Finite Group Databasebeta

← #49122923

Subgroups — #49122923

Order 36224.

258 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 55 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#31827013C211yesmaximal, characteristic
2#31826437C212yesmaximal, complemented
3#31826259C212yesmaximal, complemented
4#31826258C211yesmaximal, characteristic, complemented
5#353×C283C211yesmaximal, characteristic, complemented
6C2×#17985391C411yescharacteristic, complemented
7C2×#17985391C2211yescharacteristic
8#22583610C411yescharacteristic, complemented
9#22583597C411yescharacteristic, complemented
10#22583597C2211yescharacteristic
11#22583581—22no—
12D4×Dic283—22no—
13#22583566—22no—
14C42×D283C411yescharacteristic, complemented
15#22583525C2212yes—
16#22583524—22no—
17#127×C283C2211yescharacteristic
18#99×C283C2212yes—
19#17985391—41no—
20#17985391—41no—
21#17985391—41no—
22#17985391—41no—
23C2×Dic566C2×C411yescharacteristic
24C2×Dic566C2×C411yescharacteristic
25C2×C22⋊D283—21no—
26C2×C22⋊D283D412yes—
27C22×Dic283—21no—
28C22×Dic283D412yes—
29C2×C4×D283—21no—
30C2×C4×D283C2×C411yescharacteristic
31C4⋊Dic283—22no—
32Dic283⋊C4—42no—
33C22⋊Dic283—22no—
34C22⋊Dic283—22no—
35C22⋊Dic283—22no—
36C4×Dic283—21no—
37C4×Dic283—42no—
38C4×Dic283C2×C411yescharacteristic
39C2×D4×C283—22no—
40C2×D4×C283C2×C411yescharacteristic
41C2×D4×C283C2311yescharacteristic
42C22⋊C4×C283—22no—
43C42×C283C2×C411yescharacteristic
44C22×D283C22⋊C411yescharacteristic
45Dic566—41no—
46Dic566—41no—
47C22⋊D283—42no—
48C22⋊D283—42no—
49C2×Dic283—22no—
50C2×Dic283—22no—
51C2×Dic283—21no—
52C2×Dic283—21no—
53C2×Dic283—41no—
54C2×Dic283—41no—
55C2×Dic283—42no—
56C2×Dic283C22⋊C412yes—
57C2×Dic283C22⋊C411yescharacteristic
58C4×D283—21no—
59C4×D283—21no—
60C4×D283—42no—
61C23×C283—21no—
62C23×C283C2×D412yes—
63D4×C283—41no—
64D4×C283—41no—
65D4×C283—44no—
66C2×C4×C283—21no—
67C2×C4×C283—42no—
68C2×C4×C283C22×C411yescharacteristic
69C22×C283—22no—
70C22×C283—42no—
71C22×C283—42no—
72C22×C283—41no—
73C22×C283C2×C22⋊C411yescharacteristic
74C2×D283—21no—
75C2×D283—41no—
76C4×C283—22no—
77C4×C283—21no—
78C4×C283—82no—
79Dic283—22no—
80Dic283—41no—
81Dic283—41no—
82Dic283—82no—
83Dic283#9911yescharacteristic
84Dic283#9911yescharacteristic
85C2×C283—21no—
86C2×C283—42no—
87C2×C283—81no—
88C2×C283C2×#9911yescharacteristic
89D283—41no—
90C283#212611yescharacteristic, complemented, Sylow
91#2126—2831nomaximal, Sylow
92#532—2831no—
93#410—2832no—
94#354—2832no—
95#353—2831no—
96#353D28311yescharacteristic, complemented
97C2×#41—2831no—
98C2×#41—2831no—
99C4⋊Q8—2831no—
100#127C2×D28311yescharacteristic
101#124—2831no—
102#124—2831no—
103#121—5662no—
104C4×D4—5662no—
105#117—5662no—
106C2×C42—2831no—
107#99—2832no—
108#99—5662no—
109#99C2×D28312yes—
110#41—11321no—
111#41—11321no—
112#41—11321no—
113#41—11321no—
114C2×Q8—2831no—
115C2×Q8—2831no—
116C2×D4—22no—
117C2×D4—2832no—
118C2×D4—5661no—
119C2×D4C4×D28311yescharacteristic
120C2×D4C22×D28311yescharacteristic
121C22×C4—2832no—
122C22×C4—2831no—
123C22×C4—5661no—
124C22×C4—5661no—
125C4⋊C4—5662no—
126C4⋊C4—11322no—
127C22⋊C4—22no—
128C22⋊C4—5662no—
129C22⋊C4—5662no—
130C22⋊C4—5662no—
131C42—2831no—
132C42—5661no—
133C42—11322no—
134C42C4×D28311yescharacteristic
135C23—21no—
136C23—2831no—
137C23D4×D28312yes—
138Q8—11321no—
139Q8—11321no—
140D4—41no—
141D4—41no—
142D4—44no—
143D4—11322no—
144D4—11322no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—2831no—
148C2×C4—2832no—
149C2×C4—5661no—
150C2×C4—5661no—
151C2×C4—5662no—
152C2×C4—5662no—
153C2×C4—5661no—
154C2×C4—5661no—
155C2×C4—11322no—
156C2×C4—11321no—
157C2×C4—11321no—
158C2×C4—11322no—
159C2×C4C2×C4×D28311yescharacteristic
160C22—22no—
161C22—42no—
162C22—42no—
163C22—41no—
164C22—5661no—
165C22—11321no—
166C22C22⋊C4×D28311yescharacteristic
167C4—22no—
168C4—21no—
169C4—82no—
170C4—2831no—
171C4—2831no—
172C4—5662no—
173C4—11321no—
174C4—11321no—
175C4—22642no—
176C2—21no—
177C2—42no—
178C2—81no—
179C2—11321no—
180C2#99×D28311yescharacteristic
subgroupG/H
centerC2#99×D283
commutatorC2×C4×C283C22×C4
FrattiniC2×D4C22×D283
Fitting#353×C283C2
socleC2×C283C2×#99
prime radical#31827013C2
cosocleC2×D4×C283C23
hypercenter#353D283
hypocenterC283#2126
LCSG ▷ C2×C4×C283 ▷ C22×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #31826437C2semidirect
3. #31826259C2semidirect
4. #31826258C2semidirect
5. #353×C283C2semidirect
6. C2×#17985391C4semidirect
8. #22583610C4semidirect
9. #22583597C4semidirect
14. C42×D283C4semidirect
90. C283#2126semidirect
96. #353D283semidirect

Maximal subgroups

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

subgroupindex
#318270132
#318264372
#318262592
#318262582
#353×C2832
#2126283