Finite Group Databasebeta

← #49120280

Subgroups — #49120280

Order 36224.

288 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 53 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#31827027C211yesmaximal, characteristic
2#31826783C212yesmaximal, complemented
3#31826198C212yesmaximal, complemented
4#31826189C211yesmaximal, characteristic, complemented
5#399×C283C211yesmaximal, characteristic, complemented
6C22×C22⋊D283—21no—
7#22583594—22no—
8#22583584—21no—
9#22583584C2212yes—
10#22583580—22no—
11#22583541—22no—
12#22583530—22no—
13C22⋊(C283⋊C8)C2212yescomplemented
14C2×C4⋊C4×C283—21no—
15C2×C4⋊C4×C283C2211yescharacteristic, complemented
16C2×C22⋊C4×C283—21no—
17C2×C22⋊C4×C283C2211yescharacteristic
18#95×C283C2211yescharacteristic, complemented
19C23×D283—21no—
20C2×D1132—21no—
21C2×D1132D412yes—
22C2×C22⋊D283—22no—
23C2×C22⋊D283—22no—
24C2×C22⋊D283—42no—
25C2×C22⋊D283—42no—
26C22×Dic283—21no—
27C2×C283⋊C8—22no—
28C4⋊Dic283—21no—
29C4⋊Dic283D412yes—
30(C2×D283)⋊C4—22no—
31(C2×D283)⋊C4—22no—
32C24×C283D411yescharacteristic
33C22×C4×C283—21no—
34C22×C4×C283—21no—
35C22×C4×C283D411yescharacteristic
36C22×C4×C283C2311yescharacteristic
37C4⋊C4×C283—22no—
38C4⋊C4×C283—42no—
39C22⋊C4×C283—22no—
40C22⋊C4×C283—42no—
41C22×D283—22no—
42C22×D283—42no—
43C22×D283—81no—
44C22×D283—81no—
45D1132—42no—
46D1132—81no—
47C22⋊D283—44no—
48C22⋊D283—42no—
49C22⋊D283—42no—
50C2×Dic283—22no—
51C2×Dic283—42no—
52C283⋊C8—42no—
53C23×C283—21no—
54C23×C283—21no—
55C23×C283—41no—
56C23×C283—41no—
57C23×C283SD1612yes—
58C23×C283C2×D411yescharacteristic
59C2×C4×C283—22no—
60C2×C4×C283—21no—
61C2×C4×C283—41no—
62C2×C4×C283—41no—
63C2×C4×C283—41no—
64C2×C4×C283—41no—
65C2×C4×C283—41no—
66C2×C4×C283—41no—
67C2×C4×C283—41no—
68C2×C4×C283C2×D412yes—
69C22×C283—21no—
70C22×C283—21no—
71C22×C283—22no—
72C22×C283—42no—
73C22×C283—41no—
74C22×C283—41no—
75C22×C283—81no—
76C22×C283#12011yescharacteristic
77C22×C283C2×SD1611yescharacteristic
78C22×C283AGL(1,ℤ/8)11yescharacteristic
79C2×D283—42no—
80C2×D283—42no—
81C2×D283—42no—
82C2×D283—84no—
83C4×C283—22no—
84C4×C283—42no—
85C4×C283—41no—
86C4×C283—81no—
87Dic283—42no—
88C2×C283—21no—
89C2×C283—21no—
90C2×C283—42no—
91C2×C283#45011yescharacteristic
92C2×C283#45311yescharacteristic
93C2×C283#45811yescharacteristic
94D283—82no—
95C283#161311yescharacteristic, complemented, Sylow
96#1613—2831nomaximal, Sylow
97#534—2831no—
98#481—2832no—
99#399D28311yescharacteristic, complemented
100#327—2832no—
101#323—2831no—
102C22×D4—5661no—
103#123—5662no—
104#121—2832no—
105#121—5661no—
106#120—5662no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C2×D28311yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4C2×D28311yescharacteristic
111#106—5662no—
112#102—5662no—
113C22⋊C8—2832no—
114#95C2×D28311yescharacteristic, complemented
115C24—5661no—
116C24C22⋊D28311yescharacteristic
117C2×D4—2832no—
118C2×D4—5661no—
119C2×D4—5662no—
120C2×D4—5662no—
121C2×D4—11322no—
122C2×D4—11322no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—5661no—
126C22×C4C22⋊D28311yescharacteristic
127C22×C4C22×D28311yescharacteristic
128C2×C8—5662no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—2832no—
132C4⋊C4—5661no—
133C22⋊C4—22no—
134C22⋊C4—42no—
135C22⋊C4—5662no—
136C22⋊C4—5662no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—5662no—
142C23—11322no—
143C23—22641no—
144C23—22641no—
145C23#1798537912yes—
146C23C2×C22⋊D28311yescharacteristic
147D4—11322no—
148D4—11324no—
149D4—11322no—
150D4—11322no—
151D4—22641no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—5662no—
162C2×C4—11322no—
163C2×C4D4×D28312yes—
164C8—11322no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—11322no—
173C22—11322no—
174C22—11322no—
175C22—22644no—
176C22#2258357811yescharacteristic
177C22C2×#1798537911yescharacteristic
178C22#2258363511yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—81no—
183C4—11322no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—22642no—
188C2#3182661511yescharacteristic
189C2#3182663211yescharacteristic
190C2#3182665411yescharacteristic
subgroupG/H
centerC22#22583578
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting#399×C283C2
socleC22×C283#120
prime radical#31827027C2
hypercenter#399D283
hypocenterC283#1613
LCSG ▷ C22×C4×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #31826783C2semidirect
3. #31826198C2semidirect
4. #31826189C2semidirect
5. #399×C283C2semidirect
13. C22⋊(C283⋊C8)C22semidirect
15. C2×C4⋊C4×C283C22semidirect
18. #95×C283C22semidirect
95. C283#1613semidirect
99. #399D283semidirect
108. C2×C4⋊C4C2×D283semidirect
114. #95C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318270272
#318267832
#318261982
#318261892
#399×C2832
#1613283