Finite Group Databasebeta

← #29596285

Subgroups — #29596285

Order 15616.

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

normal subgroups only

This group has 566 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#21445743C212yesmaximal
2#21443946C214yesmaximal, complemented
3C61×#2310C211yesmaximal, characteristic
4C2×#14519696—22no—
5C2×#14519696C2214yes—
6#16839726—22no—
7C23×Q8×C61C2211yescharacteristic
8C2×#117×C61C411yescharacteristic
9C2×#117×C61C411yescharacteristic
10C61×#391—22no—
11C61×#391C412yes—
12C61×#391C2212yes—
13C61×#381—22no—
14C2×#11758194—24no—
15#14519696—28no—
16#14519696—44no—
17C22⋊#92418—44no—
18C22×Q8×C61—24no—
19C22×Q8×C61—42no—
20C22×Q8×C61—42no—
21C22×Q8×C61C2×C411yescharacteristic
22C22×Q8×C61D412yes—
23C22×Q8×C61C2311yescharacteristic
24C23×C4×C61C2×C411yescharacteristic
25C23×C4×C61D412yes—
26#117×C61—42no—
27#117×C61—42no—
28C2×C4⋊C4×C61—21no—
29C2×C4⋊C4×C61C2×C412yes—
30C2×C22⋊C4×C61—21no—
31C2×C22⋊C4×C61—21no—
32C2×C42×C61—21no—
33C2×C42×C61C2×C412yes—
34#95×C61—24no—
35#95×C61—24no—
36C24×C61C22⋊C411yescharacteristic
37C2×Q8×C61—24no—
38C2×Q8×C61—22no—
39C2×Q8×C61—24no—
40C2×Q8×C61—41no—
41C2×Q8×C61—41no—
42C2×Q8×C61—42no—
43C2×Q8×C61—41no—
44C2×Q8×C61—41no—
45C2×Q8×C61—416no—
46C2×Q8×C61#4114yes—
47C22×C4×C61—22no—
48C22×C4×C61—22no—
49C22×C4×C61—22no—
50C22×C4×C61—24no—
51C22×C4×C61—24no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—42no—
55C22×C4×C61C22⋊C412yes—
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22×C411yescharacteristic
58C22×C4×C61C2×D412yes—
59C4⋊C4×C61—42no—
60C4⋊C4×C61—42no—
61C22⋊C4×C61—42no—
62C22⋊C4×C61—42no—
63C42×C61—42no—
64C42×C61—42no—
65#11758194—44no—
66#11758194—44no—
67C2×#92418—44no—
68C23×C61—21no—
69C23×C61—21no—
70C23×C61—42no—
71C23×C61#9912yes—
72C23×C61C2×C22⋊C411yescharacteristic
73Q8×C61—44no—
74Q8×C61—48no—
75Q8×C61—44no—
76C2×C4×C61—28no—
77C2×C4×C61—21no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—22no—
84C2×C4×C61—22no—
85C2×C4×C61—22no—
86C2×C4×C61—48no—
87C2×C4×C61—44no—
88C2×C4×C61—44no—
89C2×C4×C61—42no—
90C2×C4×C61—42no—
91C2×C4×C61#9912yes—
92C2×C4×C61#11712yes—
93C2×C4×C61#12314yes—
94#92418—84no—
95#12862—611nomaximal, Sylow
96C22×C61—22no—
97C22×C61—22no—
98C22×C61—24no—
99C22×C61—42no—
100C22×C61—42no—
101C22×C61#38111yescharacteristic
102C22×C61C2×#9911yescharacteristic
103C22×C61C2×#9911yescharacteristic
104C4×C61—24no—
105C4×C61—22no—
106C4×C61—22no—
107C4×C61—42no—
108C4×C61—82no—
109C4×C61—82no—
110#2310D6111yescharacteristic
111#1784—612no—
112#1394—614no—
113C2×C61—21no—
114C2×C61—21no—
115C2×C61—42no—
116C2×C61#181911yescharacteristic
117C2×C61#213312yes—
118C23×Q8C2×D6111yescharacteristic
119C2×#117Dic6111yescharacteristic
120C2×#117Dic6111yescharacteristic
121C2×#101—614no—
122C2×#101—1222no—
123#407—1222no—
124#391—22no—
125#391Dic6112yes—
126#391C2×D6112yes—
127#381—22no—
128C61#1286211yescharacteristic, complemented, Sylow
129C22×Q8—24no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8C2×Dic6111yescharacteristic
133C22×Q8C22⋊D6112yes—
134C22×Q8C22×D6111yescharacteristic
135C23×C4C2×Dic6111yescharacteristic
136C23×C4C22⋊D6112yes—
137C2×M16—1224no—
138#117—42no—
139#117—42no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4C2×Dic6112yes—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C42—21no—
145C2×C42C2×Dic6112yes—
146#101—1228no—
147#101—2444no—
148C22⋊C8—2444no—
149#95—24no—
150#95—24no—
151C24C22⋊Dic6111yescharacteristic
152C2×Q8—24no—
153C2×Q8—22no—
154C2×Q8—24no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—42no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—416no—
161C2×Q8#1175822214yes—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4C22⋊Dic6112yes—
171C22×C4C22⋊Dic6111yescharacteristic
172C22×C4C22×Dic6111yescharacteristic
173C22×C4C2×C22⋊D6112yes—
174M16—2444no—
175M16—2444no—
176C2×C8—2444no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C42—42no—
182C42—42no—
183C23—21no—
184C23—21no—
185C23—42no—
186C23#1451969212yes—
187C23C2×C22⋊Dic6111yescharacteristic
188Q8—44no—
189Q8—48no—
190Q8—44no—
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—48no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4#1451969212yes—
207C2×C4#1451973612yes—
208C2×C4#1451976014yes—
209C8—4884no—
210C22—22no—
211C22—22no—
212C22—24no—
213C22—42no—
214C22—42no—
215C22#1683962511yescharacteristic
216C22C2×#1451969211yescharacteristic
217C22C2×#1451969211yescharacteristic
218C4—24no—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—82no—
223C4—82no—
224C2—21no—
225C2—21no—
226C2—42no—
227C2#2144589311yescharacteristic
228C2#2144750812yes—
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#2310C2
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC22×Q8×C61C23
hypercenter#2310D61
hypocenterC61#12862
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. #21443946C2semidirect
128. C61#12862semidirect

Maximal subgroups

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

subgroupindex
#214457432
#214439462
C61×#23102
#1286261