—
111C23×D61C2×D411yescharacteristic
112C2×Q8×C61—24no—
113C2×Q8×C61—82no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—24no—
117C2×D4×C61—24no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61—82no—
121C2×D4×C61—82no—
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—82no—
129C2×C22⋊D61—82no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—22no—
139C2×C4×D61—24no—
140C2×C4×D61—82no—
141C2×C4×D61#4114yes—
142C2×Dic122—24no—
143C2×Dic122—82no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C42×C61C2×D412yes—
153C42×C61C2×D412yes—
154C22⋊Dic61—44no—
155C22⋊Dic61—82no—
156C22⋊Dic61—82no—
157#11758198—24no—
158#11758198—22no—
159#11758198—22no—
160C4⋊Dic61—24no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—24no—
164C4⋊Dic61C2×D412yes—
165C4⋊Dic61C2×D412yes—
166#11758196—24no—
167#11758196—24no—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—22no—
172#11758196—22no—
173C4×Dic61—24no—
174C4×Dic61—21no—
175C4×Dic61—21no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61C22×D411yescharacteristic
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61#12412yes—
192Q8×C61—84no—
193D4×C61—44no—
194D4×C61—44no—
195D4×C61—84no—
196D4×C61—84no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61#12012yes—
204C2×C4×C61#12012yes—
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4112yes—
208C2×C4×C612−1+412yes—
209C22⋊D61—88no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—22no—
214C2×Dic61—21no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61#12412yes—
224C4×D61—24no—
225C4×D61—24no—
226C4×D61—84no—
227Dic122—84no—
228C8×C61—42no—
229#92418—42no—
230#25915—611nomaximal, Sylow
231C22×C61—82no—
232C22×C61—82no—
233C22×C61—82no—
234C22×C61—82no—
235C22×C61—82no—
236C22×C61—82no—
237C22×C61—82no—
238C22×C61C2×#12011yescharacteristic
239C22×C61C2×#12411yescharacteristic
240C22×C61#53611yescharacteristic
241C22×C61#57612yes—
242C22×C61#57712yes—
243C2×D61—24no—
244C2×D61—22no—
245C2×D61—42no—
246C2×D61—41no—
247C2×D61—41no—
248C4×C61—22no—
249C4×C61—22no—
250C4×C61—42no—
251C4×C61—42no—
252C4×C61#54812yes—
253C4×C61#54812yes—
254Dic61—22no—
255Dic61—41no—
256Dic61—42no—
257Dic61—41no—
258Dic61—42no—
259Dic61—82no—
260C2×#544—611no—
261C2×#490—611no—
262C2×#486D6111yescharacteristic, complemented
263#2619—611no—
264#2025—612no—
265#2023—612no—
266#1993—611no—
267#1991—611no—
268#1875—612no—
269#1874—612no—
270#1750—611no—
271C2×C61—162no—
272C2×C61#243611yescharacteristic
273C2×C61#300811yescharacteristic
274C2×C61#301011yescharacteristic
275C2×C61#313612yes—
276C2×C61#324112yes—
277D61—42no—
278#544—1224no—
279#544—1224no—
280C2×C4⋊Q8—611no—
281C2×C4⋊Q8—1221no—
282C2×C4⋊Q8C2×D6111yescharacteristic
283C2×#127C2×D6111yescharacteristic
284C2×#125—611no—
285C2×#125—611no—
286C2×#125—1221no—
287C2×C22⋊Q8—1221no—
288C2×C22⋊Q8—1221no—
289C2×#121—612no—
290C2×#117—611no—
291C2×#117—611no—
292C22×C4⋊C4—611no—
293#490—1224no—
294#486—24no—
295C2×#102—612no—
296C2×#102—612no—
297C2×#102C2×D6112yes—
298C2×#102C2×D6112yes—
299C2×C22⋊C8—612no—
300C2×#97—611no—
301C2×C4×C8C2×D6111yescharacteristic
302#394—1221no—
303#394—1221no—
304#392—1221no—
305#392—1221no—
306#390—612no—
307#382—611no—
308#340—612no—
309#340—612no—
310#340—612no—
311#340—612no—
312#336—612no—
313C61#2591511yescharacteristic, complemented, Sylow
314C22×Q8—21no—
315C22×Q8—1221no—
316C22×D4—21no—
317C22×D4—21no—
318C22×D4—1222no—
319C22×D4C22×D6112yes—
320C23×C4—611no—
321C23×C4—1221no—
322C22×C8—612no—
323C22×C8C22×D6112yes—
324C4⋊Q8—22no—
325C4⋊Q8—1222no—
326C4⋊Q8—1228no—
327C4⋊Q8—1222no—
328C4⋊Q8C22⋊D6114yescomplemented
329#127—22no—
330#127—22no—
331#125—614no—
332#125—1222no—
333#125—1228no—
334#125—1222no—
335#125—1222no—
336C22⋊Q8—1228no—
337C22⋊Q8—1228no—
338#121—1228no—
339#117—1222no—
340#117—1222no—
341#117—1222no—
342#117—1222no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4C22×D6111yescharacteristic
358C2×C4⋊C4C22×D6111yescharacteristic
359C2×C22⋊C4—611no—
360C2×C22⋊C4—611no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1222no—
365C2×C42—611no—
366C2×C42—1221no—
367C2×C42C22×D6111yescharacteristic
368#102—24no—
369#102—24no—
370#102—1224no—
371#102—1224no—
372C22⋊C8—1224no—
373#97—1222no—
374C4×C8—22no—
375#95—1221no—
376#95—1221no—
377C24—22no—
378C24—611no—
379C2×Q8—24no—
380C2×Q8—82no—
381C2×Q8—1224no—
382C2×Q8—4882no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—4882no—
392C2×D4—4882no—
393C2×D4—2444no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—611no—
397C22×C4—614no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—1224no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4—1222no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1222no—
407C22×C4—4882no—
408C22×C4D4×D6111yescharacteristic
409C22×C4D4×D6111yescharacteristic
410C22×C4C23×D6111yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—24no—
420C4⋊C4—612no—
421C4⋊C4—612no—
422C4⋊C4—1224no—
423C4⋊C4—1224no—
424C4⋊C4—1224no—
425C4⋊C4—1224no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1222no—
429C4⋊C4—1222no—
430C4⋊C4—1224no—
431C4⋊C4—1224no—
432C4⋊C4—1224no—
433C4⋊C4C2×C22⋊D6112yes—
434C4⋊C4C2×C22⋊D6112yes—
435C22⋊C4—1224no—
436C22⋊C4—1222no—
437C22⋊C4—1222no—
438C22⋊C4—4882no—
439C22⋊C4—4882no—
440C22⋊C4—2444no—
441C42—1224no—
442C42—1221no—
443C42—1221no—
444C42D4×D6112yes—
445C42C2×C22⋊D6112yes—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—612no—
454C23—1221no—
455C23—1222no—
456C23—1222no—
457C23—1221no—
458C23C2×D4×D6111yescharacteristic
459Q8—84no—
460Q8—4884no—
461D4—44no—
462D4—44no—
463D4—84no—
464D4—84no—
465D4—4888no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—612no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—1224no—
476C2×C4—1224no—
477C2×C4—1222no—
478C2×C4—1221no—
479C2×C4—1221no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—4884no—
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#1451982412yes—
495C8—42no—
496C8—2442no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—1222no—
505C22—1224no—
506C22—2442no—
507C22—2441no—
508C22—2441no—
509C22C2×#1451974511yescharacteristic
510C22#124×D6111yescharacteristic
511C22#1684033211yescharacteristic
512C22#1684056312yes—
513C22#1684056812yes—
514C4—22no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—1222no—
519C4—4882no—
520C4—2441no—
521C4—2442no—
522C4—2441no—
523C4—2442no—
524C4#1684041912yes—
525C4#1684042012yes—
526C2—162no—
527C2—2442no—
528C2#2144947411yescharacteristic
529C2#2145508311yescharacteristic
530C2#2145510111yescharacteristic
531C2#2145616812yes—
532C2#2145706512yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#486C2
socleC23×C61C22×D4
prime radical#21451570C2
hypercenterC2×#486D61
hypocenterC61#25915
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. C2×#16840389C2semidirect
2. C2×#16840126C2semidirect
4. #21446963C2semidirect
5. #21446953C2semidirect
6. #21446793C2semidirect
7. #21446784C2semidirect
8. #21446138C2semidirect
9. #21446133C2semidirect
10. #21445586C2semidirect
11. C2×C61×#486C2semidirect
56. #14519769D4semidirect
100. C4⋊Q8×C61D4semidirect
262. C2×#486D61semidirect
313. C61#25915semidirect
328. C4⋊Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403892
C2×#168401262
#214515702
#214469632
#214469532
#214467932
#214467842
#214461382
#214461332
#214455862
C2×C61×#4862
#2591561