Finite Group Databasebeta

← #29579401

Subgroups — #29579401

Order 15616.

408 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 115 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#21456142C211yesmaximal, characteristic
2#21445983C211yesmaximal, characteristic, complemented
3#21444384C212yesmaximal, complemented
4#21444380C212yesmaximal
5C61×#1452C211yesmaximal, characteristic, complemented
6#16840319C2211yescharacteristic
7#16840312C2211yescharacteristic
8#16840117—22no—
9#16840113C412yescomplemented
10#16840109C412yescomplemented
11C2×#14519711—21no—
12C2×#14519709—21no—
13#16839759—21no—
14#16839759—21no—
15#16839628C2212yes—
16#16839500—24no—
17#16839497—24no—
18C61×#431C2212yes—
19C2×C4×C8×C61C2211yescharacteristic
20C61×#322—22no—
21C2×#11758222—21no—
22#14519777C2×C411yescharacteristic
23#14519777C2×C412yes—
24#14519774C2×C411yescharacteristic
25#14519769—21no—
26#14519765—21no—
27#14519756—22no—
28#14519751—22no—
29C4×Dic122—22no—
30C4×D244—22no—
31#14519736—21no—
32C2×C4⋊Dic61—22no—
33#14519711—42no—
34#14519709—42no—
35#14519702—22no—
36#14519702—22no—
37#14519699—22no—
38#14519699—22no—
39#14519684—22no—
40#14519684—22no—
41C22×C8×C61D411yescharacteristic
42C22×C8×C61D411yescharacteristic
43C2×C42×C61C2311yescharacteristic
44C4⋊C8×C61—22no—
45C22⋊C8×C61D414yes—
46C4×C8×C61—22no—
47C4×C8×C61C2×C412yes—
48C4×C8×C61D412yes—
49C22×C4×C61C2×D412yes—
50C22×C4×C61C2×D411yescharacteristic
51C2×C22⋊D61—41no—
52C22×Dic61—42no—
53#11758222—41no—
54#11758222—41no—
55#11758222—41no—
56#11758222—41no—
57C2×D244—21no—
58C2×D244—22no—
59C2×C4×D61—41no—
60C2×Dic122—21no—
61C2×Dic122—22no—
62C2×C8×C61—21no—
63C2×C8×C61—21no—
64C2×C8×C61—21no—
65C2×C8×C61—21no—
66C2×C8×C61—24no—
67C2×C8×C61C22⋊C412yes—
68C2×C8×C61C22⋊C412yes—
69C42×C61C22×C411yescharacteristic
70C42×C61C2×D411yescharacteristic
71C42×C61#4112yes—
72C22⋊Dic61—41no—
73#11758198—42no—
74C4⋊Dic61—22no—
75C4⋊Dic61—22no—
76C4⋊Dic61—22no—
77C4⋊Dic61—42no—
78C4⋊Dic61—42no—
79#11758196—42no—
80C4×Dic61—41no—
81C23×C61#12711yescharacteristic
82C22×D61—41no—
83C2×C4×C61—42no—
84C2×C4×C61C2×C22⋊C411yescharacteristic
85C2×C4×C61C4×D412yes—
86C2×C4×C61#12112yes—
87C2×C4×C61#12411yescharacteristic
88C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
90C2×C4×C61#13711yescharacteristic
91C2×C4×C61#13711yescharacteristic
92C22⋊D61—81no—
93C22⋊D61—81no—
94C2×Dic61—42no—
95C2×Dic61—41no—
96C2×Dic61—84no—
97C2×Dic61—82no—
98D244—41no—
99D244—41no—
100D244—82no—
101C4×D61—81no—
102C4×D61—81no—
103Dic122—41no—
104Dic122—41no—
105Dic122—82no—
106C8×C61—22no—
107C8×C61—22no—
108C8×C61—44no—
109#9758—611nomaximal, Sylow
110C22×C61—22no—
111C22×C61#39011yescharacteristic
112C22×C61#49611yescharacteristic
113C22×C61#49711yescharacteristic
114C2×D61—81no—
115C2×D61—82no—
116C4×C61—22no—
117C4×C61—22no—
118C4×C61#41711yescharacteristic
119C4×C61#41711yescharacteristic
120C4×C61#48911yescharacteristic
121C4×C61#48911yescharacteristic
122Dic61—81no—
123Dic61—82no—
124Dic61—162no—
125#3131—611no—
126#1846—611no—
127#1518—612no—
128#1517—612no—
129#1452D6111yescharacteristic, complemented
130C2×C61—21no—
131C2×C61—21no—
132C2×C61#195911yescharacteristic
133C2×C61#197612yes—
134D61—161no—
135#533—611no—
136#532—611no—
137#488—1222no—
138#487—612no—
139#486—612no—
140#431C2×D6112yes—
141C2×#107—1221no—
142C2×#106—1221no—
143#416—1221no—
144#416—1221no—
145C2×C4×C8C2×D6111yescharacteristic
146#382—612no—
147#328—1224no—
148#327—1224no—
149#322—22no—
150C61#975811yescharacteristic, complemented, Sylow
151C2×#41—1221no—
152C22×C8D24411yescharacteristic
153C22×C8D24411yescharacteristic
154C4⋊Q8—611no—
155C4⋊Q8—612no—
156#127—611no—
157#125—1221no—
158#124—1221no—
159C22⋊Q8—1222no—
160#121—1222no—
161C4×Q8—1222no—
162C4×D4—1222no—
163#117—1221no—
164C2×C4⋊C4—1222no—
165C2×C42C22×D6111yescharacteristic
166#107—2442no—
167#106—2442no—
168C4⋊C8—22no—
169#103—1222no—
170#103—1222no—
171#102—1222no—
172#102—1222no—
173C22⋊C8D24414yes—
174C4×C8—22no—
175C4×C8C4×D6112yes—
176C4×C8C22⋊D6112yes—
177#95—1222no—
178#95—1222no—
179#41—2441no—
180#41—2441no—
181#41—2441no—
182#41—2441no—
183C2×Q8—1221no—
184C2×Q8—1222no—
185C2×D4—1221no—
186C2×D4—1222no—
187C2×D4—2441no—
188C22×C4—2442no—
189C22×C4—2441no—
190C22×C4C2×D24412yes—
191C22×C4C2×D24411yescharacteristic
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—24no—
197C2×C8#1175819812yes—
198C2×C8#1175819812yes—
199C4⋊C4—1222no—
200C4⋊C4—1222no—
201C4⋊C4—1222no—
202C4⋊C4—2442no—
203C4⋊C4—2442no—
204C4⋊C4—2442no—
205C22⋊C4—2441no—
206C22⋊C4—2442no—
207C42—2441no—
208C42C2×C4×D6111yescharacteristic
209C42#1175822212yes—
210C42C2×C22⋊D6111yescharacteristic
211C23—2441no—
212C23#1451977411yescharacteristic
213Q8—4882no—
214Q8—2441no—
215Q8—2441no—
216D4—4882no—
217D4—4881no—
218D4—4881no—
219D4—2441no—
220D4—2441no—
221C2×C4—42no—
222C2×C4—4884no—
223C2×C4—4882no—
224C2×C4—4881no—
225C2×C4—4881no—
226C2×C4—2442no—
227C2×C4—2441no—
228C2×C4C2×#1175819811yescharacteristic
229C2×C4C4×D24412yes—
230C2×C4#1451975112yes—
231C2×C4#1451976511yescharacteristic
232C2×C4#1451980511yescharacteristic
233C2×C4#1451980511yescharacteristic
234C2×C4#1451981011yescharacteristic
235C2×C4#1451981011yescharacteristic
236C8—22no—
237C8—22no—
238C8—44no—
239C22—22no—
240C22—4882no—
241C22—4881no—
242C22#1683966411yescharacteristic
243C22#1684015111yescharacteristic
244C22#1684015611yescharacteristic
245C4—22no—
246C4—22no—
247C4—9762no—
248C4—4882no—
249C4—4881no—
250C4#1683976411yescharacteristic
251C4#1683976411yescharacteristic
252C4#1684012211yescharacteristic
253C4#1684012211yescharacteristic
254C2—21no—
255C2—21no—
256C2—9761no—
257C2#2144663511yescharacteristic
258C2#2144671712yes—
subgroupG/H
centerC22#16839664
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1452C2
socleC22×C61#390
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1452D61
hypocenterC61#9758
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1452

Complements

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

NH ≅ G/Ntype
2. #21445983C2semidirect
3. #21444384C2semidirect
5. C61×#1452C2semidirect
9. #16840113C4semidirect
10. #16840109C4semidirect
129. #1452D61semidirect
150. C61#9758semidirect

Maximal subgroups

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

subgroupindex
#214561422
#214459832
#214443842
#214443802
C61×#14522
#975861