Finite Group Databasebeta

← #29661337

Subgroups — #29661337

Order 15616.

938 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16840510C211yesmaximal, characteristic
2#21456818C212yesmaximal, complemented
3#21445134C218yesmaximal, complemented
4#21444446C211yesmaximal, characteristic, complemented
5#21444416C212yesmaximal, complemented
6C61×#3166C211yesmaximal, characteristic, complemented
7#16840510—24no—
8#16840510—22no—
9#16840510C2214yes—
10C2×#14519765C2212yes—
11C2×#14519762—22no—
12C2×#14519755—22no—
13C2×#14519752—22no—
14#16840154C2214yescomplemented
15#16840124C2214yescomplemented
16#16839984—24no—
17#16839917—22no—
18#16839912—22no—
19#16839819C2212yes—
20#16839504C2214yescomplemented
21#16839501C2218yescomplemented
22C61×#533C2211yescharacteristic
23C2×#125×C61C2211yescharacteristic
24C61×#501C2214yescomplemented
25C61×#499—22no—
26C61×#428—22no—
27C4×M16×C61C2211yescharacteristic
28C22×Dic122D412yes—
29C22×D244—22no—
30C22×D244D412yes—
31C22×C4×D61—22no—
32C2×#11758202—24no—
33C2×#11758201—24no—
34#14519765—22no—
35#14519765—22no—
36#14519765D414yes—
37#14519765C2314yes—
38#14519762—28no—
39#14519762—28no—
40#14519755—28no—
41#14519755—28no—
42#14519752—28no—
43#14519752—28no—
44C2×#11758196—22no—
45C2×#11758198—22no—
46C2×#11758198—22no—
47C2×#11758198—22no—
48C4⋊#92418—22no—
49#14519700—24no—
50#14519697—24no—
51C22⋊#92418—22no—
52#14519688C2314yes—
53C2×M16×C61—21no—
54C4⋊Q8×C61C2312yes—
55#125×C61—22no—
56#125×C61—21no—
57#125×C61D412yes—
58#125×C61C2312yes—
59C22⋊Q8×C61—22no—
60C4×Q8×C61—22no—
61#117×C61—21no—
62C2×C4⋊C4×C61—22no—
63C2×C4⋊C4×C61D412yes—
64C2×C42×C61C2311yescharacteristic
65#107×C61—24no—
66#106×C61—24no—
67#97×C61C2312yes—
68C4×C8×C61—21no—
69C23×D61—22no—
70C2×Q8×C61—22no—
71C22×C4×C61—22no—
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D412yes—
74C22×Dic61—22no—
75C2×D244—28no—
76C2×D244—22no—
77C2×D244—82no—
78C2×D244—84no—
79C2×D244C2×D414yes—
80C2×C4×D61—28no—
81C2×C4×D61—84no—
82C2×Dic122—22no—
83C2×Dic122—82no—
84C2×Dic122C2×D414yes—
85M16×C61—24no—
86C2×C8×C61—22no—
87C4⋊C4×C61—24no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—24no—
90C4⋊C4×C61—22no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—42no—
93C4⋊C4×C61C2×D414yes—
94C22⋊C4×C61—41no—
95C42×C61—41no—
96C42×C61C2×D412yes—
97C42×C61C2×D411yescharacteristic
98C42×C61C2411yescharacteristic
99#11758202—84no—
100#11758201—84no—
101#11758198—28no—
102#11758198—24no—
103#11758198—24no—
104#11758198—24no—
105#11758198—24no—
106#11758196—28no—
107C2×#92418—24no—
108C23×C61#12011yescharacteristic
109C22×D61—24no—
110C22×D61—44no—
111C22×D61—44no—
112C22×D61—44no—
113Q8×C61—44no—
114C2×C4×C61—22no—
115C2×C4×C61—24no—
116C2×C4×C61—42no—
117C2×C4×C61—44no—
118C2×C4×C61—42no—
119C2×C4×C61#12011yescharacteristic
120C2×C4×C61#12012yes—
121C2×C4×C61C22×D412yes—
122C2×C4×C61C22×D411yescharacteristic
123C2×Dic61—24no—
124C2×Dic61—44no—
125D244—44no—
126D244—82no—
127D244—88no—
128C4×D61—88no—
129Dic122—44no—
130Dic122—82no—
131C8×C61—24no—
132#92418—44no—
133#19006—611nomaximal, Sylow
134C22×C61—22no—
135C22×C61#45612yes—
136C22×C61C2×#12011yescharacteristic
137C2×D61—44no—
138C2×D61—44no—
139C2×D61—44no—
140C2×D61—88no—
141C4×C61—21no—
142C4×C61—21no—
143C4×C61—22no—
144C4×C61—22no—
145C4×C61—44no—
146C4×C61—82no—
147Dic61—44no—
148C2×#562—611no—
149#3215—612no—
150#3166D6111yescharacteristic, complemented
151#1661—618no—
152#1534—611no—
153#1527—612no—
154C2×C61—22no—
155C2×C61C2×#45611yescharacteristic
156C2×C61#301912yes—
157D61—84no—
158#562—614no—
159#562—1224no—
160#562—1222no—
161#533C2×D6111yescharacteristic
162C2×#125C2×D6111yescharacteristic
163C2×#124—612no—
164C2×#123—1222no—
165C2×C22⋊Q8—1222no—
166C2×#121—1222no—
167#501C2×D6114yescomplemented
168#499—22no—
169#497—614no—
170#489—614no—
171#464—1224no—
172#451—1222no—
173#450—1222no—
174#432—612no—
175#428—22no—
176C4×M16C2×D6111yescharacteristic
177#330—614no—
178#329—618no—
179C61#1900611yescharacteristic, complemented, Sylow
180C22×Q8—612no—
181C22×D4—612no—
182C22×D4—1222no—
183C23×C4—1222no—
184C2×Q16—1224no—
185C2×SD16—1224no—
186C2×M16—21no—
187C4⋊Q8C22×D6112yes—
188#125—22no—
189#125—21no—
190#125C22⋊D6112yes—
191#125C22×D6112yes—
192#124—614no—
193#124—614no—
194#124—1222no—
195#124—1222no—
196#123—1228no—
197#123—1228no—
198C22⋊Q8—22no—
199C22⋊Q8—1228no—
200C22⋊Q8—1228no—
201#121—1228no—
202#121—1228no—
203C4×Q8—22no—
204#117—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—1222no—
207C2×C4⋊C4C22⋊D6112yes—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1222no—
210C2×C22⋊C4—1222no—
211C2×C42C22×D6111yescharacteristic
212#107—24no—
213#106—24no—
214C4⋊C8—1222no—
215#103—1224no—
216#102—1224no—
217C22⋊C8—1222no—
218#97—614no—
219#97C22×D6112yes—
220C4×C8—21no—
221C24—1222no—
222C2×Q8—22no—
223C2×Q8—614no—
224C2×Q8—1222no—
225C2×Q8—4882no—
226C2×D4—614no—
227C2×D4—1228no—
228C2×D4—1222no—
229C2×D4—4884no—
230C2×D4—4882no—
231C22×C4—22no—
232C22×C4—1228no—
233C22×C4—1222no—
234C22×C4—4884no—
235C22×C4D4×D6112yes—
236C22×C4C2×C22⋊D6111yescharacteristic
237Q16—4884no—
238SD16—4884no—
239M16—24no—
240C2×C8—22no—
241C2×C8—1224no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—1228no—
249C4⋊C4C2×C22⋊D6114yes—
250C22⋊C4—41no—
251C22⋊C4—1228no—
252C22⋊C4—1224no—
253C22⋊C4—1224no—
254C22⋊C4—1224no—
255C22⋊C4—1224no—
256C42—41no—
257C42D4×D6112yes—
258C42C2×C22⋊D6111yescharacteristic
259C42C23×D6111yescharacteristic
260C23—1224no—
261C23—2444no—
262C23—2444no—
263C23—2444no—
264C23#1451974511yescharacteristic
265Q8—44no—
266Q8—4882no—
267Q8—2444no—
268D4—4882no—
269D4—4888no—
270D4—2444no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—1224no—
277C2×C4—4888no—
278C2×C4—2444no—
279C2×C4#1451974511yescharacteristic
280C2×C4#1451974512yes—
281C2×C4C2×D4×D6112yes—
282C2×C4C22×C22⋊D6111yescharacteristic
283C8—24no—
284C8—2444no—
285C22—22no—
286C22—4888no—
287C22—2444no—
288C22—2444no—
289C22—2444no—
290C22#1683994412yes—
291C22C2×#1451974511yescharacteristic
292C4—21no—
293C4—21no—
294C4—22no—
295C4—22no—
296C4—44no—
297C4—82no—
298C4—2444no—
299C2—22no—
300C2—4884no—
301C2C2×#1683994411yescharacteristic
302C2#2145517412yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3166C2
socleC22×C61C2×#120
prime radicalC2×#16840510C2
hypercenter#3166D61
hypocenterC61#19006
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3166

Complements

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

NH ≅ G/Ntype
2. #21456818C2semidirect
3. #21445134C2semidirect
4. #21444446C2semidirect
5. #21444416C2semidirect
6. C61×#3166C2semidirect
14. #16840154C22semidirect
15. #16840124C22semidirect
20. #16839504C22semidirect
21. #16839501C22semidirect
24. C61×#501C22semidirect
150. #3166D61semidirect
167. #501C2×D61semidirect
179. C61#19006semidirect

Maximal subgroups

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

subgroupindex
C2×#168405102
#214568182
#214451342
#214444462
#214444162
C61×#31662
#1900661