Finite Group Databasebeta

← #49120347

Subgroups — #49120347

Order 36224.

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

normal subgroups only

This group has 256 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#31827118C211yesmaximal, characteristic, complemented
2#31826657C211yesmaximal, characteristic, complemented
3#31826657C211yesmaximal, characteristic, complemented
4#31826211C211yesmaximal, characteristic, complemented
5#31826194C211yesmaximal, characteristic, complemented
6#31826194C211yesmaximal, characteristic, complemented
7#486×C283C211yesmaximal, characteristic, complemented
8C2×#17985393—21no—
9C2×#17985377—21no—
10C2×#17985377—21no—
11#22583595—22no—
12#22583586—21no—
13#22583585—21no—
14C4×D1132C2211yescharacteristic
15C4×D1132C2211yescharacteristic
16C4⋊C4×D283—21no—
17C4⋊(C283⋊C8)C2211yescharacteristic
18C4⋊(C283⋊C8)C2211yescharacteristic
19#22583531—21no—
20#22583531—21no—
21C4⋊Q8×C283C2211yescharacteristic
22#127×C283C2211yescharacteristic
23#102×C283—22no—
24C4×C8×C283C2211yescharacteristic
25#17985393—42no—
26#17985393—42no—
27C2×D1132—22no—
28C2×D1132D411yescharacteristic
29C2×D1132D411yescharacteristic
30C2×C4×D283—22no—
31C2×C4×D283—21no—
32C2×C4×D283—21no—
33#17985377—42no—
34#17985377—42no—
35C2×C283⋊C8—21no—
36C2×C283⋊C8—21no—
37C4⋊Dic283D411yescharacteristic
38C4⋊Dic283D411yescharacteristic
39Dic283⋊C4—22no—
40(C2×D283)⋊C4—22no—
41(C2×D283)⋊C4—21no—
42(C2×D283)⋊C4—21no—
43C2×Q8×C283—21no—
44C2×D4×C283—21no—
45C2×D4×C283—21no—
46C2×D4×C283—21no—
47C2×C8×C283—21no—
48C4⋊C4×C283—21no—
49C4⋊C4×C283D412yes—
50C42×C283C2311yescharacteristic
51C22×D283—21no—
52C22×D283—21no—
53D1132—21no—
54D1132—21no—
55D1132—21no—
56D1132—44no—
57D1132D812yescomplemented
58C2×Dic283—21no—
59C2×Dic283—21no—
60C4×D283—44no—
61C4×D283—41no—
62C4×D283—41no—
63C4×D283—41no—
64C4×D283—41no—
65C283⋊C8—41no—
66C283⋊C8—41no—
67C23×C283—41no—
68Q8×C283—41no—
69Q8×C283—41no—
70D4×C283—42no—
71D4×C283—41no—
72D4×C283—41no—
73D4×C283—41no—
74D4×C283—41no—
75C2×C4×C283—22no—
76C2×C4×C283C2×D411yescharacteristic
77C2×C4×C283C2×D411yescharacteristic
78C2×C4×C283C2×D411yescharacteristic
79C8×C283—41no—
80C22×C283—81no—
81C22×C283—81no—
82C22×C283—81no—
83C22×C283#12011yescharacteristic
84C2×D283—22no—
85C2×D283—41no—
86C2×D283—41no—
87C2×D283—41no—
88C2×D283—41no—
89C2×D283—41no—
90C4×C283—21no—
91C4×C283—42no—
92C4×C283C2×D811yescharacteristic
93C4×C283#13511yescharacteristic
94C4×C283AGL(1,ℤ/8)11yescharacteristic
95C4×C283AGL(1,ℤ/8)11yescharacteristic
96Dic283—41no—
97Dic283—41no—
98C2×C283—161no—
99C2×C283#44711yescharacteristic
100C2×C283#44911yescharacteristic
101C2×C283#45411yescharacteristic
102D283—42no—
103D283—81no—
104C283#162511yescharacteristic, complemented, Sylow
105#1625—2831nomaximal, Sylow
106#547—2831no—
107#486D28311yescharacteristic, complemented
108#459—2831no—
109#459—2831no—
110#332—2831no—
111D4⋊C8—2831no—
112D4⋊C8—2831no—
113C2×#41—5661no—
114C2×D8—5661no—
115C2×D8—5661no—
116C4⋊Q8C2×D28311yescharacteristic
117#127C2×D28311yescharacteristic
118#123—5662no—
119C22⋊Q8—5661no—
120#121—5661no—
121C4×D4—2831no—
122C4×D4—2831no—
123C2×C4⋊C4—5661no—
124C4⋊C8—2831no—
125C4⋊C8—2831no—
126#102—22no—
127#102—5661no—
128#102—5661no—
129C4×C8C2×D28311yescharacteristic
130#41—11322no—
131#41—11322no—
132C2×Q8—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—2831no—
137C2×D4—2831no—
138C2×D4—5662no—
139C22×C4—5662no—
140C22×C4—5661no—
141C22×C4—5661no—
142D8—11322no—
143D8—11322no—
144C2×C8—21no—
145C2×C8—5661no—
146C2×C8—5661no—
147C4⋊C4—21no—
148C4⋊C4—2831no—
149C4⋊C4—2831no—
150C4⋊C4—5662no—
151C4⋊C4C22⋊D28312yes—
152C22⋊C4—5662no—
153C22⋊C4—5661no—
154C22⋊C4—5661no—
155C42C22×D28311yescharacteristic
156C23—41no—
157C23—5661no—
158C23—5661no—
159Q8—41no—
160Q8—41no—
161D4—42no—
162D4—41no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—2832no—
167D4—5661no—
168D4—5661no—
169D4—5661no—
170D4—11324no—
171C2×C4—22no—
172C2×C4—5661no—
173C2×C4—5661no—
174C2×C4—11324no—
175C2×C4—11321no—
176C2×C4—11321no—
177C2×C4—11321no—
178C2×C4—11321no—
179C2×C4D4×D28311yescharacteristic
180C2×C4D4×D28311yescharacteristic
181C2×C4C2×C22⋊D28311yescharacteristic
182C8—41no—
183C8—11321no—
184C8—11321no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—5662no—
189C22—11321no—
190C22—11321no—
191C22—11321no—
192C22—11321no—
193C22—11321no—
194C22#2258357811yescharacteristic
195C4—21no—
196C4—42no—
197C4—11321no—
198C4—11321no—
199C4D8×D28311yescharacteristic
200C4#2258363111yescharacteristic
201C4#2258363411yescharacteristic
202C4#2258363711yescharacteristic
203C2—161no—
204C2—11322no—
205C2—22641no—
206C2#3182659711yescharacteristic
207C2#3182660911yescharacteristic
208C2#3182663711yescharacteristic
subgroupG/H
centerC22#22583578
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#486×C283C2
socleC22×C283#120
hypercenter#486D283
hypocenterC283#1625
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #31827118C2semidirect
2. #31826657C2semidirect
3. #31826657C2semidirect
4. #31826211C2semidirect
5. #31826194C2semidirect
6. #31826194C2semidirect
7. #486×C283C2semidirect
57. D1132D8semidirect
104. C283#1625semidirect
107. #486D283semidirect

Maximal subgroups

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

subgroupindex
#318271182
#318266572
#318262112
#318261942
#486×C2832
#1625283