Finite Group Databasebeta

← #29577530

Subgroups — #29577530

Order 15616.

488 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 99 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#21456279C211yesmaximal, characteristic
2#21444537C212yesmaximal, complemented
3#21444454C211yesmaximal, characteristic
4#21443690C212yesmaximal, complemented
5C61×#2980C211yesmaximal, characteristic
6C2×#14519807—22no—
7C2×#14519777C2211yescharacteristic
8#16840157—22no—
9#16840141C412yes—
10#16840134C412yes—
11#16839819C2212yes—
12C4×#11758194C2211yescharacteristic
13#16839509—24no—
14#16839502—22no—
15C61×#532C2211yescharacteristic
16C61×#435—24no—
17C61×#432C2212yes—
18C61×#408—22no—
19C22×Dic122—21no—
20#14519807—44no—
21#14519807—44no—
22C2×#11758202—24no—
23C2×#11758200—24no—
24C2×#11758194—21no—
25#14519777—22no—
26#14519777—21no—
27#14519777C2×C412yes—
28C2×C4⋊Dic61—22no—
29C4⋊#92418—22no—
30C22⋊#92418—22no—
31#14519688—21no—
32#14519688—22no—
33C4×#92418C2×C412yes—
34C2×#41×C61D412yes—
35C2×M16×C61—22no—
36C22×C8×C61—22no—
37C4⋊Q8×C61C2×C411yescharacteristic
38#127×C61C2×C411yescharacteristic
39#124×C61D412yes—
40#121×C61—22no—
41C4×D4×C61—22no—
42C2×C42×C61C2311yescharacteristic
43C4⋊C8×C61D414yes—
44C22⋊C8×C61—24no—
45C22⋊C8×C61—22no—
46#97×C61—22no—
47#41×C61—42no—
48#41×C61—42no—
49C2×Q8×C61C22⋊C412yes—
50C2×D4×C61—22no—
51C2×D4×C61—22no—
52C2×D4×C61C22⋊C412yes—
53C22×C4×C61—22no—
54C22×C4×C61C2×D411yescharacteristic
55C22×C4×C61#4112yes—
56C22×Dic61—41no—
57C2×Dic122—22no—
58C2×Dic122—42no—
59C2×Dic122—42no—
60C2×Dic122—41no—
61M16×C61—82no—
62C2×C8×C61—24no—
63C2×C8×C61—82no—
64C4⋊C4×C61—22no—
65C22⋊C4×C61—24no—
66C42×C61C22×C411yescharacteristic
67C42×C61C2×D411yescharacteristic
68C42×C61C2×D412yes—
69#11758202—48no—
70#11758200—48no—
71C4⋊Dic61—24no—
72C4⋊Dic61—42no—
73#11758194—24no—
74C2×#92418—22no—
75C2×#92418—42no—
76C23×C61—22no—
77C23×C61#12311yescharacteristic
78Q8×C61—42no—
79D4×C61—42no—
80D4×C61—82no—
81D4×C61—82no—
82C2×C4×C61—22no—
83C2×C4×C61—22no—
84C2×C4×C61—42no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61C2×C22⋊C411yescharacteristic
88C2×C4×C61C4×D412yes—
89C2×C4×C61#12011yescharacteristic
90C2×C4×C61#12112yes—
91C2×Dic61—42no—
92C2×Dic61—81no—
93C2×Dic61—81no—
94Dic122—44no—
95Dic122—82no—
96C8×C61—44no—
97#92418—24no—
98#92418—82no—
99#9421—611nomaximal, Sylow
100C22×C61—21no—
101C22×C61—22no—
102C22×C61—42no—
103C22×C61—44no—
104C22×C61#38611yescharacteristic
105C4×C61—21no—
106C4×C61—22no—
107C4×C61—22no—
108C4×C61—42no—
109C4×C61#45312yes—
110Dic61—82no—
111#3152—611no—
112#2980D6111yescharacteristic
113#1556—612no—
114#1536—611no—
115#1291—612no—
116C2×C61—41no—
117C2×C61—82no—
118C2×C61#188711yescharacteristic
119C2×C61#190212yes—
120C2×#137—1222no—
121#532C2×D6111yescharacteristic
122C2×C4⋊Q8—611no—
123#497—1222no—
124#494—612no—
125#492—612no—
126#435—24no—
127#432—612no—
128#432C2×D6112yes—
129#408—22no—
130C4×M16—611no—
131#332—1224no—
132#329—1222no—
133C61#942111yescharacteristic, complemented, Sylow
134C2×#41C22⋊D6112yes—
135C22×Q8—1221no—
136#137—2444no—
137#137—2444no—
138C2×Q16—1224no—
139C2×SD16—1224no—
140C2×M16—22no—
141C2×M16—1221no—
142C22×C8—22no—
143C4⋊Q8—612no—
144C4⋊Q8—1221no—
145C4⋊Q8—1222no—
146C4⋊Q8C4×D6111yescharacteristic
147#127C4×D6111yescharacteristic
148#124D24412yes—
149#121—22no—
150C4×D4—22no—
151C2×C4⋊C4—1222no—
152C2×C42C22×D6111yescharacteristic
153C4⋊C8—1222no—
154C4⋊C8C22⋊D6114yes—
155C22⋊C8—24no—
156C22⋊C8—22no—
157C22⋊C8—1222no—
158#97—22no—
159#97—1221no—
160#97—1222no—
161C4×C8—612no—
162#41—42no—
163#41—42no—
164C2×Q8—1222no—
165C2×Q8—2442no—
166C2×Q8—2442no—
167C2×Q8—2441no—
168C2×Q8#1175819812yes—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4#1175819812yes—
172C22×C4—22no—
173C22×C4—2441no—
174C22×C4#1175822212yes—
175C22×C4C2×C22⋊D6111yescharacteristic
176Q16—2448no—
177SD16—2448no—
178M16—82no—
179M16—1224no—
180C2×C8—24no—
181C2×C8—82no—
182C2×C8—1222no—
183C2×C8—2442no—
184C4⋊C4—22no—
185C4⋊C4—1224no—
186C4⋊C4—2442no—
187C22⋊C4—24no—
188C42C2×C4×D6111yescharacteristic
189C42C2×D24411yescharacteristic
190C42C2×C22⋊D6112yes—
191C23—22no—
192C23#1451976011yescharacteristic
193Q8—42no—
194Q8—4882no—
195Q8—2444no—
196D4—42no—
197D4—82no—
198D4—82no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—4881no—
205C2×C4—4881no—
206C2×C4—2442no—
207C2×C4C2×#1175819811yescharacteristic
208C2×C4C4×C22⋊D6112yes—
209C2×C4#1451974611yescharacteristic
210C2×C4#1451975112yes—
211C8—44no—
212C8—1224no—
213C8—4882no—
214C22—21no—
215C22—22no—
216C22—42no—
217C22—44no—
218C22#1683964111yescharacteristic
219C4—21no—
220C4—22no—
221C4—22no—
222C4—42no—
223C4—4882no—
224C4#1683993012yes—
225C2—41no—
226C2—82no—
227C2#2144621711yescharacteristic
228C2#2144629212yes—
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2980C2
socleC22×C61#386
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#2980D61
hypocenterC61#9421
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
2. #21444537C2semidirect
4. #21443690C2semidirect
133. C61#9421semidirect

Maximal subgroups

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

subgroupindex
#214562792
#214445372
#214444542
#214436902
C61×#29802
#942161