Finite Group Databasebeta

← #29652382

Subgroups — #29652382

Order 15616.

1654 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1654 conjugacy classes of subgroups — too many to draw. Only its 167 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×#16840324C211yesmaximal, characteristic, complemented
2#21455071C212yesmaximal, complemented
3#21444770C214yesmaximal, complemented
4#21444736C214yesmaximal, complemented
5#21444347C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic, complemented
7C61×#2404C211yesmaximal, characteristic, complemented
8C23×C22⋊D61—21no—
9C2×#14519802—22no—
10#16840324—24no—
11#16840324—22no—
12#16840324C2214yes—
13C2×#14519761—22no—
14C2×#14519751—21no—
15C2×#14519751C2212yes—
16C2×#14519747—22no—
17#16839907—24no—
18#16839901C2214yescomplemented
19#16839896C2214yescomplemented
20#16839763—22no—
21#16839726C2212yescomplemented
22#16839495C2218yescomplemented
23#16839486C2214yescomplemented
24C22×#41×C61C2211yescharacteristic, complemented
25C2×#120×C61—21no—
26C22×C22⋊C4×C61C2211yescharacteristic
27C61×#393C2212yescomplemented
28C61×#392C2212yescomplemented
29C61×#386—21no—
30C61×#386—22no—
31C61×#381C2211yescharacteristic, complemented
32C24×D61—21no—
33C22×D244—21no—
34C22×D244D412yescomplemented
35C22×C22⋊D61—22no—
36C22×C22⋊D61—24no—
37C22×C22⋊D61—22no—
38C22×C22⋊D61—24no—
39C22×C22⋊D61—48no—
40C23×Dic61—21no—
41#14519802—48no—
42C2×#11758201—24no—
43C2×#11758199—24no—
44C2×#11758194—22no—
45#14519761—28no—
46#14519761—28no—
47#14519751—24no—
48#14519751—24no—
49#14519751—22no—
50#14519751—22no—
51#14519751D414yescomplemented
52#14519751C2314yes—
53#14519747—28no—
54#14519747—28no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61D412yescomplemented
57C2×#11758198—22no—
58C2×#11758198—22no—
59#14519701—24no—
60#14519698—24no—
61C22⋊#92418—22no—
62C22⋊#92418C2314yescomplemented
63C25×C61D411yescharacteristic
64C2×#41×C61—22no—
65C2×#41×C61—22no—
66C2×#41×C61—44no—
67C22×Q8×C61C2311yescharacteristic, complemented
68C22×D4×C61—21no—
69C22×D4×C61—22no—
70C22×D4×C61C2311yescharacteristic, complemented
71C23×C4×C61—21no—
72C23×C4×C61D411yescharacteristic
73C23×C4×C61C2311yescharacteristic
74#120×C61—44no—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—21no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61D412yes—
81C2×C22⋊C4×C61C2312yes—
82#95×C61—21no—
83#95×C61—22no—
84#95×C61C2312yescomplemented
85C24×C61—21no—
86C24×C61—21no—
87C24×C61—24no—
88C24×C61—21no—
89C24×C61—42no—
90C24×C61—81no—
91C24×C61C2×D411yescharacteristic
92C24×C61C2×D412yes—
93C23×D61—22no—
94C23×D61—24no—
95C23×D61—44no—
96C23×D61—82no—
97C23×D61—82no—
98#41×C61—28no—
99#41×C61—44no—
100#41×C61—44no—
101#41×C61—44no—
102C2×Q8×C61—22no—
103C2×Q8×C61—42no—
104C2×D4×C61—24no—
105C2×D4×C61—22no—
106C2×D4×C61—42no—
107C2×D4×C61—42no—
108C2×D4×C61—41no—
109C2×D4×C61—41no—
110C2×D4×C61—44no—
111C2×D4×C61—44no—
112C2×D4×C61—82no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—81no—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—28no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—48no—
131C2×C22⋊D61—416no—
132C2×C22⋊D61—48no—
133C2×C22⋊D61—48no—
134C2×C22⋊D61—48no—
135C22×Dic61—24no—
136C22×Dic61—22no—
137C22×Dic61—44no—
138C2×D244—24no—
139C2×D244—22no—
140C2×D244—44no—
141C2×D244—82no—
142C2×D244C2×D414yes—
143C22⋊C4×C61—24no—
144C22⋊C4×C61—24no—
145C22⋊C4×C61—44no—
146C22⋊C4×C61—42no—
147C22⋊C4×C61—82no—
148#11758201—48no—
149#11758199—48no—
150#11758198—28no—
151#11758198—24no—
152#11758198—24no—
153C4⋊Dic61—24no—
154C4⋊Dic61—22no—
155C4⋊Dic61C2×D414yes—
156#11758194—44no—
157C2×#92418—24no—
158C23×C61—24no—
159C23×C61—24no—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61—41no—
166C23×C61—42no—
167C23×C61—42no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—41no—
171C23×C61—42no—
172C23×C61—41no—
173C23×C61—42no—
174C23×C61—81no—
175C23×C61—81no—
176C23×C61—88no—
177C23×C61#12011yescharacteristic
178C23×C61#12011yescharacteristic
179C23×C61AGL(1,ℤ/8)14yescomplemented
180C23×C61C22×D411yescharacteristic
181C22×D61—24no—
182C22×D61—44no—
183C22×D61—44no—
184C22×D61—48no—
185C22×D61—44no—
186C22×D61—84no—
187C22×D61—84no—
188C22×D61—816no—
189Q8×C61—24no—
190Q8×C61—42no—
191D4×C61—24no—
192D4×C61—42no—
193D4×C61—44no—
194D4×C61—44no—
195D4×C61—88no—
196C2×C4×C61—24no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—41no—
202C2×C4×C61—48no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—81no—
211C2×C4×C61#12012yes—
212C2×C4×C61C22×D412yes—
213C22⋊D61—416no—
214C22⋊D61—48no—
215C22⋊D61—48no—
216C2×Dic61—24no—
217C2×Dic61—44no—
218C2×Dic61—48no—
219D244—44no—
220D244—82no—
221D244—84no—
222#92418—44no—
223#18132—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—24no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—44no—
233C22×C61—44no—
234C22×C61—48no—
235C22×C61—44no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—82no—
240C22×C61—82no—
241C22×C61—84no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61B(4,𝔽2)12yes—
245C22×C61C2×#12011yescharacteristic
246C22×C61C2×AGL(1,ℤ/8)12yes—
247C2×D61—44no—
248C2×D61—44no—
249C2×D61—44no—
250C2×D61—88no—
251C2×D61—816no—
252C4×C61—22no—
253C4×C61—22no—
254C4×C61—42no—
255C4×C61—42no—
256C4×C61—82no—
257Dic61—44no—
258C2×#534—611no—
259#3006—612no—
260#2404D6111yescharacteristic, complemented
261#1601—614no—
262#1595—614no—
263#1510—612no—
264#1462—611no—
265C2×C61—22no—
266C2×C61—22no—
267C2×C61—44no—
268C2×C61—41no—
269C2×C61—82no—
270C2×C61#300311yescharacteristic
271C2×C61#301911yescharacteristic
272C2×C61C2×B(4,𝔽2)11yescharacteristic
273D61—84no—
274C22×#41C2×D6111yescharacteristic, complemented
275C23×D4—1221no—
276C2×AGL(1,ℤ/8)—1222no—
277#534—614no—
278#534—1224no—
279#534—1222no—
280C2×#123—1222no—
281C2×#121—612no—
282C2×#121—1221no—
283C2×#120—21no—
284C2×#120—1222no—
285C22×C22⋊C4C2×D6111yescharacteristic
286#449—1224no—
287#448—614no—
288#447—614no—
289#417—1222no—
290#407—612no—
291#393C2×D6112yescomplemented
292#392C2×D6112yescomplemented
293#386—21no—
294#386—22no—
295#381C2×D6111yescharacteristic, complemented
296#327—618no—
297#323—614no—
298C61#1813211yescharacteristic, complemented, Sylow
299C25—1221no—
300C25C22⋊D6111yescharacteristic
301C2×#41—22no—
302C2×#41—22no—
303C2×#41—44no—
304C22×Q8C22×D6111yescharacteristic, complemented
305C22×D4—21no—
306C22×D4—22no—
307C22×D4—612no—
308C22×D4—1221no—
309C22×D4—1222no—
310C22×D4—1224no—
311C22×D4—1222no—
312C22×D4—1224no—
313C22×D4—2448no—
314C22×D4C22×D6111yescharacteristic, complemented
315C23×C4—21no—
316C23×C4—1221no—
317C23×C4C22⋊D6111yescharacteristic
318C23×C4C22×D6111yescharacteristic
319AGL(1,ℤ/8)—2448no—
320C2×SD16—1224no—
321C2×D8—1224no—
322C2×M16—1222no—
323#123—1228no—
324#123—1228no—
325#121—614no—
326#121—614no—
327#121—1224no—
328#121—1222no—
329#121—1222no—
330#121—1224no—
331#120—44no—
332#120—1228no—
333#120—1228no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—1221no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4C22⋊D6112yes—
344C2×C22⋊C4C22×D6112yes—
345#103—1224no—
346#102—1224no—
347C22⋊C8—614no—
348C22⋊C8—1222no—
349#95—21no—
350#95—22no—
351#95C22×D6112yescomplemented
352C24—21no—
353C24—21no—
354C24—24no—
355C24—21no—
356C24—42no—
357C24—81no—
358C24—1224no—
359C24—1222no—
360C24—4882no—
361C24—4882no—
362C24—2444no—
363C24C2×C22⋊D6111yescharacteristic
364C24C2×C22⋊D6112yes—
365#41—28no—
366#41—44no—
367#41—44no—
368#41—44no—
369C2×Q8—22no—
370C2×Q8—42no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—82no—
380C2×D4—614no—
381C2×D4—1224no—
382C2×D4—1228no—
383C2×D4—1224no—
384C2×D4—1224no—
385C2×D4—1222no—
386C2×D4—4882no—
387C2×D4—2444no—
388C2×D4—2448no—
389C2×D4—24416no—
390C2×D4—2448no—
391C2×D4—2448no—
392C2×D4—2448no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—81no—
402C22×C4—1224no—
403C22×C4—1222no—
404C22×C4—2444no—
405C22×C4D4×D6112yes—
406C22×C4D4×D6112yes—
407C22×C4C2×C22⋊D6112yes—
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410SD16—2448no—
411D8—2448no—
412M16—2444no—
413C2×C8—1224no—
414C4⋊C4—614no—
415C4⋊C4—1224no—
416C4⋊C4—1222no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—44no—
420C22⋊C4—42no—
421C22⋊C4—82no—
422C22⋊C4—1228no—
423C22⋊C4—1224no—
424C22⋊C4—1224no—
425C23—24no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23—81no—
442C23—81no—
443C23—88no—
444C23—1224no—
445C23—48816no—
446C23—4884no—
447C23—4884no—
448C23—2444no—
449C23—2444no—
450C23—2448no—
451C23—2444no—
452C23#1451974511yescharacteristic
453C23#1451974511yescharacteristic
454C23#1451980214yescomplemented
455C23C22×C22⋊D6111yescharacteristic
456Q8—24no—
457Q8—42no—
458D4—24no—
459D4—42no—
460D4—44no—
461D4—44no—
462D4—88no—
463D4—4882no—
464D4—4884no—
465D4—2444no—
466D4—24416no—
467D4—2448no—
468D4—2448no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—48no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—81no—
484C2×C4—1224no—
485C2×C4—2444no—
486C2×C4—2448no—
487C2×C4#1451974512yes—
488C2×C4C2×D4×D6112yes—
489C8—2444no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—24no—
497C22—21no—
498C22—44no—
499C22—44no—
500C22—48no—
501C22—44no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22—84no—
508C22—81no—
509C22—81no—
510C22—4888no—
511C22—48816no—
512C22—2444no—
513C22—2444no—
514C22—2444no—
515C22#1683994812yes—
516C22C2×#1451974511yescharacteristic
517C22C2×#1451980212yes—
518C4—22no—
519C4—22no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—2444no—
524C2—22no—
525C2—22no—
526C2—41no—
527C2—44no—
528C2—82no—
529C2—4884no—
530C2#2145504511yescharacteristic
531C2#2145517511yescharacteristic
532C2C2×#1683994811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2404C2
socleC22×C61C2×#120
hypercenter#2404D61
hypocenterC61#18132
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. C2×#16840324C2semidirect
2. #21455071C2semidirect
3. #21444770C2semidirect
4. #21444736C2semidirect
5. #21444347C2semidirect
6. #21444146C2semidirect
7. C61×#2404C2semidirect
18. #16839901C22semidirect
19. #16839896C22semidirect
21. #16839726C22semidirect
22. #16839495C22semidirect
23. #16839486C22semidirect
24. C22×#41×C61C22semidirect
27. C61×#393C22semidirect
28. C61×#392C22semidirect
31. C61×#381C22semidirect
34. C22×D244D4semidirect
51. #14519751D4semidirect
56. C2×C4⋊Dic61D4semidirect
62. C22⋊#92418C23semidirect
67. C22×Q8×C61C23semidirect
70. C22×D4×C61C23semidirect
84. #95×C61C23semidirect
179. C23×C61AGL(1,ℤ/8)semidirect
260. #2404D61semidirect
274. C22×#41C2×D61semidirect
291. #393C2×D61semidirect
292. #392C2×D61semidirect
295. #381C2×D61semidirect
298. C61#18132semidirect
304. C22×Q8C22×D61semidirect
314. C22×D4C22×D61semidirect
351. #95C22×D61semidirect
454. C23#14519802semidirect

Maximal subgroups

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

subgroupindex
C2×#168403242
#214550712
#214447702
#214447362
#214443472
#214441462
C61×#24042
#1813261