Finite Group Databasebeta

← #29714373

Subgroups — #29714373

Order 15616.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16840058C211yesmaximal, characteristic, complemented
3C2×#16839969C211yesmaximal, characteristic, complemented
4C4×#14519757C211yesmaximal, characteristic
5#21447021C212yesmaximal
6#21446828C212yesmaximal
7#21446767C211yesmaximal, characteristic, complemented
8#21446158C212yesmaximal, complemented
9#21446075C212yesmaximal, complemented
10#21445796C211yesmaximal, characteristic
11C61×#1981C211yesmaximal, characteristic, complemented
12C22×#11758202C2212yes—
13#16840317—24no—
14#16840317—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519768—21no—
17C2×#14519757—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519753C2212yes—
20C2×Q8×Dic61C2212yes—
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519742—21no—
23C2×#14519735—21no—
24C2×C42×D61C2211yescharacteristic
25#16840058—24no—
26#16839969—24no—
27C2×#14519711C2212yes—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2212yes—
30C2×#14519701C2212yes—
31C2×#14519689C2212yes—
32#16839667C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839647—21no—
35#16839636—21no—
36#16839619C2211yescharacteristic
37C4×C4⋊Dic61—21no—
38C4×#11758198—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C2×#103×C61C2212yes—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C2×#11758202—24no—
53C2×#11758202—24no—
54C2×#11758202—42no—
55C22×#92418C2312yes—
56#14519777—22no—
57#14519777D414yes—
58#14519768—28no—
59#14519757—28no—
60#14519757—22no—
61#14519757—22no—
62#14519753—24no—
63#14519753—24no—
64Q8×Dic61—24no—
65Q8×Dic61—24no—
66C4×Dic122—22no—
67C4×Dic122—22no—
68#14519742—28no—
69#14519735—28no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2312yes—
76C2×#11758196—22no—
77C2×#11758196—21no—
78C2×#11758198—22no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61—22no—
87C42×D61—22no—
88#14519711—24no—
89C4⋊#92418—22no—
90#14519702—24no—
91#14519701—24no—
92#14519689—42no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×Q8×C61C2312yes—
98C22×C8×C61C2312yes—
99C4⋊Q8×C61—22no—
100C4⋊Q8×C61D414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—22no—
108#103×C61—24no—
109#103×C61—42no—
110#95×C61—21no—
111C23×D61C2×D411yescharacteristic
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61C2×D414yes—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—22no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—41no—
140C2×C4×D61—41no—
141C2×C4×D61Q1614yescomplemented
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—24no—
147C2×Dic122—21no—
148C2×Dic122—42no—
149C2×Dic122—42no—
150C2×Dic122#4112yes—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61#4112yes—
160C42×C61—42no—
161C42×C61C2×D412yes—
162C42×C61C2×D412yes—
163#11758202—48no—
164#11758198—28no—
165#11758198—24no—
166#11758198—42no—
167C4⋊Dic61—28no—
168C4⋊Dic61—24no—
169C4⋊Dic61—24no—
170C4⋊Dic61—22no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—42no—
174C4⋊Dic61C2×D414yes—
175#11758196—28no—
176#11758196—42no—
177C4×Dic61—28no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—42no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C23×C61C22×D411yescharacteristic
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61C2×Q1612yes—
194Q8×C61—24no—
195Q8×C61—24no—
196Q8×C61—42no—
197Q8×C61—42no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—24no—
201C2×C4×C61—22no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61#12012yes—
210C2×C4×C61#12012yes—
211C2×C4×C61C2×Q1612yes—
212C2×C4×C61#13712yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61H(5,𝔽2)11yescharacteristic
218C2×C4×C612−1+411yescharacteristic
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—24no—
224C2×Dic61—21no—
225C2×Dic61—22no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61C2×Q1612yes—
237C4×D61—28no—
238C4×D61—44no—
239C4×D61—41no—
240C4×D61—41no—
241C4×D61—41no—
242C4×D61—41no—
243Dic122—22no—
244Dic122—22no—
245Dic122—48no—
246Dic122—41no—
247Dic122—41no—
248C8×C61—42no—
249#92418—42no—
250#24155—611nomaximal, Sylow
251C22×C61C2×#12011yescharacteristic
252C22×C61#52511yescharacteristic
253C22×C61#53711yescharacteristic
254C22×C61C22×Q1611yescharacteristic
255C22×C61C2×#13711yescharacteristic
256C22×C61#57511yescharacteristic
257C22×C61#57511yescharacteristic
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—41no—
262C2×D61—41no—
263C2×D61—41no—
264C2×D61—41no—
265C4×C61—22no—
266C4×C61—42no—
267C4×C61—44no—
268C4×C61—81no—
269C4×C61#45114yes—
270C4×C61#54611yescharacteristic
271C4×C61#54711yescharacteristic
272C4×C61#54811yescharacteristic
273C4×C61#55011yescharacteristic
274Dic61—22no—
275Dic61—42no—
276Dic61—41no—
277Dic61—44no—
278Dic61—41no—
279Dic61—81no—
280C2×#533—611no—
281C2×#477—611no—
282C2×#462—611no—
283C4×C22⋊Q8—611no—
284#2036—612no—
285#2001—612no—
286#1987—611no—
287#1981D6111yescharacteristic, complemented
288#1878—612no—
289#1863—612no—
290#1797—611no—
291C2×C61#240711yescharacteristic
292C2×C61C2×#45111yescharacteristic
293C2×C61#300511yescharacteristic
294C2×C61#310411yescharacteristic
295C2×C61#311611yescharacteristic
296C2×C61#324911yescharacteristic
297C2×C61#325211yescharacteristic
298D61—42no—
299C22×Q16—612no—
300#533—1224no—
301#533—1224no—
302C2×C4⋊Q8—611no—
303C2×C4⋊Q8C2×D6111yescharacteristic
304C2×#125—1221no—
305C2×C22⋊Q8—612no—
306C2×C22⋊Q8—611no—
307C2×C22⋊Q8—1221no—
308C2×C4×Q8—612no—
309C2×C4×Q8—611no—
310C2×C4×Q8—1221no—
311C2×#117—1221no—
312C22×C42—611no—
313#477—1224no—
314#462—1224no—
315C2×#107—612no—
316C2×C4⋊C8—611no—
317C2×C4⋊C8C2×D6111yescharacteristic
318C2×#103—612no—
319C2×#103—612no—
320C2×#103C2×D6112yes—
321C2×#103C2×D6112yes—
322C2×C22⋊C8—612no—
323#391—611no—
324#389—611no—
325#387—1221no—
326#385—1221no—
327#380—611no—
328C4×C4⋊C4—1221no—
329C4×C4⋊C4C2×D6111yescharacteristic
330C4×C22⋊C4—1221no—
331#340—612no—
332#340—612no—
333#336—612no—
334C61#2415511yescharacteristic, complemented, Sylow
335C22×Q8—611no—
336C22×Q8—1221no—
337C22×Q8C22×D6112yes—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—611no—
341C2×Q16—1224no—
342C2×Q16—1224no—
343C2×Q16—2442no—
344C22×C8—612no—
345C22×C8C22×D6112yes—
346C4⋊Q8—22no—
347C4⋊Q8—614no—
348C4⋊Q8—1222no—
349C4⋊Q8D24414yes—
350#125—1228no—
351C22⋊Q8—1222no—
352C22⋊Q8—1228no—
353C22⋊Q8—1224no—
354C22⋊Q8—1224no—
355C22⋊Q8—1222no—
356C4×Q8—1228no—
357C4×Q8—1224no—
358C4×Q8—1224no—
359C4×Q8—1222no—
360C4×Q8—1222no—
361#117—1228no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C42—21no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—1222no—
382C2×C42—1222no—
383C2×C42—1222no—
384C2×C42—1221no—
385C2×C42C22×D6111yescharacteristic
386#107—1224no—
387C4⋊C8—22no—
388C4⋊C8—1222no—
389#103—24no—
390#103—42no—
391#103—1224no—
392#103—1224no—
393C22⋊C8—2442no—
394#95—21no—
395#95—1221no—
396#95—1221no—
397#95—1221no—
398#95—1221no—
399C24—611no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—612no—
406C2×Q8—1221no—
407C2×Q8—1224no—
408C2×Q8—1221no—
409C2×Q8—1221no—
410C2×Q8—1221no—
411C2×Q8—1221no—
412C2×Q8—2442no—
413C2×Q8—2442no—
414C2×Q8C2×D24414yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—611no—
419C22×C4—614no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1222no—
433C22×C4—1221no—
434C22×C4—2441no—
435C22×C4—2441no—
436C22×C4D4×D6111yescharacteristic
437C22×C4D4×D6111yescharacteristic
438C22×C4C23×D6111yescharacteristic
439Q16—2448no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—21no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—614no—
452C4⋊C4—1228no—
453C4⋊C4—1228no—
454C4⋊C4—1224no—
455C4⋊C4—1222no—
456C4⋊C4—1224no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—2442no—
460C4⋊C4—2442no—
461C4⋊C4#1175822412yes—
462C22⋊C4—1228no—
463C22⋊C4—1224no—
464C22⋊C4—2442no—
465C42—42no—
466C42—1228no—
467C42—1221no—
468C42—1221no—
469C42—1222no—
470C42—1222no—
471C42—2442no—
472C42C2×D24412yes—
473C42D4×D6112yes—
474C23—612no—
475C23—1221no—
476C23—1221no—
477C23—1221no—
478C23—1221no—
479C23—1221no—
480C23—1221no—
481C23C2×D4×D6111yescharacteristic
482Q8—24no—
483Q8—24no—
484Q8—42no—
485Q8—42no—
486Q8—1222no—
487Q8—1222no—
488Q8—2448no—
489Q8—2441no—
490Q8—2441no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—612no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1228no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1224no—
509C2×C4—1222no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—2444no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4#1451974712yes—
527C2×C4#1451974712yes—
528C2×C4Q16×D6112yes—
529C2×C4#1451980612yes—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C22×D24411yescharacteristic
532C2×C4#41×D6111yescharacteristic
533C2×C4C2×#1175822411yescharacteristic
534C2×C4#1451982211yescharacteristic
535C2×C4#1451982411yescharacteristic
536C8—42no—
537C8—2442no—
538C22—1222no—
539C22—1222no—
540C22—1222no—
541C22—2441no—
542C22—2441no—
543C22—2441no—
544C22—2441no—
545C22C2×#1451974711yescharacteristic
546C22#1684027211yescharacteristic
547C22#1684034111yescharacteristic
548C22C2×Q16×D6111yescharacteristic
549C22C2×#1451980611yescharacteristic
550C22#1684056111yescharacteristic
551C22#1684056111yescharacteristic
552C4—22no—
553C4—42no—
554C4—44no—
555C4—81no—
556C4—1222no—
557C4—4881no—
558C4—2441no—
559C4—2442no—
560C4—2441no—
561C4—2444no—
562C4#1683991914yes—
563C4#1684040711yescharacteristic
564C4#1684041611yescharacteristic
565C4#1684042011yescharacteristic
566C4#1684043211yescharacteristic
567C2—2442no—
568C2#2144916211yescharacteristic
569C2C2×#1683991911yescharacteristic
570C2#2145506511yescharacteristic
571C2#2145585711yescharacteristic
572C2#2145597811yescharacteristic
573C2#2145716211yescharacteristic
574C2#2145718811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1981C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1981D61
hypocenterC61#24155
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
2. C2×#16840058C2semidirect
3. C2×#16839969C2semidirect
7. #21446767C2semidirect
8. #21446158C2semidirect
9. #21446075C2semidirect
11. C61×#1981C2semidirect
141. C2×C4×D61Q16semidirect
287. #1981D61semidirect
334. C61#24155semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168400582
C2×#168399692
C4×#145197572
#214470212
#214468282
#214467672
#214461582
#214460752
#214457962
C61×#19812
#2415561