Finite Group Databasebeta

← #29654119

Subgroups — #29654119

Order 15616.

1516 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1516 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456616C212yesmaximal, complemented
2#21450750C211yesmaximal, characteristic, complemented
3#21444839C214yesmaximal, complemented
4#21444737C214yesmaximal, complemented
5#21444335C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic, complemented
7C2×C61×#534C211yesmaximal, characteristic, complemented
8C2×#14519760—21no—
9C2×#14519751C2211yescharacteristic, complemented
10C2×#14519746—21no—
11#16840080—22no—
12#16840078C2214yescomplemented
13#16840013—24no—
14#16839896—22no—
15#16839895C2214yescomplemented
16#16839726C2212yescomplemented
17#16839698C2212yescomplemented
18#16839671C2212yescomplemented
19#16839662—22no—
20C4×C22⋊Dic61C2211yescharacteristic, complemented
21#16839496C2218yescomplemented
22#16839486C2214yescomplemented
23C23×D4×C61—21no—
24C61×#534—24no—
25C61×#534—22no—
26C61×#534C2214yes—
27C2×#123×C61—22no—
28C2×#121×C61—21no—
29C2×#121×C61C2212yes—
30C2×#120×C61—22no—
31C22×C22⋊C4×C61C2211yescharacteristic
32C22×D244C2311yescharacteristic, complemented
33C22×C22⋊D61—21no—
34C22×C22⋊D61—22no—
35C2×#11758199—24no—
36C2×#11758194—22no—
37#14519760—44no—
38#14519751—22no—
39#14519751—22no—
40#14519746—44no—
41C2×C4⋊Dic61C2311yescharacteristic, complemented
42C2×#11758196—22no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—22no—
45C2×#11758198—21no—
46C2×#11758198—22no—
47C2×C4×Dic61—21no—
48#14519708—24no—
49#14519698—24no—
50#14519697—24no—
51C22⋊#92418—22no—
52C22⋊#92418C2314yescomplemented
53#14519684C2312yescomplemented
54C25×C61—21no—
55C25×C61D411yescharacteristic
56C22×D4×C61—24no—
57C22×D4×C61—22no—
58C22×D4×C61—24no—
59C22×D4×C61—22no—
60C22×D4×C61—21no—
61C22×D4×C61—48no—
62C22×D4×C61D412yescomplemented
63C23×C4×C61—21no—
64C23×C4×C61D411yescharacteristic
65C23×C4×C61C2311yescharacteristic
66#123×C61—28no—
67#123×C61—28no—
68#121×C61—24no—
69#121×C61—22no—
70#121×C61—22no—
71#121×C61—24no—
72#121×C61D414yescomplemented
73#121×C61C2314yes—
74#120×C61—28no—
75#120×C61—28no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61D412yescomplemented
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—22no—
81C2×C22⋊C4×C61—22no—
82C2×C22⋊C4×C61D412yes—
83C2×C22⋊C4×C61C2312yes—
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—24no—
87C24×C61—24no—
88C24×C61—22no—
89C24×C61—44no—
90C24×C61—42no—
91C24×C61—81no—
92C24×C61—82no—
93C24×C61—82no—
94C24×C61C2×D411yescharacteristic
95C24×C61C2×D412yes—
96C23×D61—21no—
97C2×D4×C61—28no—
98C2×D4×C61—24no—
99C2×D4×C61—24no—
100C2×D4×C61—24no—
101C2×D4×C61—22no—
102C2×D4×C61—48no—
103C2×D4×C61—416no—
104C2×D4×C61—48no—
105C2×D4×C61—48no—
106C2×D4×C61—48no—
107C2×D4×C61—82no—
108C2×D4×C61—82no—
109C2×D4×C61C2×D414yes—
110C22×C4×C61—24no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—44no—
115C22×C4×C61—42no—
116C22×C4×C61—81no—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—44no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—44no—
128C2×C22⋊D61—82no—
129C22×Dic61—22no—
130C22×Dic61—21no—
131C2×D244—22no—
132C2×D244—81no—
133C4⋊C4×C61—24no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61C2×D414yes—
136C22⋊C4×C61—28no—
137C22⋊C4×C61—24no—
138C22⋊C4×C61—24no—
139C22⋊C4×C61—24no—
140C22⋊C4×C61—24no—
141C22⋊Dic61—24no—
142C22⋊Dic61—44no—
143#11758199—84no—
144#11758198—44no—
145#11758198—42no—
146C4⋊Dic61—22no—
147#11758196—44no—
148C4×Dic61—81no—
149#11758194—82no—
150C2×#92418—24no—
151C23×C61—24no—
152C23×C61—24no—
153C23×C61—22no—
154C23×C61—22no—
155C23×C61—22no—
156C23×C61—22no—
157C23×C61—24no—
158C23×C61—41no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61—44no—
163C23×C61—48no—
164C23×C61—44no—
165C23×C61—42no—
166C23×C61—44no—
167C23×C61—81no—
168C23×C61—81no—
169C23×C61—88no—
170C23×C61—84no—
171C23×C61—84no—
172C23×C61—816no—
173C23×C61#12011yescharacteristic
174C23×C61#12011yescharacteristic
175C23×C61AGL(1,ℤ/8)14yes—
176C23×C61C22×D411yescharacteristic
177C22×D61—41no—
178C22×D61—42no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—42no—
182D4×C61—44no—
183D4×C61—416no—
184D4×C61—48no—
185D4×C61—48no—
186D4×C61—82no—
187D4×C61—84no—
188C2×C4×C61—22no—
189C2×C4×C61—24no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—44no—
195C2×C4×C61—48no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—81no—
199C2×C4×C61#12012yes—
200C2×C4×C61C22×D412yes—
201C22⋊D61—88no—
202C22⋊D61—82no—
203C22⋊D61—82no—
204C2×Dic61—24no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210D244—44no—
211#92418—44no—
212#18316—611nomaximal, Sylow
213C22×C61—21no—
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—24no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—44no—
222C22×C61—44no—
223C22×C61—48no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—44no—
227C22×C61—44no—
228C22×C61—44no—
229C22×C61—82no—
230C22×C61—88no—
231C22×C61—84no—
232C22×C61—816no—
233C22×C61#45312yes—
234C22×C61C2×#12011yescharacteristic
235C22×C61C2×AGL(1,ℤ/8)12yes—
236C2×D61—82no—
237C2×D61—82no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C4×C61—22no—
242C4×C61—22no—
243C4×C61—44no—
244C4×C61—42no—
245Dic61—44no—
246Dic61—81no—
247C2×#534D6111yescharacteristic, complemented
248#3192—612no—
249#2539—611no—
250#1613—614no—
251#1595—614no—
252#1507—612no—
253#1462—611no—
254C2×C61—22no—
255C2×C61—22no—
256C2×C61—41no—
257C2×C61—44no—
258C2×C61—84no—
259C2×C61#300311yescharacteristic
260C2×C61C2×#45311yescharacteristic
261C2×C61#302611yescharacteristic
262D61—161no—
263C23×D4—21no—
264#534—24no—
265#534—22no—
266#534C2×D6114yes—
267C2×#123—22no—
268C2×#123—1221no—
269C2×#121—21no—
270C2×#121—611no—
271C2×#121C2×D6112yes—
272C2×#120—22no—
273C2×#120—1221no—
274C22×C22⋊C4C2×D6111yescharacteristic
275#481—614no—
276#481—1222no—
277#469—1224no—
278#447—614no—
279#447—1222no—
280#407—612no—
281#399—612no—
282#392—612no—
283#390—1222no—
284C4×C22⋊C4—611no—
285#327—618no—
286#323—614no—
287C61#1831611yescharacteristic, complemented, Sylow
288C25—21no—
289C25C22⋊D6111yescharacteristic
290C22×D4—22no—
291C22×D4—24no—
292C22×D4—22no—
293C22×D4—24no—
294C22×D4—21no—
295C22×D4—48no—
296C22×D4—611no—
297C22×D4—1222no—
298C22×D4—1221no—
299C22×D4C22⋊D6112yescomplemented
300C23×C4—21no—
301C23×C4C22⋊D6111yescharacteristic
302C23×C4C22×D6111yescharacteristic
303C2×D8—1224no—
304C2×M16—1222no—
305#123—28no—
306#123—28no—
307#123—2444no—
308#121—24no—
309#121—24no—
310#121—22no—
311#121—22no—
312#121—1222no—
313#121—1222no—
314#121C22⋊D6114yescomplemented
315#121C22×D6114yes—
316#120—28no—
317#120—28no—
318#120—2444no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4C22⋊D6112yescomplemented
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—1221no—
328C2×C22⋊C4—1222no—
329C2×C22⋊C4—1221no—
330C2×C22⋊C4—1222no—
331C2×C22⋊C4C22⋊D6112yes—
332C2×C22⋊C4C22×D6112yes—
333C2×C42—1221no—
334#106—1224no—
335#102—1224no—
336#102—1224no—
337C22⋊C8—614no—
338C22⋊C8—1222no—
339#95—612no—
340C24—21no—
341C24—21no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—44no—
346C24—42no—
347C24—81no—
348C24—82no—
349C24—82no—
350C24—1221no—
351C24C2×C22⋊D6111yescharacteristic
352C24C2×C22⋊D6112yes—
353C2×D4—28no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—48no—
359C2×D4—416no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4—1222no—
366C2×D4—4881no—
367C2×D4—4882no—
368C2×D4—2444no—
369C2×D4—2442no—
370C2×D4—2442no—
371C2×D4—2441no—
372C2×D4—2441no—
373C2×D4—2444no—
374C2×D4C2×C22⋊D6114yes—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—81no—
382C22×C4—1222no—
383C22×C4—1221no—
384C22×C4C2×C22⋊D6112yes—
385C22×C4C2×C22⋊D6112yes—
386C22×C4C2×C22⋊D6112yes—
387C22×C4C2×C22⋊D6111yescharacteristic
388C22×C4C23×D6111yescharacteristic
389D8—4884no—
390M16—4882no—
391C2×C8—1224no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—1222no—
395C4⋊C4—2444no—
396C4⋊C4C2×C22⋊D6114yes—
397C22⋊C4—28no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—1224no—
403C22⋊C4—2444no—
404C22⋊C4—2444no—
405C22⋊C4—2442no—
406C42—4881no—
407C23—24no—
408C23—24no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—24no—
414C23—41no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—44no—
419C23—48no—
420C23—44no—
421C23—42no—
422C23—44no—
423C23—81no—
424C23—81no—
425C23—88no—
426C23—84no—
427C23—84no—
428C23—816no—
429C23—2441no—
430C23—2442no—
431C23—2441no—
432C23—2441no—
433C23—2442no—
434C23#1451974611yescharacteristic
435C23#1451974611yescharacteristic
436C23#1451980314yes—
437C23C22×C22⋊D6111yescharacteristic
438D4—44no—
439D4—416no—
440D4—48no—
441D4—48no—
442D4—82no—
443D4—84no—
444D4—4882no—
445D4—4882no—
446D4—4888no—
447D4—2444no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—44no—
455C2×C4—48no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—81no—
459C2×C4—1224no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4#1451974612yes—
466C2×C4C22×C22⋊D6112yes—
467C8—2444no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—24no—
475C22—21no—
476C22—44no—
477C22—44no—
478C22—48no—
479C22—42no—
480C22—42no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—82no—
485C22—88no—
486C22—84no—
487C22—816no—
488C22—4882no—
489C22—4882no—
490C22—4881no—
491C22—4881no—
492C22—4881no—
493C22#1683993012yes—
494C22C2×#1451974611yescharacteristic
495C22C2×#1451980312yes—
496C4—22no—
497C4—22no—
498C4—42no—
499C4—44no—
500C4—4881no—
501C4—2444no—
502C2—22no—
503C2—22no—
504C2—41no—
505C2—44no—
506C2—84no—
507C2—9761no—
508C2#2145504611yescharacteristic
509C2C2×#1683993011yescharacteristic
510C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#534C2
socleC22×C61C2×#120
hypercenterC2×#534D61
hypocenterC61#18316
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. #21456616C2semidirect
2. #21450750C2semidirect
3. #21444839C2semidirect
4. #21444737C2semidirect
5. #21444335C2semidirect
6. #21444146C2semidirect
7. C2×C61×#534C2semidirect
9. C2×#14519751C22semidirect
12. #16840078C22semidirect
15. #16839895C22semidirect
16. #16839726C22semidirect
17. #16839698C22semidirect
18. #16839671C22semidirect
20. C4×C22⋊Dic61C22semidirect
21. #16839496C22semidirect
22. #16839486C22semidirect
32. C22×D244C23semidirect
41. C2×C4⋊Dic61C23semidirect
52. C22⋊#92418C23semidirect
53. #14519684C23semidirect
62. C22×D4×C61D4semidirect
72. #121×C61D4semidirect
77. C2×C4⋊C4×C61D4semidirect
247. C2×#534D61semidirect
287. C61#18316semidirect
299. C22×D4C22⋊D61semidirect
314. #121C22⋊D61semidirect
322. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214566162
#214507502
#214448392
#214447372
#214443352
#214441462
C2×C61×#5342
#1831661