Finite Group Databasebeta

← #29580486

Subgroups — #29580486

Order 15616.

1018 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 105 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#21456572C211yesmaximal, characteristic, complemented
2#21449452C211yesmaximal, characteristic
3#21446805C211yesmaximal, characteristic, complemented
4#21445750C211yesmaximal, characteristic, complemented
5#21444388C211yesmaximal, characteristic, complemented
6#21443665C211yesmaximal, characteristic, complemented
7C2×C61×#450C211yesmaximal, characteristic, complemented
8C2×#14519773—21no—
9C2×#14519751C2211yescharacteristic
10C2×#14519746—22no—
11#16840090—21no—
12#16840080—21no—
13#16839993—22no—
14#16839912—21no—
15#16839901—21no—
16C2×#14519698—22no—
17C2×C22⋊#92418C2211yescharacteristic
18#16839670—22no—
19#16839665—21no—
20#16839622C2211yescharacteristic
21C4×C22⋊Dic61C2211yescharacteristic
22#16839529—22no—
23#16839497—22no—
24C23×D4×C61C2211yescharacteristic
25C22×SD16×C61—21no—
26C2×C22⋊Q8×C61C2211yescharacteristic
27C61×#450—22no—
28C61×#450—22no—
29C2×#102×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C22×C22⋊D61—24no—
34C2×#11758201—22no—
35C22×#92418D412yescomplemented
36#14519773—44no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41#14519746—44no—
42#14519746—44no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×C22⋊Dic61—24no—
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61D412yes—
48C2×#11758198—21no—
49C2×C4×Dic61—21no—
50#14519708—22no—
51#14519701—22no—
52#14519698—44no—
53#14519697—22no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56#14519684—21no—
57#14519684—21no—
58C25×C61—21no—
59C22×Q8×C61D411yescharacteristic, complemented
60C22×D4×C61—21no—
61C22×D4×C61—21no—
62C22×D4×C61—21no—
63C22×D4×C61—21no—
64C22×D4×C61—21no—
65C22×D4×C61—44no—
66C22×D4×C61D412yes—
67C23×C4×C61C2311yescharacteristic
68C2×SD16×C61—24no—
69C2×SD16×C61—44no—
70C22×C8×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—21no—
73C22⋊Q8×C61—21no—
74C22⋊Q8×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61D411yescharacteristic, complemented
77C2×C22⋊C4×C61—21no—
78#102×C61—24no—
79C22⋊C8×C61—21no—
80C22⋊C8×C61D412yescomplemented
81C24×C61—22no—
82C24×C61—44no—
83C24×C61—41no—
84C24×C61—41no—
85C24×C61—42no—
86C24×C61—42no—
87C24×C61—82no—
88C24×C61C2×D411yescharacteristic
89C23×D61—21no—
90C2×Q8×C61—21no—
91C2×Q8×C61—21no—
92C2×Q8×C61—42no—
93C2×D4×C61—24no—
94C2×D4×C61—24no—
95C2×D4×C61—24no—
96C2×D4×C61—22no—
97C2×D4×C61—22no—
98C2×D4×C61—42no—
99C2×D4×C61—42no—
100C2×D4×C61—42no—
101C2×D4×C61—44no—
102C2×D4×C61—44no—
103C2×D4×C61—44no—
104C2×D4×C61—41no—
105C2×D4×C61—41no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—42no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C2×C22⋊D61—48no—
116C2×C22⋊D61—48no—
117C2×C22⋊D61—41no—
118C2×C22⋊D61—41no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C2×C22⋊D61—84no—
122C22×Dic61—22no—
123C22×Dic61—21no—
124C22×Dic61—41no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—42no—
128SD16×C61—48no—
129C2×C8×C61—22no—
130C2×C8×C61—42no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C22⋊Dic61—22no—
138C22⋊Dic61—48no—
139C22⋊Dic61—42no—
140C22⋊Dic61SD1614yescomplemented
141#11758201—82no—
142#11758198—41no—
143#11758198—41no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—42no—
147C4×Dic61—42no—
148C2×#92418—22no—
149C2×#92418—42no—
150C23×C61—24no—
151C23×C61—41no—
152C23×C61—41no—
153C23×C61—44no—
154C23×C61—44no—
155C23×C61—44no—
156C23×C61—42no—
157C23×C61—42no—
158C23×C61—44no—
159C23×C61—82no—
160C23×C61—82no—
161C23×C61—88no—
162C23×C61—88no—
163C23×C61—82no—
164C23×C61—81no—
165C23×C61—81no—
166C23×C61#12011yescharacteristic
167C23×C61#12111yescharacteristic
168C23×C61#12711yescharacteristic
169C23×C61C2×SD1612yes—
170C23×C61AGL(1,ℤ/8)12yes—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—41no—
174C22×D61—42no—
175Q8×C61—42no—
176Q8×C61—42no—
177D4×C61—24no—
178D4×C61—24no—
179D4×C61—42no—
180D4×C61—44no—
181D4×C61—42no—
182D4×C61—44no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—81no—
192C2×C4×C61—81no—
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C22⋊D61—816no—
198C22⋊D61—82no—
199C22⋊D61—81no—
200C22⋊D61—81no—
201C2×Dic61—24no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—41no—
205C2×Dic61—42no—
206C2×Dic61—81no—
207C2×Dic61—81no—
208C2×Dic61—81no—
209D244—42no—
210D244—81no—
211C8×C61—42no—
212#92418—42no—
213#9954—611nomaximal, Sylow
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—48no—
223C22×C61—44no—
224C22×C61—84no—
225C22×C61—84no—
226C22×C61—88no—
227C22×C61—84no—
228C22×C61—82no—
229C22×C61—82no—
230C22×C61#39211yescharacteristic
231C22×C61#45012yes—
232C22×C61#46012yes—
233C22×C61#49211yescharacteristic
234C22×C61#49611yescharacteristic
235C2×D61—84no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61B(4,𝔽2)12yes—
245Dic61—44no—
246Dic61—81no—
247Dic61—161no—
248#3187—611no—
249C2×#450D6111yescharacteristic, complemented
250#2433—611no—
251#1996—611no—
252#1786—611no—
253#1519—611no—
254#1282—611no—
255C2×C61—22no—
256C2×C61—21no—
257C2×C61—22no—
258C2×C61—21no—
259C2×C61—84no—
260C2×C61#200811yescharacteristic
261C2×C61#201411yescharacteristic
262C2×C61#202111yescharacteristic
263D61—161no—
264C23×D4C2×D6111yescharacteristic
265C22×SD16—21no—
266C2×#127—1221no—
267C2×C22⋊Q8C2×D6111yescharacteristic
268C2×#121—611no—
269C2×#120—1222no—
270#483—1221no—
271#481—1221no—
272#465—1222no—
273#450—22no—
274#450—22no—
275#450—1221no—
276#448—1221no—
277C2×#102—21no—
278C2×#102—1222no—
279C2×C22⋊C8—611no—
280C2×C22⋊C8C2×D6111yescharacteristic
281#392—1222no—
282#390—1221no—
283#380—611no—
284C4×C22⋊C4—611no—
285#342—1222no—
286#327—1222no—
287C61#995411yescharacteristic, complemented, Sylow
288C25—21no—
289C22×Q8C22⋊D6111yescharacteristic, complemented
290C22×D4—21no—
291C22×D4—21no—
292C22×D4—21no—
293C22×D4—21no—
294C22×D4—21no—
295C22×D4—44no—
296C22×D4—611no—
297C22×D4—1224no—
298C22×D4—1221no—
299C22×D4C22⋊D6112yes—
300C23×C4C22×D6111yescharacteristic
301C2×SD16—24no—
302C2×SD16—44no—
303C2×SD16—1222no—
304C22×C8—21no—
305C22×C8—612no—
306#127—2444no—
307C22⋊Q8—21no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311#121—1221no—
312#121—1221no—
313#121—1221no—
314#121—1221no—
315#120—2444no—
316#120—2444no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—611no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4C22⋊D6111yescharacteristic, complemented
321C2×C22⋊C4—21no—
322C2×C22⋊C4—612no—
323C2×C22⋊C4—1224no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C42—1221no—
327#106—1222no—
328#103—1222no—
329#102—24no—
330#102—1222no—
331#102—2444no—
332C22⋊C8—21no—
333C22⋊C8—1221no—
334C22⋊C8—1221no—
335C22⋊C8C22⋊D6112yescomplemented
336#95—1221no—
337#95—1221no—
338C24—22no—
339C24—44no—
340C24—41no—
341C24—41no—
342C24—42no—
343C24—42no—
344C24—82no—
345C24—1221no—
346C24D4×D6111yescharacteristic
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—42no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—1221no—
364C2×D4—1221no—
365C2×D4—4884no—
366C2×D4—2442no—
367C2×D4—2448no—
368C2×D4—2448no—
369C2×D4—2441no—
370C2×D4—2441no—
371C2×D4—2442no—
372C2×D4—2442no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—42no—
376C22×C4—1222no—
377C22×C4—1221no—
378C22×C4—2441no—
379C22×C4D4×D6111yescharacteristic
380C22×C4D4×D6111yescharacteristic
381C22×C4#1175822411yescharacteristic
382C22×C4C2×C22⋊D6111yescharacteristic
383C22×C4C2×C22⋊D6111yescharacteristic
384C22×C4C2×C22⋊D6111yescharacteristic
385SD16—48no—
386SD16—4882no—
387C2×C8—22no—
388C2×C8—42no—
389C2×C8—1222no—
390C2×C8—2442no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—2442no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—614no—
401C22⋊C4—1222no—
402C22⋊C4—2448no—
403C22⋊C4—2442no—
404C22⋊C4—2441no—
405C22⋊C4—2441no—
406C42—2442no—
407C23—24no—
408C23—41no—
409C23—41no—
410C23—44no—
411C23—44no—
412C23—42no—
413C23—44no—
414C23—44no—
415C23—42no—
416C23—82no—
417C23—82no—
418C23—88no—
419C23—88no—
420C23—82no—
421C23—81no—
422C23—81no—
423C23—2442no—
424C23—2442no—
425C23—2442no—
426C23—2441no—
427C23#1451974511yescharacteristic
428C23#1451974911yescharacteristic
429C23#1451977311yescharacteristic
430C23SD16×D6112yes—
431C23#1451980412yes—
432Q8—42no—
433Q8—42no—
434D4—24no—
435D4—24no—
436D4—42no—
437D4—44no—
438D4—42no—
439D4—44no—
440D4—4881no—
441D4—4882no—
442D4—4881no—
443D4—4881no—
444D4—48816no—
445D4—2442no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—1224no—
457C2×C4—4881no—
458C2×C4—4881no—
459C2×C4—4881no—
460C2×C4—2442no—
461C2×C4—2441no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4#1451974511yescharacteristic
465C2×C4#1451974611yescharacteristic
466C2×C4#1451974811yescharacteristic
467C2×C4#1451974911yescharacteristic
468C8—42no—
469C8—2442no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—42no—
477C22—42no—
478C22—48no—
479C22—44no—
480C22—84no—
481C22—84no—
482C22—88no—
483C22—84no—
484C22—82no—
485C22—82no—
486C22—4884no—
487C22—4881no—
488C22—4881no—
489C22—4881no—
490C22#1683967011yescharacteristic
491C22#1683991112yes—
492C22#1683996012yes—
493C22#1684013311yescharacteristic
494C22#1684014811yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—81no—
499C4—81no—
500C4—9761no—
501C4—4881no—
502C4—2444no—
503C4#1683994812yes—
504C2—21no—
505C2—22no—
506C2—22no—
507C2—21no—
508C2—84no—
509C2—9761no—
510C2#2144687011yescharacteristic
511C2#2144690511yescharacteristic
512C2#2144694211yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#450C2
socleC22×C61#392
prime radical#21449452C2
hypercenterC2×#450D61
hypocenterC61#9954
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. #21456572C2semidirect
3. #21446805C2semidirect
4. #21445750C2semidirect
5. #21444388C2semidirect
6. #21443665C2semidirect
7. C2×C61×#450C2semidirect
35. C22×#92418D4semidirect
59. C22×Q8×C61D4semidirect
76. C2×C4⋊C4×C61D4semidirect
80. C22⋊C8×C61D4semidirect
140. C22⋊Dic61SD16semidirect
249. C2×#450D61semidirect
287. C61#9954semidirect
289. C22×Q8C22⋊D61semidirect
320. C2×C4⋊C4C22⋊D61semidirect
335. C22⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214565722
#214494522
#214468052
#214457502
#214443882
#214436652
C2×C61×#4502
#995461