Finite Group Databasebeta

← #29714059

Subgroups — #29714059

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×#16840039C211yesmaximal, characteristic, complemented
2C2×#16839872C211yesmaximal, characteristic, complemented
3#21450720C211yesmaximal, characteristic, complemented
4#21446998C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446141C212yesmaximal, complemented
9#21446018C212yesmaximal, complemented
10#21445829C211yesmaximal, characteristic, complemented
11C2×C61×#555C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C2×#14519751C2212yes—
14C2×C4×D244C2211yescharacteristic
15#16840039—24no—
16#16839872—24no—
17C2×#14519710C2212yes—
18C2×#14519708C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2212yes—
22C2×#14519697C2211yescharacteristic
23C2×#14519697C2212yes—
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2212yes—
26C2×#14519688C2211yescharacteristic
27#16839688—21no—
28#16839670—21no—
29#16839638—22no—
30#16839628C2211yescharacteristic
31#16839622C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C61×#555—24no—
36C61×#555—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#124×C61—22no—
39C2×#121×C61—21no—
40C2×#121×C61C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C2×#117×C61—21no—
44C2×#117×C61C2212yes—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—22no—
47C22×C4×D61—21no—
48C2×#11758199—22no—
49C2×#11758199—22no—
50C2×#11758199—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2312yes—
53C22×#92418C2311yescharacteristic
54#14519751—24no—
55#14519751—42no—
56C4×D244—22no—
57C4×D244—22no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×C22⋊Dic61—22no—
61C2×#11758198—21no—
62C2×#11758198—22no—
63#14519710—24no—
64#14519708—22no—
65C4⋊#92418—22no—
66#14519698—22no—
67#14519698—42no—
68#14519697—24no—
69#14519697—22no—
70C22⋊#92418—24no—
71C22⋊#92418—42no—
72#14519688—22no—
73#14519684—22no—
74#14519684—21no—
75#14519684—21no—
76C22×Q8×C61—21no—
77C22×D4×C61—22no—
78C22×D4×C61D411yescharacteristic
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61—21no—
81C23×C4×C61C2312yes—
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61D414yescomplemented
84#124×C61—216no—
85#121×C61—28no—
86#121×C61—24no—
87#121×C61—24no—
88C4×D4×C61—22no—
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61D414yescomplemented
92#117×C61—28no—
93#117×C61—24no—
94#117×C61—24no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C4⋊C4×C61C2312yes—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61D412yes—
103C2×C42×C61—22no—
104C2×C42×C61C2311yescharacteristic
105C24×C61—21no—
106C24×C61C2×D412yes—
107C23×D61—21no—
108C2×Q8×C61—24no—
109C2×Q8×C61—82no—
110C2×D4×C61—28no—
111C2×D4×C61—22no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—42no—
118C2×D4×C61—82no—
119C2×D4×C61—82no—
120C2×D4×C61C2×D412yes—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—24no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—44no—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—44no—
136C2×C22⋊D61—44no—
137C2×C22⋊D61—82no—
138C22×Dic61—21no—
139C22×Dic61—22no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244#4112yes—
146C2×C4×D61—42no—
147C2×C4×D61—82no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61C2×D414yes—
154C22⋊C4×C61—28no—
155C22⋊C4×C61—28no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C42×C61—28no—
159C42×C61C2×D412yes—
160C42×C61C2×D412yes—
161C22⋊Dic61—82no—
162#11758199—28no—
163#11758198—44no—
164#11758198—42no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
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—
196Q8×C61—84no—
197D4×C61—22no—
198D4×C61—22no—
199D4×C61—44no—
200D4×C61—41no—
201D4×C61—41no—
202D4×C61—88no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—48no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61#12012yes—
219C2×C4×C61#12012yes—
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C22⋊D61—88no—
227C2×Dic61—41no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232D244—22no—
233D244—22no—
234D244—41no—
235D244—41no—
236C4×D61—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—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—82no—
252C22×C61—82no—
253C22×C61C2×#12011yescharacteristic
254C22×C61#54611yescharacteristic
255C22×C61#54711yescharacteristic
256C22×C61#57512yes—
257C22×C61#57612yes—
258C2×D61—42no—
259C2×D61—44no—
260C2×D61—82no—
261C2×D61—81no—
262C2×D61—81no—
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—42no—
266C4×C61—42no—
267C4×C61—44no—
268C4×C61#52511yescharacteristic
269C4×C61#53711yescharacteristic
270C4×C61#54811yescharacteristic
271C4×C61#55011yescharacteristic
272Dic61—81no—
273Dic61—82no—
274C2×#555D6111yescharacteristic, complemented
275C2×#474—611no—
276C2×#442—611no—
277#2535—611no—
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—24no—
295#555—24no—
296C2×C4⋊Q8C2×D6111yescharacteristic
297C2×#124—22no—
298C2×#121—21no—
299C2×#121—612no—
300C2×#121C2×D6112yes—
301C2×C4×D4—611no—
302C2×C4×D4C2×D6111yescharacteristic
303C2×C4×D4C2×D6111yescharacteristic
304C2×#117—21no—
305C2×#117C2×D6112yes—
306#474—1224no—
307#442—1224no—
308C2×#107—612no—
309C2×#106—611no—
310C2×C4⋊C8—611no—
311C2×#102—611no—
312C2×#102—611no—
313C2×#102—612no—
314C2×#102—612no—
315C2×C22⋊C8—612no—
316C2×C22⋊C8—612no—
317C2×#97—611no—
318#397—1221no—
319#392—1221no—
320#386—1222no—
321#382—611no—
322#380—612no—
323#340—612no—
324#340—612no—
325#340—612no—
326C61#2412911yescharacteristic, complemented, Sylow
327C22×Q8—21no—
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—22no—
343C4⋊Q8C22⋊D6114yescomplemented
344#124—216no—
345#121—28no—
346#121—24no—
347#121—24no—
348#121—1224no—
349#121—2442no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—1222no—
354C4×D4—1222no—
355C4×D4C22⋊D6114yescomplemented
356#117—28no—
357#117—24no—
358#117—24no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4C22⋊D6111yescharacteristic
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C4⋊C4C22×D6112yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1222no—
371C2×C22⋊C4C22⋊D6112yes—
372C2×C42—22no—
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—1222no—
385#95—1221no—
386#95—1221no—
387C24—21no—
388C24—1221no—
389C24C2×C22⋊D6112yes—
390C2×Q8—24no—
391C2×Q8—82no—
392C2×D4—28no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—42no—
400C2×D4—82no—
401C2×D4—82no—
402C2×D4—612no—
403C2×D4—1221no—
404C2×D4—1221no—
405C2×D4—1221no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—4882no—
409C2×D4—2444no—
410C2×D4—2444no—
411C2×D4C2×C22⋊D6112yes—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—4882no—
425C22×C4—2442no—
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—21no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—1221no—
445C4⋊C4—1221no—
446C4⋊C4—1222no—
447C4⋊C4—1222no—
448C4⋊C4C2×C22⋊D6112yes—
449C4⋊C4C2×C22⋊D6114yes—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—4882no—
455C22⋊C4—2444no—
456C22⋊C4—2442no—
457C42—28no—
458C42C2×C22⋊D6112yes—
459C42C2×C22⋊D6112yes—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23—1222no—
473C23—2441no—
474C23—2442no—
475C23—2442no—
476C23—2441no—
477C23C22×C22⋊D6111yescharacteristic
478Q8—84no—
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—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—4884no—
506C2×C4—2441no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4#1451974612yes—
512C2×C4#1451974612yes—
513C2×C4C22×C22⋊D6111yescharacteristic
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4#41×D6111yescharacteristic
516C2×C4#41×D6111yescharacteristic
517C2×C4#1451982311yescharacteristic
518C2×C4#1451982411yescharacteristic
519C8—1222no—
520C8—2441no—
521C8—2442no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—82no—
533C22—82no—
534C22—4882no—
535C22—4881no—
536C22—4881no—
537C22—2444no—
538C22—2442no—
539C22C2×#1451974611yescharacteristic
540C22#1684040011yescharacteristic
541C22#1684041111yescharacteristic
542C22#1684056012yes—
543C22#1684056512yes—
544C4—22no—
545C4—22no—
546C4—42no—
547C4—42no—
548C4—44no—
549C4—4882no—
550C4—4881no—
551C4#1684026911yescharacteristic
552C4#1684034011yescharacteristic
553C4#1684041911yescharacteristic
554C4#1684043111yescharacteristic
555C2—82no—
556C2—82no—
557C2—4882no—
558C2#2144915811yescharacteristic
559C2#2145506111yescharacteristic
560C2#2145508711yescharacteristic
561C2#2145796912yes—
562C2#2145831912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#555C2
socleC23×C61C22×D4
hypercenterC2×#555D61
hypocenterC61#24129
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#16840039C2semidirect
2. C2×#16839872C2semidirect
3. #21450720C2semidirect
4. #21446998C2semidirect
5. #21446815C2semidirect
6. #21446793C2semidirect
7. #21446618C2semidirect
8. #21446141C2semidirect
9. #21446018C2semidirect
10. #21445829C2semidirect
11. C2×C61×#555C2semidirect
83. C4⋊Q8×C61D4semidirect
91. C4×D4×C61D4semidirect
274. C2×#555D61semidirect
326. C61#24129semidirect
343. C4⋊Q8C22⋊D61semidirect
355. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
C2×#168398722
#214507202
#214469982
#214468152
#214467932
#214466182
#214461412
#214460182
#214458292
C2×C61×#5552
#2412961