Finite Group Databasebeta

← #29572366

Subgroups — #29572366

Order 15616.

820 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 107 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#21460224C211yesmaximal, characteristic
2#21444539C212yesmaximal, complemented
3#21444454C211yesmaximal, characteristic, complemented
4#21444326C212yesmaximal, complemented
5C61×#1963C211yesmaximal, characteristic, complemented
6#16840504—22no—
7#16840478C412yescomplemented
8#16840477—22no—
9D4×D244—22no—
10#16840397C412yescomplemented
11#16840328—24no—
12#16840326—22no—
13#16840312C2212yes—
14#16839819C2212yescomplemented
15#16839507—22no—
16#16839505—22no—
17#16839502—22no—
18#16839495—24no—
19C61×#532C2211yescharacteristic
20C61×#423C2211yescharacteristic, complemented
21C2×#104×C61—22no—
22C61×#376C2211yescharacteristic, complemented
23C2×#11758222—22no—
24C22×D244—22no—
25C2×D4×D61—24no—
26#14519777D412yes—
27#14519774D412yes—
28#14519773—24no—
29#14519766—24no—
30#14519765—22no—
31#14519763—28no—
32#14519752—24no—
33#14519751—24no—
34#14519751—22no—
35#14519751D414yes—
36#14519748—24no—
37#14519747—28no—
38#14519745—24no—
39#14519745—24no—
40C4×D244—22no—
41C4×D244C2×C414yes—
42#14519736—22no—
43#14519735—24no—
44C4⋊#92418—22no—
45C22⋊#92418—22no—
46#14519688—22no—
47C2×#41×C61D412yes—
48C2×M16×C61—21no—
49C4⋊Q8×C61C2×C411yescharacteristic
50#127×C61C2×C411yescharacteristic
51#124×C61D412yes—
52#121×C61—22no—
53C4×D4×C61—22no—
54C2×C42×C61—21no—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—22no—
57#104×C61—44no—
58#95×C61—22no—
59C23×D61—24no—
60#41×C61—42no—
61#41×C61—42no—
62C2×Q8×C61C22⋊C412yes—
63C2×D4×C61—22no—
64C2×D4×C61—22no—
65C2×D4×C61C22⋊C412yes—
66C22×C4×C61—22no—
67C22×C4×C61—41no—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D412yes—
70C2×C22⋊D61—28no—
71C2×C22⋊D61—24no—
72D4×D61—88no—
73#11758222—84no—
74C2×D244—22no—
75C2×D244—24no—
76C2×D244—22no—
77C2×D244—82no—
78C2×D244—82no—
79C2×D244C22⋊C414yes—
80C2×C4×D61—24no—
81C2×Dic122—22no—
82M16×C61—42no—
83C2×C8×C61—41no—
84C4⋊C4×C61—22no—
85C22⋊C4×C61—24no—
86C42×C61—22no—
87C42×C61—41no—
88C42×C61C22×C411yescharacteristic
89C42×C61C2×D411yescharacteristic
90C42×C61C2×D412yes—
91C22⋊Dic61—24no—
92#11758198—28no—
93#11758198—24no—
94#11758198—24no—
95C4⋊Dic61—22no—
96C4⋊Dic61C22⋊C414yes—
97C4×Dic61—24no—
98C2×#92418—42no—
99C23×C61—22no—
100C23×C61#12011yescharacteristic
101C22×D61—24no—
102C22×D61—84no—
103C22×D61—88no—
104Q8×C61—42no—
105D4×C61—42no—
106D4×C61—44no—
107D4×C61—82no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—22no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61—82no—
116C2×C4×C61C2×C22⋊C412yes—
117C2×C4×C61C2×C22⋊C411yescharacteristic
118C2×C4×C61#12012yes—
119C2×C4×C61#12011yescharacteristic
120C22⋊D61—84no—
121C22⋊D61—88no—
122C2×Dic61—24no—
123D244—44no—
124D244—42no—
125D244—42no—
126D244—82no—
127D244—84no—
128C4×D61—48no—
129Dic122—82no—
130C8×C61—81no—
131#92418—82no—
132#8491—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—22no—
135C22×C61—42no—
136C22×C61—44no—
137C22×C61#37911yescharacteristic
138C2×D61—44no—
139C2×D61—44no—
140C2×D61—44no—
141C2×D61—168no—
142C4×C61—21no—
143C4×C61—22no—
144C4×C61—22no—
145C4×C61—42no—
146C4×C61—82no—
147C4×C61#45311yescharacteristic
148C4×C61#45511yescharacteristic
149Dic61—44no—
150#3549—611no—
151#1963D6111yescharacteristic, complemented
152#1557—612no—
153#1536—611no—
154#1505—612no—
155C2×C61—41no—
156C2×C61—82no—
157C2×C61#179211yescharacteristic
158C2×C61#179512yes—
159D61—84no—
160H(3,𝔽4)—1222no—
161#555—612no—
162#555—1222no—
163D42—612no—
164D42—1222no—
165#535—1224no—
166#535—1222no—
167#532—612no—
168#532C2×D6111yescharacteristic
169#432—612no—
170#423C2×D6111yescharacteristic, complemented
171C2×#104—22no—
172#376C2×D6111yescharacteristic, complemented
173#332—1222no—
174#331—1222no—
175#329—1222no—
176#327—1224no—
177C61#849111yescharacteristic, complemented, Sylow
178C2×#41—1222no—
179C2×#41C22⋊D6112yes—
180C22×D4—1222no—
181C22×D4—1224no—
182C2×M16—21no—
183C4⋊Q8—612no—
184C4⋊Q8C4×D6111yescharacteristic
185#127—612no—
186#127—1224no—
187#127C4×D6111yescharacteristic
188#124—1228no—
189#124—1224no—
190#124—1222no—
191#124D24412yes—
192#121—22no—
193#121—614no—
194#121—1224no—
195#121—1224no—
196#121—1224no—
197#121—1222no—
198#120—1228no—
199#120—1224no—
200#120—1224no—
201C4×D4—22no—
202C4×D4—614no—
203C4×D4—1222no—
204#117—1224no—
205#117—1222no—
206C2×C42—21no—
207C2×C42C22×D6111yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—1222no—
210#104—44no—
211C22⋊C8—1222no—
212#97—1222no—
213#95—22no—
214C24—1224no—
215#41—42no—
216#41—42no—
217#41—4884no—
218C2×Q8—1222no—
219C2×Q8#1175819812yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—614no—
223C2×D4—1224no—
224C2×D4—1228no—
225C2×D4—1224no—
226C2×D4—1222no—
227C2×D4—1222no—
228C2×D4—4882no—
229C2×D4—4882no—
230C2×D4—4888no—
231C2×D4#1175819812yes—
232C22×C4—22no—
233C22×C4—41no—
234C22×C4—1224no—
235C22×C4D4×D6112yes—
236C22×C4C2×C22⋊D6111yescharacteristic
237M16—42no—
238C2×C8—41no—
239C2×C8—2442no—
240C4⋊C4—22no—
241C4⋊C4—614no—
242C4⋊C4—1222no—
243C22⋊C4—24no—
244C22⋊C4—1228no—
245C22⋊C4—1224no—
246C22⋊C4—1224no—
247C22⋊C4—1224no—
248C42—22no—
249C42—41no—
250C42—1224no—
251C42C2×C4×D6111yescharacteristic
252C42C2×D24411yescharacteristic
253C42D4×D6112yes—
254C23—22no—
255C23—1224no—
256C23—4884no—
257C23—4888no—
258C23#1451974511yescharacteristic
259Q8—42no—
260Q8—4882no—
261D4—42no—
262D4—44no—
263D4—82no—
264D4—4882no—
265D4—4884no—
266D4—4884no—
267D4—4888no—
268D4—2444no—
269D4—2442no—
270D4—2442no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—1224no—
280C2×C4—2448no—
281C2×C4C22⋊C4×D6112yes—
282C2×C4C2×#1175819811yescharacteristic
283C2×C4#1451974511yescharacteristic
284C2×C4#1451974712yes—
285C8—81no—
286C8—4882no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—44no—
291C22—9768no—
292C22—2444no—
293C22—2444no—
294C22—2444no—
295C22#1683961711yescharacteristic
296C4—21no—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—82no—
301C4—2444no—
302C4#1683992911yescharacteristic
303C4#1683993911yescharacteristic
304C2—41no—
305C2—82no—
306C2—4884no—
307C2#2144577211yescharacteristic
308C2#2144578812yes—
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1963C2
socleC22×C61#379
prime radical#21460224C2
cosocleC2×C42×C61C23
hypercenter#1963D61
hypocenterC61#8491
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
2. #21444539C2semidirect
3. #21444454C2semidirect
4. #21444326C2semidirect
5. C61×#1963C2semidirect
7. #16840478C4semidirect
10. #16840397C4semidirect
14. #16839819C22semidirect
20. C61×#423C22semidirect
22. C61×#376C22semidirect
151. #1963D61semidirect
170. #423C2×D61semidirect
172. #376C2×D61semidirect
177. C61#8491semidirect

Maximal subgroups

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

subgroupindex
#214602242
#214445392
#214444542
#214443262
C61×#19632
#849161