Finite Group Databasebeta

← #29740040

Subgroups — #29740040

Order 15616.

840 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839985C211yesmaximal, characteristic
2C2×C4×#11758190C211yesmaximal, characteristic
3#21449606C211yesmaximal, characteristic
4C4⋊C4×Dic122C211yesmaximal, characteristic, complemented
5#21447361C211yesmaximal, characteristic, complemented
6#21447317C211yesmaximal, characteristic, complemented
7#21447224C211yesmaximal, characteristic
8#21447061C211yesmaximal, characteristic
9#21446495C211yesmaximal, characteristic, complemented
10#21446173C211yesmaximal, characteristic
11#21446159C211yesmaximal, characteristic, complemented
12#21446159C211yesmaximal, characteristic, complemented
13#21446090C211yesmaximal, characteristic, complemented
14#21446083C211yesmaximal, characteristic
15C61×#1916C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23#16839985—24no—
24C4×#11758190—24no—
25C2×#14519709C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35#16839691—21no—
36#16839688—21no—
37#16839668C2211yescharacteristic
38#16839668C2211yescharacteristic
39#16839664C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839659—21no—
44#16839659—21no—
45#16839633—21no—
46#16839632C2211yescharacteristic
47#16839631—21no—
48#16839624—21no—
49C4⋊C4×Dic61—21no—
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×Dic244—22no—
66C2×Dic244—22no—
67C2×Dic244—22no—
68C2×#11758190—22no—
69C2×#11758190—22no—
70C2×#11758190D414yes—
71#14519765—22no—
72#14519765—41no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C4×Dic122—22no—
77C4×Dic122—41no—
78C4×Dic122—41no—
79C4×D244—22no—
80C4×D244—41no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102#14519709—22no—
103#14519709—41no—
104#14519702—22no—
105#14519702—22no—
106#14519702—22no—
107#14519702—22no—
108#14519702—41no—
109#14519699—22no—
110#14519699—22no—
111#14519699—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C4⋊C8×C61—22no—
129C4×C8×C61—22no—
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C23×D61—21no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244C2×D412yes—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—81no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122C2×D412yes—
166C2×Dic122#4112yes—
167C2×Dic122#4112yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C2×D412yes—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188#11758198—42no—
189#11758198—42no—
190#11758198—42no—
191#11758198—42no—
192#11758198—81no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—42no—
203C4⋊Dic61—81no—
204C4⋊Dic61#4112yes—
205#11758196—24no—
206#11758196—42no—
207#11758196—42no—
208#11758196—81no—
209#11758196—81no—
210#11758196—81no—
211C4×Dic61—24no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214Dic244—44no—
215#11758190—28no—
216C23×C61C22×D411yescharacteristic
217C22×D61—22no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
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—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61#13512yes—
241C2×C4×C61#13712yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C2×C4×C612−1+411yescharacteristic
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269D244—22no—
270D244—22no—
271D244—41no—
272D244—41no—
273C4×D61—84no—
274Dic122—22no—
275Dic122—22no—
276Dic122—22no—
277Dic122—41no—
278Dic122—41no—
279Dic122—41no—
280Dic122—41no—
281Dic122#12314yes—
282C8×C61—22no—
283C8×C61—41no—
284C8×C61—41no—
285C8×C61—41no—
286#26161—611nomaximal, Sylow
287C22×C61#52511yescharacteristic
288C22×C61D4211yescharacteristic
289C22×C61#54811yescharacteristic
290C22×C61C2×#13511yescharacteristic
291C22×C61C2×#13711yescharacteristic
292C22×C61#57711yescharacteristic
293C22×C61#57711yescharacteristic
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—81no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61C2×#12311yescharacteristic
308C4×C61#53911yescharacteristic
309C4×C61#54711yescharacteristic
310C4×C61#55311yescharacteristic
311Dic61—24no—
312Dic61—42no—
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317Dic61—81no—
318C2×#464—611no—
319C2×C4×SD16—611no—
320#2447—611no—
321Q8×C4⋊C4—611no—
322#2096—611no—
323#2088—611no—
324#2071—611no—
325#2042—611no—
326#1933—611no—
327#1916D6111yescharacteristic, complemented
328#1880—611no—
329#1878—611no—
330#1878—611no—
331#1865—611no—
332#1864—611no—
333C2×C61#244311yescharacteristic
334C2×C61#310811yescharacteristic
335C2×C61#312511yescharacteristic
336C2×C61#326611yescharacteristic
337C2×C61#328411yescharacteristic
338C2×C61#336511yescharacteristic
339C2×C61#336911yescharacteristic
340D61—82no—
341C22×Q16—611no—
342C22×SD16—611no—
343C2×#124—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8—611no—
346C2×C4×Q8—611no—
347C2×C4×D4—611no—
348#464—1224no—
349C4×SD16—1224no—
350C2×#106—611no—
351C2×#106—611no—
352C2×C4⋊C8C2×D6111yescharacteristic
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#102—611no—
361C2×C4×C8C2×D6111yescharacteristic
362#397—1221no—
363#397—1221no—
364#391—611no—
365#391—611no—
366#390—611no—
367#389—611no—
368#389—611no—
369#389—611no—
370#389—1221no—
371#389—1221no—
372#385—1221no—
373#384—611no—
374#384—1221no—
375#381—1221no—
376C4×C4⋊C4—1221no—
377C4×C4⋊C4C2×D6111yescharacteristic
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#336C2×D6111yescharacteristic
383#336C2×D6111yescharacteristic
384#336C2×D6111yescharacteristic
385#336C2×D6111yescharacteristic
386C61#2616111yescharacteristic, complemented, Sylow
387C22×Q8—611no—
388C22×Q8—611no—
389C22×D4—611no—
390C23×C4—1221no—
391C2×Q16—1222no—
392C2×Q16—1222no—
393C2×Q16—1222no—
394C2×SD16—614no—
395C2×SD16—1222no—
396C2×SD16—1222no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399C22×C8C22×D6111yescharacteristic
400C22×C8C22×D6111yescharacteristic
401#124—1222no—
402#124—2441no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—1222no—
407C4×Q8—2441no—
408C4×Q8—2441no—
409C4×D4—1222no—
410C4×D4—2441no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4D24411yescharacteristic
425C2×C4⋊C4D24411yescharacteristic
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42—1221no—
434C2×C42C22×D6111yescharacteristic
435C2×C42C22×D6111yescharacteristic
436C2×C42C22×D6111yescharacteristic
437#106—1222no—
438#106—2441no—
439C4⋊C8—22no—
440#103—1222no—
441#103—1222no—
442#103—1222no—
443#103—1222no—
444#103—2441no—
445#102—1222no—
446#102—1222no—
447#102—1222no—
448C4×C8—22no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95D24411yescharacteristic
457#95D24411yescharacteristic
458C24—1221no—
459C2×Q8—612no—
460C2×Q8—612no—
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×D4—612no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—4881no—
485C22×C4—2442no—
486C22×C4—2442no—
487C22×C4C2×D24411yescharacteristic
488C22×C4C2×D24411yescharacteristic
489C22×C4C2×D24411yescharacteristic
490C22×C4C2×D24411yescharacteristic
491C22×C4C2×D24411yescharacteristic
492C22×C4C2×D24411yescharacteristic
493C22×C4C23×D6111yescharacteristic
494Q16—2444no—
495SD16—1228no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8D4×D6112yes—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—612no—
511C4⋊C4—1224no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—4881no—
522C4⋊C4—4881no—
523C4⋊C4—4881no—
524C4⋊C4—4881no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C22⋊C4—4881no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C22⋊C4—2442no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—1224no—
540C42—2442no—
541C42—2442no—
542C23—1222no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23—2441no—
549C23C22×D24411yescharacteristic
550Q8—614no—
551Q8—1222no—
552Q8—1222no—
553Q8—1222no—
554Q8—2441no—
555Q8—2441no—
556Q8—2441no—
557Q8—2441no—
558D4—1222no—
559D4—1222no—
560D4—2441no—
561D4—2441no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—4884no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4#1451979912yes—
601C2×C4#1451981012yes—
602C2×C4C2×D4×D6111yescharacteristic
603C2×C4#41×D6111yescharacteristic
604C2×C4#41×D6111yescharacteristic
605C2×C4#41×D6111yescharacteristic
606C2×C4#1451982311yescharacteristic
607C2×C4#1451982511yescharacteristic
608C8—22no—
609C8—41no—
610C8—41no—
611C8—41no—
612C22—4881no—
613C22—4881no—
614C22—4881no—
615C22—4881no—
616C22—2442no—
617C22—2442no—
618C22—2442no—
619C22#1684027411yescharacteristic
620C22D4×D24411yescharacteristic
621C22#1684042411yescharacteristic
622C22C2×#1451979911yescharacteristic
623C22C2×#1451981011yescharacteristic
624C22#1684057411yescharacteristic
625C22#1684057411yescharacteristic
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—1224no—
633C4—4881no—
634C4—4881no—
635C4—4881no—
636C4—4881no—
637C4—4881no—
638C4—2442no—
639C4#123×D6111yescharacteristic
640C4#1684036011yescharacteristic
641C4#1684041411yescharacteristic
642C4#1684046311yescharacteristic
643C2—4882no—
644C2#2144955911yescharacteristic
645C2#2145590111yescharacteristic
646C2#2145609811yescharacteristic
647C2#2145733411yescharacteristic
648C2#2145755111yescharacteristic
649C2#2145851711yescharacteristic
650C2#2145856111yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1916C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1916D61
hypocenterC61#26161
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
4. C4⋊C4×Dic122C2semidirect
5. #21447361C2semidirect
6. #21447317C2semidirect
9. #21446495C2semidirect
11. #21446159C2semidirect
12. #21446159C2semidirect
13. #21446090C2semidirect
15. C61×#1916C2semidirect
327. #1916D61semidirect
386. C61#26161semidirect

Maximal subgroups

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

subgroupindex
C2×#168399852
C2×C4×#117581902
#214496062
C4⋊C4×Dic1222
#214473612
#214473172
#214472242
#214470612
#214464952
#214461732
#214461592
#214460902
#214460832
C61×#19162
#2616161