Finite Group Databasebeta

← #11790548

Subgroups — #11790548

Order 1152.

222 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 61 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#10593738C211yesmaximal, characteristic
2#10593565C212yesmaximal
3C2×#63748C211yesmaximal, characteristic
4C2×#63748C211yesmaximal, characteristic
5#10590156C212yesmaximal
6C2×#3792C2212yes—
7C2×#3792C2212yes—
8C2×#3792C2211yescharacteristic
9C22×#622C2211yescharacteristic
10#64086C2211yescharacteristic
11#63777—21no—
12C4×#622—21no—
13#63748—24no—
14#63748—21no—
15#63748—21no—
16#63748C414yescomplemented
17C22×C3⋊Dic3C2311yescharacteristic
18#3792—22no—
19#3792—21no—
20#3792—22no—
21#3792—22no—
22#3792C2×C414yes—
23#3792D412yes—
24C2×#622—22no—
25C2×#622—21no—
26C2×#622—22no—
27C2×#622C2×C412yes—
28C2×#622D411yescharacteristic
29C2×#622D411yescharacteristic
30C2×#622Q812yes—
31C32×C22⋊C4Q811yescharacteristic
32#3757Q811yescharacteristic
33#3750—21no—
34C4×C3⋊Dic3—21no—
35#1953—91nomaximal, Sylow
36#877—121no—
37C23×C32C2×Q811yescharacteristic
38C2×C32×C4—21no—
39C2×C3⋊Dic3—21no—
40C2×C3⋊Dic3C22×C411yescharacteristic
41C2×C3⋊Dic3C2×D411yescharacteristic
42C2×C3⋊Dic3C2×D411yescharacteristic
43C2×C3⋊Dic3C2×Q811yescharacteristic
44C2×C3⋊Dic3#4111yescharacteristic
45C2×C3⋊Dic3#4111yescharacteristic
46#622—24no—
47#622—22no—
48#622—42no—
49#622C4⋊C412yes—
50#622C4⋊C412yes—
51#430—92no—
52C2×#108—91no—
53C2×#108—91no—
54#405—91no—
55#344—92no—
56C22×Dic3—121no—
57C3×C22⋊C4—41no—
58#217—121no—
59Dic3⋊C4—122no—
60C4×Dic3—122no—
61C22×C32—41no—
62C22×C32C4×Q811yescharacteristic
63C22×C32C22⋊Q811yescharacteristic
64C22×C32C4⋊Q811yescharacteristic
65C32×C4—41no—
66C3⋊Dic3—41no—
67C3⋊Dic3C2×C4⋊C411yescharacteristic
68C3⋊Dic3C4×D411yescharacteristic
69C3⋊Dic3#12111yescharacteristic
70C3⋊Dic3#12511yescharacteristic
71C2×M16—92no—
72C2×M16—91no—
73C2×M16—92no—
74C22×C8—91no—
75#117—91no—
76#108—94no—
77#108—181no—
78#108—181no—
79#108—184no—
80#97—181no—
81C4×C8—181no—
82C23×C3—41no—
83C2×C3×C4—42no—
84C2×Dic3—121no—
85C2×Dic3—121no—
86C2×Dic3—121no—
87C2×Dic3—121no—
88C2×Dic3—121no—
89C2×Dic3—121no—
90C2×Dic3—122no—
91C2×C32—21no—
92C2×C32—21no—
93C2×C32—21no—
94C2×C32#38911yescharacteristic
95C22×C4—91no—
96M16—94no—
97M16—92no—
98M16—181no—
99M16—182no—
100M16—182no—
101M16—182no—
102C2×C8—91no—
103C2×C8—91no—
104C2×C8—92no—
105C2×C8—92no—
106C2×C8—181no—
107C2×C8—182no—
108C2×C8—182no—
109C4⋊C4—181no—
110C22⋊C4—91no—
111C22⋊C4PSU(3,𝔽2)11yescharacteristic
112C42—181no—
113C22×C3—41no—
114C22×C3—41no—
115C22×C3—41no—
116C22×C3—82no—
117C3×C4—82no—
118Dic3—121no—
119Dic3—121no—
120Dic3—121no—
121Dic3—121no—
122Dic3—242no—
123C32#195311yescharacteristic, complemented, Sylow
124C23C2×PSU(3,𝔽2)11yescharacteristic
125C2×C4—21no—
126C2×C4—91no—
127C2×C4—91no—
128C2×C4—91no—
129C2×C4—91no—
130C2×C4—91no—
131C2×C4—91no—
132C2×C4—181no—
133C8—92no—
134C8—92no—
135C8—184no—
136C8—182no—
137C8—362no—
138C2×C3—42no—
139C2×C3—41no—
140C2×C3—81no—
141C2×C3—81no—
142C22—41no—
143C22C4×PSU(3,𝔽2)11yescharacteristic
144C22C4⋊PSU(3,𝔽2)11yescharacteristic
145C22C22⋊PSU(3,𝔽2)11yescharacteristic
146C4—41no—
147C4—91no—
148C4—91no—
149C4—91no—
150C4—91no—
151C4—361no—
152C3—41no—
153C2—21no—
154C2—21no—
155C2—21no—
156C2#1059354811yescharacteristic
subgroupG/H
centerC2#10593548
commutatorC2×C3⋊Dic3C22×C4
FrattiniC23C2×PSU(3,𝔽2)
FittingC32×C22⋊C4Q8
socleC2×C32#389
prime radicalC23×C32C2×Q8
cosocleC22×C3⋊Dic3C23
hypercenterC22⋊C4PSU(3,𝔽2)
hypocenterC32#1953
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C23 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
16. #63748C4semidirect
123. C32#1953semidirect

Maximal subgroups

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

subgroupindex
#105937382
#105935652
C2×#637482
#105901562
#19539