Finite Group Databasebeta

← #29652422

Subgroups — #29652422

Order 15616.

1398 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1398 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#21455755C212yesmaximal, complemented
2#21450093C211yesmaximal, characteristic, complemented
3#21444839C214yesmaximal, complemented
4#21444805C214yesmaximal, complemented
5#21444348C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic, complemented
7C2×C61×#534C211yesmaximal, characteristic
8C2×#14519736C2211yescharacteristic, complemented
9#16840083—24no—
10#16840078C2214yescomplemented
11#16840073C2214yescomplemented
12#16839802—22no—
13#16839767—22no—
14#16839726C2212yescomplemented
15#16839698C2212yescomplemented
16#16839687C2212yescomplemented
17#16839650—22no—
18#16839635—21no—
19#16839625C2211yescharacteristic, complemented
20#16839618—21no—
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
32C2×#11758194—22no—
33#14519736—22no—
34#14519736—22no—
35C2×C4⋊Dic61C2311yescharacteristic, complemented
36C2×C4⋊Dic61C2311yescharacteristic, complemented
37C2×#11758196—22no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—22no—
42C2×C4×Dic61—21no—
43#14519710—24no—
44#14519708—24no—
45#14519698—24no—
46#14519698—24no—
47C22⋊#92418—22no—
48C22⋊#92418C2314yescomplemented
49#14519684—22no—
50#14519684—21no—
51#14519684C2312yescomplemented
52C25×C61—21no—
53C25×C61D411yescharacteristic
54C22×D4×C61—24no—
55C22×D4×C61—22no—
56C22×D4×C61—24no—
57C22×D4×C61—22no—
58C22×D4×C61—21no—
59C22×D4×C61—48no—
60C22×D4×C61D412yes—
61C23×C4×C61—21no—
62C23×C4×C61D411yescharacteristic
63C23×C4×C61C2311yescharacteristic
64#123×C61—28no—
65#123×C61—28no—
66#121×C61—24no—
67#121×C61—22no—
68#121×C61—22no—
69#121×C61—24no—
70#121×C61D414yes—
71#121×C61C2314yes—
72#120×C61—28no—
73#120×C61—28no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61D412yes—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61D412yes—
81C2×C22⋊C4×C61C2312yes—
82C24×C61—21no—
83C24×C61—21no—
84C24×C61—24no—
85C24×C61—24no—
86C24×C61—22no—
87C24×C61—44no—
88C24×C61—42no—
89C24×C61—81no—
90C24×C61—82no—
91C24×C61—82no—
92C24×C61C2×D411yescharacteristic
93C24×C61C2×D412yes—
94C2×D4×C61—28no—
95C2×D4×C61—24no—
96C2×D4×C61—24no—
97C2×D4×C61—24no—
98C2×D4×C61—22no—
99C2×D4×C61—44no—
100C2×D4×C61—48no—
101C2×D4×C61—416no—
102C2×D4×C61—48no—
103C2×D4×C61—48no—
104C2×D4×C61—48no—
105C2×D4×C61—82no—
106C2×D4×C61C2×D414yes—
107C22×C4×C61—24no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—42no—
112C22×C4×C61—44no—
113C22×C4×C61—81no—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—22no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C4⋊C4×C61—24no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61C2×D414yes—
125C22⋊C4×C61—28no—
126C22⋊C4×C61—24no—
127C22⋊C4×C61—24no—
128C22⋊C4×C61—24no—
129C22⋊C4×C61—24no—
130C22⋊Dic61—42no—
131C22⋊Dic61—41no—
132C22⋊Dic61—41no—
133C22⋊Dic61—44no—
134C22⋊Dic61—82no—
135C4⋊Dic61—22no—
136C4⋊Dic61—22no—
137#11758196—82no—
138C4×Dic61—42no—
139#11758194—44no—
140C2×#92418—24no—
141C23×C61—24no—
142C23×C61—24no—
143C23×C61—22no—
144C23×C61—22no—
145C23×C61—22no—
146C23×C61—22no—
147C23×C61—24no—
148C23×C61—41no—
149C23×C61—42no—
150C23×C61—42no—
151C23×C61—42no—
152C23×C61—44no—
153C23×C61—48no—
154C23×C61—44no—
155C23×C61—42no—
156C23×C61—44no—
157C23×C61—81no—
158C23×C61—81no—
159C23×C61—88no—
160C23×C61—84no—
161C23×C61—84no—
162C23×C61—816no—
163C23×C61#12011yescharacteristic
164C23×C61#12011yescharacteristic
165C23×C61AGL(1,ℤ/8)14yes—
166C23×C61C22×D411yescharacteristic
167D4×C61—44no—
168D4×C61—416no—
169D4×C61—48no—
170D4×C61—48no—
171D4×C61—82no—
172D4×C61—84no—
173C2×C4×C61—24no—
174C2×C4×C61—22no—
175C2×C4×C61—24no—
176C2×C4×C61—42no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—44no—
180C2×C4×C61—48no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—81no—
184C2×C4×C61#12012yes—
185C2×C4×C61C22×D412yes—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193#92418—44no—
194#18137—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—24no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—44no—
204C22×C61—44no—
205C22×C61—48no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61—44no—
209C22×C61—44no—
210C22×C61—44no—
211C22×C61—82no—
212C22×C61—88no—
213C22×C61—84no—
214C22×C61—816no—
215C22×C61#45812yes—
216C22×C61C2×#12011yescharacteristic
217C22×C61C2×AGL(1,ℤ/8)12yes—
218C4×C61—22no—
219C4×C61—22no—
220C4×C61—44no—
221C4×C61—42no—
222Dic61—82no—
223Dic61—81no—
224Dic61—81no—
225C2×#534D6111yescharacteristic
226#3094—612no—
227#2482—611no—
228#1613—614no—
229#1607—614no—
230#1510—612no—
231#1462—611no—
232C2×C61—22no—
233C2×C61—22no—
234C2×C61—41no—
235C2×C61—44no—
236C2×C61—84no—
237C2×C61#300311yescharacteristic
238C2×C61#301911yescharacteristic
239C2×C61C2×#45811yescharacteristic
240C23×D4—21no—
241#534—24no—
242#534—22no—
243#534C2×D6114yes—
244C2×#123—22no—
245C2×#121—21no—
246C2×#121C2×D6112yes—
247C2×#120—22no—
248C2×#117—611no—
249C22×C22⋊C4C2×D6111yescharacteristic
250#482—1224no—
251#481—614no—
252#480—614no—
253#428—1222no—
254#418—1222no—
255#407—612no—
256#399—612no—
257#397—612no—
258#388—1222no—
259#385—1221no—
260#381—611no—
261#379—1221no—
262#327—618no—
263#323—614no—
264C61#1813711yescharacteristic, complemented, Sylow
265C25—21no—
266C25C22⋊D6111yescharacteristic
267C22×D4—22no—
268C22×D4—24no—
269C22×D4—22no—
270C22×D4—24no—
271C22×D4—21no—
272C22×D4—48no—
273C22×D4C22⋊D6112yes—
274C23×C4—21no—
275C23×C4C22⋊D6111yescharacteristic
276C23×C4C22×D6111yescharacteristic
277C2×M16—1222no—
278#123—28no—
279#123—28no—
280#121—24no—
281#121—24no—
282#121—22no—
283#121—22no—
284#121C22⋊D6114yes—
285#121C22×D6114yes—
286#120—28no—
287#120—28no—
288#117—1222no—
289#117—1222no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—611no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—1222no—
294C2×C4⋊C4C22⋊D6112yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—1222no—
300C2×C22⋊C4—1221no—
301C2×C22⋊C4—1221no—
302C2×C22⋊C4—1222no—
303C2×C22⋊C4C22⋊D6112yes—
304C2×C22⋊C4C22×D6112yes—
305C2×C42—1221no—
306#107—1224no—
307#106—1224no—
308#102—1224no—
309#102—1224no—
310C22⋊C8—614no—
311C22⋊C8—1222no—
312#95—612no—
313#95—1221no—
314#95—1222no—
315C24—21no—
316C24—21no—
317C24—24no—
318C24—24no—
319C24—22no—
320C24—44no—
321C24—42no—
322C24—81no—
323C24—82no—
324C24—82no—
325C24C2×C22⋊D6111yescharacteristic
326C24C2×C22⋊D6112yes—
327C2×D4—28no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—44no—
333C2×D4—48no—
334C2×D4—416no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—82no—
339C2×D4C2×C22⋊D6114yes—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—81no—
347C22×C4—1222no—
348C22×C4—1221no—
349C22×C4—1221no—
350C22×C4C2×C22⋊D6112yes—
351C22×C4C2×C22⋊D6112yes—
352C22×C4C2×C22⋊D6112yes—
353C22×C4C2×C22⋊D6111yescharacteristic
354C22×C4C23×D6111yescharacteristic
355M16—2444no—
356C2×C8—1224no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—1222no—
360C4⋊C4—1222no—
361C4⋊C4—4882no—
362C4⋊C4C2×C22⋊D6114yes—
363C22⋊C4—28no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—4882no—
369C22⋊C4—2442no—
370C22⋊C4—2441no—
371C22⋊C4—2441no—
372C22⋊C4—2444no—
373C42—2442no—
374C23—24no—
375C23—24no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—44no—
386C23—48no—
387C23—44no—
388C23—42no—
389C23—44no—
390C23—81no—
391C23—81no—
392C23—88no—
393C23—84no—
394C23—84no—
395C23—816no—
396C23#1451974611yescharacteristic
397C23#1451974611yescharacteristic
398C23#1451980314yes—
399C23C22×C22⋊D6111yescharacteristic
400D4—44no—
401D4—416no—
402D4—48no—
403D4—48no—
404D4—82no—
405D4—84no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—81no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4—2442no—
420C2×C4—2441no—
421C2×C4—2441no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4#1451974612yes—
425C2×C4C22×C22⋊D6112yes—
426C8—2444no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—24no—
434C22—21no—
435C22—44no—
436C22—44no—
437C22—48no—
438C22—42no—
439C22—42no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—82no—
444C22—88no—
445C22—84no—
446C22—816no—
447C22#1683995212yes—
448C22C2×#1451974611yescharacteristic
449C22C2×#1451980312yes—
450C4—22no—
451C4—22no—
452C4—42no—
453C4—44no—
454C4—4882no—
455C4—4881no—
456C4—4881no—
457C2—22no—
458C2—22no—
459C2—41no—
460C2—44no—
461C2—84no—
462C2#2145504611yescharacteristic
463C2#2145517611yescharacteristic
464C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#534C2
socleC22×C61C2×#120
hypercenterC2×#534D61
hypocenterC61#18137
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. #21455755C2semidirect
2. #21450093C2semidirect
3. #21444839C2semidirect
4. #21444805C2semidirect
5. #21444348C2semidirect
6. #21444146C2semidirect
8. C2×#14519736C22semidirect
10. #16840078C22semidirect
11. #16840073C22semidirect
14. #16839726C22semidirect
15. #16839698C22semidirect
16. #16839687C22semidirect
19. #16839625C22semidirect
21. #16839496C22semidirect
22. #16839486C22semidirect
35. C2×C4⋊Dic61C23semidirect
36. C2×C4⋊Dic61C23semidirect
48. C22⋊#92418C23semidirect
51. #14519684C23semidirect
264. C61#18137semidirect

Maximal subgroups

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

subgroupindex
#214557552
#214500932
#214448392
#214448052
#214443482
#214441462
C2×C61×#5342
#1813761