Finite Group Databasebeta

← #29713240

Subgroups — #29713240

Order 15616.

1338 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840417C211yesmaximal, characteristic
2C2×#16840092C211yesmaximal, characteristic, complemented
3C2×#16839761C211yesmaximal, characteristic, complemented
4#21447065C212yesmaximal, complemented
5#21447015C212yesmaximal, complemented
6#21446250C211yesmaximal, characteristic, complemented
7#21446117C211yesmaximal, characteristic, complemented
8#21446079C212yesmaximal, complemented
9#21445757C211yesmaximal, characteristic, complemented
10#21445757C212yesmaximal, complemented
11C61×#2487C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758222C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15#16840417—24no—
16#16840417—24no—
17C2×#14519777C2211yescharacteristic
18C2×#14519759—21no—
19C2×#14519759—21no—
20C2×#14519756C2211yescharacteristic
21C2×#14519756C2211yescharacteristic
22C2×#14519748—21no—
23C2×C4×D244C2211yescharacteristic
24C2×C4⋊C4×D61—21no—
25#16840092—24no—
26C2×#14519708C2212yes—
27#16839761—24no—
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519700C2212yes—
31C2×#14519700C2212yes—
32C2×#14519697C2212yes—
33C2×#14519697C2212yes—
34C2×C22⋊#92418C2211yescharacteristic
35C2×C22⋊#92418C2212yes—
36#16839524C2212yes—
37#16839517C2212yes—
38C2×C4×D4×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C61×#400—21no—
41C61×#399—21no—
42C61×#389C2212yes—
43C61×#386—21no—
44C61×#385—21no—
45C61×#380C2212yes—
46C61×#380C2211yescharacteristic
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—24no—
50C2×#11758222—24no—
51C2×#11758222—22no—
52C2×#11758222—42no—
53C2×#11758222—42no—
54C2×#11758222D414yes—
55C22×Dic122C2311yescharacteristic
56C22×Dic122C2311yescharacteristic
57C22×D244C2311yescharacteristic
58C22×C22⋊D61—21no—
59C22×C22⋊D61—21no—
60C22×C4×D61—21no—
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C2×#11758194—22no—
64C2×#11758194—42no—
65C22×#92418C2312yes—
66C22×#92418C2312yes—
67#14519777—22no—
68#14519777—22no—
69#14519759—28no—
70#14519759—28no—
71#14519756—22no—
72#14519756—22no—
73#14519756—22no—
74#14519756D414yes—
75#14519748—28no—
76C4×D244—22no—
77C4×D244—22no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C4⋊C4×D61—28no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×#11758198—21no—
87#14519708—24no—
88#14519700—24no—
89#14519700—24no—
90#14519700—42no—
91#14519700—42no—
92#14519697—24no—
93#14519697—24no—
94C22⋊#92418—22no—
95C22⋊#92418—42no—
96C22×D4×C61D411yescharacteristic
97C23×C4×C61—21no—
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2312yes—
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#95×C61—22no—
115#95×C61—21no—
116#95×C61—22no—
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C23×D61—21no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—41no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—42no—
132C22×C4×C61—82no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—24no—
141C2×C22⋊D61—24no—
142C2×C22⋊D61—44no—
143C2×C22⋊D61—44no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147#11758222—28no—
148#11758222—44no—
149#11758222—48no—
150#11758222—48no—
151#11758222—44no—
152#11758222—44no—
153C2×D244—21no—
154C2×D244—22no—
155C2×D244C2×D412yes—
156C2×D244#4114yes—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—24no—
160C2×C4×D61—44no—
161C2×C4×D61—44no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×Dic122—22no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—22no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122C2×D412yes—
171C2×Dic122C2×D412yes—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61—82no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—82no—
183C42×C61—21no—
184C42×C61—21no—
185C22⋊Dic61—24no—
186C22⋊Dic61—24no—
187#11758198—24no—
188C4⋊Dic61—24no—
189C4⋊Dic61—24no—
190C4⋊Dic61—21no—
191C4⋊Dic61C2×D412yes—
192#11758196—24no—
193#11758196—24no—
194#11758194—44no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C23×C61—22no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—22no—
210C23×C61#12012yes—
211C23×C61C22×D411yescharacteristic
212C22×D61—22no—
213C22×D61—24no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—42no—
217D4×C61—44no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—84no—
238C2×C4×C61#12012yes—
239C2×C4×C61#13712yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4112yes—
243C2×C4×C61H(5,𝔽2)12yes—
244C22⋊D61—48no—
245C22⋊D61—48no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253D244—22no—
254D244—24no—
255D244#12314yes—
256C4×D61—48no—
257C4×D61—48no—
258C4×D61—48no—
259Dic122—22no—
260Dic122—22no—
261Dic122—24no—
262Dic122—41no—
263Dic122—41no—
264Dic122—42no—
265#92418—42no—
266#92418—42no—
267#24055—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—42no—
278C22×C61C2×#12011yescharacteristic
279C22×C61#54612yes—
280C22×C61C2×#13711yescharacteristic
281C22×C61#57511yescharacteristic
282C22×C61#57712yes—
283C2×D61—24no—
284C2×D61—44no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—44no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—82no—
293C4×C61—82no—
294C4×C61C2×#12311yescharacteristic
295C4×C61#53411yescharacteristic
296C4×C61#54612yes—
297Dic61—42no—
298Dic61—42no—
299Dic61—42no—
300C2×#547—611no—
301C2×#483—611no—
302C2×#417—611no—
303#2487D6111yescharacteristic, complemented
304#2043—612no—
305#2035—612no—
306#1894—611no—
307#1870—611no—
308#1864—612no—
309#1788—611no—
310#1788—612no—
311C2×C61—42no—
312C2×C61—81no—
313C2×C61#240511yescharacteristic
314C2×C61#300711yescharacteristic
315C2×C61#301011yescharacteristic
316C2×C61#330712yes—
317C2×C61#333112yes—
318D61—44no—
319C22×#41—611no—
320C22×#41—611no—
321C22×M16—611no—
322#547—1224no—
323#547—1224no—
324C2×C4⋊Q8—611no—
325C2×#123—1221no—
326C2×#123—1221no—
327C2×C22⋊Q8—611no—
328C2×C22⋊Q8—611no—
329C2×#121—1221no—
330C2×C4×D4—611no—
331C2×C4×D4C2×D6111yescharacteristic
332C22×C4⋊C4—1221no—
333C22×C4⋊C4C2×D6111yescharacteristic
334#483—1224no—
335C2×#106—612no—
336#417—1224no—
337C2×#103—612no—
338C2×#103—612no—
339C2×#103—612no—
340C2×#103—612no—
341C2×#102—612no—
342C2×#102—612no—
343C2×C22⋊C8—611no—
344C2×C22⋊C8—612no—
345#400—21no—
346#399—21no—
347#389C2×D6112yes—
348#386—21no—
349#385—21no—
350#380C2×D6111yescharacteristic
351#380C2×D6112yes—
352#340—612no—
353#336—612no—
354C61#2405511yescharacteristic, complemented, Sylow
355C2×#41—614no—
356C2×#41—1222no—
357C2×#41—1222no—
358C2×#41—1224no—
359C2×#41—1224no—
360C2×#41—1222no—
361C2×#41—2442no—
362C2×#41—2442no—
363C22×Q8—611no—
364C22×Q8—611no—
365C22×D4—611no—
366C22×D4—1221no—
367C22×D4—1221no—
368C22×D4C22⋊D6111yescharacteristic
369C23×C4—21no—
370C23×C4—1221no—
371C23×C4—1221no—
372C23×C4—1221no—
373C23×C4C22×D6111yescharacteristic
374C23×C4C22×D6111yescharacteristic
375C2×M16—1222no—
376C2×M16—2442no—
377C22×C8—612no—
378C22×C8—612no—
379C4⋊Q8—1222no—
380C4⋊Q8—1222no—
381#123—1228no—
382#123—1228no—
383C22⋊Q8—614no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386C22⋊Q8—1222no—
387#121—1228no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—1222no—
391C4×D4—1222no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1228no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4C22⋊D6111yescharacteristic
403C2×C4⋊C4C22×D6112yes—
404C2×C4⋊C4C22×D6112yes—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4C22⋊D6111yescharacteristic
411C2×C22⋊C4C22⋊D6111yescharacteristic
412C2×C42C22×D6111yescharacteristic
413#106—1224no—
414#103—1224no—
415#103—1224no—
416#103—2442no—
417#103—2442no—
418#102—1224no—
419#102—1224no—
420C22⋊C8—1222no—
421C22⋊C8—2442no—
422#95—22no—
423#95—21no—
424#95—22no—
425C24—1221no—
426C24C2×C22⋊D6111yescharacteristic
427C24C2×C22⋊D6111yescharacteristic
428#41—1228no—
429#41—2444no—
430#41—2448no—
431#41—2444no—
432#41—2448no—
433#41—2444no—
434C2×Q8—612no—
435C2×Q8—612no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1222no—
439C2×Q8—1222no—
440C2×Q8—1222no—
441C2×Q8—1222no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—41no—
445C2×D4—612no—
446C2×D4—614no—
447C2×D4—1224no—
448C2×D4—1224no—
449C2×D4—1221no—
450C2×D4—1222no—
451C2×D4—2444no—
452C2×D4—2444no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—82no—
463C22×C4—1224no—
464C22×C4—1224no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1224no—
469C22×C4—2444no—
470C22×C4—2444no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4D4×D6112yes—
474C22×C4D4×D6112yes—
475C22×C4C2×C22⋊D6111yescharacteristic
476C22×C4C2×C22⋊D6111yescharacteristic
477C22×C4C2×C22⋊D6111yescharacteristic
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C23×D6111yescharacteristic
480M16—2444no—
481C2×C8—1222no—
482C2×C8—1222no—
483C2×C8—1222no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—82no—
494C4⋊C4—612no—
495C4⋊C4—1224no—
496C4⋊C4—1224no—
497C4⋊C4—1224no—
498C4⋊C4—1221no—
499C4⋊C4—1224no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—42no—
503C22⋊C4—82no—
504C22⋊C4—1224no—
505C22⋊C4—1224no—
506C22⋊C4—1224no—
507C42—21no—
508C42—21no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—22no—
517C23—22no—
518C23—1222no—
519C23—1224no—
520C23—2441no—
521C23—2441no—
522C23—2442no—
523C23#1451974512yes—
524C23C22×C22⋊D6111yescharacteristic
525Q8—1222no—
526Q8—1222no—
527Q8—1224no—
528Q8—2441no—
529Q8—2441no—
530Q8—2442no—
531D4—44no—
532D4—614no—
533D4—1222no—
534D4—1224no—
535D4—2448no—
536D4—2448no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—2448no—
561C2×C4—2448no—
562C2×C4—2448no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4#1451974512yes—
568C2×C4#1451980812yes—
569C2×C4C2×D4×D6111yescharacteristic
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4#41×D6112yes—
572C2×C4#1451982312yes—
573C8—2442no—
574C8—2442no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—41no—
582C22—42no—
583C22—41no—
584C22—41no—
585C22—1224no—
586C22—2444no—
587C22—2444no—
588C22—2442no—
589C22—2442no—
590C22C2×#1451974511yescharacteristic
591C22#1684040012yes—
592C22C2×#1451980811yescharacteristic
593C22#1684056011yescharacteristic
594C22#1684057312yes—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—82no—
600C4—82no—
601C4—2442no—
602C4—2442no—
603C4—2442no—
604C4#123×D6111yescharacteristic
605C4#1684032311yescharacteristic
606C4#1684040412yes—
607C2—42no—
608C2—81no—
609C2—2444no—
610C2#2144914311yescharacteristic
611C2#2145507711yescharacteristic
612C2#2145510311yescharacteristic
613C2#2145778912yes—
614C2#2145812912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2487C2
socleC23×C61C22×D4
prime radicalC2×#16840417C2
hypercenter#2487D61
hypocenterC61#24055
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. C2×#16840092C2semidirect
3. C2×#16839761C2semidirect
4. #21447065C2semidirect
5. #21447015C2semidirect
6. #21446250C2semidirect
7. #21446117C2semidirect
8. #21446079C2semidirect
9. #21445757C2semidirect
10. #21445757C2semidirect
11. C61×#2487C2semidirect
303. #2487D61semidirect
354. C61#24055semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
C2×#168400922
C2×#168397612
#214470652
#214470152
#214462502
#214461172
#214460792
#214457572
C61×#24872
#2405561