Finite Group Databasebeta

← #49122934

Subgroups — #49122934

Order 36224.

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

normal subgroups only

This group has 238 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#31827035C211yesmaximal, characteristic
2#31826437C212yesmaximal
3#31826260C211yesmaximal, characteristic, complemented
4#31826259C212yesmaximal, complemented
5#354×C283C211yesmaximal, characteristic, complemented
6C2×#17985391C2211yescharacteristic
7C2×Q8×D283C411yescharacteristic, complemented
8#22583610C411yescharacteristic, complemented
9#22583608C411yescharacteristic, complemented
10#22583597C2211yescharacteristic
11#22583592—22no—
12#22583587—22no—
13#22583567C411yescharacteristic, complemented
14#22583566—22no—
15#22583525C2212yes—
16#22583524—22no—
17#124×C283C2211yescharacteristic
18#99×C283C2212yes—
19#17985391—42no—
20C2×Dic566—21no—
21C2×Dic566C2×C411yescharacteristic
22C2×Dic566C2×C411yescharacteristic
23Q8×D283—42no—
24C2×C22⋊D283D412yes—
25C22×Dic283D412yes—
26C2×C4×D283—21no—
27C2×C4×D283C2×C411yescharacteristic
28C4⋊Dic283—22no—
29Dic283⋊C4—21no—
30Dic283⋊C4—22no—
31Dic283⋊C4—42no—
32C22⋊Dic283—22no—
33C22⋊Dic283—22no—
34(C2×D283)⋊C4—21no—
35C4×Dic283—42no—
36C4×Dic283C2×C411yescharacteristic
37C2×Q8×C283C2×C411yescharacteristic
38C2×D4×C283C2311yescharacteristic
39C22⋊C4×C283—22no—
40C22⋊C4×C283—22no—
41C42×C283C2×C411yescharacteristic
42C22×D283C22⋊C411yescharacteristic
43Dic566—41no—
44Dic566—42no—
45Dic566—41no—
46C22⋊D283—42no—
47C2×Dic283—22no—
48C2×Dic283—22no—
49C2×Dic283—21no—
50C2×Dic283—21no—
51C2×Dic283—42no—
52C2×Dic283C22⋊C412yes—
53C2×Dic283C22⋊C411yescharacteristic
54C4×D283—22no—
55C4×D283—42no—
56C23×C283C2×D412yes—
57Q8×C283—41no—
58D4×C283—41no—
59C2×C4×C283—21no—
60C2×C4×C283—21no—
61C2×C4×C283—42no—
62C2×C4×C283C22×C411yescharacteristic
63C22×C283—22no—
64C22×C283—42no—
65C22×C283C2×C22⋊C411yescharacteristic
66C2×D283—21no—
67C2×D283—41no—
68C4×C283—21no—
69C4×C283—41no—
70C4×C283—41no—
71C4×C283—82no—
72Dic283—22no—
73Dic283—41no—
74Dic283—41no—
75Dic283—82no—
76Dic283#9912yes—
77C2×C283—21no—
78C2×C283—42no—
79C2×C283C2×#9911yescharacteristic
80D283—41no—
81C283#212911yescharacteristic, complemented, Sylow
82#2129—2831nomaximal, Sylow
83#536—2831no—
84#410—2832no—
85#354—2831no—
86#354—2832no—
87#354D28311yescharacteristic, complemented
88C2×#41—2831no—
89C22×Q8—2831no—
90C4⋊Q8—2831no—
91C4⋊Q8—2831no—
92#124—2831no—
93#124C2×D28311yescharacteristic
94#123—5662no—
95C22⋊Q8—5662no—
96#117—2831no—
97#117—5662no—
98#99—2832no—
99#99—5662no—
100#99C2×D28312yes—
101#41—11322no—
102C2×Q8—2831no—
103C2×Q8—2831no—
104C2×Q8—5661no—
105C2×Q8—11322no—
106C2×Q8C4×D28311yescharacteristic
107C2×D4—2832no—
108C2×D4C22×D28311yescharacteristic
109C22×C4—2832no—
110C22×C4—2831no—
111C22×C4—5661no—
112C4⋊C4—5661no—
113C4⋊C4—5662no—
114C4⋊C4—5662no—
115C4⋊C4—11322no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—5661no—
119C22⋊C4—5662no—
120C22⋊C4—5662no—
121C42—2831no—
122C42—11322no—
123C42C4×D28311yescharacteristic
124C23—2831no—
125C23D4×D28312yes—
126Q8—41no—
127Q8—11321no—
128Q8—11321no—
129Q8—11322no—
130D4—41no—
131D4—11322no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—2831no—
136C2×C4—2832no—
137C2×C4—5662no—
138C2×C4—5662no—
139C2×C4—5662no—
140C2×C4—5661no—
141C2×C4—5661no—
142C2×C4—11322no—
143C2×C4—11322no—
144C2×C4C2×C4×D28311yescharacteristic
145C22—22no—
146C22—42no—
147C22—5661no—
148C22—11321no—
149C22C22⋊C4×D28311yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—2832no—
155C4—5662no—
156C4—11321no—
157C4—11321no—
158C4—22642no—
159C2—21no—
160C2—42no—
161C2—11321no—
162C2#99×D28311yescharacteristic
subgroupG/H
centerC2#99×D283
commutatorC2×C4×C283C22×C4
FrattiniC2×D4C22×D283
Fitting#354×C283C2
socleC2×C283C2×#99
prime radicalC2×#17985391C22
cosocleC2×D4×C283C23
hypercenter#354D283
hypocenterC283#2129
LCSG ▷ C2×C4×C283 ▷ C22×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #31826260C2semidirect
4. #31826259C2semidirect
5. #354×C283C2semidirect
7. C2×Q8×D283C4semidirect
8. #22583610C4semidirect
9. #22583608C4semidirect
13. #22583567C4semidirect
81. C283#2129semidirect
87. #354D283semidirect

Maximal subgroups

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

subgroupindex
#318270352
#318264372
#318262602
#318262592
#354×C2832
#2129283