Finite Group Databasebeta

← #29583102

Subgroups — #29583102

Order 15616.

454 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 97 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#21454197C211yesmaximal, characteristic, complemented
2#21446647C211yesmaximal, characteristic
3#21444380C212yesmaximal
4#21443678C212yesmaximal, complemented
5C61×#1276C211yesmaximal, characteristic, complemented
6#16840312C2211yescharacteristic
7C2×#14519777—21no—
8#16839786C2211yescharacteristic
9C2×#14519705—21no—
10C2×#14519705—21no—
11#16839668—24no—
12#16839628—21no—
13#16839628C2212yes—
14#16839497—24no—
15C61×#431C2212yes—
16C43×C61C2211yescharacteristic
17C2×#11758222—21no—
18C22×Dic122—22no—
19C2×#11758194—21no—
20#14519777—42no—
21#14519777—42no—
22#14519777Q811yescharacteristic, complemented
23#14519774Q811yescharacteristic, complemented
24#14519765—21no—
25#14519751—22no—
26C4×D244—22no—
27C2×C4⋊Dic61—22no—
28C2×C4⋊Dic61—22no—
29C4⋊#92418—22no—
30#14519705—42no—
31#14519705—42no—
32#14519684—24no—
33#14519684—22no—
34#14519684D414yes—
35C2×C42×C61—22no—
36C2×C42×C61D411yescharacteristic
37C2×C42×C61D411yescharacteristic
38C2×C42×C61C2311yescharacteristic
39C4⋊C8×C61—22no—
40C22⋊C8×C61—22no—
41C4×C8×C61—22no—
42C22×C4×C61—21no—
43C22×C4×C61—21no—
44C22×C4×C61C2×D412yes—
45C22×C4×C61C2×D411yescharacteristic
46C2×C22⋊D61—41no—
47C22×Dic61—24no—
48#11758222—41no—
49#11758222—41no—
50#11758222—41no—
51#11758222—41no—
52C2×D244—21no—
53C2×D244—22no—
54C2×C4×D61—41no—
55C2×Dic122—21no—
56C2×Dic122—48no—
57C2×Dic122—44no—
58C2×C8×C61—42no—
59C42×C61—21no—
60C42×C61—21no—
61C42×C61—44no—
62C42×C61D812yes—
63C42×C61SD1612yes—
64C42×C61C2×Q811yescharacteristic
65C42×C61#4112yes—
66C42×C61#4111yescharacteristic
67#11758198—42no—
68C4⋊Dic61—22no—
69C4⋊Dic61—44no—
70C4⋊Dic61—42no—
71C4⋊Dic61—42no—
72#11758194—41no—
73#11758194—41no—
74C2×#92418—41no—
75C23×C61#12011yescharacteristic
76C22×D61—41no—
77C2×C4×C61—24no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—42no—
81C2×C4×C61—42no—
82C2×C4×C61C22⋊Q812yes—
83C2×C4×C61C22⋊Q811yescharacteristic
84C2×C4×C61#12412yes—
85C2×C4×C61#12411yescharacteristic
86C2×C4×C61C2×D811yescharacteristic
87C2×C4×C61C2×SD1611yescharacteristic
88C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C61#13711yescharacteristic
90C22⋊D61—81no—
91C22⋊D61—81no—
92C2×Dic61—48no—
93C2×Dic61—41no—
94C2×Dic61—44no—
95D244—41no—
96D244—41no—
97D244—82no—
98C4×D61—81no—
99C4×D61—81no—
100Dic122—41no—
101Dic122—41no—
102Dic122—88no—
103C8×C61—82no—
104#92418—81no—
105#10362—611nomaximal, Sylow
106C22×C61—22no—
107C22×C61#39311yescharacteristic
108C22×C61#44711yescharacteristic
109C22×C61#44811yescharacteristic
110C2×D61—81no—
111C2×D61—82no—
112C4×C61—24no—
113C4×C61—22no—
114C4×C61—22no—
115C4×C61—42no—
116C4×C61#47411yescharacteristic
117C4×C61#47611yescharacteristic
118C4×C61#48611yescharacteristic
119C4×C61#48911yescharacteristic
120Dic61—84no—
121Dic61—81no—
122#2862—611no—
123#1962—611no—
124#1517—612no—
125#1286—612no—
126#1276D6111yescharacteristic, complemented
127C2×C61—21no—
128C2×C61—21no—
129C2×C61#202311yescharacteristic
130C2×C61#203112yes—
131D61—161no—
132#532—611no—
133C2×C4⋊Q8—1221no—
134#431C2×D6112yes—
135#423—611no—
136C2×#104—1221no—
137C2×#104—1221no—
138#391—1224no—
139#382—612no—
140#382—1221no—
141C43C2×D6111yescharacteristic
142#327—1224no—
143C61#1036211yescharacteristic, complemented, Sylow
144C2×#41—1221no—
145C22×Q8—1222no—
146C2×M16—1221no—
147C4⋊Q8—611no—
148C4⋊Q8—2442no—
149C4⋊Q8—2442no—
150#127—611no—
151#124—1221no—
152#121—1222no—
153C4×D4—1222no—
154C2×C4⋊C4—1222no—
155C2×C4⋊C4—1222no—
156C2×C42—22no—
157C2×C42D24411yescharacteristic
158C2×C42D24411yescharacteristic
159C2×C42C22×D6111yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—1222no—
162#104—2442no—
163#104—2442no—
164C22⋊C8—22no—
165C4×C8—22no—
166#95—614no—
167#95—1224no—
168#95—1222no—
169#41—2441no—
170#41—2441no—
171#41—2441no—
172#41—2441no—
173C2×Q8—1221no—
174C2×Q8—2444no—
175C2×Q8—2448no—
176C2×D4—1222no—
177C2×D4—1221no—
178C2×D4—2441no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—1224no—
182C22×C4—2441no—
183C22×C4C2×D24411yescharacteristic
184C22×C4D4×D6112yes—
185M16—2441no—
186M16—2441no—
187C2×C8—42no—
188C2×C8—2441no—
189C4⋊C4—1222no—
190C4⋊C4—2444no—
191C4⋊C4—2442no—
192C4⋊C4—2442no—
193C22⋊C4—2442no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42#1175819012yes—
198C42D48812yes—
199C42#1175822212yes—
200C42#1175822411yescharacteristic
201C42Q8×D6111yescharacteristic
202C23—2441no—
203C23#1451974711yescharacteristic
204Q8—4888no—
205Q8—2441no—
206Q8—2441no—
207D4—4882no—
208D4—4881no—
209D4—4881no—
210D4—2441no—
211D4—2441no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—4881no—
218C2×C4—4881no—
219C2×C4—2448no—
220C2×C4—2444no—
221C2×C4—2441no—
222C2×C4#1451975512yes—
223C2×C4#1451975711yescharacteristic
224C2×C4#1451976312yes—
225C2×C4#1451976511yescharacteristic
226C2×C4C2×D48811yescharacteristic
227C2×C4C2×#1175819011yescharacteristic
228C2×C4#1451980511yescharacteristic
229C2×C4#1451981011yescharacteristic
230C8—82no—
231C8—4881no—
232C22—22no—
233C22—4881no—
234C22—4882no—
235C22#1683967511yescharacteristic
236C22#1683989811yescharacteristic
237C22#1683990311yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—4884no—
243C4—4881no—
244C4#1684004311yescharacteristic
245C4#1684005311yescharacteristic
246C4#1684010911yescharacteristic
247C4#1684012211yescharacteristic
248C2—21no—
249C2—21no—
250C2—9761no—
251C2#2144695611yescharacteristic
252C2#2144699212yes—
subgroupG/H
centerC22#16839675
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1276C2
socleC22×C61#393
prime radical#16840312C22
hypercenter#1276D61
hypocenterC61#10362
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #21454197C2semidirect
4. #21443678C2semidirect
5. C61×#1276C2semidirect
22. #14519777Q8semidirect
23. #14519774Q8semidirect
126. #1276D61semidirect
143. C61#10362semidirect

Maximal subgroups

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

subgroupindex
#214541972
#214466472
#214443802
#214436782
C61×#12762
#1036261