Finite Group Databasebeta

← #29714061

Subgroups — #29714061

Order 15616.

1204 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840477C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3C2×#16839872C211yesmaximal, characteristic, complemented
4#21447001C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446791C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446142C212yesmaximal, complemented
9#21446016C212yesmaximal, complemented
10#21445826C211yesmaximal, characteristic, complemented
11C61×#2535C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13#16840477—24no—
14#16840477—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519763—22no—
17C2×#14519751C2212yes—
18C2×#14519748—21no—
19C2×C4×D244C2211yescharacteristic
20C2×#14519736C2212yes—
21C2×#14519735—21no—
22#16840042—24no—
23#16839872—24no—
24C2×#14519710C2212yes—
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2212yes—
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×C22⋊#92418C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37C2×#121×C61C2212yes—
38C2×C4×D4×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#397—21no—
41C61×#392—21no—
42C61×#386—22no—
43C61×#382C2211yescharacteristic
44C61×#380C2212yes—
45C22×Dic122—21no—
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—22no—
48C22×C4×D61—21no—
49C2×#11758199—22no—
50C2×#11758199—22no—
51C2×#11758199—22no—
52C22×#92418C2311yescharacteristic
53C22×#92418C2312yes—
54C22×#92418C2311yescharacteristic
55#14519777—22no—
56#14519777D414yescomplemented
57#14519763—216no—
58#14519751—24no—
59#14519751—24no—
60#14519748—28no—
61C4×D244—22no—
62C4×D244D414yescomplemented
63#14519736—24no—
64#14519736—24no—
65#14519735—28no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C22⋊Dic61—22no—
70C2×#11758198—22no—
71C2×#11758198—21no—
72C2×C4×Dic61—22no—
73#14519710—24no—
74#14519708—22no—
75C4⋊#92418—22no—
76#14519698—24no—
77#14519698—22no—
78#14519697—22no—
79#14519697—42no—
80C22⋊#92418—24no—
81C22⋊#92418—42no—
82#14519688—22no—
83C22×D4×C61—22no—
84C22×D4×C61D411yescharacteristic
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61—21no—
87C23×C4×C61C2312yes—
88#121×C61—24no—
89#121×C61—42no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61D412yes—
101C2×C42×C61C2311yescharacteristic
102#95×C61—22no—
103#95×C61—21no—
104#95×C61—21no—
105C24×C61—21no—
106C24×C61C2×D412yes—
107C23×D61—21no—
108C2×D4×C61—22no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—42no—
115C2×D4×C61—44no—
116C2×D4×C61—44no—
117C2×D4×C61—82no—
118C2×D4×C61#4112yes—
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×C61—21no—
127C22×C4×C61—42no—
128C22×C4×C61—82no—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—28no—
134C2×C22⋊D61—82no—
135C2×C22⋊D61—82no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C2×D412yes—
144C2×C4×D61—24no—
145C2×C4×D61—44no—
146C2×Dic122—24no—
147C2×Dic122—82no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—44no—
155C22⋊C4×C61—42no—
156C22⋊C4×C61—82no—
157C42×C61C2×D412yes—
158C42×C61C2×D412yes—
159C22⋊Dic61—28no—
160#11758199—28no—
161#11758198—28no—
162#11758198—24no—
163C4⋊Dic61—24no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61C2×D412yes—
167C4⋊Dic61C2×D414yes—
168C4×Dic61—28no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—21no—
175C2×#92418—22no—
176C2×#92418—21no—
177C2×#92418#4112yes—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61C22×D411yescharacteristic
191C22×D61—22no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—42no—
195C22×D61—42no—
196D4×C61—22no—
197D4×C61—22no—
198D4×C61—44no—
199D4×C61—41no—
200D4×C61—41no—
201D4×C61—88no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—84no—
217C2×C4×C61#12012yes—
218C2×C4×C61#12012yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C22⋊D61—88no—
226C2×Dic61—22no—
227C2×Dic61—24no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231D244—22no—
232D244—22no—
233D244—41no—
234D244—41no—
235C4×D61—48no—
236Dic122—84no—
237#92418—22no—
238#92418—41no—
239#92418—42no—
240#24129—611nomaximal, Sylow
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—44no—
249C22×C61—42no—
250C22×C61—82no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61C2×#12011yescharacteristic
254C22×C61#54611yescharacteristic
255C22×C61#54711yescharacteristic
256C22×C61#57512yes—
257C22×C61#57612yes—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—82no—
262C2×D61—82no—
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—42no—
266C4×C61—81no—
267C4×C61—82no—
268C4×C61#52511yescharacteristic
269C4×C61#53711yescharacteristic
270C4×C61#54811yescharacteristic
271C4×C61#55011yescharacteristic
272Dic61—42no—
273Dic61—44no—
274C2×#555—611no—
275C2×#474—611no—
276C2×#442—611no—
277#2535D6111yescharacteristic, complemented
278#2033—612no—
279#1999—612no—
280#1993—611no—
281#1955—611no—
282#1875—612no—
283#1854—612no—
284#1804—611no—
285C2×C61—82no—
286C2×C61—82no—
287C2×C61#240711yescharacteristic
288C2×C61#300511yescharacteristic
289C2×C61#300811yescharacteristic
290C2×C61#331912yes—
291C2×C61#334712yes—
292D61—82no—
293C22×D8—611no—
294#555—1224no—
295#555—1224no—
296C2×C4⋊Q8—611no—
297C2×#124—1222no—
298C2×#121—612no—
299C2×#121—1221no—
300C2×#121C2×D6112yes—
301C2×C4×D4—611no—
302C2×C4×D4C2×D6111yescharacteristic
303C2×C4×D4C2×D6111yescharacteristic
304C2×#117—612no—
305C2×#117—1221no—
306#474—1224no—
307#442—1224no—
308C2×#107—612no—
309C2×#106—611no—
310C2×C4⋊C8—611no—
311C2×#102—611no—
312C2×#102—612no—
313C2×#102—611no—
314C2×#102—612no—
315C2×C22⋊C8—612no—
316C2×C22⋊C8—612no—
317C2×#97—611no—
318#397—21no—
319#392—21no—
320#386—22no—
321#382C2×D6111yescharacteristic
322#380C2×D6112yes—
323#340—612no—
324#340—612no—
325#340—612no—
326C61#2412911yescharacteristic, complemented, Sylow
327C22×Q8—1221no—
328C22×D4—22no—
329C22×D4—611no—
330C22×D4—1222no—
331C22×D4C22⋊D6111yescharacteristic
332C22×D4C22×D6111yescharacteristic
333C23×C4—21no—
334C23×C4—1221no—
335C23×C4C22×D6112yes—
336C2×D8—1222no—
337C2×D8—1222no—
338C2×D8—1222no—
339C22×C8—611no—
340C22×C8—611no—
341C22×C8—612no—
342C4⋊Q8—614no—
343C4⋊Q8—1222no—
344#124—12216no—
345#121—24no—
346#121—42no—
347#121—1224no—
348#121—1224no—
349#121—1228no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—614no—
355C4×D4—1222no—
356#117—1228no—
357#117—1224no—
358#117—1224no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—612no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4C22⋊D6111yescharacteristic
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C4⋊C4C22×D6112yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4C22⋊D6112yes—
372C2×C42—1222no—
373C2×C42C22×D6111yescharacteristic
374#107—1224no—
375#106—1222no—
376C4⋊C8—1222no—
377#102—1222no—
378#102—1222no—
379#102—1224no—
380#102—2442no—
381C22⋊C8—1224no—
382C22⋊C8—2442no—
383#97—1222no—
384#95—22no—
385#95—21no—
386#95—21no—
387C24—21no—
388C24—1221no—
389C24C2×C22⋊D6112yes—
390C2×Q8—1224no—
391C2×Q8—4882no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—82no—
402C2×D4—612no—
403C2×D4—1228no—
404C2×D4—1221no—
405C2×D4—1221no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1221no—
409C2×D4—4882no—
410C2×D4—4882no—
411C2×D4#1175822412yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—42no—
421C22×C4—82no—
422C22×C4—1221no—
423C22×C4—1222no—
424C22×C4—1224no—
425C22×C4—2444no—
426C22×C4C2×C22⋊D6112yes—
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430D8—1228no—
431C2×C8—612no—
432C2×C8—1221no—
433C2×C8—1221no—
434C2×C8—1221no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1221no—
438C2×C8—1222no—
439C2×C8—1221no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—612no—
446C4⋊C4—614no—
447C4⋊C4—1221no—
448C4⋊C4—1222no—
449C4⋊C4—1224no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—82no—
454C22⋊C4—1228no—
455C22⋊C4—1228no—
456C22⋊C4—1224no—
457C42—1228no—
458C42D4×D6112yes—
459C42D4×D6112yes—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—1222no—
473C23—2442no—
474C23—2442no—
475C23—2441no—
476C23—2441no—
477C23C22×C22⋊D6111yescharacteristic
478Q8—4884no—
479D4—22no—
480D4—22no—
481D4—44no—
482D4—41no—
483D4—41no—
484D4—88no—
485D4—1222no—
486D4—1222no—
487D4—4888no—
488D4—2441no—
489D4—2441no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—84no—
505C2×C4—1222no—
506C2×C4—1224no—
507C2×C4—2448no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4#1451974512yes—
512C2×C4#1451974512yes—
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4#41×D6111yescharacteristic
516C2×C4C2×#1175822411yescharacteristic
517C2×C4#1451982211yescharacteristic
518C2×C4#1451982411yescharacteristic
519C8—1222no—
520C8—2441no—
521C8—2442no—
522C22—42no—
523C22—44no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—82no—
532C22—81no—
533C22—81no—
534C22—4882no—
535C22—4882no—
536C22—2442no—
537C22—2442no—
538C22—2442no—
539C22C2×#1451974511yescharacteristic
540C22#1684040611yescharacteristic
541C22#1684041111yescharacteristic
542C22#1684056012yes—
543C22#1684056512yes—
544C4—22no—
545C4—22no—
546C4—42no—
547C4—82no—
548C4—81no—
549C4—2442no—
550C4—2444no—
551C4#1684027211yescharacteristic
552C4#1684034111yescharacteristic
553C4#1684041811yescharacteristic
554C4#1684043011yescharacteristic
555C2—82no—
556C2—82no—
557C2—4882no—
558C2#2144916011yescharacteristic
559C2#2145505911yescharacteristic
560C2#2145508511yescharacteristic
561C2#2145797512yes—
562C2#2145831312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2535C2
socleC23×C61C22×D4
hypercenter#2535D61
hypocenterC61#24129
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. C2×#16840477C2semidirect
2. C2×#16840042C2semidirect
3. C2×#16839872C2semidirect
4. #21447001C2semidirect
5. #21446815C2semidirect
6. #21446791C2semidirect
7. #21446618C2semidirect
8. #21446142C2semidirect
9. #21446016C2semidirect
10. #21445826C2semidirect
11. C61×#2535C2semidirect
56. #14519777D4semidirect
62. C4×D244D4semidirect
277. #2535D61semidirect
326. C61#24129semidirect

Maximal subgroups

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

subgroupindex
C2×#168404772
C2×#168400422
C2×#168398722
#214470012
#214468152
#214467912
#214466182
#214461422
#214460162
#214458262
C61×#25352
#2412961