Finite Group Databasebeta

← #29743406

Subgroups — #29743406

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840056C211yesmaximal, characteristic, complemented
2C2×#16839864C211yesmaximal, characteristic
3#21449670C211yesmaximal, characteristic
4#21448733C211yesmaximal, characteristic, complemented
5#21447344C211yesmaximal, characteristic
6#21447294C211yesmaximal, characteristic, complemented
7#21447188C211yesmaximal, characteristic
8#21447146C211yesmaximal, characteristic, complemented
9#21446383C211yesmaximal, characteristic, complemented
10#21446175C211yesmaximal, characteristic
11#21446175C211yesmaximal, characteristic
12#21446084C211yesmaximal, characteristic, complemented
13#21446072C211yesmaximal, characteristic, complemented
14#21446065C211yesmaximal, characteristic
15C61×#1944C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519755—21no—
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23#16840056—24no—
24#16839864—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35#16839686—21no—
36#16839675—21no—
37#16839663C2211yescharacteristic
38#16839663C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839659—21no—
41#16839658C2211yescharacteristic
42#16839657—21no—
43#16839633—21no—
44#16839631—21no—
45#16839631C2211yescharacteristic
46C4×#11758196C2211yescharacteristic
47#16839613—21no—
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×#11758190—22no—
66C2×#11758190—22no—
67C2×#11758190Q814yescomplemented
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519776—22no—
71#14519776—41no—
72#14519755—44no—
73C4×Dic122—22no—
74C4×Dic122—41no—
75#14519742—22no—
76#14519742—22no—
77#14519742—22no—
78#14519742—41no—
79C4×D244—22no—
80C4×D244—41no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196D411yescharacteristic
92C2×#11758196D411yescharacteristic
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103#14519711—22no—
104#14519710—22no—
105#14519710—41no—
106#14519706—22no—
107#14519702D414yes—
108#14519700—22no—
109#14519700—22no—
110#14519699—22no—
111#14519697—22no—
112#14519697—41no—
113#14519684—21no—
114#14519684—21no—
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126#107×C61—22no—
127#106×C61—22no—
128#97×C61—22no—
129#95×C61—21no—
130#95×C61—21no—
131C23×D61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244C2×Q812yes—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61—81no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—42no—
159C2×Dic122—42no—
160C2×Dic122—81no—
161C2×Dic122C2×D412yes—
162C2×Dic122C2×Q812yes—
163C2×Dic122#4112yes—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61C2×D412yes—
169C2×C8×C61C2×Q812yes—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61—81no—
179C4⋊C4×C61#4112yes—
180C42×C61—21no—
181C42×C61—21no—
182#11758198—42no—
183#11758198—42no—
184#11758198—42no—
185#11758198—42no—
186#11758198—81no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—81no—
191C4⋊Dic61C2×D412yes—
192#11758196—24no—
193#11758196—22no—
194#11758196—22no—
195#11758196—42no—
196#11758196—42no—
197#11758196—42no—
198#11758196—42no—
199#11758196—42no—
200#11758196—81no—
201C4×Dic61—24no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—42no—
207C4×Dic61—42no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214#11758190—28no—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
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×C61C22×Q811yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61AGL(1,ℤ/8)12yes—
267C2×Dic61#13712yes—
268D244—22no—
269D244—22no—
270D244—41no—
271D244—41no—
272C4×D61—84no—
273Dic122—22no—
274Dic122—22no—
275Dic122—22no—
276Dic122—84no—
277Dic122C22⋊Q814yescomplemented
278C8×C61—22no—
279C8×C61—22no—
280#92418—41no—
281#92418—41no—
282#26397—611nomaximal, Sylow
283C22×C61#52511yescharacteristic
284C22×C61D4211yescharacteristic
285C22×C61D4×Q811yescharacteristic
286C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C61C2×#13711yescharacteristic
288C22×C61#57611yescharacteristic
289C22×C61#57811yescharacteristic
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C4×C61—42no—
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61—81no—
303C4×C61C2×C22⋊Q811yescharacteristic
304C4×C61#53911yescharacteristic
305C4×C61#54711yescharacteristic
306C4×C61#55411yescharacteristic
307Dic61—22no—
308Dic61—24no—
309Dic61—22no—
310Dic61—41no—
311Dic61—41no—
312Dic61—42no—
313Dic61—81no—
314C2×#477—611no—
315C2×#440—611no—
316#2452—611no—
317Q8×C4⋊C4—611no—
318#2093—611no—
319#2083—611no—
320#2065—611no—
321#2057—611no—
322#1944D6111yescharacteristic, complemented
323#1917—611no—
324#1880—611no—
325#1880—611no—
326#1865—611no—
327#1863—611no—
328#1862—611no—
329C2×C61#244911yescharacteristic
330C2×C61#311211yescharacteristic
331C2×C61#312411yescharacteristic
332C2×C61#327011yescharacteristic
333C2×C61#327411yescharacteristic
334C2×C61#338511yescharacteristic
335C2×C61#338611yescharacteristic
336D61—82no—
337C22×SD16—611no—
338C2×C4⋊Q8—611no—
339C2×C22⋊Q8—1221no—
340C2×C4×Q8—611no—
341C2×C4×Q8—611no—
342C2×C4×Q8—611no—
343C2×C4×D4—611no—
344#477—1224no—
345#440—1224no—
346C2×#107—611no—
347C2×#107—611no—
348C2×#107—611no—
349C2×#107C2×D6111yescharacteristic
350C2×#106C2×D6111yescharacteristic
351C2×C4⋊C8—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#97C2×D6111yescharacteristic
359#396—1221no—
360#393—1221no—
361#390—611no—
362#390—611no—
363#389—611no—
364#389—611no—
365#389—1221no—
366#389—1221no—
367#389C2×D6111yescharacteristic
368#389C2×D6111yescharacteristic
369#385—1221no—
370#384—611no—
371#384—1221no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—1221no—
374#340—611no—
375#340—611no—
376#340C2×D6111yescharacteristic
377#340C2×D6111yescharacteristic
378#336—611no—
379#336—611no—
380#336—611no—
381#336—611no—
382C61#2639711yescharacteristic, complemented, Sylow
383C22×Q8—611no—
384C22×Q8—1221no—
385C22×D4—611no—
386C23×C4—1221no—
387C2×SD16—614no—
388C2×SD16—1222no—
389C2×SD16—1222no—
390C22×C8—611no—
391C22×C8—611no—
392C22×C8C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394C4⋊Q8—1222no—
395C4⋊Q8—2441no—
396C22⋊Q8—2444no—
397C4×Q8—1222no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—2441no—
402C4×Q8—2441no—
403C4×D4—1222no—
404C4×D4—2441no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
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⋊C4—1221no—
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—1221no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42C22×D6111yescharacteristic
434#107—22no—
435#107—1222no—
436#107—1222no—
437#107—2441no—
438#106—22no—
439C4⋊C8—1222no—
440#103—614no—
441#103—1222no—
442#103—1222no—
443#102—1222no—
444#102—1222no—
445#102—2441no—
446#97—22no—
447#95—21no—
448#95—21no—
449#95—611no—
450#95—611no—
451#95—1221no—
452#95—1221no—
453C24—1221no—
454C2×Q8—612no—
455C2×Q8—612no—
456C2×Q8—612no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—4881no—
461C2×Q8—2442no—
462C2×Q8—2442no—
463C2×D4—612no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—4881no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4D4×D6111yescharacteristic
484C22×C4D4×D6111yescharacteristic
485C22×C4C23×D6111yescharacteristic
486SD16—1228no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8D4×D6112yes—
498C2×C8Q8×D6112yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—612no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1224no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—4881no—
516C4⋊C4—4881no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4#1175822212yes—
523C22⋊C4—4881no—
524C22⋊C4—2442no—
525C22⋊C4—2442no—
526C22⋊C4—2442no—
527C22⋊C4—2442no—
528C42—21no—
529C42—21no—
530C42—1224no—
531C42—1221no—
532C42—1221no—
533C42—1221no—
534C42—1221no—
535C42—2442no—
536C42—2442no—
537C23—1222no—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23C2×D4×D6111yescharacteristic
545Q8—614no—
546Q8—1222no—
547Q8—1222no—
548Q8—1222no—
549Q8—4884no—
550D4—1222no—
551D4—1222no—
552D4—2441no—
553D4—2441no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
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—612no—
573C2×C4—612no—
574C2×C4—1221no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1222no—
588C2×C4—4884no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C2×Q8×D6111yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C2×C4#1451982311yescharacteristic
599C2×C4#1451982511yescharacteristic
600C8—22no—
601C8—22no—
602C8—2441no—
603C8—2441no—
604C22—4881no—
605C22—4881no—
606C22—4881no—
607C22—4881no—
608C22—2442no—
609C22—2442no—
610C22—2442no—
611C22#1684027111yescharacteristic
612C22#1684039611yescharacteristic
613C22#1684042711yescharacteristic
614C22AGL(1,ℤ/8)×D6111yescharacteristic
615C22#137×D6111yescharacteristic
616C22#1684056611yescharacteristic
617C22#1684057811yescharacteristic
618C4—41no—
619C4—42no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—1222no—
625C4—1224no—
626C4—1222no—
627C4—4881no—
628C4—2441no—
629C4—2442no—
630C4—2441no—
631C4C22⋊Q8×D6111yescharacteristic
632C4#1684036011yescharacteristic
633C4#1684041311yescharacteristic
634C4#1684047211yescharacteristic
635C2—4882no—
636C2#2144963511yescharacteristic
637C2#2145593911yescharacteristic
638C2#2145608511yescharacteristic
639C2#2145739611yescharacteristic
640C2#2145743611yescharacteristic
641C2#2145872611yescharacteristic
642C2#2145873711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1944C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1944D61
hypocenterC61#26397
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
1. C2×#16840056C2semidirect
4. #21448733C2semidirect
6. #21447294C2semidirect
8. #21447146C2semidirect
9. #21446383C2semidirect
12. #21446084C2semidirect
13. #21446072C2semidirect
15. C61×#1944C2semidirect
67. C2×#11758190Q8semidirect
277. Dic122C22⋊Q8semidirect
322. #1944D61semidirect
382. C61#26397semidirect

Maximal subgroups

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

subgroupindex
C2×#168400562
C2×#168398642
#214496702
#214487332
#214473442
#214472942
#214471882
#214471462
#214463832
#214461752
#214460842
#214460722
#214460652
C61×#19442
#2639761