Finite Group Databasebeta

← #29688586

Subgroups — #29688586

Order 15616.

1094 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840153C211yesmaximal, characteristic
2C2×#16839870C211yesmaximal, characteristic, complemented
3C2×#16839867C211yesmaximal, characteristic, complemented
4C2×#16839862C211yesmaximal, characteristic
5C4×#14519775C211yesmaximal, characteristic, complemented
6C4×#14519764C211yesmaximal, characteristic
7#21446678C211yesmaximal, characteristic, complemented
8#21446677C211yesmaximal, characteristic, complemented
9#21446675C211yesmaximal, characteristic
10#21446674C211yesmaximal, characteristic
11#21446637C211yesmaximal, characteristic, complemented
12#21446634C211yesmaximal, characteristic
13#21445986C211yesmaximal, characteristic, complemented
14#97×Dic61C211yesmaximal, characteristic, complemented
15C2×C61×#440C211yesmaximal, characteristic
16C22×Dic244C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C22×#11758200C2211yescharacteristic
19C2×#14519775C2211yescharacteristic
20C2×#14519764C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×Dic122C2211yescharacteristic
23#16840153C418yescomplemented
24#16839870—24no—
25#16839867—24no—
26#16839862—24no—
27C2×#14519710C2211yescharacteristic
28C2×#14519709C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519698C2211yescharacteristic
36C2×#14519688C2211yescharacteristic
37C2×#14519687C2211yescharacteristic
38C2×#14519687C2211yescharacteristic
39C2×C8×Dic61C2211yescharacteristic
40C2×C8×Dic61C2211yescharacteristic
41#16839669C2211yescharacteristic
42#16839667C2211yescharacteristic
43#16839667C2211yescharacteristic
44#16839665C2211yescharacteristic
45#16839657—22no—
46#16839656—22no—
47#16839631C2211yescharacteristic
48#16839627C2211yescharacteristic
49C4×#11758196—22no—
50C4×C22⋊Dic61—22no—
51C42×Dic61C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C61×#440—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×Dic244—22no—
63C2×Dic244—22no—
64C2×Dic244C2×C414yescomplemented
65C2×#11758202—22no—
66C2×#11758202—22no—
67C2×#11758202C2×C414yescomplemented
68C2×#11758200—22no—
69C2×#11758200—22no—
70C2×#11758200C2×C414yes—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519775—22no—
74#14519775C2×C414yescomplemented
75#14519764—22no—
76#14519764C2×C414yes—
77C4×Dic122—22no—
78C4×Dic122—22no—
79C4×Dic122—22no—
80C4×Dic122—22no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—22no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—22no—
87C2×C22⋊Dic61—22no—
88C2×C22⋊Dic61—22no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61D412yes—
92C2×C4×Dic61D412yes—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519710—22no—
96#14519709—22no—
97#14519708—22no—
98#14519702—22no—
99#14519702—22no—
100#14519701—22no—
101#14519700—22no—
102#14519700—22no—
103#14519698—22no—
104#14519688D414yes—
105#14519687—22no—
106#14519687C2×C414yescomplemented
107C8×Dic61—22no—
108C8×Dic61—22no—
109#14519684—22no—
110#14519684—22no—
111#14519684—22no—
112#14519684—22no—
113C22×Q8×C61C2311yescharacteristic
114C22×D4×C61C2311yescharacteristic
115C23×C4×C61—21no—
116C2×SD16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61C2×C414yes—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C4×Q8×C61—22no—
122C4×Q8×C61—22no—
123C4×D4×C61—22no—
124C4×D4×C61—22no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—21no—
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#107×C61—22no—
132#103×C61—22no—
133#102×C61—22no—
134#97×C61D414yes—
135C24×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61C22×C412yes—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61C22×C412yes—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—44no—
152C22×C4×C61—42no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122C22×C412yes—
173C2×Dic122C22×C412yes—
174SD16×C61—28no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C22×C412yes—
178C2×C8×C61C2×D412yes—
179C2×C8×C61C2×D412yes—
180C2×C8×C61#4112yes—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—42no—
186C22⋊C4×C61—42no—
187C42×C61—42no—
188C42×C61C2×D412yes—
189C42×C61C2×D412yes—
190C22⋊Dic61—28no—
191C22⋊Dic61—44no—
192#11758202—28no—
193#11758200—28no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198#11758196—28no—
199#11758196—44no—
200#11758196—42no—
201#11758196—42no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—24no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C4×Dic61—42no—
208C4×Dic61C22×C412yes—
209C4×Dic61#4112yes—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C22×C412yes—
213C2×#92418C2×D412yes—
214C2×#92418C2×D412yes—
215C2×#92418#4112yes—
216Dic244—28no—
217C23×C61—22no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—42no—
221C23×C61—42no—
222C23×C61C22×D411yescharacteristic
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—41no—
230D4×C61—41no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—84no—
241C2×C4×C61#12712yes—
242C2×C4×C61#12712yes—
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61C4×D414yes—
264C2×Dic61#13714yes—
265Dic122—22no—
266Dic122—22no—
267Dic122—22no—
268Dic122—22no—
269Dic122—41no—
270Dic122—41no—
271Dic122—41no—
272Dic122—41no—
273C8×C61—22no—
274C8×C61C4×D414yes—
275#92418—22no—
276#92418C4×D414yes—
277#21702—611nomaximal, Sylow
278C22×C61—42no—
279C22×C61—44no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—82no—
283C22×C61C2×C4×D411yescharacteristic
284C22×C61C2×#12711yescharacteristic
285C22×C61#53211yescharacteristic
286C22×C61C2×#13712yes—
287C22×C61#57512yes—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—42no—
291C4×C61—81no—
292C4×C61—81no—
293C4×C61C2×C4×D411yescharacteristic
294C4×C61C2×C4×D411yescharacteristic
295C4×C61#53211yescharacteristic
296C4×C61#53211yescharacteristic
297Dic61—24no—
298Dic61—24no—
299Dic61—42no—
300Dic61—42no—
301Dic61—81no—
302Dic61—81no—
303C2×#497—611no—
304C2×#441—611no—
305C2×#441—611no—
306C2×#440—611no—
307C2×#440D6111yescharacteristic
308C4×C4⋊Q8—611no—
309C4×#124—611no—
310#1969—611no—
311#1969—611no—
312#1968—611no—
313#1968—611no—
314#1960—611no—
315#1959—611no—
316#1847—611no—
317C4×#97—611no—
318C2×C61—82no—
319C2×C61C4×#12711yescharacteristic
320C2×C61C4×#13712yes—
321C2×C61#315211yescharacteristic
322C2×C61#315311yescharacteristic
323C2×C61#318112yes—
324C22×Q16—611no—
325C22×Q16—611no—
326C22×SD16—611no—
327C22×SD16C2×D6111yescharacteristic
328C2×C4⋊Q8—611no—
329C2×#124—611no—
330C2×C4×Q8—611no—
331C2×C4×Q8—611no—
332C2×C4×Q8C2×D6111yescharacteristic
333C2×C4×D4C2×D6111yescharacteristic
334#497—618no—
335#441—1224no—
336#441—1224no—
337#440—24no—
338#440—1224no—
339C2×#107—611no—
340C2×#107C2×D6111yescharacteristic
341C2×#106—611no—
342C2×#106—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103—611no—
348C2×#103C2×D6111yescharacteristic
349C2×#102—611no—
350C2×#102C2×D6111yescharacteristic
351C2×#97—611no—
352C2×#97—611no—
353C2×#97—611no—
354C2×#97C2×D6111yescharacteristic
355C2×C4×C8—611no—
356C2×C4×C8—611no—
357#391—611no—
358#391—611no—
359#391—611no—
360#390—611no—
361#389—1222no—
362#388—1222no—
363#384—611no—
364#382—611no—
365C4×C4⋊C4—1222no—
366C4×C22⋊C4—1222no—
367C43—611no—
368C61#2170211yescharacteristic, complemented, Sylow
369C22×Q8—611no—
370C22×Q8—611no—
371C22×Q8C22×D6111yescharacteristic
372C22×D4C22×D6111yescharacteristic
373C23×C4—21no—
374C2×Q16—614no—
375C2×Q16—614no—
376C2×Q16—1222no—
377C2×Q16—1222no—
378C2×Q16—1222no—
379C2×Q16—1222no—
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16—614no—
383C2×SD16—1222no—
384C2×SD16—1222no—
385C2×SD16C4×D6114yes—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8C22×D6111yescharacteristic
389C22×C8C22×D6111yescharacteristic
390C4⋊Q8—614no—
391C4⋊Q8—1222no—
392#124—614no—
393#124—1222no—
394C4×Q8—22no—
395C4×Q8—22no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×D4—22no—
401C4×D4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1222no—
413C2×C22⋊C4—1222no—
414C2×C42—21no—
415C2×C42—611no—
416C2×C42—612no—
417C2×C42—611no—
418C2×C42—612no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42C22×D6111yescharacteristic
422#107—22no—
423#107—1222no—
424#106—1222no—
425#106—1222no—
426#103—22no—
427#103—1222no—
428#103—1222no—
429#103—1222no—
430#103—1222no—
431#103—1222no—
432#102—22no—
433#102—1222no—
434#97—614no—
435#97—614no—
436#97—1222no—
437#97C22⋊D6114yes—
438C4×C8—1222no—
439C4×C8—1222no—
440#95—1222no—
441#95—1222no—
442#95—1222no—
443#95—1222no—
444C24—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—612no—
451C2×Q8—612no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8C2×C4×D6112yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×C4×D6112yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4D4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4C23×D6111yescharacteristic
483Q16—1228no—
484Q16—1228no—
485SD16—28no—
486SD16—1228no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—612no—
490C2×C8—612no—
491C2×C8—612no—
492C2×C8—612no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8C2×C4×D6112yes—
496C2×C8#1175822212yes—
497C2×C8C2×C22⋊D6112yes—
498C2×C8C2×C22⋊D6112yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—1228no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—2444no—
510C4⋊C4—2442no—
511C4⋊C4—2442no—
512C22⋊C4—42no—
513C22⋊C4—1228no—
514C22⋊C4—2444no—
515C42—42no—
516C42—612no—
517C42—612no—
518C42—1224no—
519C42—1221no—
520C42—1221no—
521C42—2442no—
522C42—2442no—
523C42—2442no—
524C42D4×D6112yes—
525C42C2×C22⋊D6112yes—
526C23—22no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23C2×D4×D6111yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—1222no—
537Q8—1222no—
538Q8—1222no—
539Q8—1222no—
540Q8—2441no—
541Q8—2441no—
542Q8—2441no—
543Q8—2441no—
544D4—22no—
545D4—22no—
546D4—41no—
547D4—41no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—84no—
558C2×C4—614no—
559C2×C4—614no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—2442no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2442no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4#1451977312yes—
575C2×C4#1451977312yes—
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C2×C4#41×D6111yescharacteristic
581C2×C4C2×#1175822211yescharacteristic
582C8—22no—
583C8—614no—
584C8—1222no—
585C8C4×C22⋊D6114yes—
586C22—44no—
587C22—42no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22C4×D4×D6111yescharacteristic
592C22C2×#1451977311yescharacteristic
593C22#1684030911yescharacteristic
594C22#137×D6112yes—
595C22#1684055812yes—
596C4—22no—
597C4—22no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—1224no—
602C4—1224no—
603C4—4881no—
604C4—4881no—
605C4—2442no—
606C4—2442no—
607C4C4×D4×D6111yescharacteristic
608C4C2×C4×C22⋊D6111yescharacteristic
609C4#1684030811yescharacteristic
610C4#1684030911yescharacteristic
611C2—82no—
612C2C4×#1451977311yescharacteristic
613C2#2145464112yes—
614C2#2145627411yescharacteristic
615C2#2145628111yescharacteristic
616C2#2145652412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#440C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC2×#440D61
hypocenterC61#21702
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. C2×#16839870C2semidirect
3. C2×#16839867C2semidirect
5. C4×#14519775C2semidirect
7. #21446678C2semidirect
8. #21446677C2semidirect
11. #21446637C2semidirect
13. #21445986C2semidirect
14. #97×Dic61C2semidirect
23. #16840153C4semidirect
64. C2×Dic244C2×C4semidirect
67. C2×#11758202C2×C4semidirect
74. #14519775C2×C4semidirect
106. #14519687C2×C4semidirect
368. C61#21702semidirect

Maximal subgroups

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

subgroupindex
C2×#168401532
C2×#168398702
C2×#168398672
C2×#168398622
C4×#145197752
C4×#145197642
#214466782
#214466772
#214466752
#214466742
#214466372
#214466342
#214459862
#97×Dic612
C2×C61×#4402
#2170261