Finite Group Databasebeta

← #11799718

Subgroups — #11799718

Order 1152.

305 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 72 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#10592895C211yesmaximal, characteristic
2#10590900C211yesmaximal, characteristic, complemented
3C3×#4566C212yesmaximal
4#10589136C212yesmaximal, complemented
5#10589135C211yesmaximal, characteristic, complemented
6#69246—32nomaximal
7C32×#135C2211yescharacteristic
8#64157—21no—
9#63919—21no—
10#63664—21no—
11#63646C2211yescharacteristic
12C8×C3⋊Dic3C2211yescharacteristic
13C3×#817C2212yes—
14C3×#812C2212yes—
15#63574—21no—
16#5196—32no—
17#5196—61no—
18#4855—32no—
19C3×#360—32no—
20#4566D312yes—
21#4513—32no—
22#4511—32no—
23#4487—32no—
24C32×#41—21no—
25C2×#627—41no—
26C32×Q16Q811yescharacteristic, complemented
27C32×SD16—21no—
28C32×D8Q811yescharacteristic, complemented
29C2×C32×C8C2311yescharacteristic
30C4×C3⋊Dic3D411yescharacteristic
31#3748—21no—
32#3748—41no—
33C2×C3⋊(C3⋊C8)—41no—
34C2×C3⋊(C3⋊C8)D411yescharacteristic
35C3×C4⋊Dic3—22no—
36#3724—41no—
37C3×C3⋊C16—22no—
38#2230—91noSylow
39C3×#135—21no—
40C3×#135C2×D312yes—
41#948—62no—
42#948—62no—
43#888—62no—
44C3×M32—32no—
45C3×#106—32no—
46#837—62no—
47#837—62no—
48#819—32no—
49#819—61no—
50#817C2×D312yes—
51C8×Dic3—32no—
52C8×Dic3—61no—
53#812C2×D312yes—
54#808—62no—
55C32×Q8—21no—
56C32×D4—21no—
57C32×D4—41no—
58C2×C32×C4—41no—
59C2×C32×C4C2×D411yescharacteristic
60C2×C3⋊Dic3—21no—
61C2×C3×Dic3—42no—
62#627—82no—
63C32×C8C2×Q811yescharacteristic
64C32×C8#4111yescharacteristic
65C3⋊(C3⋊C8)—41no—
66C3⋊(C3⋊C8)SD1611yescharacteristic
67C3⋊(C3⋊C8)SD1611yescharacteristic
68#498—91no—
69#443—91no—
70#365—92no—
71#364—91no—
72#360—32no—
73C3×#41—22no—
74C3×#41—22no—
75C2×Dic6—122no—
76C3×Q16—21no—
77C3×Q16Dic612yescomplemented
78C3×SD16—22no—
79C3×SD16—22no—
80C3×D8—21no—
81C3×D8Dic612yescomplemented
82C2×C3×C8—21no—
83C2×C3×C8C22×D312yes—
84C3×C4⋊C4—62no—
85C4⋊Dic3—22no—
86Dic3⋊C4—122no—
87C4×Dic3—32no—
88C4×Dic3—61no—
89#208—62no—
90#208—62no—
91#208—122no—
92#208—122no—
93C2×C3⋊C8—32no—
94C2×C3⋊C8—61no—
95C2×C3⋊C8—122no—
96C2×C3⋊C8—122no—
97C3×C16—62no—
98C3⋊C16—22no—
99C22×C32—41no—
100C22×C32AGL(1,ℤ/8)11yescharacteristic
101C32×C4—41no—
102C32×C4C22⋊Q811yescharacteristic
103C32×C4C2×SD1611yescharacteristic
104C3⋊Dic3—22no—
105C3×Dic3—82no—
106#135D3211yescharacteristic
107#131—181no—
108C4⋊Q8—181no—
109M32—32no—
110#108—91no—
111#106—32no—
112#106—181no—
113#104—181no—
114C4×C8—91no—
115C3×Q8—22no—
116C3×Q8—22no—
117C3×D4—22no—
118C3×D4—22no—
119C3×D4—42no—
120C3×D4—42no—
121C2×C3×C4—21no—
122C2×C3×C4—42no—
123C2×C3×C4—42no—
124C2×C3×C4—122no—
125C2×C3×C4D3×D412yes—
126C2×Dic3—42no—
127C2×Dic3—62no—
128C2×Dic3—121no—
129Dic6—244no—
130C3×C8—21no—
131C3×C8—21no—
132C3×C8C2×Dic612yes—
133C3×C8#23712yes—
134C3⋊C8—32no—
135C3⋊C8—32no—
136C3⋊C8—61no—
137C3⋊C8—61no—
138C3⋊C8—122no—
139C3⋊C8—122no—
140C2×C32—21no—
141C2×C32—81no—
142C2×C32#47611yescharacteristic
143#41—21no—
144C2×Q8—361no—
145Q16#62711yescharacteristic, complemented
146SD16—21no—
147D8#62711yescharacteristic, complemented
148M16—181no—
149M16—361no—
150C2×C8—91no—
151C2×C8—361no—
152C2×C8C2×D3211yescharacteristic
153C4⋊C4—62no—
154C4⋊C4—361no—
155C42—91no—
156C16—62no—
157C22×C3—21no—
158C22×C3—42no—
159C22×C3—42no—
160C22×C3#91112yes—
161C3×C4—21no—
162C3×C4—21no—
163C3×C4—42no—
164C3×C4—42no—
165C3×C4—242no—
166C3×C4#87812yes—
167C3×C4D3×SD1612yes—
168Dic3—64no—
169Dic3—82no—
170Dic3—122no—
171C32#223011yescharacteristic, complemented, Sylow
172Q8—21no—
173Q8—722no—
174D4—21no—
175D4—41no—
176C2×C4—41no—
177C2×C4—122no—
178C2×C4—181no—
179C2×C4#381111yescharacteristic
180C8—91no—
181C8—91no—
182C8—361no—
183C8#380511yescharacteristic
184C8C2×#62711yescharacteristic
185C2×C3—22no—
186C2×C3—21no—
187C2×C3—41no—
188C2×C3—82no—
189C2×C3—82no—
190C2×C3#476112yes—
191C22—41no—
192C22#6392711yescharacteristic
193C4—41no—
194C4—182no—
195C4—242no—
196C4#6393011yescharacteristic
197C4#6398211yescharacteristic
198C3—21no—
199C3#6924612yescomplemented
200C2—21no—
201C2—81no—
202C2#1059149511yescharacteristic
subgroupG/H
centerC2#10591495
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
Fitting, prime radicalC32×#135C22
socleC2×C32#476
hypercenter#135D32
hypocenterC32#2230
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #135

Complements

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

NH ≅ G/Ntype
2. #10590900C2semidirect
4. #10589136C2semidirect
5. #10589135C2semidirect
26. C32×Q16Q8semidirect
28. C32×D8Q8semidirect
77. C3×Q16Dic6semidirect
81. C3×D8Dic6semidirect
145. Q16#627semidirect
147. D8#627semidirect
171. C32#2230semidirect
199. C3#69246semidirect

Maximal subgroups

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

subgroupindex
#105928952
#105909002
C3×#45662
#105891362
#105891352
#692463