Finite Group Databasebeta

← #29595751

Subgroups — #29595751

Order 15616.

928 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21447568C211yesmaximal, characteristic, complemented
2C2×#16839575C211yesmaximal, characteristic, complemented
3#21447555C211yesmaximal, characteristic, complemented
4#21447535C211yesmaximal, characteristic, complemented
5#21447447C211yesmaximal, characteristic, complemented
6#21445873C211yesmaximal, characteristic, complemented
7C2×C61×#576C211yesmaximal, characteristic
8C2×#14519717C2211yescharacteristic, complemented
9C2×#14519717C2211yescharacteristic, complemented
10C2×#14519713C2211yescharacteristic, complemented
11#16839800C2211yescharacteristic, complemented
12#16839779—21no—
13#16839578—21no—
14#16839575—22no—
15#16839575—21no—
16#16839572—22no—
17#16839563—22no—
18#16839546—22no—
19#16839534—21no—
20C2×H(5,𝔽2)×C61—21no—
21C61×#576—22no—
22C61×#576—22no—
23C61×#576—22no—
24C61×#576C414yescomplemented
25C2×AGL(1,ℤ/8)×C61—22no—
26C2×AGL(1,ℤ/8)×C61—21no—
27C2×#135×C61—21no—
28C2×#135×C61C2211yescharacteristic
29C22×D8×C61—21no—
30C22×D8×C61C2211yescharacteristic
31C2×#131×C61C2211yescharacteristic
32C2×#11758194—21no—
33#14519736—21no—
34#14519717—21no—
35#14519717—21no—
36#14519717—21no—
37#14519717—21no—
38C2×#11758185—21no—
39C2×#11758185—21no—
40#14519713—21no—
41#14519713—21no—
42#14519711—21no—
43#14519709—21no—
44#14519704—42no—
45H(5,𝔽2)×C61—24no—
46H(5,𝔽2)×C61—24no—
47H(5,𝔽2)×C61—42no—
48H(5,𝔽2)×C61—42no—
49C2×#41×C61—22no—
50C2×#41×C61—21no—
51C2×#41×C61—41no—
52C2×#41×C61D411yescharacteristic
53C22×D4×C61—22no—
54C22×D4×C61—21no—
55C22×D4×C61—42no—
56C22×D4×C61—41no—
57AGL(1,ℤ/8)×C61—24no—
58AGL(1,ℤ/8)×C61—28no—
59AGL(1,ℤ/8)×C61—28no—
60AGL(1,ℤ/8)×C61—24no—
61#135×C61—24no—
62#135×C61—24no—
63#135×C61—21no—
64#135×C61—21no—
65#135×C61C2×C412yes—
66#135×C61D412yes—
67C2×Q16×C61D411yescharacteristic
68C2×SD16×C61—22no—
69C2×SD16×C61—21no—
70C2×D8×C61—22no—
71C2×D8×C61—24no—
72C2×D8×C61—24no—
73C2×D8×C61—21no—
74C2×D8×C61—21no—
75C2×D8×C61—21no—
76C2×D8×C61C2×C412yes—
77C2×D8×C61D411yescharacteristic
78C2×D8×C61D412yes—
79C2×D8×C61D412yes—
80#131×C61—21no—
81#131×C61—21no—
82#131×C61C2×C412yes—
83#131×C61D412yes—
84C2×M16×C61—21no—
85C2×M16×C61D411yescharacteristic
86C22×C8×C61—21no—
87C22×C8×C61C2311yescharacteristic
88C24×C61—42no—
89C24×C61—41no—
90#41×C61—21no—
91#41×C61—21no—
92#41×C61—28no—
93#41×C61—22no—
94#41×C61—22no—
95#41×C61—44no—
96#41×C61—44no—
97#41×C61—44no—
98#41×C61—41no—
99#41×C61—41no—
100#41×C61C22⋊C412yes—
101#41×C61D812yes—
102C2×Q8×C61—21no—
103C2×Q8×C61SD1611yescharacteristic
104C2×D4×C61—21no—
105C2×D4×C61—28no—
106C2×D4×C61—22no—
107C2×D4×C61—21no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—42no—
111C2×D4×C61—44no—
112C2×D4×C61—48no—
113C2×D4×C61—48no—
114C2×D4×C61—44no—
115C2×D4×C61—42no—
116C2×D4×C61—41no—
117C2×D4×C61—44no—
118C2×D4×C61—41no—
119C2×D4×C61—41no—
120C2×D4×C61—41no—
121C2×D4×C61SD1611yescharacteristic
122C22×C4×C61—21no—
123C22×C4×C61—42no—
124C22×C4×C61—41no—
125C22×C4×C61C2×D411yescharacteristic
126Q16×C61—21no—
127Q16×C61C22⋊C412yes—
128SD16×C61—28no—
129SD16×C61—22no—
130SD16×C61—22no—
131D8×C61—28no—
132D8×C61—22no—
133D8×C61—21no—
134D8×C61—22no—
135D8×C61—22no—
136D8×C61C22⋊C412yes—
137D8×C61C22⋊C414yes—
138M16×C61—21no—
139M16×C61—22no—
140M16×C61—22no—
141M16×C61C22⋊C412yes—
142C2×C8×C61—21no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61C22×C411yescharacteristic
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61C2×D411yescharacteristic
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61C2×D411yescharacteristic
151C22⋊Dic61—41no—
152C4⋊Dic61—41no—
153C4×Dic61—41no—
154#11758194—41no—
155#11758194—41no—
156C2×#92418—41no—
157#11758185—41no—
158#11758185—41no—
159C23×C61—21no—
160C23×C61—48no—
161C23×C61—42no—
162C23×C61—41no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—162no—
166C23×C61—164no—
167C23×C61AGL(1,ℤ/8)11yescharacteristic
168Q8×C61—21no—
169Q8×C61—22no—
170Q8×C61—41no—
171Q8×C61#10212yes—
172D4×C61—22no—
173D4×C61—22no—
174D4×C61—22no—
175D4×C61—24no—
176D4×C61—21no—
177D4×C61—48no—
178D4×C61—48no—
179D4×C61—42no—
180D4×C61—42no—
181D4×C61—42no—
182D4×C61—41no—
183D4×C61—42no—
184D4×C61—42no—
185D4×C61#10212yes—
186C2×C4×C61—21no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—48no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C2×SD1611yescharacteristic
198C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
199C2×Dic61—81no—
200C8×C61—22no—
201C8×C61C2×C22⋊C411yescharacteristic
202C8×C61C2×C22⋊C411yescharacteristic
203C8×C61#12011yescharacteristic
204C8×C61#12011yescharacteristic
205#92418—81no—
206#12750—611nomaximal, Sylow
207C22×C61—22no—
208C22×C61—41no—
209C22×C61—42no—
210C22×C61—44no—
211C22×C61—42no—
212C22×C61—84no—
213C22×C61—82no—
214C22×C61—162no—
215C22×C61—162no—
216C22×C61—168no—
217C22×C61#41811yescharacteristic
218C22×C61#44711yescharacteristic
219C22×C61#45011yescharacteristic
220C4×C61—22no—
221C4×C61—42no—
222C4×C61#37911yescharacteristic
223C4×C61C2×#10211yescharacteristic
224C4×C61#44711yescharacteristic
225C4×C61#45011yescharacteristic
226Dic61—161no—
227C2×#576D6111yescharacteristic
228#2148—611no—
229C2×#361—611no—
230#2145—611no—
231#2140—611no—
232#2117—611no—
233#1814—611no—
234C2×C61—21no—
235C2×C61—21no—
236C2×C61—21no—
237C2×C61—42no—
238C2×C61—84no—
239C2×C61—82no—
240C2×C61#178611yescharacteristic
241C2×H(5,𝔽2)—21no—
242#576—22no—
243#576—22no—
244#576—22no—
245#576Dic6114yescomplemented
246C2×AGL(1,ℤ/8)—22no—
247C2×AGL(1,ℤ/8)—21no—
248C2×#135—21no—
249C2×#135C2×D6111yescharacteristic
250C22×D8—21no—
251C22×D8C2×D6111yescharacteristic
252C2×#131C2×D6111yescharacteristic
253C2×M32—611no—
254C2×M32—611no—
255C2×#108—611no—
256#427—611no—
257#421—1221no—
258#362—1221no—
259#361—1222no—
260#361—1221no—
261#360—1222no—
262#357—1222no—
263#350—1222no—
264#344—1221no—
265C61#1275011yescharacteristic, complemented, Sylow
266H(5,𝔽2)—24no—
267H(5,𝔽2)—24no—
268H(5,𝔽2)—42no—
269H(5,𝔽2)—42no—
270C2×#41—22no—
271C2×#41—21no—
272C2×#41—41no—
273C2×#41C22⋊D6111yescharacteristic
274C22×D4—22no—
275C22×D4—21no—
276C22×D4—42no—
277C22×D4—41no—
278AGL(1,ℤ/8)—24no—
279AGL(1,ℤ/8)—28no—
280AGL(1,ℤ/8)—28no—
281AGL(1,ℤ/8)—24no—
282#135—24no—
283#135—24no—
284#135—21no—
285#135—21no—
286#135C2×Dic6112yes—
287#135C22⋊D6112yes—
288C2×Q16C22⋊D6111yescharacteristic
289C2×SD16—22no—
290C2×SD16—21no—
291C2×D8—22no—
292C2×D8—24no—
293C2×D8—24no—
294C2×D8—21no—
295C2×D8—21no—
296C2×D8—21no—
297C2×D8C2×Dic6112yes—
298C2×D8C22⋊D6111yescharacteristic
299C2×D8C22⋊D6112yes—
300C2×D8C22⋊D6112yes—
301#131—21no—
302#131—21no—
303#131C2×Dic6112yes—
304#131C22⋊D6112yes—
305C2×M16—21no—
306C2×M16—1221no—
307C2×M16C22⋊D6111yescharacteristic
308C22×C8—21no—
309C22×C8C22×D6111yescharacteristic
310#117—1221no—
311M32—1221no—
312M32—1221no—
313M32—1221no—
314M32—1221no—
315C2×C16—1221no—
316C2×C16—1221no—
317#108—1221no—
318#108—1221no—
319#107—1221no—
320#106—1221no—
321#104—2442no—
322C24—42no—
323C24—41no—
324#41—21no—
325#41—21no—
326#41—28no—
327#41—22no—
328#41—22no—
329#41—44no—
330#41—44no—
331#41—44no—
332#41—41no—
333#41—41no—
334#41#1175819912yes—
335#41C22⋊Dic6112yes—
336C2×Q8—21no—
337C2×Q8#1175820011yescharacteristic
338C2×D4—21no—
339C2×D4—28no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—42no—
345C2×D4—44no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—44no—
349C2×D4—42no—
350C2×D4—41no—
351C2×D4—44no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4#1175820011yescharacteristic
356C22×C4—21no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4C2×C22⋊D6111yescharacteristic
360Q16—21no—
361Q16C22⋊Dic6112yes—
362SD16—28no—
363SD16—22no—
364SD16—22no—
365D8—28no—
366D8—22no—
367D8—21no—
368D8—22no—
369D8—22no—
370D8C22⋊Dic6112yes—
371D8C22⋊Dic6114yes—
372M16—21no—
373M16—22no—
374M16—22no—
375M16—2441no—
376M16—2441no—
377M16C22⋊Dic6112yes—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—2441no—
382C2×C8C22×Dic6111yescharacteristic
383C2×C8C2×C22⋊D6111yescharacteristic
384C2×C8C2×C22⋊D6111yescharacteristic
385C2×C8C2×C22⋊D6111yescharacteristic
386C2×C8C2×C22⋊D6111yescharacteristic
387C2×C8C2×C22⋊D6111yescharacteristic
388C4⋊C4—2441no—
389C22⋊C4—2441no—
390C42—2441no—
391C16—2441no—
392C16—2441no—
393C23—21no—
394C23—48no—
395C23—42no—
396C23—41no—
397C23—42no—
398C23—42no—
399C23—162no—
400C23—164no—
401C23#1451980311yescharacteristic
402Q8—21no—
403Q8—22no—
404Q8—41no—
405Q8#1451969812yes—
406D4—22no—
407D4—22no—
408D4—22no—
409D4—24no—
410D4—21no—
411D4—48no—
412D4—48no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—41no—
417D4—42no—
418D4—42no—
419D4#1451969812yes—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—48no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—4881no—
428C2×C4C2×C22⋊Dic6111yescharacteristic
429C2×C4#1451974611yescharacteristic
430C2×C4#1451974611yescharacteristic
431C2×C4C2×#1175819911yescharacteristic
432C2×C4C2×#1175820011yescharacteristic
433C2×C4#1451980311yescharacteristic
434C8—22no—
435C8—4881no—
436C8C2×C22⋊Dic6111yescharacteristic
437C8C2×C22⋊Dic6111yescharacteristic
438C8#1451974611yescharacteristic
439C8#1451974611yescharacteristic
440C22—22no—
441C22—41no—
442C22—42no—
443C22—44no—
444C22—42no—
445C22—84no—
446C22—82no—
447C22—162no—
448C22—162no—
449C22—168no—
450C22#1683976711yescharacteristic
451C22#1683989611yescharacteristic
452C22#1683991311yescharacteristic
453C4—22no—
454C4—42no—
455C4—9761no—
456C4#1683961811yescharacteristic
457C4C2×#1451969811yescharacteristic
458C4#1683989611yescharacteristic
459C4#1683991311yescharacteristic
460C2—21no—
461C2—21no—
462C2—21no—
463C2—42no—
464C2—84no—
465C2—82no—
466C2#2144575011yescharacteristic
subgroupG/H
centerC2#21445750
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
Fitting, prime radicalC2×C61×#576C2
socleC2×C61#1786
cosocleC22×C8×C61C23
hypercenterC2×#576D61
hypocenterC61#12750
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#576

Complements

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

NH ≅ G/Ntype
1. #21447568C2semidirect
2. C2×#16839575C2semidirect
3. #21447555C2semidirect
4. #21447535C2semidirect
5. #21447447C2semidirect
6. #21445873C2semidirect
8. C2×#14519717C22semidirect
9. C2×#14519717C22semidirect
10. C2×#14519713C22semidirect
11. #16839800C22semidirect
24. C61×#576C4semidirect
245. #576Dic61semidirect
265. C61#12750semidirect

Maximal subgroups

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

subgroupindex
#214475682
C2×#168395752
#214475552
#214475352
#214474472
#214458732
C2×C61×#5762
#1275061