Finite Group Databasebeta

← #29665252

Subgroups — #29665252

Order 15616.

728 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456937C211yesmaximal, characteristic, complemented
2#21455279C211yesmaximal, characteristic
3C2×#16839968C211yesmaximal, characteristic, complemented
4#21454842C211yesmaximal, characteristic, complemented
5#21445255C214yesmaximal, complemented
6#21445207C214yesmaximal, complemented
7#21444593C211yesmaximal, characteristic, complemented
8#21444490C211yesmaximal, characteristic, complemented
9C61×#1540C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C22×#11758190—21no—
12C2×#14519777C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840165C2212yescomplemented
15#16840108C2212yescomplemented
16#16840073—22no—
17#16840042—22no—
18#16839974—22no—
19#16839968—22no—
20#16839968C2214yes—
21#16839964C2212yes—
22#16839954C2212yes—
23C8×D244—22no—
24#16839827C2214yes—
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839763—21no—
28C2×#14519702—21no—
29C22⋊C8×D61—22no—
30#16839514C2214yescomplemented
31#16839507C2214yescomplemented
32C61×#532C2211yescharacteristic
33C61×#431C2211yescharacteristic
34C2×C4⋊C8×C61C2211yescharacteristic
35C61×#332C2212yescomplemented
36C61×#331C2212yescomplemented
37C22×Dic122—21no—
38C22×D244D411yescharacteristic
39C22×C4×D61—21no—
40#14519802—44no—
41C2×#11758190—24no—
42C2×#11758190—44no—
43C2×#11758201—22no—
44C2×#11758199—22no—
45C2×#11758194—21no—
46C2×C8×D61—24no—
47#14519777—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777C2312yes—
51C4×D244—21no—
52C4×D244—21no—
53C4×D244D412yes—
54C4×D244C2312yes—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61D411yescharacteristic
58C2×#11758198—21no—
59#14519710—24no—
60C4⋊#92418—21no—
61C4⋊#92418D412yes—
62C4⋊#92418C2312yes—
63#14519702—24no—
64#14519701—22no—
65#14519698—22no—
66#14519698—24no—
67C22⋊#92418D412yes—
68#14519689—24no—
69C4×#92418C2312yes—
70C2×#41×C61—21no—
71C22×C8×C61—21no—
72C4⋊Q8×C61C2311yescharacteristic
73#127×C61C2311yescharacteristic
74#124×C61—21no—
75#121×C61—21no—
76#121×C61—21no—
77C4×D4×C61—21no—
78C4×D4×C61—21no—
79C2×C42×C61C2311yescharacteristic
80C4⋊C8×C61—21no—
81C4⋊C8×C61C2312yes—
82C4⋊C8×C61C2312yes—
83C22⋊C8×C61—21no—
84C4×C8×C61—21no—
85C23×D61—21no—
86#41×C61—42no—
87#41×C61—42no—
88C2×Q8×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C22×Dic61—21no—
98C22×Dic61—41no—
99C2×D244—21no—
100C2×D244—22no—
101C2×D244D814yes—
102C2×D244C2×D412yes—
103C2×C4×D61—22no—
104C2×C4×D61—22no—
105C2×C4×D61—82no—
106C2×Dic122—22no—
107C2×Dic122—42no—
108C2×Dic122—42no—
109C2×Dic122—41no—
110C2×C8×C61—22no—
111C2×C8×C61—22no—
112C2×C8×C61—42no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61#4111yescharacteristic
120C42×C61C2411yescharacteristic
121#11758201—44no—
122#11758199—44no—
123#11758198—22no—
124#11758198—22no—
125C4⋊Dic61—21no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130C4⋊Dic61C2×D412yes—
131#11758194—42no—
132C2×#92418—22no—
133C2×#92418C2×D414yes—
134#11758190—48no—
135C8×D61—84no—
136C23×C61—41no—
137C23×C61#12111yescharacteristic
138C22×D61—22no—
139C22×D61—24no—
140C22×D61—81no—
141C22×D61—81no—
142Q8×C61—41no—
143Q8×C61—41no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—42no—
147D4×C61—81no—
148D4×C61—82no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—82no—
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61C2×D812yes—
157C2×C4×C61#13512yes—
158C2×C4×C61C22×D411yescharacteristic
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C2×#4111yescharacteristic
161C2×Dic61—22no—
162C2×Dic61—42no—
163C2×Dic61—81no—
164C2×Dic61—82no—
165D244—22no—
166D244—24no—
167D244C2×D814yes—
168C4×D61—44no—
169C4×D61—82no—
170Dic122—42no—
171Dic122—42no—
172Dic122—82no—
173C8×C61—42no—
174C8×C61—42no—
175#92418—24no—
176#92418—42no—
177#19392—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61#46612yes—
184C22×C61C2×#12111yescharacteristic
185C2×D61—24no—
186C2×D61—42no—
187C2×D61—42no—
188C2×D61—84no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—81no—
193C4×C61#46612yes—
194C4×C61C22×D811yescharacteristic
195C4×C61C2×#13511yescharacteristic
196C4×C61#57712yes—
197Dic61—42no—
198Dic61—82no—
199#3226—611no—
200#3036—611no—
201C2×#461—611no—
202#2972—611no—
203#1681—614no—
204#1673—614no—
205#1571—611no—
206#1546—611no—
207#1540D6111yescharacteristic, complemented
208C2×C61—22no—
209C2×C61—161no—
210C2×C61#304211yescharacteristic
211C2×C61C2×#46611yescharacteristic
212C2×C61#306911yescharacteristic
213D61—44no—
214C2×AGL(1,ℤ/8)—1221no—
215C22×SD16—1221no—
216#532C2×D6111yescharacteristic
217C2×C4⋊Q8—611no—
218C2×C4×D4—611no—
219#500—612no—
220#486—612no—
221#480—1222no—
222#474—1222no—
223#463—1222no—
224#461—612no—
225#461—614no—
226#461—1222no—
227#459—612no—
228C8×D4—614no—
229C8×D4—1222no—
230#431—611no—
231#431C2×D6111yescharacteristic
232#423—611no—
233C2×C4⋊C8C2×D6111yescharacteristic
234#417—1221no—
235C2×#103—1221no—
236C2×C22⋊C8—1222no—
237#335—614no—
238#332—614no—
239#332C2×D6112yescomplemented
240#331C2×D6112yescomplemented
241C61#1939211yescharacteristic, complemented, Sylow
242C2×#41—21no—
243C22×Q8—1221no—
244C22×D4—611no—
245C23×C4—1221no—
246AGL(1,ℤ/8)—2444no—
247C2×SD16—1224no—
248C2×SD16—1222no—
249C2×SD16—2444no—
250C2×D8—1222no—
251C2×M16—1221no—
252C22×C8—21no—
253C22×C8—1224no—
254C4⋊Q8—612no—
255C4⋊Q8—1221no—
256C4⋊Q8—1221no—
257C4⋊Q8—1221no—
258C4⋊Q8C22×D6111yescharacteristic
259#127C22×D6111yescharacteristic
260#124—21no—
261#121—21no—
262#121—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—612no—
266C4×D4—612no—
267C4×D4—1221no—
268C4×D4—1221no—
269C2×C4⋊C4—611no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C42C22×D6111yescharacteristic
274#107—1224no—
275C4⋊C8—21no—
276C4⋊C8—612no—
277C4⋊C8—612no—
278C4⋊C8—1221no—
279C4⋊C8C22×D6112yes—
280C4⋊C8C22×D6112yes—
281#103—1224no—
282#103—1222no—
283#102—1222no—
284#102—1224no—
285C22⋊C8—21no—
286C22⋊C8—612no—
287C22⋊C8—1224no—
288C4×C8—21no—
289C4×C8—612no—
290C24—1221no—
291#41—42no—
292#41—42no—
293C2×Q8—21no—
294C2×Q8—1222no—
295C2×Q8—2442no—
296C2×Q8—2442no—
297C2×Q8—2441no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—612no—
303C2×D4—614no—
304C2×D4—1221no—
305C2×D4—1222no—
306C22×C4—41no—
307C22×C4—1221no—
308C22×C4—1222no—
309C22×C4—1222no—
310C22×C4—4882no—
311C22×C4—2441no—
312C22×C4#1175822211yescharacteristic
313C22×C4D4×D6111yescharacteristic
314C22×C4D4×D6111yescharacteristic
315SD16—2448no—
316SD16—2444no—
317D8—2444no—
318M16—2442no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—42no—
322C2×C8—614no—
323C2×C8—1222no—
324C2×C8—4884no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—612no—
328C4⋊C4—1221no—
329C4⋊C4—1222no—
330C4⋊C4—1222no—
331C4⋊C4—2441no—
332C4⋊C4—2441no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—1222no—
336C22⋊C4—1222no—
337C42#1175822211yescharacteristic
338C42D4×D6111yescharacteristic
339C42D4×D6111yescharacteristic
340C42C23×D6111yescharacteristic
341C23—41no—
342C23—1224no—
343C23—1222no—
344C23—4881no—
345C23—4881no—
346C23#1451975011yescharacteristic
347Q8—41no—
348Q8—41no—
349Q8—4882no—
350Q8—2442no—
351Q8—2442no—
352D4—41no—
353D4—41no—
354D4—42no—
355D4—81no—
356D4—82no—
357D4—614no—
358D4—1224no—
359D4—1222no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—82no—
364C2×C4—1222no—
365C2×C4—4882no—
366C2×C4—4882no—
367C2×C4—4881no—
368C2×C4—2442no—
369C2×C4—2444no—
370C2×C4#1451975011yescharacteristic
371C2×C4#1451975011yescharacteristic
372C2×C4#1451975011yescharacteristic
373C2×C4D8×D6112yes—
374C2×C4#1451979812yes—
375C2×C4C2×D4×D6111yescharacteristic
376C2×C4C2×D4×D6111yescharacteristic
377C2×C4C2×#1175822211yescharacteristic
378C8—42no—
379C8—42no—
380C8—1224no—
381C8—2442no—
382C22—21no—
383C22—21no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—1224no—
388C22—4884no—
389C22—2442no—
390C22—2442no—
391C22#1683999712yes—
392C22C2×#1451975011yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—4882no—
398C4—2442no—
399C4#1683999712yes—
400C4C2×D8×D6111yescharacteristic
401C4C2×#1451979811yescharacteristic
402C4#1684056912yes—
403C2—22no—
404C2—161no—
405C2—2444no—
406C2#2145533411yescharacteristic
407C2C2×#1683999711yescharacteristic
408C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1540C2
socleC22×C61C2×#121
prime radical#21455279C2
hypercenter#1540D61
hypocenterC61#19392
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
1. #21456937C2semidirect
3. C2×#16839968C2semidirect
4. #21454842C2semidirect
5. #21445255C2semidirect
6. #21445207C2semidirect
7. #21444593C2semidirect
8. #21444490C2semidirect
9. C61×#1540C2semidirect
14. #16840165C22semidirect
15. #16840108C22semidirect
30. #16839514C22semidirect
31. #16839507C22semidirect
35. C61×#332C22semidirect
36. C61×#331C22semidirect
207. #1540D61semidirect
239. #332C2×D61semidirect
240. #331C2×D61semidirect
241. C61#19392semidirect

Maximal subgroups

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

subgroupindex
#214569372
#214552792
C2×#168399682
#214548422
#214452552
#214452072
#214445932
#214444902
C61×#15402
#1939261