Finite Group Databasebeta

← #29740174

Subgroups — #29740174

Order 15616.

868 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839967C211yesmaximal, characteristic
2C2×#16839853C211yesmaximal, characteristic
3#21449597C211yesmaximal, characteristic, complemented
4#21448724C211yesmaximal, characteristic
5#21447364C211yesmaximal, characteristic, complemented
6#21447320C211yesmaximal, characteristic, complemented
7#21447241C211yesmaximal, characteristic, complemented
8#21447066C211yesmaximal, characteristic, complemented
9#21446493C211yesmaximal, characteristic, complemented
10#21446379C211yesmaximal, characteristic, complemented
11#21446151C211yesmaximal, characteristic
12#21446151C211yesmaximal, characteristic
13#21446151C211yesmaximal, characteristic
14#21446070C211yesmaximal, characteristic, complemented
15C61×#1862C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519777C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22#16839967—24no—
23#16839853—24no—
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839704—21no—
34#16839695—21no—
35#16839667C2211yescharacteristic
36#16839667C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839649—21no—
41#16839648—21no—
42#16839635—21no—
43#16839634—21no—
44#16839627C2211yescharacteristic
45#16839620—21no—
46C4×#11758196C2211yescharacteristic
47C2×#14519684—21no—
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C23×Dic61—21no—
65C2×#11758200—22no—
66C2×#11758200—22no—
67C2×#11758200—22no—
68C2×#11758200—22no—
69C2×#11758200—22no—
70C2×#11758200D414yes—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519777—22no—
74#14519777—41no—
75#14519742—22no—
76#14519742—41no—
77D4×Dic61—22no—
78D4×Dic61—22no—
79D4×Dic61—22no—
80D4×Dic61—41no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196D411yescharacteristic
89C2×#11758196D411yescharacteristic
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519708—22no—
96C4⋊#92418—22no—
97#14519702—22no—
98#14519702—22no—
99#14519702—22no—
100#14519702—41no—
101#14519700—22no—
102#14519700—22no—
103C8×Dic61—22no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×D4×C61C2311yescharacteristic
117C23×C4×C61—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—41no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#107×C61—41no—
129#102×C61—22no—
130#102×C61—22no—
131#95×C61—21no—
132C24×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61C2×D412yes—
137C2×D4×C61#4112yes—
138C2×D4×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—81no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—24no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—42no—
154C22×Dic61—42no—
155C22×Dic61—81no—
156C22×Dic61—81no—
157C22×Dic61—81no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
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×Dic122—21no—
173C2×Dic122C2×D412yes—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—42no—
187C22⋊C4×C61—42no—
188C42×C61—21no—
189C42×C61—21no—
190C22⋊Dic61—24no—
191C22⋊Dic61—42no—
192#11758200—28no—
193#11758200—44no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—42no—
198C4⋊Dic61—42no—
199C4⋊Dic61#4112yes—
200#11758196—22no—
201#11758196—22no—
202#11758196—42no—
203#11758196—81no—
204#11758196—81no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—42no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D412yes—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61C22×D411yescharacteristic
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—22no—
227D4×C61#12314yes—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—84no—
243C2×C4×C61#13512yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—84no—
274C2×Dic61—84no—
275C2×Dic61#13712yes—
276Dic122—22no—
277Dic122—22no—
278Dic122—41no—
279Dic122—41no—
280Dic122—41no—
281Dic122—41no—
282Dic122—41no—
283Dic122—41no—
284C8×C61—41no—
285C8×C61—41no—
286#92418—22no—
287#92418—41no—
288#26170—611nomaximal, Sylow
289C22×C61—24no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61#52511yescharacteristic
297C22×C61D4211yescharacteristic
298C22×C61#54811yescharacteristic
299C22×C61C2×#13511yescharacteristic
300C22×C61C2×#13711yescharacteristic
301C22×C61#57711yescharacteristic
302C22×C61#57711yescharacteristic
303C4×C61—22no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×#12311yescharacteristic
309C4×C61#53911yescharacteristic
310C4×C61#54711yescharacteristic
311C4×C61#55311yescharacteristic
312Dic61—22no—
313Dic61—41no—
314Dic61—41no—
315Dic61—42no—
316Dic61—42no—
317Dic61—41no—
318Dic61—81no—
319Dic61—81no—
320C2×#461—611no—
321C2×C4×SD16—611no—
322#2446—611no—
323#2349—611no—
324#2097—611no—
325#2088—611no—
326#2074—611no—
327#2043—611no—
328#1933—611no—
329#1916—611no—
330#1877—611no—
331#1877—611no—
332#1877—611no—
333#1863—611no—
334#1862D6111yescharacteristic
335C2×C61—44no—
336C2×C61#244311yescharacteristic
337C2×C61#310711yescharacteristic
338C2×C61#312311yescharacteristic
339C2×C61#326611yescharacteristic
340C2×C61#328211yescharacteristic
341C2×C61#336011yescharacteristic
342C2×C61#337211yescharacteristic
343C22×SD16—611no—
344C22×SD16—611no—
345C2×C4⋊Q8—611no—
346C2×C4×Q8—611no—
347C2×C4×D4—611no—
348C2×C4×D4—611no—
349C2×C4×D4C2×D6111yescharacteristic
350#461—1224no—
351C4×SD16—1224no—
352C2×#107C2×D6111yescharacteristic
353C2×#106—611no—
354C2×C4⋊C8—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102C2×D6111yescharacteristic
362C2×#102C2×D6111yescharacteristic
363C2×C4×C8—611no—
364#400—1221no—
365#398—1221no—
366#391—611no—
367#391—611no—
368#391—611no—
369#389—611no—
370#389—611no—
371#389C2×D6111yescharacteristic
372#387—1221no—
373#387—1221no—
374#385—1221no—
375#385—1221no—
376#382—611no—
377#380—1221no—
378C4×C4⋊C4—611no—
379C2×#95—1221no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340C2×D6111yescharacteristic
384#336—611no—
385#336—611no—
386#336—611no—
387#336C2×D6111yescharacteristic
388C61#2617011yescharacteristic, complemented, Sylow
389C22×Q8—611no—
390C22×Q8—611no—
391C22×D4C22×D6111yescharacteristic
392C23×C4—21no—
393C23×C4—1221no—
394C23×C4—1221no—
395C2×SD16—614no—
396C2×SD16—1222no—
397C2×SD16—1222no—
398C2×SD16—1222no—
399C2×SD16—1222no—
400C2×SD16—1222no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405C4⋊Q8—1222no—
406C4⋊Q8—2441no—
407C4×Q8—1222no—
408C4×Q8—2441no—
409C4×D4—22no—
410C4×D4—41no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—2441no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—1221no—
434C2×C42C22×D6111yescharacteristic
435#107—41no—
436#106—1222no—
437C4⋊C8—1222no—
438#103—1222no—
439#103—1222no—
440#103—1222no—
441#103—1222no—
442#103—1222no—
443#103—2441no—
444#102—22no—
445#102—22no—
446C4×C8—1222no—
447#95—21no—
448#95—611no—
449#95—611no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460C24—21no—
461C2×Q8—612no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4#1175822212yes—
477C2×D4D4×D6112yes—
478C2×D4#1175822612yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—1224no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—4881no—
495C22×C4—4881no—
496C22×C4—4881no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501C22×C4C23×D6111yescharacteristic
502SD16—1228no—
503SD16—2444no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—612no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—612no—
524C4⋊C4—1222no—
525C4⋊C4—1221no—
526C4⋊C4—1222no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—4881no—
530C4⋊C4—4881no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C22⋊C4—42no—
535C22⋊C4—1224no—
536C22⋊C4—2442no—
537C42—21no—
538C42—21no—
539C42—1221no—
540C42—1221no—
541C42—1221no—
542C42—1221no—
543C42—2442no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C2×D4×D6111yescharacteristic
552Q8—1222no—
553Q8—1222no—
554Q8—2441no—
555Q8—2441no—
556Q8—2441no—
557Q8—2441no—
558Q8—2441no—
559Q8—2441no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4#1451976214yes—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—612no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1222no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—4884no—
594C2×C4—4884no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4#1451979612yes—
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4#41×D6111yescharacteristic
608C2×C4C2×#1175822211yescharacteristic
609C2×C4C2×#1175822611yescharacteristic
610C2×C4#1451982211yescharacteristic
611C2×C4#1451982411yescharacteristic
612C8—41no—
613C8—41no—
614C8—1222no—
615C8—2441no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22#1684027511yescharacteristic
624C22#1684039411yescharacteristic
625C22#1684042111yescharacteristic
626C22C2×#1451979611yescharacteristic
627C22#137×D6111yescharacteristic
628C22#1684056911yescharacteristic
629C22#1684057211yescharacteristic
630C4—22no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—1222no—
636C4—4881no—
637C4—4881no—
638C4—2441no—
639C4—2442no—
640C4—2441no—
641C4—2441no—
642C4—2442no—
643C4C2×#1451976211yescharacteristic
644C4#1684036111yescharacteristic
645C4#1684041011yescharacteristic
646C4#1684046111yescharacteristic
647C2—44no—
648C2#2144955811yescharacteristic
649C2#2145589211yescharacteristic
650C2#2145607311yescharacteristic
651C2#2145732711yescharacteristic
652C2#2145753011yescharacteristic
653C2#2145846611yescharacteristic
654C2#2145860111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1862C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1862D61
hypocenterC61#26170
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
3. #21449597C2semidirect
5. #21447364C2semidirect
6. #21447320C2semidirect
7. #21447241C2semidirect
8. #21447066C2semidirect
9. #21446493C2semidirect
10. #21446379C2semidirect
14. #21446070C2semidirect
388. C61#26170semidirect

Maximal subgroups

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

subgroupindex
C2×#168399672
C2×#168398532
#214495972
#214487242
#214473642
#214473202
#214472412
#214470662
#214464932
#214463792
#214461512
#214460702
C61×#18622
#2617061