Finite Group Databasebeta

← #49120464

Subgroups — #49120464

Order 36224.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#31827137C211yesmaximal, characteristic
2#31826812C211yesmaximal, characteristic, complemented
3#31826750C211yesmaximal, characteristic, complemented
4#31826671C211yesmaximal, characteristic, complemented
5#31826208C211yesmaximal, characteristic, complemented
6#31826194C211yesmaximal, characteristic, complemented
7Q8⋊C8×C283C211yesmaximal, characteristic, complemented
8C2×#17985393—21no—
9C2×#17985381—21no—
10#22583610C2211yescharacteristic
11#22583607—21no—
12#22583607C2211yescharacteristic
13#22583585—22no—
14#22583585—21no—
15C4×D1132—21no—
16C4×D1132C2211yescharacteristic
17#22583541—21no—
18C4⋊(C283⋊C8)C2211yescharacteristic
19#22583535—21no—
20#22583534—21no—
21#22583532—22no—
22C4×Q8×C283C2211yescharacteristic
23C4⋊C8×C283C2211yescharacteristic
24C4×C8×C283C2211yescharacteristic
25#17985393—42no—
26#17985393—42no—
27C2×Dic566—21no—
28C2×D1132—22no—
29C2×D1132—22no—
30C2×D1132—21no—
31C2×D1132—21no—
32C2×D1132D411yescharacteristic
33C2×D1132D412yes—
34C2×C4×D283—22no—
35C2×C4×D283—21no—
36#17985381—42no—
37C2×C283⋊C8—21no—
38C4⋊Dic283—21no—
39C4⋊Dic283—21no—
40C4⋊Dic283D411yescharacteristic
41(C2×D283)⋊C4—22no—
42(C2×D283)⋊C4—21no—
43C2×Q8×C283D411yescharacteristic
44C2×C8×C283—21no—
45C2×C8×C283—21no—
46C4⋊C4×C283—21no—
47C4⋊C4×C283D411yescharacteristic
48C42×C283—21no—
49C42×C283C2311yescharacteristic
50C22×D283—21no—
51C22×D283—22no—
52Dic566—41no—
53Dic566—41no—
54D1132—21no—
55D1132—21no—
56D1132—42no—
57D1132—44no—
58D1132—41no—
59D1132—41no—
60D1132—41no—
61D1132—41no—
62D1132—82no—
63C2×Dic283—21no—
64C2×Dic283—41no—
65C4×D283—44no—
66C4×D283—41no—
67C4×D283—41no—
68C283⋊C8—41no—
69Q8×C283—21no—
70Q8×C283SD1612yescomplemented
71C2×C4×C283—21no—
72C2×C4×C283—21no—
73C2×C4×C283C2×D411yescharacteristic
74C2×C4×C283C2×D411yescharacteristic
75C2×C4×C283C2×D411yescharacteristic
76C8×C283—41no—
77C8×C283—41no—
78C22×C283#12011yescharacteristic
79C2×D283—41no—
80C2×D283—42no—
81C2×D283—42no—
82C2×D283—41no—
83C2×D283—42no—
84C2×D283—41no—
85C4×C283—22no—
86C4×C283—21no—
87C4×C283—41no—
88C4×C283C2×SD1611yescharacteristic
89C4×C283#13511yescharacteristic
90C4×C283AGL(1,ℤ/8)11yescharacteristic
91C4×C283AGL(1,ℤ/8)11yescharacteristic
92Dic283—41no—
93Dic283—81no—
94C2×C283#44911yescharacteristic
95C2×C283#45011yescharacteristic
96C2×C283#45511yescharacteristic
97D283—81no—
98D283—82no—
99C283#164211yescharacteristic, complemented, Sylow
100#1642—2831nomaximal, Sylow
101#550—2831no—
102#486—2831no—
103#475—2831no—
104#461—2831no—
105#331—2831no—
106Q8⋊C8D28311yescharacteristic, complemented
107D4⋊C8—2831no—
108C2×#41—5661no—
109C2×SD16—5661no—
110C4⋊Q8—2831no—
111#127—2831no—
112#127—5661no—
113#121—5662no—
114#121—5661no—
115C4×Q8C2×D28311yescharacteristic
116C4×D4—2831no—
117C4×D4—5661no—
118#106—5661no—
119C4⋊C8—2831no—
120C4⋊C8C2×D28311yescharacteristic
121#103—5661no—
122#103—5661no—
123#102—5662no—
124C4×C8C2×D28311yescharacteristic
125#41—11322no—
126#41—11322no—
127C2×Q8—5661no—
128C2×Q8D113211yescharacteristic
129C2×D4—2831no—
130C2×D4—2832no—
131C2×D4—5662no—
132C2×D4—5662no—
133C2×D4—5661no—
134C2×D4—5661no—
135C22×C4—5662no—
136C22×C4—5661no—
137SD16—11322no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—5661no—
141C4⋊C4—21no—
142C4⋊C4—2831no—
143C4⋊C4—5661no—
144C4⋊C4—5661no—
145C4⋊C4D113211yescharacteristic
146C22⋊C4—5662no—
147C22⋊C4—5661no—
148C42—21no—
149C42C22×D28311yescharacteristic
150C23—5661no—
151C23—5662no—
152Q8—21no—
153Q8—11321no—
154Q8—11321no—
155Q8#1798538112yescomplemented
156D4—5661no—
157D4—5661no—
158D4—11322no—
159D4—11324no—
160D4—11321no—
161D4—11321no—
162D4—11321no—
163D4—11321no—
164D4—22642no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—5661no—
168C2×C4—11324no—
169C2×C4—11321no—
170C2×C4—11321no—
171C2×C4—11321no—
172C2×C4D4×D28311yescharacteristic
173C2×C4D4×D28311yescharacteristic
174C2×C4C2×D113211yescharacteristic
175C8—41no—
176C8—41no—
177C8—11321no—
178C22—11321no—
179C22—11322no—
180C22—11322no—
181C22—11321no—
182C22—11322no—
183C22—11321no—
184C22#2258358011yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—11321no—
189C4—22641no—
190C4C2×#1798538111yescharacteristic
191C4#2258363311yescharacteristic
192C4#2258363711yescharacteristic
193C4#2258363811yescharacteristic
194C2—22641no—
195C2—22642no—
196C2#3182661311yescharacteristic
197C2#3182661811yescharacteristic
198C2#3182664411yescharacteristic
subgroupG/H
centerC22#22583580
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
FittingQ8⋊C8×C283C2
socleC22×C283#120
prime radical#31827137C2
hypercenterQ8⋊C8D283
hypocenterC283#1642
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #31826812C2semidirect
3. #31826750C2semidirect
4. #31826671C2semidirect
5. #31826208C2semidirect
6. #31826194C2semidirect
7. Q8⋊C8×C283C2semidirect
70. Q8×C283SD16semidirect
99. C283#1642semidirect
106. Q8⋊C8D283semidirect
155. Q8#17985381semidirect

Maximal subgroups

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

subgroupindex
#318271372
#318268122
#318267502
#318266712
#318262082
#318261942
Q8⋊C8×C2832
#1642283