Finite Group Databasebeta

← #29740458

Subgroups — #29740458

Order 15616.

812 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839983C211yesmaximal, characteristic
2C2×#16839868C211yesmaximal, characteristic, complemented
3#21449620C211yesmaximal, characteristic
4#21448736C211yesmaximal, characteristic, complemented
5#21447318C211yesmaximal, characteristic, complemented
6#21447238C211yesmaximal, characteristic, complemented
7#21447058C211yesmaximal, characteristic
8#21446513C211yesmaximal, characteristic
9#21446384C211yesmaximal, characteristic, complemented
10#21446177C211yesmaximal, characteristic
11#21446171C211yesmaximal, characteristic
12#21446159C211yesmaximal, characteristic, complemented
13#21446070C211yesmaximal, characteristic, complemented
14#21446070C211yesmaximal, characteristic, complemented
15C61×#2092C211yesmaximal, characteristic
16C22×Dic244C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×#14519742C2211yescharacteristic
23#16839983—24no—
24#16839868—24no—
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33#16839698—21no—
34#16839690—21no—
35#16839668C2211yescharacteristic
36#16839665C2211yescharacteristic
37#16839659—21no—
38#16839659—21no—
39#16839658C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839656—21no—
43#16839655—21no—
44#16839626—21no—
45C4×C4⋊Dic61C2211yescharacteristic
46#16839613—21no—
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#382C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C2×Dic244—22no—
64C2×Dic244—22no—
65C2×Dic244D414yes—
66C2×#11758200—22no—
67C2×#11758200—22no—
68C2×#11758200—22no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519764—22no—
72#14519764—41no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75#14519742—22no—
76#14519742—22no—
77#14519742—22no—
78#14519742—41no—
79#14519742—41no—
80#14519742—41no—
81C2×C4⋊Dic61D411yescharacteristic
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—21no—
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102#14519706—22no—
103#14519702—22no—
104#14519702—22no—
105#14519700—22no—
106#14519700—22no—
107#14519700—22no—
108#14519698—22no—
109#14519687—22no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117C22×D4×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#107×C61—41no—
127#106×C61—22no—
128#102×C61—22no—
129#102×C61—41no—
130#95×C61—21no—
131#95×C61—21no—
132C24×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122C2×D412yes—
162C2×Dic122C2×D412yes—
163C2×Dic122#4112yes—
164C2×Dic122#4112yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D412yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C22⋊C4×C61—42no—
178C22⋊C4×C61—81no—
179C42×C61—21no—
180C42×C61—21no—
181C22⋊Dic61—42no—
182C22⋊Dic61—42no—
183C22⋊Dic61—42no—
184C22⋊Dic61—81no—
185#11758200—44no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—41no—
190C4⋊Dic61—41no—
191C4⋊Dic61#4112yes—
192#11758196—24no—
193#11758196—42no—
194#11758196—42no—
195#11758196—42no—
196#11758196—42no—
197#11758196—42no—
198#11758196—81no—
199C4×Dic61—24no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213Dic244—28no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61C22×D411yescharacteristic
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
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×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271Dic122—22no—
272Dic122—22no—
273Dic122—22no—
274Dic122—22no—
275Dic122—22no—
276Dic122—41no—
277Dic122—41no—
278Dic122#12314yes—
279C8×C61—22no—
280C8×C61—41no—
281#92418—41no—
282#92418—41no—
283#26189—611nomaximal, Sylow
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61#52511yescharacteristic
292C22×C61D4211yescharacteristic
293C22×C61#54811yescharacteristic
294C22×C61#57511yescharacteristic
295C22×C61#57511yescharacteristic
296C22×C61#57711yescharacteristic
297C22×C61#57811yescharacteristic
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61—81no—
303C4×C61C2×#12311yescharacteristic
304C4×C61#53911yescharacteristic
305C4×C61#54711yescharacteristic
306C4×C61#55311yescharacteristic
307Dic61—24no—
308Dic61—42no—
309Dic61—42no—
310Dic61—41no—
311Dic61—41no—
312Dic61—41no—
313Dic61—41no—
314Dic61—81no—
315C2×#464—611no—
316C2×#441—611no—
317#2448—611no—
318#2351—611no—
319#2092D6111yescharacteristic
320#2088—611no—
321#2074—611no—
322#2042—611no—
323#1936—611no—
324#1917—611no—
325#1880—611no—
326#1880—611no—
327#1878—611no—
328#1863—611no—
329#1863—611no—
330C2×C61—161no—
331C2×C61#244311yescharacteristic
332C2×C61#311111yescharacteristic
333C2×C61#312111yescharacteristic
334C2×C61#327111yescharacteristic
335C2×C61#327711yescharacteristic
336C2×C61#337111yescharacteristic
337C2×C61#337711yescharacteristic
338C22×Q16—611no—
339C22×SD16—611no—
340C2×#124—611no—
341C2×C4×Q8—611no—
342C2×C4×Q8—611no—
343C2×C4×Q8—611no—
344C2×C4×Q8—611no—
345#464—1224no—
346#441—1224no—
347C2×#107C2×D6111yescharacteristic
348C2×#106C2×D6111yescharacteristic
349C2×C4⋊C8—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#102—611no—
356C2×#102C2×D6111yescharacteristic
357C2×#102C2×D6111yescharacteristic
358C2×#97—611no—
359#399—1221no—
360#397—1221no—
361#391—611no—
362#390—611no—
363#390C2×D6111yescharacteristic
364#389—611no—
365#389—611no—
366#389—611no—
367#389—1221no—
368#389—1221no—
369#388—1221no—
370#388—1221no—
371#382—1221no—
372#382C2×D6111yescharacteristic
373C4×C4⋊C4—611no—
374C4×C4⋊C4—1221no—
375#340—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#336—611no—
380#336—611no—
381#336—611no—
382#336C2×D6111yescharacteristic
383C61#2618911yescharacteristic, complemented, Sylow
384C22×Q8—611no—
385C22×Q8—611no—
386C22×D4C22×D6111yescharacteristic
387C2×Q16—614no—
388C2×Q16—1222no—
389C2×Q16—1222no—
390C2×SD16—1222no—
391C2×SD16—1222no—
392C2×SD16—1222no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397#124—1222no—
398#124—2441no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—2441no—
405C4×Q8—2441no—
406C4×Q8—2441no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—1221no—
430C2×C42—1221no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42C22×D6111yescharacteristic
434#107—41no—
435#106—22no—
436C4⋊C8—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#103—1222no—
441#103—1222no—
442#102—22no—
443#102—41no—
444#102—1222no—
445#97—1222no—
446#95—21no—
447#95—21no—
448#95—611no—
449#95—611no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455C24—21no—
456C2×Q8—612no—
457C2×Q8—612no—
458C2×Q8—612no—
459C2×Q8—612no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4D4×D6111yescharacteristic
486C22×C4D4×D6111yescharacteristic
487C22×C4C23×D6111yescharacteristic
488Q16—1228no—
489SD16—2444no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8D4×D6112yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—612no—
509C4⋊C4—1224no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—4881no—
514C4⋊C4—2441no—
515C4⋊C4—2441no—
516C4⋊C4—2442no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C22⋊C4—42no—
522C22⋊C4—81no—
523C22⋊C4—4881no—
524C22⋊C4—2442no—
525C22⋊C4—2442no—
526C22⋊C4—2442no—
527C42—21no—
528C42—21no—
529C42—1224no—
530C42—1221no—
531C42—1221no—
532C42—1221no—
533C42—1221no—
534C42—2442no—
535C42—2442no—
536C42—2442no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×D4×D6111yescharacteristic
545Q8—614no—
546Q8—1222no—
547Q8—1222no—
548Q8—1222no—
549Q8—1222no—
550Q8—1222no—
551Q8—2441no—
552Q8—2441no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—41no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1222no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4#41×D6111yescharacteristic
598C2×C4#41×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4#1451982211yescharacteristic
601C2×C4#1451982511yescharacteristic
602C8—22no—
603C8—41no—
604C8—2441no—
605C8—2441no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#1684027011yescharacteristic
614C22#1684039711yescharacteristic
615C22#1684042211yescharacteristic
616C22#1684055811yescharacteristic
617C22#1684055911yescharacteristic
618C22#1684057211yescharacteristic
619C22#1684057611yescharacteristic
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—1224no—
626C4—4881no—
627C4—2442no—
628C4—2441no—
629C4—2441no—
630C4—2442no—
631C4—2441no—
632C4—2441no—
633C4#123×D6111yescharacteristic
634C4#1684035811yescharacteristic
635C4#1684041411yescharacteristic
636C4#1684045711yescharacteristic
637C2—161no—
638C2#2144955711yescharacteristic
639C2#2145592311yescharacteristic
640C2#2145604311yescharacteristic
641C2#2145740311yescharacteristic
642C2#2145747411yescharacteristic
643C2#2145858511yescharacteristic
644C2#2145866411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2092C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2092D61
hypocenterC61#26189
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2092

Complements

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

NH ≅ G/Ntype
2. C2×#16839868C2semidirect
4. #21448736C2semidirect
5. #21447318C2semidirect
6. #21447238C2semidirect
9. #21446384C2semidirect
12. #21446159C2semidirect
13. #21446070C2semidirect
14. #21446070C2semidirect
383. C61#26189semidirect

Maximal subgroups

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

subgroupindex
C2×#168399832
C2×#168398682
#214496202
#214487362
#214473182
#214472382
#214470582
#214465132
#214463842
#214461772
#214461712
#214461592
#214460702
C61×#20922
#2618961