Finite Group Databasebeta

← #11785511

Subgroups — #11785511

Order 1152.

756 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#64116C211yesmaximal, characteristic
2#10592831C212yesmaximal
3C2×#63649C212yesmaximal, complemented
4C32×#338C211yesmaximal, characteristic, complemented
5#10589426C211yesmaximal, characteristic, complemented
6#68724—34nomaximal
7C2×#3828C2211yescharacteristic
8C2×C32×M16C2212yes—
9C2×C32×M16C2211yescharacteristic
10C2×#3748C2212yes—
11C22×C3⋊(C3⋊C8)C2211yescharacteristic
12#64116—24no—
13C2×#3743C412yes—
14C2×C8×C3⋊D3C412yes—
15#63651—22no—
16#63650—22no—
17#63649—24no—
18C2×#907—34no—
19#5132—38no—
20C2×#822—38no—
21C3×#338D314yescomplemented
22C3⋊#338—34no—
23C22×C32×C4C2311yescharacteristic
24C2×#638Q812yes—
25#3828—22no—
26#3828—22no—
27C2×C4⋊(C3⋊D3)D411yescharacteristic
28C2×C4×C3⋊D3C2×C412yes—
29C2×#634D411yescharacteristic
30C32×M16—22no—
31C32×M16—22no—
32C32×M16—22no—
33C2×C32×C8—22no—
34C2×C32×C8C2×C412yes—
35#3748—22no—
36#3748—22no—
37C2×C3⋊(C3⋊C8)—22no—
38C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
39C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
40C2×C3⋊(C3⋊C8)D414yes—
41#3743—24no—
42C8×C3⋊D3—24no—
43#1813—91noSylow
44C2×#235—34no—
45C2×C3×M16C2×D318yes—
46C2×C3×M16C2×D314yes—
47C2×#208—38no—
48C22×C3⋊C8—34no—
49#907—616no—
50C2×ΓL(1,𝔽25)—38no—
51C2×D3×C8—38no—
52#824—68no—
53#823—68no—
54#822—616no—
55C23×C32C2×Q811yescharacteristic
56C22×C3⋊D3C4⋊C412yes—
57C2×C32×C4C22×C411yescharacteristic
58C2×C32×C4C2×D412yes—
59C2×C32×C4C2×D411yescharacteristic
60C2×C32×C4#4112yes—
61#638—24no—
62C2×C3⋊Dic3C4⋊C412yes—
63C4⋊(C3⋊D3)—22no—
64C4×C3⋊D3—22no—
65C4×C3⋊D3—22no—
66#634—22no—
67C32×C8—22no—
68C32×C8—42no—
69C3⋊(C3⋊C8)—42no—
70C3⋊(C3⋊C8)C22⋊C412yes—
71C3⋊(C3⋊C8)C22⋊C412yes—
72C2×#131—91no—
73C2×#108—92no—
74#413—92no—
75#338—91no—
76#338C3⋊D311yescharacteristic, complemented
77C22×C3×C4C22×D314yes—
78C2×#67—38no—
79#235—68no—
80#235—68no—
81C2×D12—34no—
82C2×C4×D3—38no—
83C2×Dic6—34no—
84C3×M16—28no—
85C3×M16—28no—
86C3×M16—28no—
87C2×C3×C8—28no—
88C2×C3×C8C4×D318yes—
89#208—68no—
90#208—68no—
91C2×C3⋊C8—34no—
92C2×C3⋊C8—34no—
93C2×C3⋊C8—316no—
94C2×C3⋊C8—68no—
95ΓL(1,𝔽25)—616no—
96D3×C8—616no—
97C22×C32—41no—
98C22×C32C2×C4⋊C411yescharacteristic
99C22×C32C22⋊Q812yes—
100C2×C3⋊D3—22no—
101C2×C3⋊D3—42no—
102C32×C4C2×C22⋊C411yescharacteristic
103C32×C4#11711yescharacteristic
104C32×C4#12112yes—
105C3⋊Dic3—22no—
106C2×#41—91no—
107#131—184no—
108C2×M16—92no—
109C2×M16—92no—
110C2×M16C2×C3⋊D312yes—
111C2×M16C2×C3⋊D311yescharacteristic
112C22×C8—91no—
113C22×C8—92no—
114#108—184no—
115#101—182no—
116#100—182no—
117C23×C3D3×Q814yes—
118C22×D3—38no—
119C2×C3×C4C2×C4×D314yes—
120C2×C3×C4#23518yes—
121C2×C3×C4D3×D418yes—
122C2×C3×C4D3×D414yes—
123#67—616no—
124C2×Dic3—38no—
125D12—68no—
126C4×D3—68no—
127C4×D3—68no—
128Dic6—68no—
129C3×C8—28no—
130C3×C8—48no—
131C3⋊C8—38no—
132C3⋊C8—38no—
133C3⋊C8—128no—
134C2×C32—21no—
135C2×C32—22no—
136C2×C32#38011yescharacteristic
137C3⋊D3—42no—
138#41—182no—
139#41—182no—
140C2×Q8—91no—
141C2×D4—92no—
142C2×D4—91no—
143C22×C4—92no—
144C22×C4C22×C3⋊D311yescharacteristic
145M16—22no—
146M16—22no—
147M16—22no—
148M16—184no—
149M16—182no—
150M16—182no—
151C2×C8—22no—
152C2×C8—91no—
153C2×C8—91no—
154C2×C8—94no—
155C2×C8—184no—
156C2×C8—182no—
157C2×C8C4×C3⋊D312yes—
158C22×C3—44no—
159C22×C3D3×C4⋊C414yes—
160C22×C3#89618yes—
161C2×D3—68no—
162C2×D3—128no—
163C3×C4#87214yes—
164C3×C4D3×C22⋊C414yes—
165C3×C4#88418yes—
166Dic3—68no—
167C32#181311yescharacteristic, complemented, Sylow
168C23—92no—
169C23Q8×C3⋊D311yescharacteristic
170Q8—182no—
171D4—184no—
172D4—182no—
173C2×C4—92no—
174C2×C4—182no—
175C2×C4—182no—
176C2×C4C2×C4×C3⋊D311yescharacteristic
177C2×C4#382812yes—
178C2×C4D4×C3⋊D312yes—
179C2×C4D4×C3⋊D311yescharacteristic
180C8—22no—
181C8—42no—
182C8—92no—
183C8—92no—
184C8—362no—
185C2×C3—24no—
186C2×C3—28no—
187C2×C3#466714yes—
188D3—128no—
189C22—41no—
190C22—182no—
191C22—362no—
192C22C4⋊C4×C3⋊D311yescharacteristic
193C22#6410512yes—
194C4—182no—
195C4#6408111yescharacteristic
196C4C22⋊C4×C3⋊D311yescharacteristic
197C4#6409312yes—
198C3#6872414yescomplemented
199C2—21no—
200C2—22no—
201C2—362no—
202C2#1059236611yescharacteristic
subgroupG/H
centerC4#64081
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC32×#338C2
socleC2×C32#380
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenter#338C3⋊D3
hypocenterC32#1813
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×M16 ◁ #338

Complements

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

NH ≅ G/Ntype
3. C2×#63649C2semidirect
4. C32×#338C2semidirect
5. #10589426C2semidirect
21. C3×#338D3semidirect
76. #338C3⋊D3semidirect
167. C32#1813semidirect
198. C3#68724semidirect

Maximal subgroups

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

subgroupindex
C2×#641162
#105928312
C2×#636492
C32×#3382
#105894262
#687243