Finite Group Databasebeta

← #71959

Subgroups — #71959

Order 384.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C3×#563C211yesmaximal, characteristic, complemented
2#5095C212yesmaximal, complemented
3#5062C211yesmaximal, characteristic
4#4548C212yesmaximal, complemented
5#4542C211yesmaximal, characteristic, complemented
6#2203—31nomaximal, Sylow
7#1007C2211yescharacteristic
8C3×C4⋊Q8C2212yes—
9C3×#126—22no—
10C3×#125—21no—
11C3×#124C2211yescharacteristic
12C3×#123—21no—
13C3×C22⋊Q8—22no—
14#942—22no—
15#932—21no—
16#836C2212yes—
17#836C2211yescharacteristic
18#805—21no—
19#805—22no—
20#563D311yescharacteristic, complemented
21#456—32no—
22#454—31no—
23#356—32no—
24#355—31no—
25C2×C3×Q8D412yes—
26C2×C3×Q8C2311yescharacteristic
27C2×C3×D4D411yescharacteristic
28C22×C3×C4—21no—
29#239—41no—
30D3×Q8—41no—
31#235—21no—
32#235—22no—
33C3×C4⋊C4—21no—
34C3×C4⋊C4—22no—
35C3×C4⋊C4—21no—
36C3×C4⋊C4—22no—
37C3×C4⋊C4—22no—
38C3×C22⋊C4—22no—
39C3×C22⋊C4—21no—
40C3×C22⋊C4—22no—
41C3×C42D411yescharacteristic
42C3×C42D412yes—
43#216—42no—
44#215—42no—
45#214—41no—
46#213—41no—
47#208—21no—
48#208—22no—
492−1+4—31no—
50#137—62no—
51AGL(1,ℤ/8)—61no—
52C4⋊Q8C2×D312yes—
53#126—22no—
54#125—21no—
55#124C2×D311yescharacteristic
56#123—21no—
57C22⋊Q8—22no—
58#104—61no—
59#104—62no—
60#101—32no—
61#101—31no—
62C23×C3—21no—
63C3×Q8—41no—
64C3×Q8—42no—
65C3×D4—41no—
66C2×C3×C4—21no—
67C2×C3×C4—22no—
68C2×C3×C4—22no—
69C2×C3×C4—81no—
70C2×C3×C4C2×D412yes—
71C2×C3×C4C2×D411yescharacteristic
72#67—41no—
73D12—21no—
74D12—22no—
75C4×D3—41no—
76C4×D3—42no—
77Dic6—21no—
78Dic6—22no—
79C3⋊C8—41no—
80C3⋊C8—42no—
81#41—61no—
82#41—62no—
83#41—121no—
84C2×Q8—121no—
85C2×Q8#6712yes—
86C2×Q8C22×D311yescharacteristic
87C2×D4#6711yescharacteristic
88C22×C4—21no—
89Q16—122no—
90SD16—121no—
91SD16—122no—
92D8—121no—
93M16—61no—
94M16—62no—
95C4⋊C4—21no—
96C4⋊C4—22no—
97C4⋊C4—21no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C22⋊C4—22no—
101C22⋊C4—21no—
102C22⋊C4—22no—
103C42#6711yescharacteristic
104C42#6712yes—
105C22×C3—42no—
106C22×C3—41no—
107C22×C3#12011yescharacteristic
108C2×D3—41no—
109C3×C4—21no—
110C3×C4—22no—
111C3×C4—41no—
112C3×C4—42no—
113C3×C4—42no—
114Dic3—41no—
115C23—21no—
116Q8—41no—
117Q8—42no—
118Q8—61no—
119Q8—62no—
120D4—41no—
121D4—61no—
122D4—62no—
123D4—121no—
124C2×C4—21no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—81no—
128C2×C4—121no—
129C2×C4—122no—
130C2×C4C2×#6712yes—
131C2×C4C2×#6711yescharacteristic
132C8—121no—
133C8—122no—
134C2×C3—21no—
135C2×C3—81no—
136C2×C3B(4,𝔽2)11yescharacteristic
137D3—81no—
138C22—42no—
139C22—41no—
140C22—121no—
141C22#95311yescharacteristic
142C4—21no—
143C4—22no—
144C4—41no—
145C4—42no—
146C4—42no—
147C4—121no—
148C3#220311yescharacteristic, complemented, Sylow
149C2—21no—
150C2—81no—
151C2—241no—
152C2#524411yescharacteristic
subgroupG/H
centerC2#5244
commutator, cosocleC2×C3×Q8C23
FrattiniC2×Q8C22×D3
FittingC3×#563C2
socleC2×C3B(4,𝔽2)
prime radical#5062C2
hypercenter#563D3
hypocenterC3#2203
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. C3×#563C2semidirect
2. #5095C2semidirect
4. #4548C2semidirect
5. #4542C2semidirect
20. #563D3semidirect
148. C3#2203semidirect

Maximal subgroups

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

subgroupindex
C3×#5632
#50952
#50622
#45482
#45422
#22033