Finite Group Databasebeta

← #29736185

Subgroups — #29736185

Order 15616.

1338 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D61C211yesmaximal, characteristic
2C2×#16840111C212yesmaximal, complemented
3#21454170C211yesmaximal, characteristic
4#21447066C211yesmaximal, characteristic, complemented
5#21447065C211yesmaximal, characteristic, complemented
6#21447062C211yesmaximal, characteristic, complemented
7#21446619C212yesmaximal, complemented
8#21445752C212yesmaximal
9#21445752C212yesmaximal
10#21445576C211yesmaximal, characteristic
11C61×#2043C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519776—21no—
15C2×#14519776—21no—
16C2×#14519776C2211yescharacteristic
17C2×#14519775C2212yes—
18C4⋊Q8×D61—28no—
19C2×#14519755—21no—
20C2×C4⋊C4×D61—21no—
21C2×C4⋊C4×D61—21no—
22C2×C42×D61C2211yescharacteristic
23#16840111—28no—
24C2×#14519702C2212yes—
25C2×#14519702C2212yes—
26C2×#14519701C2212yes—
27C2×#14519701C2212yes—
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519689C2212yes—
31C2×C8×Dic61C2212yes—
32#16839666—21no—
33#16839660C2211yescharacteristic
34#16839657—21no—
35#16839631C2211yescharacteristic
36#16839621—21no—
37#16839619C2211yescharacteristic
38#16839619C2211yescharacteristic
39#16839517C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C2×#103×C61C2212yes—
43C61×#389C2211yescharacteristic
44C61×#336C2211yescharacteristic
45C22×Dic122—22no—
46C22×Dic122—21no—
47C22×Dic122C2312yes—
48C2×Q8×D61—24no—
49C2×Q8×D61—24no—
50C2×Q8×D61—24no—
51C2×Q8×D61—24no—
52C2×Q8×D61—42no—
53C2×Q8×D61—42no—
54C22×C4×D61—22no—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C22×#92418C2312yes—
59#14519777—24no—
60#14519776—28no—
61#14519776—28no—
62#14519776—24no—
63#14519775—24no—
64#14519775D418yes—
65#14519755—44no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C4⋊C4×D61—28no—
77C4⋊C4×D61—28no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C42×D61—24no—
84#14519702—24no—
85#14519702—42no—
86#14519701—24no—
87#14519701—42no—
88#14519700—24no—
89#14519700—42no—
90#14519689—24no—
91C8×Dic61—24no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95C22×Q8×C61C2312yes—
96C22×C8×C61C2312yes—
97C4⋊Q8×C61—24no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C42×C61C2311yescharacteristic
104#103×C61—24no—
105#103×C61—42no—
106#95×C61—21no—
107C23×D61C2×D411yescharacteristic
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61C2×D414yes—
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×Dic61—21no—
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126Q8×D61—48no—
127Q8×D61—48no—
128Q8×D61—48no—
129Q8×D61—48no—
130C2×C4×D61—22no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—28no—
138C2×C4×D61—44no—
139C2×C4×D61—44no—
140C2×C4×D61SD1614yes—
141C2×C4×D61Q1614yes—
142C2×C4×D61#4114yes—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—28no—
148C2×Dic122—22no—
149C2×Dic122—44no—
150C2×Dic122—44no—
151C2×Dic122—42no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122C2×D414yes—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—42no—
164C42×C61—22no—
165#11758198—42no—
166#11758198—42no—
167C4⋊Dic61—24no—
168C4⋊Dic61—24no—
169C4⋊Dic61—22no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172#11758196—28no—
173#11758196—28no—
174#11758196—42no—
175#11758196—42no—
176#11758196—42no—
177#11758196—42no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61C2×D414yes—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C23×C61C22×D411yescharacteristic
185C22×D61—21no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61#12412yes—
190C22×D61C2×SD1612yes—
191C22×D61C2×Q1612yes—
192Q8×C61—24no—
193Q8×C61—24no—
194Q8×C61—42no—
195Q8×C61—42no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—24no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61C2×SD1612yes—
210C2×C4×C61C2×Q1612yes—
211C2×C4×C61C22×D412yes—
212C2×C4×C61C2×#4112yes—
213C2×C4×C61H(5,𝔽2)11yescharacteristic
214C2×C4×C61H(5,𝔽2)11yescharacteristic
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—24no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61#12012yes—
227C2×Dic61#12012yes—
228C2×Dic61#12412yes—
229C2×Dic61C2×SD1612yes—
230C2×Dic61C2×Q1612yes—
231C2×Dic61#13712yes—
232C2×Dic61#13712yes—
233C4×D61—24no—
234C4×D61—24no—
235C4×D61—28no—
236C4×D61—28no—
237C4×D61—416no—
238Dic122—24no—
239Dic122—24no—
240Dic122—416no—
241Dic122—42no—
242Dic122—42no—
243Dic122—84no—
244C8×C61—42no—
245#92418—42no—
246#25856—611nomaximal, Sylow
247C22×C61C2×#12011yescharacteristic
248C22×C61C2×#12411yescharacteristic
249C22×C61#53511yescharacteristic
250C22×C61C22×SD1611yescharacteristic
251C22×C61C22×Q1611yescharacteristic
252C22×C61C2×#13711yescharacteristic
253C22×C61C2×#13711yescharacteristic
254C2×D61—22no—
255C2×D61—22no—
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61#48714yescomplemented
261C4×C61—22no—
262C4×C61—22no—
263C4×C61—44no—
264C4×C61—81no—
265C4×C61—81no—
266C4×C61#54612yes—
267C4×C61#54612yes—
268Dic61—22no—
269Dic61—44no—
270Dic61—81no—
271Dic61—81no—
272Dic61#44814yes—
273Dic61#45114yes—
274C22×C4⋊Q8—611no—
275C2×#487—612no—
276#2857—611no—
277#2043—611no—
278#2043—611no—
279#2043—611no—
280#2043D6111yescharacteristic, complemented
281#1956—612no—
282#1787—612no—
283#1787—612no—
284C4×C22⋊C8—611no—
285C2×C61#243511yescharacteristic
286C2×C61C2×#44811yescharacteristic
287C2×C61C2×#45111yescharacteristic
288C2×C61C2×#48711yescharacteristic
289C2×C61#313911yescharacteristic
290C2×C61#320611yescharacteristic
291C2×C61#320911yescharacteristic
292D61—24no—
293C23×Q8—612no—
294C2×C4⋊Q8—611no—
295C2×C4⋊Q8—612no—
296C2×C4⋊Q8—611no—
297C2×C4⋊Q8—1221no—
298C2×C4⋊Q8—1221no—
299C2×C4⋊Q8—1228no—
300C2×C4⋊Q8C2×D6111yescharacteristic
301C2×C22⋊Q8—1221no—
302C22×C4⋊C4—1221no—
303C22×C4⋊C4—1221no—
304C22×C42—611no—
305#487—1228no—
306C2×#103—612no—
307C2×#103—612no—
308C2×#103—612no—
309C2×#103—612no—
310C2×#103—612no—
311C2×#103—612no—
312C2×#103C2×D6112yes—
313C2×#103C2×D6112yes—
314C2×C22⋊C8—612no—
315C2×C4×C8—612no—
316#391—1221no—
317#389—611no—
318#389—1221no—
319#389C2×D6111yescharacteristic
320#384—611no—
321#380—611no—
322#380—611no—
323#380—1221no—
324#336—611no—
325#336C2×D6111yescharacteristic
326C61#2585611yescharacteristic, complemented, Sylow
327C22×Q8—612no—
328C22×Q8—1222no—
329C22×Q8—1221no—
330C22×Q8—1224no—
331C22×Q8—1224no—
332C22×Q8—1224no—
333C22×Q8—1224no—
334C22×Q8—2442no—
335C22×Q8—2442no—
336C22×Q8C22×D6112yes—
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—1222no—
341C22×C8—612no—
342C22×C8C22×D6112yes—
343C4⋊Q8—24no—
344C4⋊Q8—618no—
345C4⋊Q8—1224no—
346C4⋊Q8—1224no—
347C4⋊Q8—1224no—
348C4⋊Q8—1228no—
349C4⋊Q8—1228no—
350C22⋊Q8—2444no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—1228no—
357C2×C4⋊C4—1228no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4C22×D6111yescharacteristic
367C2×C4⋊C4C22×D6111yescharacteristic
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C42—611no—
371C2×C42—611no—
372C2×C42—611no—
373C2×C42—1224no—
374C2×C42C22×D6111yescharacteristic
375#103—24no—
376#103—42no—
377#103—1224no—
378#103—1224no—
379#103—1224no—
380#103—2442no—
381#103—2442no—
382#103—2442no—
383C22⋊C8—1224no—
384C4×C8—1224no—
385#95—21no—
386#95—1221no—
387#95—1221no—
388#95—1221no—
389C24—611no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—614no—
396C2×Q8—1228no—
397C2×Q8—1222no—
398C2×Q8—1222no—
399C2×Q8—1222no—
400C2×Q8—1222no—
401C2×Q8—1222no—
402C2×Q8—2442no—
403C2×Q8—2442no—
404C2×Q8—2442no—
405C2×Q8—2444no—
406C2×Q8—2444no—
407C2×Q8—2448no—
408C2×Q8—2448no—
409C2×Q8—2448no—
410C2×Q8—2448no—
411C2×Q8D4×D6114yes—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—614no—
419C22×C4—614no—
420C22×C4—614no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1228no—
426C22×C4—1222no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—2444no—
433C22×C4—2444no—
434C22×C4D4×D6111yescharacteristic
435C22×C4D4×D6111yescharacteristic
436C22×C4C23×D6111yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—42no—
449C4⋊C4—1228no—
450C4⋊C4—1228no—
451C4⋊C4—1224no—
452C4⋊C4—1224no—
453C4⋊C4—1222no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—2442no—
457C4⋊C4—2442no—
458C4⋊C4—2442no—
459C4⋊C4—2442no—
460C22⋊C4—2442no—
461C22⋊C4—2442no—
462C42—22no—
463C42—614no—
464C42—1222no—
465C42—1222no—
466C23—612no—
467C23—612no—
468C23—612no—
469C23—1221no—
470C23—1221no—
471C23—1221no—
472C23—1221no—
473C23C2×D4×D6111yescharacteristic
474Q8—24no—
475Q8—24no—
476Q8—42no—
477Q8—42no—
478Q8—1224no—
479Q8—1224no—
480Q8—4884no—
481Q8—24416no—
482Q8—2442no—
483Q8—2442no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—612no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—612no—
503C2×C4—612no—
504C2×C4—1228no—
505C2×C4—1228no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—1224no—
511C2×C4—24416no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4SD16×D6112yes—
521C2×C4Q16×D6112yes—
522C2×C4C2×D4×D6112yes—
523C2×C4#41×D6112yes—
524C2×C4#1451982211yescharacteristic
525C2×C4#1451982311yescharacteristic
526C8—42no—
527C8—2442no—
528C22—614no—
529C22—1222no—
530C22—1222no—
531C22—1222no—
532C22—1222no—
533C22—1222no—
534C22—1222no—
535C22#120×D6111yescharacteristic
536C22#124×D6111yescharacteristic
537C22#1684032811yescharacteristic
538C22C2×SD16×D6111yescharacteristic
539C22C2×Q16×D6111yescharacteristic
540C22#137×D6111yescharacteristic
541C22#137×D6111yescharacteristic
542C4—22no—
543C4—22no—
544C4—44no—
545C4—81no—
546C4—81no—
547C4—614no—
548C4—614no—
549C4—1222no—
550C4—4881no—
551C4—4881no—
552C4—2444no—
553C4#1684040112yes—
554C4#1684040412yes—
555C2—1224no—
556C2#2144946511yescharacteristic
557C2#448×D6111yescharacteristic
558C2#451×D6111yescharacteristic
559C2#487×D6111yescharacteristic
560C2#2145619811yescharacteristic
561C2#2145674311yescharacteristic
562C2#2145676611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2043C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2043D61
hypocenterC61#25856
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
2. C2×#16840111C2semidirect
4. #21447066C2semidirect
5. #21447065C2semidirect
6. #21447062C2semidirect
7. #21446619C2semidirect
11. C61×#2043C2semidirect
260. C2×D61#487semidirect
280. #2043D61semidirect
326. C61#25856semidirect

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×#168401112
#214541702
#214470662
#214470652
#214470622
#214466192
#214457522
#214455762
C61×#20432
#2585661