Finite Group Databasebeta

← #29736668

Subgroups — #29736668

Order 15616.

1320 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D61C211yesmaximal, characteristic, complemented
2C2×#16840124C211yesmaximal, characteristic, complemented
3C2×#16840122C211yesmaximal, characteristic, complemented
4#21454121C211yesmaximal, characteristic
5#21446959C212yesmaximal, complemented
6#21446954C212yesmaximal, complemented
7#21446791C211yesmaximal, characteristic, complemented
8#21446135C212yesmaximal
9#21446125C212yesmaximal, complemented
10#21445582C211yesmaximal, characteristic, complemented
11C61×#1993C211yesmaximal, 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×#14519765C2211yescharacteristic
20C2×#14519757C2211yescharacteristic
21C2×#14519752C2211yescharacteristic
22C2×C4⋊C4×D61—21no—
23C2×C4⋊C4×D61C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16840124—24no—
26#16840122—24no—
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519699C2212yes—
30C2×#14519697C2212yes—
31C2×#14519689C2212yes—
32C2×#14519688C2211yescharacteristic
33#16839667C2211yescharacteristic
34#16839663C2211yescharacteristic
35#16839654—21no—
36#16839654—21no—
37#16839640—21no—
38#16839640—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C2×#97×C61C2211yescharacteristic
44C61×#382C2211yescharacteristic
45C61×#340C2212yes—
46C61×#340C2212yes—
47C22×Dic122—21no—
48C22×Dic122—22no—
49C22×Dic122C2311yescharacteristic
50C2×Q8×D61—28no—
51C2×Q8×D61—44no—
52C2×Q8×D61—44no—
53C22×D244—21no—
54C22×D244C2311yescharacteristic
55C22×C4×D61—22no—
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#14519765—22no—
67#14519765—22no—
68#14519757—24no—
69#14519752—24no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×#11758196—22no—
75C4⋊C4×D61—28no—
76C4⋊C4×D61—24no—
77C4⋊C4×D61—24no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87C42×D61—22no—
88C42×D61—22no—
89#14519702—24no—
90#14519700—24no—
91#14519699—24no—
92#14519697—24no—
93#14519689—24no—
94#14519688—22no—
95#14519684—21no—
96#14519684—21no—
97C22×Q8×C61—21no—
98C22×C8×C61C2312yes—
99C4⋊Q8×C61—22no—
100C4⋊Q8×C61D414yescomplemented
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2312yes—
105C2×C42×C61C2311yescharacteristic
106#97×C61—22no—
107#95×C61—21no—
108#95×C61—21no—
109C23×D61—21no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—24no—
112C2×Q8×C61—42no—
113C2×Q8×C61—42no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C22×Dic61—22no—
121C22×Dic61—21no—
122C22×Dic61C2×D411yescharacteristic
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125Q8×D61—432no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—42no—
131C2×D244—42no—
132C2×D244—42no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—28no—
143C2×C4×D61—44no—
144C2×C4×D61—44no—
145C2×C4×D61#4114yes—
146C2×Dic122—28no—
147C2×Dic122—24no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—44no—
153C2×Dic122—42no—
154C2×Dic122—42no—
155C2×Dic122—44no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61C2×D414yes—
164C42×C61C2×D412yes—
165C42×C61C2×D412yes—
166#11758198—42no—
167#11758198—42no—
168#11758198—42no—
169#11758198—42no—
170#11758198—42no—
171#11758198—42no—
172C4⋊Dic61—24no—
173C4⋊Dic61—22no—
174C4⋊Dic61—42no—
175C4⋊Dic61C2×D414yes—
176#11758196—28no—
177#11758196—28no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C23×C61C22×D411yescharacteristic
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61#12412yes—
198C22×D61AGL(1,ℤ/8)12yes—
199C22×D61#13712yes—
200Q8×C61—48no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—24no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61#12012yes—
215C2×C4×C61#12012yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4112yes—
219C2×C4×C612−1+412yes—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—24no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61#12412yes—
233C2×Dic61AGL(1,ℤ/8)12yes—
234C2×Dic61#13712yes—
235D244—42no—
236D244—42no—
237D244—84no—
238C4×D61—28no—
239C4×D61—28no—
240C4×D61—24no—
241C4×D61—24no—
242C4×D61—416no—
243Dic122—416no—
244Dic122—48no—
245Dic122—42no—
246Dic122—42no—
247C8×C61—42no—
248#92418—42no—
249#25908—611nomaximal, Sylow
250C22×C61C2×#12011yescharacteristic
251C22×C61C2×#12411yescharacteristic
252C22×C61#53611yescharacteristic
253C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C61C2×#13711yescharacteristic
256C22×C61C2×#13711yescharacteristic
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61#48914yescomplemented
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—44no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61#54812yes—
277C4×C61#54812yes—
278Dic61—22no—
279Dic61—22no—
280Dic61—22no—
281Dic61—44no—
282Dic61—81no—
283C22×C4⋊Q8—611no—
284C2×#489—611no—
285C2×#489—611no—
286#2850—611no—
287#2024—612no—
288#2023—612no—
289#1993—611no—
290#1993D6111yescharacteristic, complemented
291#1874—612no—
292#1872—612no—
293#1749—611no—
294C2×C61#243611yescharacteristic
295C2×C61#300311yescharacteristic
296C2×C61#300411yescharacteristic
297C2×C61C2×#48911yescharacteristic
298C2×C61#314111yescharacteristic
299C2×C61#323811yescharacteristic
300C2×C61#323911yescharacteristic
301D61—24no—
302D61—161no—
303C23×Q8—1221no—
304C2×C4⋊Q8—612no—
305C2×C4⋊Q8—611no—
306C2×C4⋊Q8—1221no—
307C2×C4⋊Q8—1221no—
308C2×C4⋊Q8—1224no—
309C2×C4⋊Q8—1224no—
310C2×C4⋊Q8C2×D6111yescharacteristic
311C2×#124—611no—
312C2×C22⋊Q8—611no—
313C2×#121—611no—
314C22×C4⋊C4—612no—
315C22×C4⋊C4—1221no—
316C22×C42—611no—
317#489—1224no—
318#489—1224no—
319C2×#103—612no—
320C2×#103—612no—
321C2×#102—612no—
322C2×#102—612no—
323C2×C22⋊C8—612no—
324C2×#97—611no—
325C2×#97C2×D6111yescharacteristic
326#391—611no—
327#390—611no—
328#388—1221no—
329#388—1221no—
330#386—1221no—
331#386—1221no—
332#382C2×D6111yescharacteristic
333#340—612no—
334#340—612no—
335#340C2×D6112yes—
336#340C2×D6112yes—
337#336—612no—
338C61#2590811yescharacteristic, complemented, Sylow
339C22×Q8—21no—
340C22×Q8—611no—
341C22×Q8—1222no—
342C22×Q8—1228no—
343C22×Q8—1221no—
344C22×Q8—2444no—
345C22×Q8—2444no—
346C22×D4—611no—
347C22×D4—1221no—
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—1222no—
352C22×C8—612no—
353C22×C8C22×D6112yes—
354C4⋊Q8—22no—
355C4⋊Q8—614no—
356C4⋊Q8—1224no—
357C4⋊Q8—1228no—
358C4⋊Q8—1228no—
359C4⋊Q8—1224no—
360C4⋊Q8—1222no—
361C4⋊Q8D24414yescomplemented
362#124—1222no—
363#124—1222no—
364C22⋊Q8—1224no—
365#121—1224no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1224no—
369C2×C4⋊C4—1228no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1224no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4C22×D6111yescharacteristic
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C4⋊C4C22×D6112yes—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—1222no—
388C2×C42—1222no—
389C2×C42C22×D6111yescharacteristic
390#103—1224no—
391#103—1224no—
392#102—1224no—
393#102—1224no—
394C22⋊C8—1224no—
395#97—22no—
396#97—1222no—
397#95—21no—
398#95—21no—
399#95—1221no—
400#95—1221no—
401C24—611no—
402C24—1221no—
403C2×Q8—24no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—1228no—
407C2×Q8—1224no—
408C2×Q8—1221no—
409C2×Q8—1221no—
410C2×Q8—1222no—
411C2×Q8—1222no—
412C2×Q8—2442no—
413C2×Q8—2442no—
414C2×Q8—2444no—
415C2×Q8—2444no—
416C2×Q8—24432no—
417C2×D4—1221no—
418C2×D4—1221no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—2442no—
422C2×D4—2442no—
423C2×D4—2442no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—614no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1228no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1221no—
443C22×C4—2444no—
444C22×C4—2444no—
445C22×C4D4×D6111yescharacteristic
446C22×C4D4×D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1222no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—614no—
459C4⋊C4—1228no—
460C4⋊C4—1228no—
461C4⋊C4—1222no—
462C4⋊C4—1224no—
463C4⋊C4—2442no—
464C4⋊C4C2×D24414yes—
465C22⋊C4—2442no—
466C22⋊C4—2442no—
467C22⋊C4—2442no—
468C22⋊C4—2442no—
469C22⋊C4—2442no—
470C22⋊C4—2442no—
471C42—1221no—
472C42—1221no—
473C42—1222no—
474C42—1222no—
475C42C2×D24412yes—
476C42D4×D6112yes—
477C23—612no—
478C23—612no—
479C23—612no—
480C23—1221no—
481C23—1221no—
482C23—1221no—
483C23—1221no—
484C23—2441no—
485C23—2441no—
486C23—2441no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23C2×D4×D6111yescharacteristic
492Q8—48no—
493Q8—2448no—
494Q8—24416no—
495Q8—2442no—
496Q8—2442no—
497D4—4884no—
498D4—2442no—
499D4—2442no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—1228no—
517C2×C4—1228no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—24416no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4#1451974712yes—
534C2×C4#1451974712yes—
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4C22×D24411yescharacteristic
537C2×C4#41×D6112yes—
538C2×C4#1451982512yes—
539C8—42no—
540C8—2442no—
541C22—614no—
542C22—1222no—
543C22—1222no—
544C22—1222no—
545C22—1222no—
546C22—1222no—
547C22—1222no—
548C22—4881no—
549C22—4881no—
550C22—4881no—
551C22—4881no—
552C22—4881no—
553C22—4881no—
554C22—4881no—
555C22C2×#1451974711yescharacteristic
556C22#124×D6111yescharacteristic
557C22#1684033711yescharacteristic
558C22AGL(1,ℤ/8)×D6111yescharacteristic
559C22AGL(1,ℤ/8)×D6111yescharacteristic
560C22#137×D6111yescharacteristic
561C22#137×D6111yescharacteristic
562C4—22no—
563C4—22no—
564C4—44no—
565C4—81no—
566C4—81no—
567C4—1222no—
568C4—1222no—
569C4—1222no—
570C4—4881no—
571C4—2444no—
572C4#1684042212yes—
573C4#1684042412yes—
574C2—1224no—
575C2—9761no—
576C2#2144947611yescharacteristic
577C2#2145504811yescharacteristic
578C2#2145505511yescharacteristic
579C2#489×D6111yescharacteristic
580C2#2145621111yescharacteristic
581C2#2145704111yescharacteristic
582C2#2145705011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1993C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#1993D61
hypocenterC61#25908
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D61C2semidirect
2. C2×#16840124C2semidirect
3. C2×#16840122C2semidirect
5. #21446959C2semidirect
6. #21446954C2semidirect
7. #21446791C2semidirect
9. #21446125C2semidirect
10. #21445582C2semidirect
11. C61×#1993C2semidirect
61. #14519777D4semidirect
100. C4⋊Q8×C61D4semidirect
270. C2×D61#489semidirect
290. #1993D61semidirect
338. C61#25908semidirect
361. C4⋊Q8D244semidirect

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×#168401242
C2×#168401222
#214541212
#214469592
#214469542
#214467912
#214461352
#214461252
#214455822
C61×#19932
#2590861