Finite Group Databasebeta

← #29740033

Subgroups — #29740033

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×#16839982C211yesmaximal, characteristic
2C2×SD16×Dic61C211yesmaximal, characteristic
3#21449604C211yesmaximal, characteristic
4Q8×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21447361C211yesmaximal, characteristic, complemented
6#21447319C211yesmaximal, characteristic, complemented
7#21447222C211yesmaximal, characteristic
8#21447059C211yesmaximal, characteristic
9#21446496C211yesmaximal, characteristic, complemented
10#21446377C211yesmaximal, characteristic, complemented
11#21446160C211yesmaximal, characteristic, complemented
12#21446158C211yesmaximal, characteristic, complemented
13#21446089C211yesmaximal, characteristic, complemented
14#21446082C211yesmaximal, characteristic
15C61×#1880C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21#16839982—24no—
22SD16×Dic61—24no—
23C2×#14519709C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839687—21no—
34#16839687—21no—
35#16839668C2211yescharacteristic
36#16839667C2211yescharacteristic
37#16839661—21no—
38#16839661—21no—
39#16839661C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839661C2211yescharacteristic
42#16839637—21no—
43#16839632—21no—
44#16839632C2211yescharacteristic
45#16839623—21no—
46C4×C4⋊Dic61—21no—
47C4×C4⋊Dic61C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C23×Dic61—21no—
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202—22no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519764—22no—
69#14519764—41no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—41no—
73Q8×Dic61—41no—
74D4×Dic61—22no—
75D4×Dic61—41no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61D411yescharacteristic
84C2×C4⋊Dic61D411yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519709—22no—
96#14519709—41no—
97#14519706—22no—
98#14519702—22no—
99#14519702—41no—
100#14519701—22no—
101#14519700—22no—
102#14519698—22no—
103#14519698—22no—
104C8×Dic61—22no—
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
116C2×SD16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61D414yes—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C4×Q8×C61—22no—
122C4×Q8×C61—22no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61—21no—
129C2×C42×C61C2311yescharacteristic
130#103×C61—22no—
131#102×C61—22no—
132C24×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61C2×D412yes—
137C2×Q8×C61#4112yes—
138C2×Q8×C61#4112yes—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61C2×D412yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—42no—
156C22×Dic61—42no—
157C22×Dic61—81no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168SD16×C61—28no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61C2×D412yes—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61#4112yes—
180C22⋊C4×C61—42no—
181C22⋊C4×C61—81no—
182C42×C61—24no—
183C42×C61—21no—
184C42×C61—21no—
185C22⋊Dic61—42no—
186C22⋊Dic61—42no—
187C22⋊Dic61—42no—
188#11758202—44no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—42no—
198C4⋊Dic61—42no—
199C4⋊Dic61—42no—
200C4⋊Dic61—81no—
201C4⋊Dic61—81no—
202C4⋊Dic61—81no—
203C4⋊Dic61—81no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—42no—
209C4×Dic61—42no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C23×C61—22no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61C22×D411yescharacteristic
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61#12314yes—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61#13512yes—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C612−1+411yescharacteristic
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
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—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—84no—
277C2×Dic61#13712yes—
278Dic122—41no—
279Dic122—41no—
280Dic122—41no—
281Dic122—41no—
282C8×C61—22no—
283C8×C61—41no—
284#92418—41no—
285#92418—41no—
286#26161—611nomaximal, Sylow
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61#52511yescharacteristic
295C22×C61D4211yescharacteristic
296C22×C61#54811yescharacteristic
297C22×C61C2×#13511yescharacteristic
298C22×C61C2×#13711yescharacteristic
299C22×C61#57711yescharacteristic
300C22×C61#57711yescharacteristic
301C4×C61—22no—
302C4×C61—24no—
303C4×C61—41no—
304C4×C61—42no—
305C4×C61—81no—
306C4×C61C2×#12311yescharacteristic
307C4×C61#53911yescharacteristic
308C4×C61#54711yescharacteristic
309C4×C61#55311yescharacteristic
310Dic61—22no—
311Dic61—41no—
312Dic61—41no—
313Dic61—41no—
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#1916—611no—
328#1880D6111yescharacteristic
329#1878—611no—
330#1878—611no—
331#1865—611no—
332#1864—611no—
333C2×C61—82no—
334C2×C61#244311yescharacteristic
335C2×C61#310811yescharacteristic
336C2×C61#312511yescharacteristic
337C2×C61#326611yescharacteristic
338C2×C61#328411yescharacteristic
339C2×C61#336511yescharacteristic
340C2×C61#336911yescharacteristic
341C22×Q16—611no—
342C22×SD16C2×D6111yescharacteristic
343C2×#124—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8—611no—
346C2×C4×Q8C2×D6111yescharacteristic
347C2×C4×D4—611no—
348#464—1224no—
349C4×SD16—1224no—
350C2×#106—611no—
351C2×#106—611no—
352C2×C4⋊C8—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103C2×D6111yescharacteristic
358C2×#102—611no—
359C2×#102—611no—
360C2×#102C2×D6111yescharacteristic
361C2×C4×C8—611no—
362#397—1221no—
363#397—1221no—
364#391—611no—
365#391—611no—
366#390C2×D6111yescharacteristic
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—611no—
377C4×C4⋊C4—1221no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340C2×D6111yescharacteristic
382#336—611no—
383#336—611no—
384#336—611no—
385#336C2×D6111yescharacteristic
386C61#2616111yescharacteristic, complemented, Sylow
387C22×Q8—611no—
388C22×Q8C22×D6111yescharacteristic
389C22×D4C22×D6111yescharacteristic
390C23×C4—1221no—
391C2×Q16—1222no—
392C2×Q16—1222no—
393C2×Q16—1222no—
394C2×SD16—22no—
395C2×SD16—22no—
396C2×SD16D24414yes—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8C22×D6111yescharacteristic
400C22×C8C22×D6111yescharacteristic
401#124—1222no—
402#124—2441no—
403C4×Q8—22no—
404C4×Q8—22no—
405C4×Q8—1222no—
406C4×Q8—1222no—
407C4×Q8—2441no—
408C4×Q8—2441no—
409C4×D4—1222no—
410C4×D4—2441no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
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⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C42—21no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42C22×D6111yescharacteristic
437#106—1222no—
438#106—2441no—
439C4⋊C8—1222no—
440#103—22no—
441#103—1222no—
442#103—1222no—
443#103—1222no—
444#103—2441no—
445#102—22no—
446#102—1222no—
447#102—1222no—
448C4×C8—1222no—
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—
458C24—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8C2×D24412yes—
469C2×Q8#1175822412yes—
470C2×Q8#1175822412yes—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4C2×D24412yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—4881no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4C23×D6111yescharacteristic
494Q16—2444no—
495SD16—28no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8C2×D24412yes—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—4881no—
521C4⋊C4—4881no—
522C4⋊C4—4881no—
523C4⋊C4—4881no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4#1175822412yes—
528C22⋊C4—42no—
529C22⋊C4—81no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C22⋊C4—2442no—
533C42—24no—
534C42—21no—
535C42—21no—
536C42—1221no—
537C42—1221no—
538C42—1221no—
539C42—1221no—
540C42—2442no—
541C42—2442no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×D4×D6111yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8—2441no—
554Q8—2441no—
555Q8—2441no—
556Q8—2441no—
557Q8#1451976114yes—
558D4—22no—
559D4—22no—
560D4—41no—
561D4—41no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—612no—
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—4884no—
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—2441no—
601C2×C4#1451979812yes—
602C2×C4C22×D24411yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C2×C4C2×#1175822411yescharacteristic
605C2×C4C2×#1175822411yescharacteristic
606C2×C4#1451982311yescharacteristic
607C2×C4#1451982411yescharacteristic
608C8—22no—
609C8—41no—
610C8—2441no—
611C8—2441no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22#1684026811yescharacteristic
620C22D4×D24411yescharacteristic
621C22#1684041811yescharacteristic
622C22C2×#1451979811yescharacteristic
623C22#137×D6111yescharacteristic
624C22#1684056911yescharacteristic
625C22#1684057211yescharacteristic
626C4—22no—
627C4—24no—
628C4—41no—
629C4—42no—
630C4—81no—
631C4—1222no—
632C4—4881no—
633C4—4881no—
634C4—4881no—
635C4—4881no—
636C4—2441no—
637C4—2441no—
638C4—2441no—
639C4C2×#1451976111yescharacteristic
640C4#1684035611yescharacteristic
641C4#1684041611yescharacteristic
642C4#1684046411yescharacteristic
643C2—82no—
644C2#2144955211yescharacteristic
645C2#2145589411yescharacteristic
646C2#2145609711yescharacteristic
647C2#2145733111yescharacteristic
648C2#2145755011yescharacteristic
649C2#2145851911yescharacteristic
650C2#2145855411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1880C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1880D61
hypocenterC61#26161
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
4. Q8×C4⋊Dic61C2semidirect
5. #21447361C2semidirect
6. #21447319C2semidirect
9. #21446496C2semidirect
10. #21446377C2semidirect
11. #21446160C2semidirect
12. #21446158C2semidirect
13. #21446089C2semidirect
386. C61#26161semidirect

Maximal subgroups

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

subgroupindex
C2×#168399822
C2×SD16×Dic612
#214496042
Q8×C4⋊Dic612
#214473612
#214473192
#214472222
#214470592
#214464962
#214463772
#214461602
#214461582
#214460892
#214460822
C61×#18802
#2616161