Finite Group Databasebeta

← #29740176

Subgroups — #29740176

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×#16839965C211yesmaximal, characteristic
2C2×#16839853C211yesmaximal, characteristic
3#21449598C211yesmaximal, characteristic, complemented
4#21448729C211yesmaximal, characteristic
5#21447367C211yesmaximal, characteristic, complemented
6#21447322C211yesmaximal, characteristic, complemented
7#21447239C211yesmaximal, characteristic, complemented
8#21447064C211yesmaximal, characteristic, complemented
9#21446497C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446155C211yesmaximal, characteristic
12#21446153C211yesmaximal, characteristic
13#21446074C211yesmaximal, characteristic, complemented
14#21446064C211yesmaximal, characteristic
15C61×#1877C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21#16839965—24no—
22#16839853—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839703—21no—
34#16839696—21no—
35#16839669C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839645—21no—
41#16839644—21no—
42#16839637—21no—
43#16839637—21no—
44#16839628C2211yescharacteristic
45#16839622—21no—
46C4×C4⋊Dic61C2211yescharacteristic
47C2×#14519684—21no—
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C23×Dic61—21no—
63C23×Dic61—21no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200D414yes—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519775—22no—
70#14519775—41no—
71#14519742—22no—
72#14519742—41no—
73D4×Dic61—22no—
74D4×Dic61—22no—
75D4×Dic61—41no—
76D4×Dic61—41no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61D411yescharacteristic
80C2×C4⋊Dic61D411yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519710—41no—
94#14519708—22no—
95#14519706—22no—
96#14519702—22no—
97#14519702—22no—
98#14519701—22no—
99#14519701—41no—
100#14519700—22no—
101#14519698—22no—
102C8×Dic61—22no—
103#14519684—21no—
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#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114C22×Q8×C61C2311yescharacteristic
115C22×D4×C61C2311yescharacteristic
116C23×C4×C61—21no—
117C2×SD16×C61—22no—
118C2×SD16×C61—22no—
119C2×SD16×C61—22no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C4×D4×C61—22no—
123C4×D4×C61—22no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#103×C61—22no—
129#102×C61—22no—
130#95×C61—21no—
131#95×C61—21no—
132C24×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61C2×D412yes—
143C2×D4×C61#4112yes—
144C2×D4×C61#4112yes—
145C22×C4×C61—21no—
146C22×C4×C61—24no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—81no—
150C22×C4×C61—81no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61—81no—
163C22×Dic61—81no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C2×D412yes—
174SD16×C61—44no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61#4112yes—
185C22⋊C4×C61—24no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—42no—
189C22⋊Dic61—42no—
190#11758200—28no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—81no—
200C4⋊Dic61—81no—
201#11758196—42no—
202#11758196—42no—
203#11758196—42no—
204#11758196—42no—
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
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—41no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61#12314yes—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—84no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×C4×C612−1+411yescharacteristic
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
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—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—84no—
277C2×Dic61—84no—
278C2×Dic61#13512yes—
279C2×Dic61#13712yes—
280Dic122—22no—
281Dic122—22no—
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—41no—
304C4×C61—41no—
305C4×C61—42no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×#12311yescharacteristic
309C4×C61#53911yescharacteristic
310C4×C61#54711yescharacteristic
311C4×C61#55311yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—42no—
315Dic61—41no—
316Dic61—41no—
317Dic61—81no—
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#1877D6111yescharacteristic
333#1863—611no—
334#1862—611no—
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×SD16C2×D6111yescharacteristic
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×#107—611no—
353C2×#106—611no—
354C2×C4⋊C8—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103C2×D6111yescharacteristic
361C2×#102—611no—
362C2×#102C2×D6111yescharacteristic
363C2×C4×C8—611no—
364#400—1221no—
365#398—1221no—
366#391—611no—
367#391—611no—
368#391C2×D6111yescharacteristic
369#389—611no—
370#389—611no—
371#389—611no—
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×Q8C22×D6111yescharacteristic
391C22×D4C22×D6111yescharacteristic
392C23×C4—21no—
393C23×C4—1221no—
394C23×C4—1221no—
395C2×SD16—22no—
396C2×SD16—22no—
397C2×SD16—22no—
398C2×SD16—614no—
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—22no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—2441no—
414C4×D4—2441no—
415C2×C4⋊C4—611no—
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⋊C4—1221no—
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—2441no—
436#106—1222no—
437C4⋊C8—1222no—
438#103—22no—
439#103—1222no—
440#103—1222no—
441#103—1222no—
442#103—1222no—
443#103—2441no—
444#102—22no—
445#102—1222no—
446C4×C8—1222no—
447#95—21no—
448#95—21no—
449#95—611no—
450#95—611no—
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—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—612no—
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#1175822412yes—
479C22×C4—21no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—81no—
484C22×C4—81no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—4881no—
494C22×C4—4881no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501C22×C4C23×D6111yescharacteristic
502SD16—44no—
503SD16—1228no—
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—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—4881no—
528C4⋊C4—4881no—
529C4⋊C4—2442no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4#1175822412yes—
534C22⋊C4—24no—
535C22⋊C4—2442no—
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—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556Q8—1222no—
557Q8—1222no—
558Q8—2441no—
559Q8—2441no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4#1451975914yes—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
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—2441no—
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4C2×#1175822411yescharacteristic
608C2×C4C2×#1175822411yescharacteristic
609C2×C4C2×#1175822211yescharacteristic
610C2×C4#1451982311yescharacteristic
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#1684026811yescharacteristic
624C22#1684039611yescharacteristic
625C22#1684042111yescharacteristic
626C22#135×D6111yescharacteristic
627C22#137×D6111yescharacteristic
628C22#1684056811yescharacteristic
629C22#1684057211yescharacteristic
630C4—41no—
631C4—41no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—1222no—
636C4—1222no—
637C4—4881no—
638C4—4881no—
639C4—4881no—
640C4—2442no—
641C4—2441no—
642C4—2441no—
643C4C2×#1451975911yescharacteristic
644C4#1684035611yescharacteristic
645C4#1684041211yescharacteristic
646C4#1684045811yescharacteristic
647C2—44no—
648C2#2144956011yescharacteristic
649C2#2145588311yescharacteristic
650C2#2145606511yescharacteristic
651C2#2145733511yescharacteristic
652C2#2145753211yescharacteristic
653C2#2145846911yescharacteristic
654C2#2145860111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1877C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1877D61
hypocenterC61#26170
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
3. #21449598C2semidirect
5. #21447367C2semidirect
6. #21447322C2semidirect
7. #21447239C2semidirect
8. #21447064C2semidirect
9. #21446497C2semidirect
10. #21446375C2semidirect
13. #21446074C2semidirect
388. C61#26170semidirect

Maximal subgroups

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

subgroupindex
C2×#168399652
C2×#168398532
#214495982
#214487292
#214473672
#214473222
#214472392
#214470642
#214464972
#214463752
#214461552
#214461532
#214460742
#214460642
C61×#18772
#2617061