Finite Group Databasebeta

← #29654118

Subgroups — #29654118

Order 15616.

1516 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1516 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#21456616C212yesmaximal, complemented
3#21444838C214yesmaximal, complemented
4#21444736C214yesmaximal, complemented
5#21444334C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic, complemented
7C61×#2539C211yesmaximal, characteristic, complemented
8C23×C22⋊D61—21no—
9#16840324—24no—
10#16840324—22no—
11#16840324C2214yes—
12C2×#14519761—22no—
13C2×#14519751—21no—
14C2×#14519751C2212yes—
15C2×#14519747—22no—
16#16840080C2214yescomplemented
17#16840078—22no—
18#16840013—24no—
19#16839896C2214yescomplemented
20#16839895—22no—
21#16839726C2212yescomplemented
22#16839495C2218yescomplemented
23#16839486C2214yescomplemented
24C2×#123×C61—21no—
25C2×#121×C61C2211yescharacteristic, complemented
26C2×#120×C61—21no—
27C22×C22⋊C4×C61C2211yescharacteristic
28C61×#399C2212yescomplemented
29C61×#392C2212yescomplemented
30C61×#390—22no—
31C4×C22⋊C4×C61C2211yescharacteristic, 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—
41C2×#11758199—24no—
42C2×#11758194—22no—
43#14519761—28no—
44#14519761—28no—
45#14519751—24no—
46#14519751—24no—
47#14519751—22no—
48#14519751—22no—
49#14519751D414yescomplemented
50#14519751C2314yes—
51#14519747—28no—
52#14519747—28no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61D412yescomplemented
55C2×#11758198—22no—
56C2×#11758198—22no—
57#14519708—24no—
58#14519698—24no—
59#14519697—24no—
60C22⋊#92418—22no—
61C22⋊#92418C2314yescomplemented
62C25×C61D411yescharacteristic
63C22×D4×C61—21no—
64C22×D4×C61—22no—
65C22×D4×C61C2311yescharacteristic, complemented
66C23×C4×C61D411yescharacteristic
67C23×C4×C61C2311yescharacteristic
68#123×C61—44no—
69#121×C61—22no—
70#121×C61—22no—
71#120×C61—44no—
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61C2311yescharacteristic, complemented
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—21no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61D412yes—
81C2×C22⋊C4×C61C2312yes—
82C2×C42×C61—21no—
83#95×C61C2312yescomplemented
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—24no—
87C24×C61—21no—
88C24×C61—42no—
89C24×C61—81no—
90C24×C61C2×D411yescharacteristic
91C24×C61C2×D412yes—
92C23×D61—22no—
93C23×D61—24no—
94C23×D61—44no—
95C23×D61—82no—
96C23×D61—82no—
97C2×D4×C61—22no—
98C2×D4×C61—44no—
99C2×D4×C61—42no—
100C2×D4×C61—42no—
101C2×D4×C61—41no—
102C2×D4×C61—41no—
103C2×D4×C61—44no—
104C2×D4×C61—81no—
105C2×D4×C61—82no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—42no—
111C22×C4×C61—81no—
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—28no—
118C2×C22⋊D61—24no—
119C2×C22⋊D61—24no—
120C2×C22⋊D61—48no—
121C2×C22⋊D61—416no—
122C2×C22⋊D61—48no—
123C2×C22⋊D61—48no—
124C2×C22⋊D61—48no—
125C22×Dic61—24no—
126C22×Dic61—22no—
127C22×Dic61—44no—
128C2×D244—24no—
129C2×D244—22no—
130C2×D244—82no—
131C2×D244—82no—
132C2×D244C2×D414yes—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—44no—
135C22⋊C4×C61—24no—
136C22⋊C4×C61—24no—
137C22⋊C4×C61—24no—
138C22⋊C4×C61—44no—
139C22⋊C4×C61—44no—
140C22⋊C4×C61—42no—
141C42×C61—81no—
142#11758199—84no—
143#11758198—28no—
144#11758198—24no—
145#11758198—24no—
146C4⋊Dic61—24no—
147C4⋊Dic61—22no—
148C4⋊Dic61C2×D414yes—
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—41no—
158C23×C61—42no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61—41no—
162C23×C61—41no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—41no—
166C23×C61—42no—
167C23×C61—81no—
168C23×C61—81no—
169C23×C61—88no—
170C23×C61#12011yescharacteristic
171C23×C61#12011yescharacteristic
172C23×C61AGL(1,ℤ/8)14yes—
173C23×C61C22×D411yescharacteristic
174C22×D61—24no—
175C22×D61—44no—
176C22×D61—44no—
177C22×D61—48no—
178C22×D61—44no—
179C22×D61—84no—
180C22×D61—84no—
181C22×D61—816no—
182D4×C61—44no—
183D4×C61—88no—
184D4×C61—82no—
185D4×C61—82no—
186C2×C4×C61—24no—
187C2×C4×C61—22no—
188C2×C4×C61—42no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—81no—
200C2×C4×C61#12012yes—
201C2×C4×C61C22×D412yes—
202C22⋊D61—416no—
203C22⋊D61—48no—
204C22⋊D61—48no—
205C2×Dic61—24no—
206C2×Dic61—44no—
207C2×Dic61—48no—
208D244—44no—
209D244—82no—
210D244—84no—
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—82no—
227C22×C61—82no—
228C22×C61—82no—
229C22×C61—84no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61#45312yes—
234C22×C61C2×#12011yescharacteristic
235C22×C61C2×AGL(1,ℤ/8)12yes—
236C2×D61—44no—
237C2×D61—44no—
238C2×D61—44no—
239C2×D61—88no—
240C2×D61—816no—
241C4×C61—22no—
242C4×C61—22no—
243C4×C61—44no—
244C4×C61—42no—
245C4×C61—81no—
246Dic61—44no—
247C2×#534—611no—
248#3192—612no—
249#2539D6111yescharacteristic, complemented
250#1613—614no—
251#1595—614no—
252#1507—612no—
253#1462—611no—
254C2×C61—22no—
255C2×C61—22no—
256C2×C61—44no—
257C2×C61—41no—
258C2×C61—161no—
259C2×C61#300311yescharacteristic
260C2×C61C2×#45311yescharacteristic
261C2×C61#302611yescharacteristic
262D61—84no—
263C23×D4—1221no—
264#534—614no—
265#534—1224no—
266#534—1222no—
267C2×#123—21no—
268C2×#123—1222no—
269C2×#121—612no—
270C2×#121—1221no—
271C2×#121C2×D6111yescharacteristic, complemented
272C2×#120—21no—
273C2×#120—1222no—
274C22×C22⋊C4C2×D6111yescharacteristic
275#481—614no—
276#481—1222no—
277#469—1224no—
278#447—614no—
279#447—1222no—
280#407—612no—
281#399C2×D6112yescomplemented
282#392C2×D6112yescomplemented
283#390—22no—
284C4×C22⋊C4C2×D6111yescharacteristic, complemented
285#327—618no—
286#323—614no—
287C61#1831611yescharacteristic, complemented, Sylow
288C25—1221no—
289C25C22⋊D6111yescharacteristic
290C22×D4—21no—
291C22×D4—22no—
292C22×D4—612no—
293C22×D4—1221no—
294C22×D4—1222no—
295C22×D4—1224no—
296C22×D4—1222no—
297C22×D4—1224no—
298C22×D4—2448no—
299C22×D4C22×D6111yescharacteristic, complemented
300C23×C4—1221no—
301C23×C4C22⋊D6111yescharacteristic
302C23×C4C22×D6111yescharacteristic
303C2×D8—1224no—
304C2×M16—1222no—
305#123—44no—
306#123—1228no—
307#123—1228no—
308#121—22no—
309#121—22no—
310#121—614no—
311#121—614no—
312#121—1224no—
313#121—1222no—
314#121—1222no—
315#121—1224no—
316#120—44no—
317#120—1228no—
318#120—1228no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—612no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4C22×D6111yescharacteristic, complemented
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—1222no—
330C2×C22⋊C4—1222no—
331C2×C22⋊C4C22⋊D6112yes—
332C2×C22⋊C4C22×D6112yes—
333C2×C42—21no—
334#106—1224no—
335#102—1224no—
336#102—1224no—
337C22⋊C8—614no—
338C22⋊C8—1222no—
339#95C22×D6112yescomplemented
340C24—21no—
341C24—21no—
342C24—24no—
343C24—21no—
344C24—42no—
345C24—81no—
346C24—1224no—
347C24—1222no—
348C24—4882no—
349C24—4882no—
350C24—2444no—
351C24C2×C22⋊D6111yescharacteristic
352C24C2×C22⋊D6112yes—
353C2×D4—22no—
354C2×D4—44no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—44no—
360C2×D4—81no—
361C2×D4—82no—
362C2×D4—614no—
363C2×D4—1224no—
364C2×D4—1228no—
365C2×D4—1224no—
366C2×D4—1224no—
367C2×D4—1222no—
368C2×D4—4882no—
369C2×D4—4882no—
370C2×D4—2448no—
371C2×D4—24416no—
372C2×D4—2448no—
373C2×D4—2448no—
374C2×D4—2448no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—42no—
380C22×C4—81no—
381C22×C4—1224no—
382C22×C4—1222no—
383C22×C4—2444no—
384C22×C4D4×D6112yes—
385C22×C4D4×D6112yes—
386C22×C4C2×C22⋊D6112yes—
387C22×C4C2×C22⋊D6111yescharacteristic
388C22×C4C23×D6111yescharacteristic
389D8—4884no—
390M16—4882no—
391C2×C8—1224no—
392C4⋊C4—22no—
393C4⋊C4—44no—
394C4⋊C4—614no—
395C4⋊C4—1224no—
396C4⋊C4—1222no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—42no—
403C22⋊C4—1228no—
404C22⋊C4—1224no—
405C22⋊C4—1224no—
406C42—81no—
407C23—24no—
408C23—24no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—41no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23—41no—
422C23—42no—
423C23—81no—
424C23—81no—
425C23—88no—
426C23—1224no—
427C23—48816no—
428C23—4884no—
429C23—4884no—
430C23—2444no—
431C23—2444no—
432C23—2448no—
433C23—2444no—
434C23#1451974511yescharacteristic
435C23#1451974511yescharacteristic
436C23#1451980214yes—
437C23C22×C22⋊D6111yescharacteristic
438D4—44no—
439D4—88no—
440D4—82no—
441D4—82no—
442D4—4882no—
443D4—4884no—
444D4—2444no—
445D4—24416no—
446D4—2448no—
447D4—2448no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—81no—
462C2×C4—1224no—
463C2×C4—2444no—
464C2×C4—2448no—
465C2×C4#1451974512yes—
466C2×C4C2×D4×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—82no—
482C22—82no—
483C22—82no—
484C22—84no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—4888no—
489C22—48816no—
490C22—2444no—
491C22—2444no—
492C22—2444no—
493C22#1683992912yes—
494C22C2×#1451974511yescharacteristic
495C22C2×#1451980212yes—
496C4—22no—
497C4—22no—
498C4—44no—
499C4—42no—
500C4—81no—
501C4—2444no—
502C2—22no—
503C2—22no—
504C2—41no—
505C2—44no—
506C2—161no—
507C2—4884no—
508C2#2145504511yescharacteristic
509C2C2×#1683992911yescharacteristic
510C2#2145521511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2539C2
socleC22×C61C2×#120
hypercenter#2539D61
hypocenterC61#18316
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
1. C2×#16840324C2semidirect
2. #21456616C2semidirect
3. #21444838C2semidirect
4. #21444736C2semidirect
5. #21444334C2semidirect
6. #21444146C2semidirect
7. C61×#2539C2semidirect
16. #16840080C22semidirect
19. #16839896C22semidirect
21. #16839726C22semidirect
22. #16839495C22semidirect
23. #16839486C22semidirect
25. C2×#121×C61C22semidirect
28. C61×#399C22semidirect
29. C61×#392C22semidirect
31. C4×C22⋊C4×C61C22semidirect
34. C22×D244D4semidirect
49. #14519751D4semidirect
54. C2×C4⋊Dic61D4semidirect
61. C22⋊#92418C23semidirect
65. C22×D4×C61C23semidirect
73. C2×C4⋊C4×C61C23semidirect
83. #95×C61C23semidirect
249. #2539D61semidirect
271. C2×#121C2×D61semidirect
281. #399C2×D61semidirect
282. #392C2×D61semidirect
284. C4×C22⋊C4C2×D61semidirect
287. C61#18316semidirect
299. C22×D4C22×D61semidirect
322. C2×C4⋊C4C22×D61semidirect
339. #95C22×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403242
#214566162
#214448382
#214447362
#214443342
#214441462
C61×#25392
#1831661