Finite Group Databasebeta

← #49120389

Subgroups — #49120389

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#31827135C211yesmaximal, characteristic
2#31826837C211yesmaximal, characteristic, complemented
3#31826761C211yesmaximal, characteristic, complemented
4#31826756C211yesmaximal, characteristic, complemented
5#31826195C211yesmaximal, characteristic, complemented
6#31826192C211yesmaximal, characteristic, complemented
7#331×C283C211yesmaximal, characteristic, complemented
8C2×#17985391—21no—
9C2×#17985380—22no—
10#22583610C2211yescharacteristic
11#22583606—21no—
12#22583582—22no—
13#22583581—21no—
14C4×Dic566C2211yescharacteristic
15C4×D1132C2211yescharacteristic
16D4×Dic283—21no—
17#22583544—21no—
18#22583542—21no—
19#22583535—21no—
20#22583532—21no—
21C4×C283⋊C8C2211yescharacteristic
22#127×C283C2211yescharacteristic
23C4⋊C8×C283C2211yescharacteristic
24C4⋊C8×C283C2211yescharacteristic
25#17985391—42no—
26#17985391—42no—
27C2×Dic566—21no—
28C2×Dic566D411yescharacteristic
29C2×D1132D411yescharacteristic
30C2×C22⋊D283—22no—
31C2×C22⋊D283—22no—
32C22×Dic283—22no—
33C2×C4×D283—21no—
34#17985380—44no—
35C2×C283⋊C8—21no—
36C4⋊Dic283—21no—
37C4⋊Dic283—21no—
38C4⋊Dic283D411yescharacteristic
39C4⋊Dic283D411yescharacteristic
40Dic283⋊C4—21no—
41C22⋊Dic283—22no—
42(C2×D283)⋊C4—21no—
43C4×Dic283—21no—
44C2×D4×C283—21no—
45C2×D4×C283—21no—
46C2×D4×C283D412yes—
47C2×C8×C283—21no—
48C2×C8×C283—21no—
49C42×C283C2311yescharacteristic
50C22×D283—21no—
51Dic566—21no—
52Dic566—42no—
53Dic566SD1612yescomplemented
54D1132—21no—
55D1132—21no—
56C22⋊D283—44no—
57C22⋊D283—82no—
58C2×Dic283—21no—
59C2×Dic283—21no—
60C2×Dic283—21no—
61C2×Dic283—44no—
62C2×Dic283—41no—
63C4×D283—41no—
64C4×D283—41no—
65C283⋊C8—22no—
66C23×C283—22no—
67D4×C283—42no—
68D4×C283—41no—
69D4×C283—41no—
70D4×C283—41no—
71D4×C283—41no—
72C2×C4×C283C2×D411yescharacteristic
73C2×C4×C283C2×D411yescharacteristic
74C2×C4×C283C2×D411yescharacteristic
75C8×C283—41no—
76C8×C283—41no—
77C22×C283—42no—
78C22×C283—42no—
79C22×C283—42no—
80C22×C283#12011yescharacteristic
81C2×D283—41no—
82C2×D283—41no—
83C2×D283—41no—
84C4×C283—21no—
85C4×C283C2×SD1611yescharacteristic
86C4×C283#13511yescharacteristic
87C4×C283AGL(1,ℤ/8)11yescharacteristic
88C4×C283#13711yescharacteristic
89Dic283—22no—
90Dic283—41no—
91Dic283—41no—
92Dic283—81no—
93C2×C283—82no—
94C2×C283#45011yescharacteristic
95C2×C283#45211yescharacteristic
96C2×C283#45311yescharacteristic
97D283—81no—
98C283#163111yescharacteristic, complemented, Sylow
99#1631—2831nomaximal, Sylow
100#550—2831no—
101#492—2831no—
102#477—2831no—
103#476—2831no—
104#331D28311yescharacteristic, complemented
105Q8⋊C8—2831no—
106D4⋊C8—2831no—
107C2×#41—5661no—
108C2×SD16—5662no—
109C4⋊Q8—2831no—
110#127—5661no—
111#127C2×D28311yescharacteristic
112#121—5661no—
113#121—5662no—
114C4×Q8—2831no—
115C4×D4—2831no—
116C4×D4—5661no—
117#107—5661no—
118#106—5661no—
119C4⋊C8C2×D28311yescharacteristic
120C4⋊C8C2×D28311yescharacteristic
121#103—5661no—
122#102—5661no—
123C4×C8—2831no—
124#41—11322no—
125#41—11322no—
126C2×Q8—2831no—
127C2×Q8—5661no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—2831no—
131C2×D4—5662no—
132C2×D4—5662no—
133C2×D4D113212yes—
134C22×C4—5662no—
135C22×C4—5661no—
136SD16—11324no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—5661no—
140C4⋊C4—2831no—
141C4⋊C4—2831no—
142C4⋊C4—5661no—
143C4⋊C4—5661no—
144C4⋊C4—5661no—
145C22⋊C4—5662no—
146C22⋊C4—5661no—
147C42—5661no—
148C42C22×D28311yescharacteristic
149C23—22no—
150C23—5661no—
151Q8—2832no—
152Q8—5661no—
153Q8—11322no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—5661no—
160D4—5661no—
161D4—11324no—
162D4—22642no—
163C2×C4—5661no—
164C2×C4—5661no—
165C2×C4—5661no—
166C2×C4—11324no—
167C2×C4—11321no—
168C2×C4—11321no—
169C2×C4—11321no—
170C2×C4D4×D28311yescharacteristic
171C2×C4D4×D28311yescharacteristic
172C2×C4C2×D113211yescharacteristic
173C8—41no—
174C8—41no—
175C8—5662no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—11321no—
180C22—11321no—
181C22—11321no—
182C22#2258358011yescharacteristic
183C4—21no—
184C4—5662no—
185C4—11321no—
186C4—11321no—
187C4—22641no—
188C4SD16×D28311yescharacteristic
189C4#2258362911yescharacteristic
190C4#2258363411yescharacteristic
191C4#2258363911yescharacteristic
192C2—82no—
193C2—22641no—
194C2#3182661711yescharacteristic
195C2#3182662811yescharacteristic
196C2#3182663511yescharacteristic
subgroupG/H
centerC22#22583580
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#331×C283C2
socleC22×C283#120
prime radical#31827135C2
hypercenter#331D283
hypocenterC283#1631
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #31826837C2semidirect
3. #31826761C2semidirect
4. #31826756C2semidirect
5. #31826195C2semidirect
6. #31826192C2semidirect
7. #331×C283C2semidirect
53. Dic566SD16semidirect
98. C283#1631semidirect
104. #331D283semidirect

Maximal subgroups

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

subgroupindex
#318271352
#318268372
#318267612
#318267562
#318261952
#318261922
#331×C2832
#1631283