Finite Group Databasebeta

← #49122945

Subgroups — #49122945

Order 36224.

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

normal subgroups only

This group has 250 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#31827016C211yesmaximal, characteristic
2#31826448C212yesmaximal, complemented
3#31826264C211yesmaximal, characteristic, complemented
4#31826261C212yesmaximal, complemented
5#356×C283C211yesmaximal, characteristic, complemented
6C2×#17985393C411yescharacteristic
7C2×#17985393C2211yescharacteristic
8M16×D283—22no—
9#22583607C411yescharacteristic
10#22583599C411yescharacteristic
11#22583599C2211yescharacteristic
12#22583585—22no—
13#22583572—22no—
14C42×D283C411yescharacteristic
15#22583529C2212yes—
16#22583526—22no—
17C4⋊Q8×C283C2211yescharacteristic
18#101×C283C2212yes—
19#17985393—41no—
20#17985393—41no—
21#17985393—41no—
22#17985393—41no—
23C2×D1132—22no—
24C2×D1132—21no—
25C2×D1132C2×C411yescharacteristic
26C2×D1132C2×C411yescharacteristic
27C2×D1132D412yes—
28C2×C4×D283—21no—
29C2×C4×D283—21no—
30C2×C4×D283C2×C411yescharacteristic
31C2×C4×D283D412yes—
32#17985376—22no—
33#17985375—42no—
34C8×D283—42no—
35(C2×D283)⋊C4—22no—
36(C2×D283)⋊C4—22no—
37C4×Dic283—21no—
38C4×Dic283C2×C411yescharacteristic
39C2×Q8×C283C2×C411yescharacteristic
40C2×Q8×C283C2311yescharacteristic
41M16×C283—22no—
42C4⋊C4×C283—22no—
43C42×C283C2×C411yescharacteristic
44C22×D283—21no—
45C22×D283C22⋊C412yes—
46C22×D283C22⋊C411yescharacteristic
47D1132—41no—
48D1132—42no—
49D1132—41no—
50D1132—44no—
51D1132—42no—
52C2×Dic283—21no—
53C2×Dic283C22⋊C411yescharacteristic
54C4×D283—22no—
55C4×D283—22no—
56C4×D283—21no—
57C4×D283—21no—
58C4×D283—42no—
59C4×D283—41no—
60C4×D283—41no—
61C283⋊C8—42no—
62Q8×C283—41no—
63Q8×C283—41no—
64C2×C4×C283—21no—
65C2×C4×C283—21no—
66C2×C4×C283C22×C411yescharacteristic
67C2×C4×C283C2×D412yes—
68C8×C283—42no—
69C22×C283C2×C22⋊C411yescharacteristic
70C2×D283—21no—
71C2×D283—22no—
72C2×D283—41no—
73C2×D283—42no—
74C2×D283—42no—
75C2×D283—41no—
76C4×C283—22no—
77C4×C283—22no—
78C4×C283—21no—
79C4×C283—41no—
80Dic283—41no—
81Dic283#9911yescharacteristic
82Dic283#9911yescharacteristic
83C2×C283—21no—
84C2×C283C2×#9911yescharacteristic
85D283—41no—
86D283—42no—
87D283—81no—
88C283#213211yescharacteristic, complemented, Sylow
89#2132—2831nomaximal, Sylow
90#532—2831no—
91#413—2832no—
92#356—2831no—
93#356D28311yescharacteristic, complemented
94#355—2832no—
95C2×#41—2831no—
96C2×#41—2831no—
97C2×M16—5662no—
98C4⋊Q8C2×D28311yescharacteristic
99#127—2831no—
100#124—2831no—
101#124—2831no—
102#121—5662no—
103C4×D4—5662no—
104C2×C42—2831no—
105#101—2832no—
106#101C2×D28312yes—
107#100—5662no—
108#41—11321no—
109#41—11321no—
110#41—11321no—
111#41—11321no—
112C2×Q8C4×D28311yescharacteristic
113C2×Q8C22×D28311yescharacteristic
114C2×D4—2831no—
115C2×D4—2831no—
116C2×D4—2832no—
117C2×D4—5662no—
118C2×D4—5661no—
119C22×C4—2831no—
120C22×C4—2832no—
121C22×C4—5661no—
122C22×C4—5661no—
123M16—22no—
124M16—5662no—
125M16—11322no—
126C2×C8—11322no—
127C4⋊C4—22no—
128C22⋊C4—5662no—
129C22⋊C4—5662no—
130C42—2831no—
131C42—5661no—
132C42C4×D28311yescharacteristic
133C23—2832no—
134C23—2831no—
135C23—5661no—
136Q8—41no—
137Q8—41no—
138D4—11322no—
139D4—11321no—
140D4—11321no—
141D4—11324no—
142D4—11322no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—2831no—
146C2×C4—5662no—
147C2×C4—5662no—
148C2×C4—5661no—
149C2×C4—5661no—
150C2×C4—5661no—
151C2×C4—11322no—
152C2×C4—11321no—
153C2×C4—11321no—
154C2×C4C2×C4×D28311yescharacteristic
155C2×C4D4×D28312yes—
156C8—42no—
157C8—11322no—
158C22—5661no—
159C22—5662no—
160C22—11321no—
161C22—11322no—
162C22—11322no—
163C22—11321no—
164C22C22⋊C4×D28311yescharacteristic
165C4—22no—
166C4—22no—
167C4—21no—
168C4—41no—
169C4—2831no—
170C4—2831no—
171C4—11321no—
172C2—21no—
173C2—11321no—
174C2—11322no—
175C2—22641no—
176C2#99×D28311yescharacteristic
subgroupG/H
centerC2#99×D283
commutatorC2×C4×C283C22×C4
FrattiniC2×Q8C22×D283
Fitting#356×C283C2
socleC2×C283C2×#99
prime radical#31827016C2
cosocleC2×Q8×C283C23
hypercenter#356D283
hypocenterC283#2132
LCSG ▷ C2×C4×C283 ▷ C22×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #31826448C2semidirect
3. #31826264C2semidirect
4. #31826261C2semidirect
5. #356×C283C2semidirect
88. C283#2132semidirect
93. #356D283semidirect

Maximal subgroups

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

subgroupindex
#318270162
#318264482
#318262642
#318262612
#356×C2832
#2132283