Finite Group Databasebeta

← #29714377

Subgroups — #29714377

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×#16839972C211yesmaximal, characteristic, complemented
3C4×#14519753C211yesmaximal, characteristic
4#21447024C212yesmaximal
5#21446828C212yesmaximal
6#21446768C211yesmaximal, characteristic, complemented
7#21446741C211yesmaximal, characteristic, complemented
8#21446157C212yesmaximal, complemented
9#21446070C212yesmaximal, complemented
10#21445796C211yesmaximal, characteristic
11C2×C61×#477C211yesmaximal, characteristic, complemented
12C22×#11758202C2212yes—
13#16840317—24no—
14#16840317—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519768—21no—
17C2×#14519757C2212yes—
18C2×#14519753—21no—
19C2×#14519753C2211yescharacteristic
20C2×Q8×Dic61—21no—
21C2×#14519742C2212yes—
22C2×#14519735—21no—
23C2×C42×D61C2211yescharacteristic
24#16839972—24no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519702C2212yes—
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519689C2212yes—
30#16839669C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839643—21no—
33#16839633—21no—
34#16839619C2211yescharacteristic
35C4×#11758196—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37C4×#11758198—21no—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C61×#477—24no—
44C2×#107×C61C2212yes—
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×#103×C61C2212yes—
47C22×Dic122C2312yes—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758202—24no—
52C2×#11758202—24no—
53C2×#11758202—42no—
54C22×#92418C2312yes—
55#14519777—22no—
56#14519777D414yes—
57#14519768—28no—
58#14519757—24no—
59#14519757—24no—
60#14519753—28no—
61#14519753—22no—
62#14519753—22no—
63Q8×Dic61—28no—
64#14519742—24no—
65#14519742—24no—
66#14519735—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—22no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758198—22no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C42×D61—22no—
84C42×D61—22no—
85C4⋊#92418—22no—
86#14519702—24no—
87#14519702—42no—
88#14519700—24no—
89#14519689—42no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95C22×Q8×C61—21no—
96C22×Q8×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C4⋊Q8×C61—22no—
99C4⋊Q8×C61D414yes—
100C4×Q8×C61—22no—
101C4×Q8×C61—22no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2312yes—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108#107×C61—24no—
109C4⋊C8×C61—22no—
110#103×C61—24no—
111C23×D61C2×D411yescharacteristic
112C2×Q8×C61—24no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—42no—
119C2×Q8×C61—42no—
120C2×Q8×C61#4112yes—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—22no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—41no—
143C2×C4×D61—41no—
144C2×C4×D61Q1614yescomplemented
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122C2×D414yes—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61C2×D414yes—
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—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61#4112yes—
174#11758196—28no—
175#11758196—24no—
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—22no—
195Q8×C61—22no—
196Q8×C61—48no—
197Q8×C61—41no—
198Q8×C61—41no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—24no—
202C2×C4×C61—22no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61#12012yes—
211C2×C4×C61#12012yes—
212C2×C4×C61C2×Q1612yes—
213C2×C4×C61#13712yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61H(5,𝔽2)11yescharacteristic
219C2×C4×C612−1+411yescharacteristic
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—24no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61C2×Q1612yes—
238C4×D61—28no—
239C4×D61—44no—
240C4×D61—41no—
241C4×D61—41no—
242C4×D61—41no—
243C4×D61—41no—
244Dic122—24no—
245Dic122—24no—
246Dic122—42no—
247Dic122—42no—
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—44no—
277Dic61—41no—
278Dic61—41no—
279Dic61—81no—
280C2×#533—611no—
281C2×#477D6111yescharacteristic, complemented
282C2×#462—611no—
283C4×C22⋊Q8—611no—
284#2036—612no—
285#2001—612no—
286#1987—611no—
287#1981—611no—
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—611no—
306C2×C22⋊Q8—612no—
307C2×C22⋊Q8—1221no—
308C2×C4×Q8—612no—
309C2×C4×Q8—1221no—
310C2×C4×Q8C2×D6111yescharacteristic
311C2×#117—1221no—
312C22×C42—611no—
313#477—24no—
314#462—1224no—
315C2×#107C2×D6112yes—
316C2×C4⋊C8—611no—
317C2×C4⋊C8C2×D6111yescharacteristic
318C2×#103—612no—
319C2×#103—612no—
320C2×#103—612no—
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—611no—
329C4×C4⋊C4—1221no—
330C4×C22⋊C4—1221no—
331#340—612no—
332#340—612no—
333#336—612no—
334C61#2415511yescharacteristic, complemented, Sylow
335C22×Q8—21no—
336C22×Q8—612no—
337C22×Q8C22×D6111yescharacteristic
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⋊Q8C22⋊D6114yes—
350#125—1228no—
351C22⋊Q8—1224no—
352C22⋊Q8—1222no—
353C22⋊Q8—1228no—
354C22⋊Q8—1222no—
355C22⋊Q8—1224no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×Q8—1224no—
359C4×Q8—1228no—
360C4×Q8—1224no—
361#117—1228no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—611no—
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×D6112yes—
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—24no—
387C4⋊C8—22no—
388C4⋊C8—1222no—
389#103—24no—
390#103—1224no—
391#103—1224no—
392#103—2442no—
393C22⋊C8—2442no—
394#95—1221no—
395#95—1221no—
396#95—1221no—
397#95—1221no—
398#95—1221no—
399C24—611no—
400C2×Q8—21no—
401C2×Q8—24no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—614no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—1222no—
412C2×Q8—1222no—
413C2×Q8—1222no—
414C2×Q8#1175822612yes—
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—22no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—612no—
452C4⋊C4—1228no—
453C4⋊C4—1224no—
454C4⋊C4—1228no—
455C4⋊C4—1221no—
456C4⋊C4—1224no—
457C4⋊C4—1224no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—2442no—
461C4⋊C4C2×C22⋊D6114yes—
462C22⋊C4—1228no—
463C22⋊C4—1224no—
464C22⋊C4—2442no—
465C42—42no—
466C42—1228no—
467C42—1221no—
468C42—1221no—
469C42—1222no—
470C42—1222no—
471C42—2442no—
472C42D4×D6112yes—
473C42C2×C22⋊D6112yes—
474C23—612no—
475C23—1221no—
476C23—1221no—
477C23—1221no—
478C23—1221no—
479C23—1221no—
480C23—1221no—
481C23C2×D4×D6111yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—48no—
485Q8—41no—
486Q8—41no—
487Q8—1224no—
488Q8—1224no—
489Q8—2442no—
490Q8—2442no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
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—2442no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2442no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4#1451974512yes—
527C2×C4#1451974512yes—
528C2×C4Q16×D6112yes—
529C2×C4#1451980912yes—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4#41×D6111yescharacteristic
533C2×C4C2×#1175822611yescharacteristic
534C2×C4#1451982311yescharacteristic
535C2×C4#1451982511yescharacteristic
536C8—42no—
537C8—2442no—
538C22—1222no—
539C22—1222no—
540C22—1222no—
541C22—2441no—
542C22—2441no—
543C22—2441no—
544C22—2441no—
545C22C2×#1451974511yescharacteristic
546C22#1684027511yescharacteristic
547C22#1684034211yescharacteristic
548C22C2×Q16×D6111yescharacteristic
549C22C2×#1451980911yescharacteristic
550C22#1684055811yescharacteristic
551C22#1684055811yescharacteristic
552C4—22no—
553C4—42no—
554C4—44no—
555C4—81no—
556C4—1222no—
557C4—4881no—
558C4—2442no—
559C4—2441no—
560C4—2444no—
561C4—2441no—
562C4#1683991614yes—
563C4#1684040311yescharacteristic
564C4#1684041411yescharacteristic
565C4#1684042311yescharacteristic
566C4#1684043111yescharacteristic
567C2—2442no—
568C2#2144916611yescharacteristic
569C2C2×#1683991611yescharacteristic
570C2#2145505811yescharacteristic
571C2#2145586011yescharacteristic
572C2#2145598411yescharacteristic
573C2#2145716311yescharacteristic
574C2#2145719511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#477C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#477D61
hypocenterC61#24155
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×#16839972C2semidirect
6. #21446768C2semidirect
7. #21446741C2semidirect
8. #21446157C2semidirect
9. #21446070C2semidirect
11. C2×C61×#477C2semidirect
144. C2×C4×D61Q16semidirect
281. C2×#477D61semidirect
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×#168399722
C4×#145197532
#214470242
#214468282
#214467682
#214467412
#214461572
#214460702
#214457962
C2×C61×#4772
#2415561