Finite Group Databasebeta

← #29716074

Subgroups — #29716074

Order 15616.

1378 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D61C211yesmaximal, characteristic, complemented
2C2×#16840043C211yesmaximal, characteristic, complemented
3C2×#16840039C211yesmaximal, characteristic, complemented
4C4×#14519752C211yesmaximal, characteristic
5#21447146C212yesmaximal, complemented
6#21446954C212yesmaximal, complemented
7#21446771C211yesmaximal, characteristic, complemented
8#21446135C212yesmaximal
9#21446008C212yesmaximal, complemented
10#21445796C211yesmaximal, characteristic, complemented
11C61×#1988C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C2×#14519777C2211yescharacteristic
14C2×#14519776—21no—
15C2×#14519776C2212yes—
16C2×#14519775—21no—
17C4⋊Q8×D61—24no—
18C4⋊Q8×D61—24no—
19C2×#14519752C2211yescharacteristic
20C2×C4×D244—21no—
21C2×C4×D244C2211yescharacteristic
22C2×C4⋊C4×D61—21no—
23C2×C4⋊C4×D61C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16840043—24no—
26#16840039—24no—
27C2×#14519711C2212yes—
28C2×#14519710C2212yes—
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519699C2212yes—
31C2×#14519697C2212yes—
32C2×#14519689C2212yes—
33#16839663C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839654—21no—
36#16839640—21no—
37#16839619C2211yescharacteristic
38C4×#11758198—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C61×#340C2212yes—
47C22×Dic122—21no—
48C22×Dic122—22no—
49C2×Q8×D61—28no—
50C2×Q8×D61—44no—
51C2×Q8×D61—44no—
52C22×D244—21no—
53C22×D244C2311yescharacteristic
54C22×C4×D61—22no—
55C22×C4×D61—21no—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C22×#92418C2312yes—
60#14519777—22no—
61#14519777D414yescomplemented
62#14519776—28no—
63#14519776—24no—
64#14519776—24no—
65#14519775—28no—
66#14519752—22no—
67#14519752Q814yescomplemented
68C4×D244—22no—
69C4×D244—22no—
70C4×D244—44no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C4⋊C4×D61—28no—
78C4⋊C4×D61—24no—
79C4⋊C4×D61—24no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61—22no—
89C42×D61—22no—
90#14519711—24no—
91#14519710—24no—
92C4⋊#92418—22no—
93#14519699—24no—
94#14519697—24no—
95#14519689—24no—
96#14519684—21no—
97#14519684—21no—
98C22×Q8×C61—21no—
99C22×C8×C61C2312yes—
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61D414yescomplemented
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2312yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—22no—
109#95×C61—21no—
110C23×D61—21no—
111C23×D61C2×D411yescharacteristic
112C2×Q8×C61—24no—
113C2×Q8×C61—42no—
114C2×Q8×C61—42no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—22no—
122C22×Dic61—21no—
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126Q8×D61—432no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—24no—
132C2×D244—21no—
133C2×D244—42no—
134C2×D244—42no—
135C2×D244C2×Q812yescomplemented
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—22no—
145C2×C4×D61—28no—
146C2×C4×D61—44no—
147C2×C4×D61—44no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61—42no—
151C2×C4×D61D814yes—
152C2×C4×D61C2×Q812yes—
153C2×C4×D61#4112yes—
154C2×Dic122—28no—
155C2×Dic122—24no—
156C2×Dic122—44no—
157C2×Dic122—42no—
158C2×Dic122—42no—
159C2×Dic122—44no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61C2×D414yes—
169C4⋊C4×C61C2×Q812yescomplemented
170C42×C61—42no—
171C42×C61C2×D412yes—
172C42×C61C2×D412yes—
173#11758198—24no—
174#11758198—42no—
175#11758198—42no—
176#11758198—42no—
177#11758198—42no—
178C4⋊Dic61—24no—
179C4⋊Dic61—22no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—42no—
183C4⋊Dic61C2×D414yes—
184#11758196—28no—
185#11758196—28no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C23×C61C22×D411yescharacteristic
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—22no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61C22⋊Q812yes—
206C22×D61C2×D812yes—
207C22×D61#13712yes—
208Q8×C61—48no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—24no—
212C2×C4×C61—22no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61#12012yes—
221C2×C4×C61#12012yes—
222C2×C4×C61C2×D812yes—
223C2×C4×C61AGL(1,ℤ/8)12yes—
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×Q811yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C2×C4×C612−1+411yescharacteristic
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—24no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61C22⋊Q812yes—
243C2×Dic61C2×D812yes—
244C2×Dic61#13712yes—
245D244—22no—
246D244—22no—
247D244—48no—
248D244—41no—
249D244—41no—
250C4×D61—28no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—28no—
256C4×D61—416no—
257C4×D61—84no—
258Dic122—416no—
259Dic122—48no—
260C8×C61—42no—
261#92418—42no—
262#24298—611nomaximal, Sylow
263C22×C61C2×#12011yescharacteristic
264C22×C61C2×C22⋊Q811yescharacteristic
265C22×C61#53611yescharacteristic
266C22×C61C22×D811yescharacteristic
267C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C61C2×#13711yescharacteristic
269C22×C61C2×#13711yescharacteristic
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61#47414yescomplemented
284C4×C61—22no—
285C4×C61—44no—
286C4×C61—42no—
287C4×C61—81no—
288C4×C61#44714yes—
289C4×C61#54811yescharacteristic
290C4×C61#54811yescharacteristic
291C4×C61D4×Q811yescharacteristic
292C4×C61D4×Q811yescharacteristic
293Dic61—22no—
294Dic61—22no—
295Dic61—22no—
296Dic61—44no—
297Dic61—81no—
298C22×C4⋊Q8—611no—
299C2×#474—611no—
300C2×#474—611no—
301C4×#121—611no—
302#2057—612no—
303#2023—612no—
304#1988—611no—
305#1988D6111yescharacteristic, complemented
306#1874—612no—
307#1852—612no—
308#1797—611no—
309C2×C61#240911yescharacteristic
310C2×C61C2×#44711yescharacteristic
311C2×C61#300411yescharacteristic
312C2×C61C2×#47411yescharacteristic
313C2×C61#307911yescharacteristic
314C2×C61#322611yescharacteristic
315C2×C61#323911yescharacteristic
316D61—24no—
317D61—82no—
318C23×Q8—1221no—
319C2×C4⋊Q8—612no—
320C2×C4⋊Q8—611no—
321C2×C4⋊Q8—1221no—
322C2×C4⋊Q8—1221no—
323C2×C4⋊Q8—1224no—
324C2×C4⋊Q8—1224no—
325C2×C4⋊Q8C2×D6111yescharacteristic
326C2×#121—611no—
327C2×C4×D4—611no—
328C2×C4×D4—1221no—
329C22×C4⋊C4—612no—
330C22×C4⋊C4—1221no—
331C22×C42—611no—
332#474—1224no—
333#474—1224no—
334C2×#107—612no—
335C2×#107—612no—
336C2×C4⋊C8—611no—
337C2×C4⋊C8C2×D6111yescharacteristic
338C2×#102—612no—
339C2×#102—612no—
340C2×C22⋊C8—612no—
341#390—611no—
342#389—611no—
343#388—1221no—
344#386—1221no—
345#380—611no—
346C4×C4⋊C4C2×D6111yescharacteristic
347C4×C22⋊C4—1221no—
348#340—612no—
349#340—612no—
350#340C2×D6112yes—
351#340C2×D6112yes—
352#336—612no—
353C61#2429811yescharacteristic, complemented, Sylow
354C22×Q8—21no—
355C22×Q8—1222no—
356C22×Q8—1228no—
357C22×Q8—1221no—
358C22×Q8—2444no—
359C22×Q8—2444no—
360C22×D4—611no—
361C22×D4—1221no—
362C23×C4—611no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—1222no—
366C23×C4—1221no—
367C22×C8—612no—
368C22×C8C22×D6112yes—
369C4⋊Q8—22no—
370C4⋊Q8—614no—
371C4⋊Q8—1224no—
372C4⋊Q8—1228no—
373C4⋊Q8—1228no—
374C4⋊Q8—1224no—
375C4⋊Q8—1222no—
376C4⋊Q8D24414yescomplemented
377#121—614no—
378#121—1222no—
379C4×D4—1222no—
380C4×D4—1222no—
381C4×D4—2444no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4—1228no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1224no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6112yes—
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C42—21no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—1222no—
405C2×C42—1222no—
406C2×C42C22×D6111yescharacteristic
407#107—1224no—
408#107—1224no—
409C4⋊C8—22no—
410C4⋊C8—1222no—
411#102—1224no—
412#102—1224no—
413C22⋊C8—1224no—
414#95—21no—
415#95—1221no—
416#95—1221no—
417C24—611no—
418C24—1221no—
419C2×Q8—24no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—1228no—
423C2×Q8—1224no—
424C2×Q8—2444no—
425C2×Q8—2444no—
426C2×Q8—24432no—
427C2×Q8—2442no—
428C2×Q8—2442no—
429C2×D4—612no—
430C2×D4—1224no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—2442no—
437C2×D4—2442no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—614no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1228no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1221no—
459C22×C4—2444no—
460C22×C4—2444no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4D4×D6111yescharacteristic
465C22×C4D4×D6111yescharacteristic
466C22×C4C23×D6111yescharacteristic
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—614no—
479C4⋊C4—1228no—
480C4⋊C4—1228no—
481C4⋊C4—1222no—
482C4⋊C4—1224no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—2442no—
486C4⋊C4C2×D24414yes—
487C4⋊C4Q8×D6112yescomplemented
488C22⋊C4—1224no—
489C22⋊C4—2442no—
490C22⋊C4—2442no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C42—42no—
494C42—1221no—
495C42—1221no—
496C42—1222no—
497C42—1222no—
498C42C2×D24412yes—
499C42D4×D6112yes—
500C23—612no—
501C23—612no—
502C23—612no—
503C23—1221no—
504C23—1221no—
505C23—1221no—
506C23—1221no—
507C23—1222no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23C2×D4×D6111yescharacteristic
515Q8—48no—
516Q8—24416no—
517Q8—2448no—
518D4—1222no—
519D4—1222no—
520D4—2448no—
521D4—2441no—
522D4—2441no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—612no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—1228no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1228no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1224no—
550C2×C4—4884no—
551C2×C4—24416no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4#1451974712yes—
558C2×C4#1451974712yes—
559C2×C4D8×D6112yes—
560C2×C4#1451980412yes—
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C22×D24411yescharacteristic
563C2×C4C2×Q8×D6111yescharacteristic
564C2×C4#41×D6111yescharacteristic
565C2×C4#1451982411yescharacteristic
566C2×C4#1451982511yescharacteristic
567C8—42no—
568C8—2442no—
569C22—614no—
570C22—1222no—
571C22—1222no—
572C22—1222no—
573C22—1222no—
574C22—1222no—
575C22—1222no—
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—2442no—
581C22—2442no—
582C22—2442no—
583C22C2×#1451974711yescharacteristic
584C22C22⋊Q8×D6111yescharacteristic
585C22#1684033511yescharacteristic
586C22C2×D8×D6111yescharacteristic
587C22C2×#1451980411yescharacteristic
588C22#137×D6111yescharacteristic
589C22#137×D6111yescharacteristic
590C4—22no—
591C4—44no—
592C4—42no—
593C4—81no—
594C4—1222no—
595C4—1222no—
596C4—1222no—
597C4—4881no—
598C4—2444no—
599C4#1683989714yes—
600C4#1684042011yescharacteristic
601C4#1684042411yescharacteristic
602C4#1684042711yescharacteristic
603C4Q8×D24411yescharacteristic
604C2—1224no—
605C2—4882no—
606C2#2144918211yescharacteristic
607C2C2×#1683989711yescharacteristic
608C2#2145505511yescharacteristic
609C2#474×D6111yescharacteristic
610C2#2145562911yescharacteristic
611C2#2145693911yescharacteristic
612C2#2145705311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1988C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#1988D61
hypocenterC61#24298
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D61C2semidirect
2. C2×#16840043C2semidirect
3. C2×#16840039C2semidirect
5. #21447146C2semidirect
6. #21446954C2semidirect
7. #21446771C2semidirect
9. #21446008C2semidirect
10. #21445796C2semidirect
11. C61×#1988C2semidirect
61. #14519777D4semidirect
67. #14519752Q8semidirect
101. C4⋊Q8×C61D4semidirect
135. C2×D244C2×Q8semidirect
169. C4⋊C4×C61C2×Q8semidirect
283. C2×D61#474semidirect
305. #1988D61semidirect
353. C61#24298semidirect
376. C4⋊Q8D244semidirect
487. C4⋊C4Q8×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
C2×#168400432
C2×#168400392
C4×#145197522
#214471462
#214469542
#214467712
#214461352
#214460082
#214457962
C61×#19882
#2429861