Finite Group Databasebeta

← #29691226

Subgroups — #29691226

Order 15616.

1080 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839901C211yesmaximal, characteristic
2#21448325C211yesmaximal, characteristic
3C2×#16839669C211yesmaximal, characteristic, complemented
4#21446697C212yesmaximal, complemented
5#21446506C212yesmaximal, complemented
6#21446463C212yesmaximal, complemented
7#21446150C212yesmaximal, complemented
8#21446134C211yesmaximal, characteristic
9#21446006C211yesmaximal, characteristic, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1787C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519751C2211yescharacteristic
14C2×#14519739C2211yescharacteristic
15C2×#14519737—21no—
16C22×C4⋊Dic61C2211yescharacteristic
17C22×C4×Dic61C2211yescharacteristic
18#16839901C418yescomplemented
19C2×#14519710C2212yes—
20C2×#14519701C2212yes—
21C2×#14519699C2212yes—
22C2×C22⋊#92418C2211yescharacteristic
23C2×#14519687C2212yes—
24#16839669—21no—
25#16839669—21no—
26#16839669C2212yes—
27#16839669C2212yes—
28#16839663C2211yescharacteristic
29#16839658C2211yescharacteristic
30#16839651—21no—
31#16839639—21no—
32C4⋊C4×Dic61C2211yescharacteristic
33#16839609—21no—
34C2×#14519684—21no—
35C2×#14519684—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839516C2212yes—
39C23×Q8×C61C2211yescharacteristic
40C2×#103×C61C2212yes—
41C2×#103×C61C2212yes—
42C2×C22⋊C8×C61C2211yescharacteristic
43C61×#380C2211yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—21no—
46C23×Dic61—21no—
47C23×Dic61D411yescharacteristic
48C23×Dic61D411yescharacteristic
49C22×C4×D61—21no—
50C2×#11758201—42no—
51C2×#11758201—42no—
52C2×#11758201C2×C418yescomplemented
53C22×#92418C2312yes—
54#14519751—22no—
55#14519751C2×C414yes—
56#14519739—42no—
57#14519737—44no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—42no—
60C2×C4⋊Dic61C2312yes—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758196—21no—
63C2×#11758198—21no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—42no—
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61D411yescharacteristic
71C2×C4×Dic61C2311yescharacteristic
72C2×C4×Dic61C2311yescharacteristic
73#14519710—42no—
74#14519701C2×C418yescomplemented
75#14519699—42no—
76C22⋊#92418C2×C414yescomplemented
77#14519687—42no—
78#14519684—22no—
79#14519684—22no—
80#14519684—22no—
81#14519684—22no—
82#14519684—21no—
83#14519684—21no—
84C22×Q8×C61—21no—
85C22×Q8×C61—22no—
86C22×Q8×C61—42no—
87C22×Q8×C61—42no—
88C22×Q8×C61C2×C414yescomplemented
89C22×Q8×C61C2312yes—
90C23×C4×C61—21no—
91C23×C4×C61C2311yescharacteristic
92C22×C8×C61C2312yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61—21no—
96#103×C61—42no—
97#103×C61—42no—
98C22⋊C8×C61—22no—
99#95×C61—21no—
100C24×C61C2×D411yescharacteristic
101C23×D61—21no—
102C2×Q8×C61—24no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—48no—
109C2×Q8×C61—42no—
110C2×Q8×C61—42no—
111C2×Q8×C61—48no—
112C2×Q8×C61C22×C414yes—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—24no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4112yes—
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—24no—
126C2×C22⋊D61—42no—
127C2×C22⋊D61—42no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—21no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—41no—
134C22×Dic61—41no—
135C22×Dic61—41no—
136C22×Dic61—42no—
137C22×Dic61—42no—
138C22×Dic61—42no—
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—21no—
146C2×D244C22×C412yes—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C22⋊C4×C61—42no—
156#11758201—48no—
157#11758198—24no—
158#11758198—42no—
159#11758198—42no—
160#11758198—42no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—42no—
165C4⋊Dic61C22×C412yes—
166#11758196—42no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C4×Dic61—41no—
170C4×Dic61—41no—
171C4×Dic61—42no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418C22×C414yes—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61C4×D412yes—
180C23×C61AGL(1,ℤ/8)12yes—
181C23×C61#13712yes—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190Q8×C61—24no—
191Q8×C61—24no—
192Q8×C61—48no—
193Q8×C61—42no—
194Q8×C61—42no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—48no—
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×C61C4×D412yes—
210C2×C4×C61C23×C411yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61H(5,𝔽2)12yes—
213C2×C4×C61H(5,𝔽2)12yes—
214C22⋊D61—48no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—22no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—44no—
228C2×Dic61—44no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—84no—
235C2×Dic61#13712yes—
236D244—24no—
237D244—42no—
238C4×D61—84no—
239C8×C61—42no—
240#92418—24no—
241#22047—611nomaximal, Sylow
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61#44014yes—
249C22×C61C2×C4×D411yescharacteristic
250C22×C61#53411yescharacteristic
251C22×C61#53511yescharacteristic
252C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
253C22×C61C2×#13711yescharacteristic
254C22×C61C2×#13711yescharacteristic
255C22×C61#57511yescharacteristic
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C4×C61—22no—
264C4×C61—44no—
265C4×C61—81no—
266C4×C61—81no—
267C4×C61#51812yes—
268C4×C61#53812yes—
269Dic61—22no—
270Dic61—41no—
271Dic61—42no—
272Dic61—41no—
273Dic61—41no—
274Dic61—82no—
275C2×#448—611no—
276C4×#121—611no—
277C2×#391—611no—
278#1973—612no—
279#1935—612no—
280#1928—612no—
281#1877—612no—
282#1874—611no—
283#1852—611no—
284#1787D6111yescharacteristic, complemented
285#1749—611no—
286C2×C61—24no—
287C2×C61#232111yescharacteristic
288C2×C61C2×#44011yescharacteristic
289C2×C61C4×#13711yescharacteristic
290C2×C61#319311yescharacteristic
291C2×C61#319611yescharacteristic
292C2×C61#321311yescharacteristic
293C2×C61#321811yescharacteristic
294D61—82no—
295C23×Q8C2×D6111yescharacteristic
296C22×SD16—612no—
297C2×#121—611no—
298C2×C4×D4—611no—
299C2×C4×D4—1221no—
300C22×C4⋊C4—611no—
301C22×C42—611no—
302#448—618no—
303C2×#107—612no—
304C2×#103—612no—
305C2×#103C2×D6112yes—
306C2×#103C2×D6112yes—
307C2×#102—612no—
308C2×C22⋊C8—611no—
309C2×C22⋊C8C2×D6111yescharacteristic
310C2×#97—612no—
311#391—612no—
312#391—612no—
313#391—1221no—
314#391—1221no—
315#390—611no—
316#389—611no—
317#388—1221no—
318#386—1221no—
319#380C2×D6111yescharacteristic
320C4×C4⋊C4—611no—
321C4×C22⋊C4—1221no—
322C2×#95—1221no—
323C2×#95—1221no—
324#340—612no—
325#340—612no—
326#336—612no—
327C61#2204711yescharacteristic, complemented, Sylow
328C22×Q8—21no—
329C22×Q8—22no—
330C22×Q8—42no—
331C22×Q8—42no—
332C22×Q8C4×D6114yescomplemented
333C22×Q8C22×D6112yes—
334C22×D4—611no—
335C22×D4—1221no—
336C23×C4—21no—
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—1221no—
340C23×C4—1221no—
341C23×C4C22×D6111yescharacteristic
342C2×SD16—618no—
343C2×SD16—2442no—
344C2×SD16—2442no—
345C22×C8—612no—
346C22×C8C22×D6112yes—
347#121—614no—
348#121—1222no—
349C4×D4—2442no—
350C4×D4—2444no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—2442no—
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C4⋊C4C22×D6111yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—1221no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1221no—
363C2×C42—611no—
364C2×C42—611no—
365C2×C42—611no—
366C2×C42—611no—
367C2×C42—1221no—
368C2×C42—2442no—
369#107—2442no—
370#103—42no—
371#103—42no—
372#103—618no—
373#102—2442no—
374C22⋊C8—22no—
375C22⋊C8—614no—
376#97—2442no—
377#95—21no—
378#95—1222no—
379#95—1222no—
380#95—1222no—
381#95—1222no—
382#95—1221no—
383#95—1221no—
384C24—1221no—
385C24D4×D6111yescharacteristic
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—48no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—48no—
396C2×Q8C2×C4×D6114yes—
397C2×D4—612no—
398C2×D4—1224no—
399C2×D4—1221no—
400C2×D4—1222no—
401C2×D4—1222no—
402C2×D4—2442no—
403C2×D4—2442no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1221no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—2442no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4—2441no—
425C22×C4—2441no—
426C22×C4—2441no—
427C22×C4—2442no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4C2×C4×D6112yes—
431C22×C4D4×D6111yescharacteristic
432C22×C4#1175822412yes—
433C22×C4C23×D6111yescharacteristic
434SD16—2448no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—614no—
439C2×C8—1222no—
440C2×C8—1222no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—612no—
444C4⋊C4—1221no—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C4⋊C4—2442no—
448C4⋊C4—2442no—
449C22⋊C4—42no—
450C22⋊C4—1224no—
451C22⋊C4—2442no—
452C22⋊C4—2442no—
453C22⋊C4—2442no—
454C42—1222no—
455C42—1222no—
456C42—2441no—
457C42—2441no—
458C42—2442no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—1222no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23—2441no—
470C23#1451973712yes—
471C23#1451980412yes—
472C23#1451980912yes—
473C23C2×D4×D6111yescharacteristic
474Q8—24no—
475Q8—24no—
476Q8—48no—
477Q8—42no—
478Q8—42no—
479D4—1224no—
480D4—2448no—
481D4—2442no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—612no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1222no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—4884no—
510C2×C4—4884no—
511C2×C4—2444no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2444no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4#1451973712yes—
519C2×C4C22×C4×D6111yescharacteristic
520C2×C4C2×#1175822411yescharacteristic
521C2×C4#1451982212yes—
522C2×C4#1451982212yes—
523C8—42no—
524C8—1224no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—4881no—
532C22—4881no—
533C22—4881no—
534C22—4881no—
535C22—2442no—
536C22—2442no—
537C22—2442no—
538C22#1683986514yes—
539C22C2×#1451973711yescharacteristic
540C22#1684032211yescharacteristic
541C22#1684032711yescharacteristic
542C22C2×#1451980411yescharacteristic
543C22#137×D6111yescharacteristic
544C22C2×#1451980911yescharacteristic
545C22#1684056111yescharacteristic
546C4—22no—
547C4—44no—
548C4—81no—
549C4—81no—
550C4—1222no—
551C4—4882no—
552C4—2442no—
553C4—2441no—
554C4—2441no—
555C4—2441no—
556C4#1684023512yes—
557C4#1684034712yes—
558C2—24no—
559C2—4882no—
560C2#2144847411yescharacteristic
561C2C2×#1683986511yescharacteristic
562C2#2145463411yescharacteristic
563C2#2145662311yescharacteristic
564C2#2145665211yescharacteristic
565C2#2145680511yescharacteristic
566C2#2145684911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#1787D61
hypocenterC61#22047
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
3. C2×#16839669C2semidirect
4. #21446697C2semidirect
5. #21446506C2semidirect
6. #21446463C2semidirect
7. #21446150C2semidirect
9. #21446006C2semidirect
10. #21445583C2semidirect
11. C61×#1787C2semidirect
18. #16839901C4semidirect
52. C2×#11758201C2×C4semidirect
74. #14519701C2×C4semidirect
76. C22⋊#92418C2×C4semidirect
88. C22×Q8×C61C2×C4semidirect
284. #1787D61semidirect
327. C61#22047semidirect
332. C22×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399012
#214483252
C2×#168396692
#214466972
#214465062
#214464632
#214461502
#214461342
#214460062
#214455832
C61×#17872
#2204761