Finite Group Databasebeta

← #29580554

Subgroups — #29580554

Order 15616.

944 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C2×#16839901C211yesmaximal, characteristic, complemented
2#21451923C211yesmaximal, characteristic
3C2×#16839497C211yesmaximal, characteristic, complemented
4C2×#16839496C211yesmaximal, characteristic, complemented
5#21444090C211yesmaximal, characteristic, complemented
6#21443661C211yesmaximal, characteristic, complemented
7C2×C61×#448C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×#14519751C2211yescharacteristic
10C2×#14519749—22no—
11#16839901—22no—
12#16839901—22no—
13C2×#14519701—21no—
14C2×C22⋊#92418C2211yescharacteristic
15#16839698—21no—
16#16839697—21no—
17#16839671—21no—
18#16839670—21no—
19C2×#14519684—22no—
20C2×#14519684C2211yescharacteristic
21C2×#14519684C2211yescharacteristic
22#16839497—22no—
23#16839496—22no—
24C23×Q8×C61C2211yescharacteristic
25C22×SD16×C61—21no—
26C2×#121×C61C2211yescharacteristic
27C61×#448—22no—
28C61×#448—22no—
29C2×#103×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C22×C22⋊D61—22no—
34C23×Dic61—21no—
35C23×Dic61D412yes—
36C2×#11758201—24no—
37C2×#11758201—44no—
38C22×#92418—21no—
39#14519751—21no—
40#14519751—21no—
41#14519751—21no—
42#14519751—21no—
43#14519749—44no—
44#14519749—44no—
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—22no—
47C2×C22⋊Dic61—22no—
48C2×#11758198—22no—
49C2×#11758198—21no—
50#14519701—24no—
51C22⋊#92418—21no—
52C22⋊#92418D412yescomplemented
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—44no—
58C22×Q8×C61—21no—
59C22×Q8×C61—21no—
60C22×Q8×C61—21no—
61C22×Q8×C61—21no—
62C22×Q8×C61—21no—
63C22×Q8×C61—44no—
64C22×Q8×C61D412yescomplemented
65C22×D4×C61—21no—
66C22×D4×C61D411yescharacteristic
67C23×C4×C61—21no—
68C23×C4×C61C2311yescharacteristic
69C2×SD16×C61—24no—
70C2×SD16×C61—44no—
71C22×C8×C61—21no—
72#121×C61—21no—
73#121×C61—21no—
74#121×C61—21no—
75#121×C61—21no—
76C2×C4⋊C4×C61D411yescharacteristic
77C2×C22⋊C4×C61—21no—
78#103×C61—24no—
79C22⋊C8×C61—21no—
80C22⋊C8×C61D412yescomplemented
81C24×C61—21no—
82C24×C61C2×D411yescharacteristic
83C23×D61—21no—
84C2×Q8×C61—24no—
85C2×Q8×C61—24no—
86C2×Q8×C61—22no—
87C2×Q8×C61—22no—
88C2×Q8×C61—44no—
89C2×Q8×C61—42no—
90C2×Q8×C61—42no—
91C2×Q8×C61—44no—
92C2×Q8×C61—44no—
93C2×Q8×C61—44no—
94C2×Q8×C61—41no—
95C2×Q8×C61—41no—
96C2×D4×C61—21no—
97C2×D4×C61—21no—
98C2×D4×C61—42no—
99C2×D4×C61—41no—
100C2×D4×C61—41no—
101C2×D4×C61—42no—
102C2×D4×C61—42no—
103C22×C4×C61—24no—
104C22×C4×C61—22no—
105C22×C4×C61—21no—
106C22×C4×C61—42no—
107C22×C4×C61—42no—
108C22×C4×C61—41no—
109C22×C4×C61—41no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C2×C22⋊D61—44no—
117C2×C22⋊D61—41no—
118C2×C22⋊D61—41no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C2×C22⋊D61—82no—
122C2×C22⋊D61—82no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—21no—
127C22×Dic61—42no—
128C22×Dic61—42no—
129C22×Dic61—41no—
130C22×Dic61—42no—
131C22×Dic61—41no—
132C22×Dic61—42no—
133C22×Dic61SD1614yescomplemented
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—42no—
137SD16×C61—48no—
138C2×C8×C61—22no—
139C2×C8×C61—42no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C22⋊C4×C61—41no—
143C22⋊C4×C61—41no—
144C22⋊Dic61—44no—
145#11758201—48no—
146#11758198—44no—
147#11758198—41no—
148#11758198—41no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151#11758196—44no—
152C2×#92418—22no—
153C2×#92418—42no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—22no—
157C23×C61—41no—
158C23×C61—42no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61—42no—
162C23×C61#12011yescharacteristic
163C23×C61#12111yescharacteristic
164C23×C61#12711yescharacteristic
165C23×C61C2×SD1612yes—
166C23×C61AGL(1,ℤ/8)12yes—
167C22×D61—41no—
168C22×D61—42no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—42no—
172Q8×C61—24no—
173Q8×C61—24no—
174Q8×C61—42no—
175Q8×C61—42no—
176Q8×C61—44no—
177Q8×C61—44no—
178D4×C61—42no—
179D4×C61—81no—
180D4×C61—82no—
181D4×C61—81no—
182D4×C61—81no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—24no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—44no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—44no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61#12111yescharacteristic
200C22⋊D61—88no—
201C22⋊D61—82no—
202C22⋊D61—81no—
203C22⋊D61—81no—
204C2×Dic61—24no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—44no—
208C2×Dic61—44no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—42no—
212C2×Dic61—82no—
213C2×Dic61—82no—
214C2×Dic61—81no—
215C2×Dic61—81no—
216D244—42no—
217D244—81no—
218C8×C61—42no—
219#92418—42no—
220#9964—611nomaximal, Sylow
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—22no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—82no—
230C22×C61—82no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61#39211yescharacteristic
235C22×C61#45012yes—
236C22×C61#45511yescharacteristic
237C22×C61#45611yescharacteristic
238C22×C61B(4,𝔽2)12yes—
239C22×C61#46012yes—
240C22×C61#49211yescharacteristic
241C22×C61#49611yescharacteristic
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—44no—
252C4×C61—81no—
253Dic61—44no—
254Dic61—82no—
255Dic61—81no—
256C2×#448—611no—
257C2×#448D6111yescharacteristic, complemented
258#2656—611no—
259C2×#327—611no—
260C2×#327—611no—
261#1446—611no—
262#1280—611no—
263C2×C61—21no—
264C2×C61—22no—
265C2×C61—21no—
266C2×C61—161no—
267C2×C61#160112yes—
268C2×C61#160412yes—
269C2×C61#200711yescharacteristic
270C2×C61#201411yescharacteristic
271C2×C61#202111yescharacteristic
272D61—161no—
273C23×Q8C2×D6111yescharacteristic
274C22×SD16—21no—
275C22×SD16—1221no—
276C2×#121—611no—
277C2×#121—1222no—
278C2×#121C2×D6111yescharacteristic
279#448—22no—
280#448—22no—
281#448—1222no—
282#448—1222no—
283C2×#103—21no—
284C2×#103—1221no—
285C2×C22⋊C8—611no—
286C2×C22⋊C8C2×D6111yescharacteristic
287#399—1221no—
288#399—1221no—
289#392—1221no—
290#392—1221no—
291C2×#95—611no—
292C2×#95—611no—
293C2×#95—1222no—
294#327—1222no—
295#327—1222no—
296C61#996411yescharacteristic, complemented, Sylow
297C22×Q8—21no—
298C22×Q8—21no—
299C22×Q8—21no—
300C22×Q8—21no—
301C22×Q8—21no—
302C22×Q8—44no—
303C22×Q8C22⋊D6112yescomplemented
304C22×D4—21no—
305C22×D4—611no—
306C22×D4—1222no—
307C22×D4—1221no—
308C22×D4C22⋊D6111yescharacteristic
309C23×C4—21no—
310C23×C4—612no—
311C23×C4—1221no—
312C23×C4C22×D6111yescharacteristic
313C2×SD16—24no—
314C2×SD16—44no—
315C2×SD16—1224no—
316C2×SD16—2444no—
317C22×C8—21no—
318C22×C8—1221no—
319#121—21no—
320#121—21no—
321#121—21no—
322#121—21no—
323#121—1221no—
324#121—1221no—
325#121—1221no—
326#121—1221no—
327#121—2444no—
328#121—2444no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4C22⋊D6111yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1222no—
335C2×C22⋊C4—1221no—
336#103—24no—
337#103—1224no—
338C22⋊C8—21no—
339C22⋊C8—612no—
340C22⋊C8—1221no—
341C22⋊C8C22⋊D6112yescomplemented
342#95—1221no—
343#95—1221no—
344#95—1221no—
345#95—1221no—
346#95—2444no—
347C24—21no—
348C24—1221no—
349C24D4×D6111yescharacteristic
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—44no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—42no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—4882no—
372C2×D4—4882no—
373C2×D4—2442no—
374C2×D4—2444no—
375C2×D4—2441no—
376C2×D4—2441no—
377C2×D4—2442no—
378C2×D4—2442no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—614no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—1221no—
391C22×C4—2442no—
392C22×C4—2441no—
393C22×C4—2441no—
394C22×C4—2442no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4D4×D6111yescharacteristic
398C22×C4D4×D6111yescharacteristic
399C22×C4#1175822411yescharacteristic
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403SD16—48no—
404SD16—2448no—
405C2×C8—22no—
406C2×C8—42no—
407C2×C8—1222no—
408C2×C8—2442no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4—2444no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—2444no—
417C22⋊C4—2444no—
418C22⋊C4—2441no—
419C22⋊C4—2441no—
420C23—21no—
421C23—21no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—2441no—
429C23—2441no—
430C23—2442no—
431C23—2442no—
432C23—2441no—
433C23#1451974511yescharacteristic
434C23#1451974911yescharacteristic
435C23#1451977311yescharacteristic
436C23SD16×D6112yes—
437C23#1451980412yes—
438Q8—24no—
439Q8—24no—
440Q8—42no—
441Q8—42no—
442Q8—44no—
443Q8—44no—
444D4—42no—
445D4—81no—
446D4—82no—
447D4—81no—
448D4—81no—
449D4—4881no—
450D4—4882no—
451D4—4881no—
452D4—4881no—
453D4—4888no—
454D4—2442no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—1224no—
469C2×C4—4881no—
470C2×C4—4882no—
471C2×C4—4881no—
472C2×C4—4882no—
473C2×C4—2444no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4—2444no—
480C2×C4#1451974511yescharacteristic
481C2×C4#1451974611yescharacteristic
482C2×C4#1451974811yescharacteristic
483C2×C4#1451974911yescharacteristic
484C8—42no—
485C8—2442no—
486C22—21no—
487C22—21no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—21no—
493C22—21no—
494C22—81no—
495C22—82no—
496C22—82no—
497C22—81no—
498C22—81no—
499C22—4881no—
500C22—4882no—
501C22—4882no—
502C22—4881no—
503C22—4881no—
504C22#1683967011yescharacteristic
505C22#1683991112yes—
506C22#1683993811yescharacteristic
507C22#1683994311yescharacteristic
508C22#1683994812yes—
509C22#1683996012yes—
510C22#1684013311yescharacteristic
511C22#1684014811yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—44no—
517C4—81no—
518C4—4882no—
519C4—4881no—
520C4—2444no—
521C2—21no—
522C2—22no—
523C2—21no—
524C2—161no—
525C2—9761no—
526C2#2144476912yes—
527C2#2144478612yes—
528C2#2144686411yescharacteristic
529C2#2144690511yescharacteristic
530C2#2144694211yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#448C2
socleC23×C61#120
prime radical#21451923C2
hypercenterC2×#448D61
hypocenterC61#9964
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C2×#16839901C2semidirect
3. C2×#16839497C2semidirect
4. C2×#16839496C2semidirect
5. #21444090C2semidirect
6. #21443661C2semidirect
7. C2×C61×#448C2semidirect
52. C22⋊#92418D4semidirect
64. C22×Q8×C61D4semidirect
80. C22⋊C8×C61D4semidirect
133. C22×Dic61SD16semidirect
257. C2×#448D61semidirect
296. C61#9964semidirect
303. C22×Q8C22⋊D61semidirect
341. C22⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399012
#214519232
C2×#168394972
C2×#168394962
#214440902
#214436612
C2×C61×#4482
#996461