Finite Group Databasebeta

← #29730101

Subgroups — #29730101

Order 15616.

840 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840120C211yesmaximal, characteristic
2C2×#16839983C211yesmaximal, characteristic
3#21451025C211yesmaximal, characteristic, complemented
4#21448892C211yesmaximal, characteristic
5#21447228C211yesmaximal, characteristic, complemented
6#21447225C211yesmaximal, characteristic
7#21447142C211yesmaximal, characteristic, complemented
8#21447135C211yesmaximal, characteristic
9#21446769C211yesmaximal, characteristic, complemented
10#21446539C211yesmaximal, characteristic, complemented
11#21446161C211yesmaximal, characteristic, complemented
12#21446153C211yesmaximal, characteristic
13#21445823C211yesmaximal, characteristic, complemented
14#21445648C211yesmaximal, characteristic, complemented
15C61×#1802C211yesmaximal, characteristic
16C22×Dic244C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519776C2211yescharacteristic
19C2×#14519764C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22#16840120—24no—
23#16839983—24no—
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33#16839682—21no—
34#16839682—21no—
35#16839668C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839665C2211yescharacteristic
39#16839657—21no—
40#16839657—21no—
41#16839656—21no—
42#16839650—21no—
43#16839635—21no—
44#16839631C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4×#11758196C2211yescharacteristic
47C4×C22⋊Dic61—21no—
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#384C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C22×Dic122C2311yescharacteristic
64C2×Dic244—22no—
65C2×Dic244—22no—
66C2×Dic244—41no—
67C2×#11758200—22no—
68C2×#11758200—22no—
69C2×#11758200—41no—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519776—22no—
73#14519776—22no—
74#14519764—22no—
75#14519764D414yes—
76#14519742—22no—
77#14519742—22no—
78#14519742—22no—
79#14519742—41no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196D411yescharacteristic
88C2×#11758196D411yescharacteristic
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98#14519706—22no—
99#14519706—41no—
100#14519702—22no—
101#14519702—22no—
102#14519700—22no—
103#14519700—22no—
104#14519700—22no—
105#14519698—22no—
106#14519687—22no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×D4×C61C2311yescharacteristic
117C23×C4×C61—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—41no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C22⋊C4×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—41no—
130#102×C61—22no—
131#102×C61—22no—
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×C61C2×D412yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—81no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
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×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—42no—
167C2×Dic122—81no—
168C2×Dic122—81no—
169C2×Dic122C2×D412yes—
170C2×Dic122#4112yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—81no—
183C4⋊C4×C61—81no—
184C4⋊C4×C61#4112yes—
185C22⋊C4×C61—42no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—24no—
189C22⋊Dic61—24no—
190C22⋊Dic61—42no—
191C22⋊Dic61—81no—
192#11758200—44no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197#11758196—22no—
198#11758196—22no—
199#11758196—42no—
200#11758196—42no—
201#11758196—42no—
202#11758196—81no—
203#11758196—81no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—42no—
207C4×Dic61—42no—
208C4×Dic61—81no—
209C4×Dic61C2×D412yes—
210C4×Dic61#4112yes—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217Dic244—44no—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61C22×D411yescharacteristic
226D4×C61—22no—
227D4×C61—22no—
228D4×C61—41no—
229D4×C61—41no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—84no—
245C2×C4×C61#13712yes—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—41no—
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#12112yes—
274C2×Dic61#12112yes—
275C2×Dic61#13712yes—
276Dic122—22no—
277Dic122—22no—
278Dic122—22no—
279Dic122—22no—
280Dic122—41no—
281Dic122—41no—
282Dic122—41no—
283Dic122—41no—
284Dic122—84no—
285C8×C61—41no—
286C8×C61—41no—
287#92418—41no—
288#92418—41no—
289#25336—611nomaximal, Sylow
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61C2×#12111yescharacteristic
298C22×C61#54611yescharacteristic
299C22×C61#54811yescharacteristic
300C22×C61C2×#13711yescharacteristic
301C22×C61C2×#13711yescharacteristic
302C22×C61#57611yescharacteristic
303C22×C61#57711yescharacteristic
304C4×C61—22no—
305C4×C61—42no—
306C4×C61—41no—
307C4×C61—81no—
308C4×C61—81no—
309C4×C61#52911yescharacteristic
310C4×C61#54211yescharacteristic
311C4×C61#54611yescharacteristic
312C4×C61#54711yescharacteristic
313Dic61—22no—
314Dic61—22no—
315Dic61—22no—
316Dic61—42no—
317Dic61—42no—
318Dic61—41no—
319Dic61—81no—
320Dic61—81no—
321C2×#489—611no—
322C2×#464—611no—
323#2567—611no—
324#2367—611no—
325#2072—611no—
326#2071—611no—
327#2056—611no—
328#2055—611no—
329#1988—611no—
330#1940—611no—
331#1878—611no—
332#1877—611no—
333#1803—611no—
334#1802D6111yescharacteristic
335#1763—611no—
336C2×C61—82no—
337C2×C61#242711yescharacteristic
338C2×C61#303911yescharacteristic
339C2×C61#304511yescharacteristic
340C2×C61#331411yescharacteristic
341C2×C61#331611yescharacteristic
342C2×C61#336911yescharacteristic
343C2×C61#337511yescharacteristic
344C22×Q16—611no—
345C22×SD16—611no—
346C2×C4⋊Q8—611no—
347C2×#124—611no—
348C2×C4×Q8—611no—
349C2×C4×Q8—611no—
350C2×C4×D4C2×D6111yescharacteristic
351#489—1224no—
352#464—1224no—
353C2×C4⋊C8—611no—
354C2×C4⋊C8—611no—
355C2×C4⋊C8C2×D6111yescharacteristic
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102—611no—
362C2×#102C2×D6111yescharacteristic
363C2×#102C2×D6111yescharacteristic
364C2×#97—611no—
365#395—1221no—
366#395—1221no—
367#391—611no—
368#391—611no—
369#391—611no—
370#390—611no—
371#389—1221no—
372#389—1221no—
373#388—1221no—
374#388—1221no—
375#385—1221no—
376#384—611no—
377#384C2×D6111yescharacteristic
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C22⋊C4—1221no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340C2×D6111yescharacteristic
386#340C2×D6111yescharacteristic
387#336—611no—
388#336—611no—
389C61#2533611yescharacteristic, complemented, Sylow
390C22×Q8—611no—
391C22×Q8—611no—
392C22×Q8—1221no—
393C22×D4C22×D6111yescharacteristic
394C23×C4—21no—
395C2×Q16—1222no—
396C2×Q16—1222no—
397C2×Q16—2441no—
398C2×SD16—1222no—
399C2×SD16—1222no—
400C2×SD16—2441no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405C4⋊Q8—1222no—
406C4⋊Q8—1222no—
407#124—614no—
408#124—1222no—
409C4×Q8—1222no—
410C4×Q8—1222no—
411C4×Q8—1222no—
412C4×Q8—2441no—
413C4×D4—22no—
414C4×D4—41no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C22⋊C4—21no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42—1221no—
439C2×C42C22×D6111yescharacteristic
440C4⋊C8—41no—
441C4⋊C8—1222no—
442C4⋊C8—2441no—
443#103—1222no—
444#103—1222no—
445#103—1222no—
446#103—1222no—
447#103—1222no—
448#102—22no—
449#102—22no—
450#102—1222no—
451#97—1222no—
452#95—611no—
453#95—611no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461C24—21no—
462C2×Q8—612no—
463C2×Q8—612no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—4881no—
475C2×Q8—4881no—
476C2×Q8—2442no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4D4×D6112yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—81no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4D4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4C23×D6111yescharacteristic
500Q16—2444no—
501SD16—2444no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—4881no—
528C4⋊C4—4881no—
529C4⋊C4—2442no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4#1175822212yes—
533C22⋊C4—42no—
534C22⋊C4—1224no—
535C22⋊C4—1224no—
536C22⋊C4—4881no—
537C22⋊C4—2442no—
538C42—21no—
539C42—21no—
540C42—612no—
541C42—612no—
542C42—1221no—
543C42—1221no—
544C42—4881no—
545C42—2442no—
546C42—2442no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C2×D4×D6111yescharacteristic
555Q8—1222no—
556Q8—1222no—
557Q8—1222no—
558Q8—1222no—
559Q8—4884no—
560Q8—2441no—
561Q8—2441no—
562Q8—2441no—
563Q8—2441no—
564D4—22no—
565D4—22no—
566D4—41no—
567D4—41no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1222no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4#1451980612yes—
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4#41×D6111yescharacteristic
611C2×C4C2×#1175822211yescharacteristic
612C2×C4#1451982211yescharacteristic
613C2×C4#1451982511yescharacteristic
614C8—41no—
615C8—41no—
616C8—2441no—
617C8—2441no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22#121×D6111yescharacteristic
626C22#1684040211yescharacteristic
627C22#1684042211yescharacteristic
628C22#137×D6111yescharacteristic
629C22C2×#1451980611yescharacteristic
630C22#1684056311yescharacteristic
631C22#1684057211yescharacteristic
632C4—22no—
633C4—41no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—1222no—
638C4—1222no—
639C4—1222no—
640C4—4881no—
641C4—4881no—
642C4—2442no—
643C4—2442no—
644C4—2441no—
645C4#1684029011yescharacteristic
646C4#1684038211yescharacteristic
647C4#1684040111yescharacteristic
648C4#1684041311yescharacteristic
649C2—82no—
650C2#2144939511yescharacteristic
651C2#2145530111yescharacteristic
652C2#2145535311yescharacteristic
653C2#2145789611yescharacteristic
654C2#2145793111yescharacteristic
655C2#2145855811yescharacteristic
656C2#2145863611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1802D61
hypocenterC61#25336
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
3. #21451025C2semidirect
5. #21447228C2semidirect
7. #21447142C2semidirect
9. #21446769C2semidirect
10. #21446539C2semidirect
11. #21446161C2semidirect
13. #21445823C2semidirect
14. #21445648C2semidirect
389. C61#25336semidirect

Maximal subgroups

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

subgroupindex
C2×#168401202
C2×#168399832
#214510252
#214488922
#214472282
#214472252
#214471422
#214471352
#214467692
#214465392
#214461612
#214461532
#214458232
#214456482
C61×#18022
#2533661