Finite Group Databasebeta

← #29682193

Subgroups — #29682193

Order 15616.

974 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839801C211yesmaximal, characteristic
2#389×D61C211yesmaximal, characteristic
3C4×#14519735C211yesmaximal, characteristic
4#21446414C211yesmaximal, characteristic, complemented
5#21446410C211yesmaximal, characteristic, complemented
6#21446406C211yesmaximal, characteristic, complemented
7#21446402C211yesmaximal, characteristic, complemented
8#21446397C211yesmaximal, characteristic, complemented
9#21446391C211yesmaximal, characteristic, complemented
10#21446387C211yesmaximal, characteristic, complemented
11#21445537C211yesmaximal, characteristic
12#21445537C211yesmaximal, characteristic
13#21445537C211yesmaximal, characteristic
14#21445511C211yesmaximal, characteristic
15C61×#1919C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C42×D61C2211yescharacteristic
24#16839801C418yescomplemented
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28#16839661C2211yescharacteristic
29#16839660C2211yescharacteristic
30#16839659—21no—
31#16839658C2211yescharacteristic
32#16839657—21no—
33C4⋊C4×Dic61C2211yescharacteristic
34C4⋊C4×Dic61C2211yescharacteristic
35C4×C4⋊Dic61C2211yescharacteristic
36C4×#11758198—21no—
37#16839609—21no—
38#16839607—21no—
39#95×D61—21no—
40C42×Dic61—21no—
41#16839525C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C4×C4⋊C4×C61C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61—21no—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C2×#11758189—22no—
66C2×#11758189—22no—
67C2×#11758189—22no—
68C2×#11758189—22no—
69C2×#11758189—22no—
70C2×#11758189C2×C414yes—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519735—22no—
74#14519735—22no—
75#14519735C2×C414yes—
76#14519735Q814yes—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—44no—
90C4⋊C4×D61C2×C414yes—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61D411yescharacteristic
99C2×C4×Dic61D411yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101C42×D61—22no—
102C42×D61—22no—
103#14519711C2×C414yescomplemented
104#14519710C2×C414yescomplemented
105#14519708C2×C414yescomplemented
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#106×C61C2×C414yescomplemented
122#95×C61—21no—
123#95×C61—21no—
124C23×D61C2×D411yescharacteristic
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C2×C4×D61—24no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61—42no—
151C2×C4×D61C22×C412yes—
152C2×C4×D61C2×Q812yes—
153C2×C4×D61#4112yes—
154C2×C4×D61#4112yes—
155C2×C4×D61#4112yes—
156C2×C4×D61#4112yes—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61C22×C412yes—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—42no—
169C4⋊C4×C61C22×C412yes—
170C4⋊C4×C61C22×C412yes—
171C4⋊C4×C61C2×Q812yes—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—42no—
175#11758198—24no—
176#11758198—24no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—42no—
183C4⋊Dic61C22×C412yes—
184C4⋊Dic61C22×C412yes—
185C4⋊Dic61C2×Q812yes—
186#11758196—24no—
187#11758196—42no—
188#11758196—42no—
189#11758196—42no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—24no—
193C4×Dic61—24no—
194C4×Dic61—22no—
195C4×Dic61—22no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198C4×Dic61—42no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C22×C412yes—
205#11758189—28no—
206#11758189—44no—
207C23×C61C22×D411yescharacteristic
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61C4×D412yes—
213C22×D61C22⋊Q812yes—
214C22×D61#12312yes—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61AGL(1,ℤ/8)12yes—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C22×Q811yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61C4×D412yes—
257C2×Dic61C22⋊Q812yes—
258C2×Dic61#12312yes—
259C2×Dic61#13712yes—
260C4×D61—44no—
261C4×D61—44no—
262C4×D61—48no—
263C4×D61—84no—
264C4×D61#11714yes—
265C4×D61C4×Q814yes—
266C4×D61#12514yes—
267C4×D61#12614yes—
268C8×C61—22no—
269C8×C61—41no—
270#92418—22no—
271#92418—41no—
272#21034—611nomaximal, Sylow
273C22×C61C2×C4×D411yescharacteristic
274C22×C61C2×C22⋊Q811yescharacteristic
275C22×C61C2×#12311yescharacteristic
276C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57511yescharacteristic
279C22×C61#57511yescharacteristic
280C2×D61—22no—
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61#38714yes—
287C4×C61—22no—
288C4×C61—42no—
289C4×C61—41no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61—81no—
293C4×C61C2×#11711yescharacteristic
294C4×C61C2×C4×Q811yescharacteristic
295C4×C61C2×#12511yescharacteristic
296C4×C61C2×#12611yescharacteristic
297Dic61—22no—
298Dic61—42no—
299Dic61—41no—
300Dic61—42no—
301Dic61—42no—
302Dic61—81no—
303Dic61#38714yes—
304C2×#428—611no—
305C2×#389—611no—
306C4×#117—611no—
307#1921—611no—
308#1921—611no—
309#1920—611no—
310#1920—611no—
311#1919—611no—
312#1919D6111yescharacteristic, complemented
313#1918—611no—
314#1918—611no—
315#1736—611no—
316#1736—611no—
317#1736—611no—
318#1728—611no—
319C2×C61C2×#38711yescharacteristic
320C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
321C2×C61C4×#13711yescharacteristic
322C2×C61#307911yescharacteristic
323C2×C61#308011yescharacteristic
324C2×C61#309411yescharacteristic
325C2×C61#309511yescharacteristic
326D61—24no—
327C22×M16—611no—
328C22×M16—611no—
329C2×#117—611no—
330C2×#117—611no—
331C22×C4⋊C4—611no—
332C22×C4⋊C4—611no—
333C22×C4⋊C4—1221no—
334C22×C42—611no—
335#428—618no—
336C2×#107—611no—
337C2×#107—611no—
338C2×#106—611no—
339C2×#106C2×D6111yescharacteristic
340#389—611no—
341#389—611no—
342#389—611no—
343#389—1221no—
344#389—1221no—
345#389C2×D6111yescharacteristic
346C4×C4⋊C4—611no—
347C4×C4⋊C4—611no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4C2×D6111yescharacteristic
350C4×C22⋊C4—1221no—
351C4×C22⋊C4—1221no—
352#376—1221no—
353C2×#95—1221no—
354C43—1221no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340C2×D6111yescharacteristic
365#340C2×D6111yescharacteristic
366#340C2×D6111yescharacteristic
367#336—611no—
368#336—611no—
369#336—611no—
370#336C2×D6111yescharacteristic
371C61#2103411yescharacteristic, complemented, Sylow
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—1221no—
376C23×C4—1221no—
377C2×M16—614no—
378C2×M16—1222no—
379C2×M16—1222no—
380C2×M16—1222no—
381C2×M16—1222no—
382C2×M16—1222no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8C22×D6111yescharacteristic
386C22×C8C22×D6111yescharacteristic
387#117—614no—
388#117—614no—
389#117—1222no—
390#117—1222no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—614no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1222no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—2444no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C42—21no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—1222no—
417C2×C42—1222no—
418C2×C42—1221no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42—1221no—
422C2×C42—1221no—
423C2×C42C22×D6111yescharacteristic
424#107—614no—
425#107—614no—
426#106—614no—
427#106C4×D6114yescomplemented
428#95—21no—
429#95—21no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436C24—611no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—611no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1224no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1222no—
458C22×C4—1221no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4—2442no—
466C22×C4D4×D6111yescharacteristic
467C22×C4D4×D6111yescharacteristic
468C22×C4C23×D6111yescharacteristic
469M16—1228no—
470M16—2444no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—612no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8C2×C4×D6112yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—612no—
490C4⋊C4—612no—
491C4⋊C4—612no—
492C4⋊C4—1224no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4C2×C4×D6112yes—
503C4⋊C4C2×C4×D6112yes—
504C4⋊C4Q8×D6112yes—
505C22⋊C4—1224no—
506C22⋊C4—1224no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—1222no—
511C42—1222no—
512C42—1224no—
513C42—1224no—
514C42—1221no—
515C42—1221no—
516C42—2442no—
517C42—2442no—
518C42—2442no—
519C23—612no—
520C23—612no—
521C23—612no—
522C23—1221no—
523C23—1221no—
524C23—1221no—
525C23—1221no—
526C23C2×D4×D6111yescharacteristic
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—614no—
548C2×C4—614no—
549C2×C4—614no—
550C2×C4—614no—
551C2×C4—612no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—4884no—
561C2×C4—2444no—
562C2×C4—2444no—
563C2×C4—2448no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4#1451980412yes—
574C2×C4C22×C4×D6111yescharacteristic
575C2×C4C2×Q8×D6111yescharacteristic
576C2×C4#41×D6111yescharacteristic
577C2×C4#41×D6111yescharacteristic
578C2×C4#41×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C8—22no—
581C8—41no—
582C8—1222no—
583C8—2441no—
584C22—614no—
585C22—1222no—
586C22—1222no—
587C22—1222no—
588C22—1222no—
589C22—1222no—
590C22—1222no—
591C22C4×D4×D6111yescharacteristic
592C22C22⋊Q8×D6111yescharacteristic
593C22#123×D6111yescharacteristic
594C22C2×#1451980411yescharacteristic
595C22#137×D6111yescharacteristic
596C22#1684055811yescharacteristic
597C22#1684055911yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—614no—
605C4—1222no—
606C4—4881no—
607C4—2442no—
608C4—2442no—
609C4—2442no—
610C4—2441no—
611C4#117×D6111yescharacteristic
612C4C4×Q8×D6111yescharacteristic
613C4#125×D6111yescharacteristic
614C4#126×D6111yescharacteristic
615C2—1224no—
616C2#387×D6111yescharacteristic
617C2C4×#1451980411yescharacteristic
618C2#2145464111yescharacteristic
619C2#2145562911yescharacteristic
620C2#2145563611yescharacteristic
621C2#2145575611yescharacteristic
622C2#2145576611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1919C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1919D61
hypocenterC61#21034
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. #21446414C2semidirect
5. #21446410C2semidirect
6. #21446406C2semidirect
7. #21446402C2semidirect
8. #21446397C2semidirect
9. #21446391C2semidirect
10. #21446387C2semidirect
15. C61×#1919C2semidirect
24. #16839801C4semidirect
103. #14519711C2×C4semidirect
104. #14519710C2×C4semidirect
105. #14519708C2×C4semidirect
121. #106×C61C2×C4semidirect
312. #1919D61semidirect
371. C61#21034semidirect
427. #106C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398012
#389×D612
C4×#145197352
#214464142
#214464102
#214464062
#214464022
#214463972
#214463912
#214463872
#214455372
#214455112
C61×#19192
#2103461