Finite Group Databasebeta

← #29699206

Subgroups — #29699206

Order 15616.

902 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839853C211yesmaximal, characteristic
2#21448772C211yesmaximal, characteristic, complemented
3D4×#11758196C211yesmaximal, characteristic
4#21446509C211yesmaximal, characteristic, complemented
5#21446507C211yesmaximal, characteristic, complemented
6#21446404C211yesmaximal, characteristic, complemented
7#21446402C211yesmaximal, characteristic, complemented
8#21446153C211yesmaximal, characteristic
9#21446151C211yesmaximal, characteristic
10#21445945C211yesmaximal, characteristic, complemented
11#21445823C211yesmaximal, characteristic, complemented
12#21445817C211yesmaximal, characteristic
13#21445687C211yesmaximal, characteristic, complemented
14#21445646C211yesmaximal, characteristic
15C61×#1762C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519742C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C22×#11758196—21no—
21#16839853C418yescomplemented
22C2×#14519708C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839668C2211yescharacteristic
31#16839667C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839646—21no—
35#16839644—21no—
36#16839635—21no—
37#16839634—21no—
38#16839631C2211yescharacteristic
39#16839631C2211yescharacteristic
40#16839626—21no—
41#16839620—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C23×Dic61—21no—
62C23×Dic61—21no—
63C2×#11758200—22no—
64C2×#11758200—22no—
65C2×#11758200C2×C414yes—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519742—22no—
69#14519742C2×C414yescomplemented
70D4×Dic61—22no—
71D4×Dic61—22no—
72D4×Dic61—41no—
73D4×Dic61C2×C414yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—44no—
85C2×#11758196D411yescharacteristic
86C2×#11758196D411yescharacteristic
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519708C2×C414yescomplemented
94#14519706—22no—
95#14519702—41no—
96#14519702C2×C414yescomplemented
97#14519700—41no—
98#14519698—22no—
99#14519687—41no—
100C8×Dic61C2×C414yescomplemented
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C23×C4×C61—21no—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C4×D4×C61—22no—
115C4×D4×C61—41no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#102×C61—22no—
123#102×C61C2×C414yes—
124#97×C61—41no—
125#95×C61—21no—
126C24×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61C22×C412yes—
131C2×D4×C61#4112yes—
132C2×D4×C61#4112yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61—81no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—24no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61—81no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122C22×C412yes—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61C22×C412yes—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22×C412yes—
174C22⋊C4×C61—42no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—81no—
178C22⋊Dic61—24no—
179C22⋊Dic61—42no—
180#11758200—28no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61C22×C412yes—
187#11758196—24no—
188#11758196—22no—
189#11758196—22no—
190#11758196—42no—
191#11758196—42no—
192#11758196—42no—
193#11758196—42no—
194#11758196—81no—
195#11758196—81no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—24no—
199C4×Dic61—81no—
200C4×Dic61C22×C412yes—
201C4×Dic61#4112yes—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C22×C412yes—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61C22×D411yescharacteristic
216D4×C61—22no—
217D4×C61—22no—
218D4×C61—22no—
219D4×C61#11714yes—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—84no—
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×C4×C612−1+411yescharacteristic
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—48no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—84no—
264C2×Dic61C4×D412yes—
265C2×Dic61C4×D412yes—
266C2×Dic61AGL(1,ℤ/8)12yes—
267C2×Dic61#13712yes—
268Dic122—22no—
269Dic122—22no—
270Dic122—41no—
271Dic122—41no—
272C8×C61—22no—
273C8×C61—41no—
274#92418—22no—
275#92418—41no—
276#22762—611nomaximal, Sylow
277C22×C61—24no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61C2×C4×D411yescharacteristic
285C22×C61#54611yescharacteristic
286C22×C61#54811yescharacteristic
287C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C22×C61#57711yescharacteristic
290C22×C61#57711yescharacteristic
291C4×C61—41no—
292C4×C61—42no—
293C4×C61—41no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61C2×#11711yescharacteristic
297C4×C61#52011yescharacteristic
298C4×C61#55211yescharacteristic
299C4×C61#55311yescharacteristic
300Dic61—22no—
301Dic61—41no—
302Dic61—42no—
303Dic61—42no—
304Dic61—41no—
305Dic61—81no—
306Dic61—81no—
307Dic61#44014yes—
308C2×C4×SD16—611no—
309#2355—611no—
310D4×C4⋊C4—611no—
311#1935—611no—
312#1935—611no—
313#1920—611no—
314#1920—611no—
315#1877—611no—
316#1877—611no—
317#1834—611no—
318#1803—611no—
319#1802—611no—
320#1770—611no—
321#1762—611no—
322#1762D6111yescharacteristic
323C2×C61—44no—
324C2×C61#235211yescharacteristic
325C2×C61C2×#44011yescharacteristic
326C2×C61#295211yescharacteristic
327C2×C61#332411yescharacteristic
328C2×C61#333111yescharacteristic
329C2×C61#337211yescharacteristic
330C2×C61#337311yescharacteristic
331C22×SD16—611no—
332C2×C4×Q8—611no—
333C2×C4×D4—611no—
334C2×C4×D4—611no—
335C2×C4×D4C2×D6111yescharacteristic
336C22×C4⋊C4—1221no—
337C4×SD16—618no—
338C2×#106—611no—
339C2×C4⋊C8—611no—
340C2×#103—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#102—611no—
344C2×#102C2×D6111yescharacteristic
345C2×#102C2×D6111yescharacteristic
346C2×#97—611no—
347C2×#97C2×D6111yescharacteristic
348C2×C4×C8—611no—
349#391—611no—
350#391—611no—
351#389—611no—
352#389—611no—
353#387—1221no—
354#387—1221no—
355#385—1221no—
356#385—1221no—
357#384—611no—
358#384—611no—
359#382—1221no—
360#380—1221no—
361C4×C4⋊C4—611no—
362C4×C4⋊C4—611no—
363C4×C4⋊C4—1221no—
364C4×C4⋊C4C2×D6111yescharacteristic
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340C2×D6111yescharacteristic
372#340C2×D6111yescharacteristic
373#336—611no—
374#336—611no—
375C61#2276211yescharacteristic, complemented, Sylow
376C22×Q8—611no—
377C22×D4C22×D6111yescharacteristic
378C23×C4—21no—
379C23×C4—1221no—
380C23×C4—1221no—
381C2×SD16—614no—
382C2×SD16—1222no—
383C2×SD16—1222no—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388C4×Q8—614no—
389C4×Q8—1222no—
390C4×D4—22no—
391C4×D4—41no—
392C4×D4—614no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4—2441no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—2444no—
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C42—21no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—1221no—
419C2×C42—1221no—
420C2×C42C22×D6111yescharacteristic
421#106—614no—
422C4⋊C8—1222no—
423#103—614no—
424#103—2441no—
425#103—2441no—
426#102—22no—
427#102—1222no—
428#102C4×D6114yes—
429#97—41no—
430#97—2441no—
431C4×C8—614no—
432#95—21no—
433#95—611no—
434#95—611no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442C24—21no—
443C2×Q8—612no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D6112yes—
453C2×D4#1175822212yes—
454C2×D4#1175822212yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—1224no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—4881no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4D4×D6111yescharacteristic
476C22×C4D4×D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478SD16—1228no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—612no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8C2×C4×D6112yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—612no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1224no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—4881no—
506C4⋊C4—4881no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C4⋊C4C2×C4×D6112yes—
512C22⋊C4—42no—
513C22⋊C4—1224no—
514C22⋊C4—2442no—
515C42—21no—
516C42—21no—
517C42—81no—
518C42—612no—
519C42—612no—
520C42—1224no—
521C42—1221no—
522C42—1221no—
523C42—4881no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D4×D6111yescharacteristic
532Q8—1222no—
533Q8—1222no—
534Q8—2441no—
535Q8—2441no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1451973414yes—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—84no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—612no—
559C2×C4—612no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—4884no—
571C2×C4—2448no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4C22×C4×D6111yescharacteristic
583C2×C4#41×D6111yescharacteristic
584C2×C4C2×#1175822211yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4#1451982211yescharacteristic
587C2×C4#1451982411yescharacteristic
588C8—22no—
589C8—41no—
590C8—1222no—
591C8—2441no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C4×D4×D6111yescharacteristic
600C22#1684040211yescharacteristic
601C22#1684042111yescharacteristic
602C22AGL(1,ℤ/8)×D6111yescharacteristic
603C22#137×D6111yescharacteristic
604C22#1684057211yescharacteristic
605C22#1684057211yescharacteristic
606C4—42no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4—81no—
611C4—614no—
612C4—1222no—
613C4—4881no—
614C4—4881no—
615C4—2442no—
616C4—2442no—
617C4—2441no—
618C4—2441no—
619C4C2×#1451973411yescharacteristic
620C4#1684024111yescharacteristic
621C4#1684044511yescharacteristic
622C4#1684046111yescharacteristic
623C2—44no—
624C2#2144874211yescharacteristic
625C2#440×D6111yescharacteristic
626C2#2145470111yescharacteristic
627C2#2145804611yescharacteristic
628C2#2145813111yescharacteristic
629C2#2145860111yescharacteristic
630C2#2145861211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#22762
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #21448772C2semidirect
4. #21446509C2semidirect
5. #21446507C2semidirect
6. #21446404C2semidirect
7. #21446402C2semidirect
10. #21445945C2semidirect
11. #21445823C2semidirect
13. #21445687C2semidirect
21. #16839853C4semidirect
69. #14519742C2×C4semidirect
93. #14519708C2×C4semidirect
96. #14519702C2×C4semidirect
100. C8×Dic61C2×C4semidirect
375. C61#22762semidirect

Maximal subgroups

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

subgroupindex
C2×#168398532
#214487722
D4×#117581962
#214465092
#214465072
#214464042
#214464022
#214461532
#214461512
#214459452
#214458232
#214458172
#214456872
#214456462
C61×#17622
#2276261