Finite Group Databasebeta

← #29709835

Subgroups — #29709835

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839803C211yesmaximal, characteristic, complemented
2C2×#16839801C211yesmaximal, characteristic, complemented
3C2×#16839762C211yesmaximal, characteristic
4C2×#16839762C211yesmaximal, characteristic
5#21448308C211yesmaximal, characteristic
6#382×D61C211yesmaximal, characteristic, complemented
7#21447369C211yesmaximal, characteristic, complemented
8#21447367C211yesmaximal, characteristic, complemented
9#21447365C211yesmaximal, characteristic, complemented
10#21447343C211yesmaximal, characteristic, complemented
11#21447341C211yesmaximal, characteristic, complemented
12#21447339C211yesmaximal, characteristic, complemented
13#21447337C211yesmaximal, characteristic, complemented
14#21445511C211yesmaximal, characteristic, complemented
15C61×#2097C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C22×#11758189C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×C4⋊C4×D61C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C42×D61C2211yescharacteristic
23#16839803—24no—
24#16839801—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519709C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31#16839762—24no—
32#16839762—24no—
33C2×#14519701C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37C2×#14519697C2211yescharacteristic
38C2×#14519697C2211yescharacteristic
39#16839669C2211yescharacteristic
40#16839664C2211yescharacteristic
41#16839663C2211yescharacteristic
42#16839640—22no—
43#16839628C2211yescharacteristic
44#16839627C2211yescharacteristic
45#16839627C2211yescharacteristic
46#16839626—22no—
47#16839609—22no—
48#95×D61—22no—
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#107×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#382C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C2×#11758226—22no—
60C2×#11758226—22no—
61C2×#11758226—22no—
62C2×#11758226—22no—
63C2×#11758226—22no—
64C2×#11758226Q814yes—
65C22×D244—21no—
66C22×D244C2311yescharacteristic
67C22×C4×D61—21no—
68C22×C4×D61—21no—
69C22×C4×D61—21no—
70C22×C4×D61D412yes—
71C22×C4×D61C2311yescharacteristic
72C2×#11758189—22no—
73C2×#11758189—22no—
74C2×#11758189—22no—
75C2×#11758189—22no—
76C2×#11758189D414yes—
77C2×#11758189D414yes—
78C22×#92418C2311yescharacteristic
79C22×#92418C2311yescharacteristic
80#14519735—22no—
81#14519735—22no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61—22no—
91C2×#11758198—21no—
92C2×#11758198—22no—
93C2×#11758198—22no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61D412yes—
96C2×C4×Dic61C2311yescharacteristic
97C42×D61—22no—
98C42×D61—22no—
99#14519711—22no—
100#14519710—22no—
101#14519710—22no—
102#14519709—22no—
103#14519708—22no—
104#14519708—22no—
105#14519701—22no—
106#14519701—22no—
107#14519699—22no—
108#14519699—22no—
109#14519697—22no—
110#14519697—22no—
111#14519684—22no—
112#14519684—22no—
113#14519684—22no—
114#14519684—22no—
115C22×Q8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122#107×C61—22no—
123#106×C61—22no—
124#103×C61—22no—
125#103×C61—22no—
126#95×C61—22no—
127#95×C61—22no—
128C23×D61—21no—
129C23×D61C2×D411yescharacteristic
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61C2×Q812yes—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61C2×D412yes—
146C22×Dic61C2×D411yescharacteristic
147#11758226—28no—
148#11758226—28no—
149#11758226—44no—
150#11758226—44no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—24no—
156C2×D244—21no—
157C2×D244—42no—
158C2×D244—42no—
159C2×D244C2×Q812yes—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—44no—
167C2×C4×D61—44no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61C2×D412yes—
171C2×C4×D61C2×D412yes—
172C2×C4×D61C2×Q812yes—
173C2×C4×D61#4112yes—
174C2×C4×D61#4112yes—
175C2×C4×D61#4112yes—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61C2×D412yes—
181C2×C8×C61C2×D412yes—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190#11758198—44no—
191#11758198—44no—
192#11758198—42no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199#11758196—42no—
200#11758196—42no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—24no—
204C4×Dic61—42no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D412yes—
210C2×#92418C2×D412yes—
211#11758189—28no—
212#11758189—28no—
213C23×C61C22×D411yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61C22⋊Q812yes—
222C22×D61#12312yes—
223C22×D61#12712yes—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—41no—
227Q8×C61—41no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×Q811yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61C22⋊Q812yes—
256C2×Dic61#12312yes—
257C2×Dic61#12712yes—
258D244—22no—
259D244—22no—
260D244—48no—
261D244—41no—
262D244—41no—
263C4×D61—48no—
264C4×D61—48no—
265C4×D61—84no—
266C4×D61—84no—
267C4×D61C22⋊Q814yes—
268C4×D61C22⋊Q814yes—
269C4×D61#12314yes—
270C4×D61#12314yes—
271C8×C61—22no—
272C8×C61—22no—
273#92418—22no—
274#92418—22no—
275#23733—611nomaximal, Sylow
276C22×C61C2×C22⋊Q811yescharacteristic
277C22×C61C2×#12311yescharacteristic
278C22×C61C2×#12711yescharacteristic
279C22×C61#57512yes—
280C22×C61#57512yes—
281C2×D61—24no—
282C2×D61—24no—
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—44no—
286C2×D61—42no—
287C2×D61—82no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61#39914yes—
291C4×C61—42no—
292C4×C61—42no—
293C4×C61—81no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61C2×C22⋊Q811yescharacteristic
297C4×C61C2×C22⋊Q811yescharacteristic
298C4×C61C2×#12311yescharacteristic
299C4×C61C2×#12311yescharacteristic
300Dic61—42no—
301Dic61—81no—
302Dic61—81no—
303Dic61—81no—
304Dic61#39914yes—
305C2×#428—611no—
306C2×#428—611no—
307C2×#417—611no—
308C2×#417—611no—
309#2297—611no—
310C2×#382—611no—
311#2097—611no—
312#2097—611no—
313#2097—611no—
314#2097D6111yescharacteristic, complemented
315#2092—611no—
316#2092—611no—
317#2092—611no—
318#2092—611no—
319#1728—611no—
320C2×C61C2×#39911yescharacteristic
321C2×C61#308012yes—
322C2×C61#309312yes—
323C2×C61#315311yescharacteristic
324C2×C61#315311yescharacteristic
325D61—24no—
326D61—82no—
327C22×#41—611no—
328C22×M16—611no—
329C22×M16—611no—
330C2×#117—611no—
331C22×C4⋊C4—611no—
332C22×C4⋊C4—611no—
333C22×C42—611no—
334#428—1224no—
335#428—1224no—
336C2×#107—611no—
337C2×#107—611no—
338C2×#107—611no—
339C2×#107C2×D6111yescharacteristic
340C2×#106—611no—
341C2×#106—611no—
342C2×#106—611no—
343C2×#106C2×D6111yescharacteristic
344#417—1224no—
345#417—1224no—
346C2×#103—611no—
347C2×#103—611no—
348C2×#103C2×D6111yescharacteristic
349C2×#103C2×D6111yescharacteristic
350C2×#102—611no—
351C2×#102—611no—
352C2×#102—611no—
353C2×#102—611no—
354#391—611no—
355#391C2×D6111yescharacteristic
356#390—611no—
357#390—611no—
358#386—1222no—
359#382—611no—
360#382—611no—
361#382—611no—
362#382—1222no—
363#382C2×D6111yescharacteristic
364C4×C22⋊C4—1222no—
365C2×#95—1222no—
366#336—611no—
367#336—611no—
368#336—611no—
369#336C2×D6111yescharacteristic
370C61#2373311yescharacteristic, complemented, Sylow
371C2×#41—614no—
372C2×#41—1222no—
373C2×#41—1222no—
374C2×#41—1222no—
375C2×#41—1222no—
376C2×#41—1222no—
377C22×Q8C22×D6111yescharacteristic
378C22×D4—611no—
379C22×D4—1221no—
380C23×C4—611no—
381C23×C4—612no—
382C23×C4—1221no—
383C23×C4—1221no—
384C23×C4—1221no—
385C2×M16—614no—
386C2×M16—614no—
387C2×M16—1222no—
388C2×M16—1222no—
389C2×M16—1222no—
390C2×M16—1222no—
391C22×C8—611no—
392C22×C8—611no—
393C22×C8C22×D6111yescharacteristic
394C22×C8C22×D6111yescharacteristic
395#117—1222no—
396#117—1222no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4—1222no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C22⋊C4—1222no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1222no—
412C2×C42—612no—
413C2×C42—611no—
414C2×C42—1222no—
415C2×C42—1222no—
416C2×C42—1221no—
417C2×C42C22×D6111yescharacteristic
418#107—22no—
419#107—1222no—
420#107—1222no—
421#107—1222no—
422#106—22no—
423#106—1222no—
424#106—1222no—
425#106—1222no—
426#103—22no—
427#103—22no—
428#103—1222no—
429#103—1222no—
430#102—1222no—
431#102—1222no—
432#102—1222no—
433#102—1222no—
434#95—22no—
435#95—22no—
436#95—1222no—
437#95—1222no—
438#95—1222no—
439#95—1222no—
440C24—611no—
441C24—1221no—
442#41—1228no—
443#41—1228no—
444#41—2444no—
445#41—2444no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8Q8×D6112yes—
452C2×D4—612no—
453C2×D4—1224no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—2442no—
460C2×D4—2442no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—611no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—612no—
470C22×C4—612no—
471C22×C4—612no—
472C22×C4—612no—
473C22×C4—1224no—
474C22×C4—1224no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4—2442no—
484C22×C4—2444no—
485C22×C4—2444no—
486C22×C4D4×D6112yes—
487C22×C4C23×D6111yescharacteristic
488M16—1228no—
489M16—1228no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—612no—
495C2×C8—612no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8D4×D6112yes—
501C2×C8D4×D6112yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—2442no—
515C4⋊C4—2442no—
516C22⋊C4—2444no—
517C22⋊C4—2444no—
518C22⋊C4—2442no—
519C42—21no—
520C42—21no—
521C42—1224no—
522C42—1221no—
523C42—1221no—
524C42—2442no—
525C23—612no—
526C23—612no—
527C23—612no—
528C23—1222no—
529C23—1222no—
530C23—1222no—
531C23—2441no—
532C23—2441no—
533C23—2442no—
534C23—2442no—
535C23C2×D4×D6111yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—1222no—
541D4—1222no—
542D4—2448no—
543D4—2441no—
544D4—2441no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—612no—
558C2×C4—612no—
559C2×C4—612no—
560C2×C4—614no—
561C2×C4—614no—
562C2×C4—614no—
563C2×C4—614no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—4884no—
568C2×C4—4884no—
569C2×C4—2448no—
570C2×C4—2448no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C2×Q8×D6111yescharacteristic
580C2×C4#41×D6111yescharacteristic
581C2×C4#41×D6111yescharacteristic
582C2×C4#41×D6111yescharacteristic
583C8—22no—
584C8—22no—
585C8—1222no—
586C8—1222no—
587C22—614no—
588C22—1224no—
589C22—1224no—
590C22—1222no—
591C22—1222no—
592C22—4882no—
593C22—4881no—
594C22—4881no—
595C22—2444no—
596C22—2442no—
597C22C22⋊Q8×D6111yescharacteristic
598C22#123×D6111yescharacteristic
599C22#127×D6111yescharacteristic
600C22#1684055912yes—
601C22#1684056112yes—
602C4—42no—
603C4—42no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—614no—
608C4—4881no—
609C4—4881no—
610C4—4881no—
611C4—2442no—
612C4C22⋊Q8×D6111yescharacteristic
613C4C22⋊Q8×D6111yescharacteristic
614C4#123×D6111yescharacteristic
615C4#123×D6111yescharacteristic
616C2—1224no—
617C2—4882no—
618C2#399×D6111yescharacteristic
619C2#2145563812yes—
620C2#2145575112yes—
621C2#2145628011yescharacteristic
622C2#2145628211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2097C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
hypercenter#2097D61
hypocenterC61#23733
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2097

Complements

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

NH ≅ G/Ntype
1. C2×#16839803C2semidirect
2. C2×#16839801C2semidirect
6. #382×D61C2semidirect
7. #21447369C2semidirect
8. #21447367C2semidirect
9. #21447365C2semidirect
10. #21447343C2semidirect
11. #21447341C2semidirect
12. #21447339C2semidirect
13. #21447337C2semidirect
14. #21445511C2semidirect
15. C61×#2097C2semidirect
314. #2097D61semidirect
370. C61#23733semidirect

Maximal subgroups

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

subgroupindex
C2×#168398032
C2×#168398012
C2×#168397622
#214483082
#382×D612
#214473692
#214473672
#214473652
#214473432
#214473412
#214473392
#214473372
#214455112
C61×#20972
#2373361