Finite Group Databasebeta

← #11782423

Subgroups — #11782423

Order 1152.

499 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 114 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#10592775C211yesmaximal, characteristic
2C3×#5076C212yesmaximal
3#10590309C211yesmaximal, characteristic, complemented
4#10590303C212yesmaximal, complemented
5C3⋊C82C211yesmaximal, characteristic, complemented
6#71254—32nomaximal
7C32×#127C2211yescharacteristic
8C2×#3754—22no—
9#64078C2211yescharacteristic
10C2×C3×#214—24no—
11C3×#869C2212yes—
12C4×C3⋊(C3⋊C8)C2211yescharacteristic
13C3×C4×C3⋊C8C2212yes—
14#63574—22no—
15#63573—24no—
16C3×#492—32no—
17#5080—32no—
18#5080—32no—
19#5076—32no—
20#5076—61no—
21#5076D312yes—
22#4680—32no—
23C8×C3⋊C8—32no—
24C2×C32×D4—21no—
25C2×C32×D4—22no—
26C2×#634—21no—
27C2×C3×Dic6—22no—
28C32×C42C2311yescharacteristic
29#3754—44no—
30C4⋊(C3⋊Dic3)—22no—
31C2×C3⋊(C3⋊C8)D412yes—
32C3×#214—48no—
33C3×C4⋊Dic3—22no—
34C3×C4⋊Dic3—22no—
35C2×C3×C3⋊C8D414yes—
36#1705—91noSylow
37C2×C3×SD16—64no—
38C3×C4⋊Q8—32no—
39C3×#127—21no—
40C3×#127C2×D312yes—
41C2×#214—24no—
42C2×#214—64no—
43C2×#214—64no—
44#869—32no—
45#869—61no—
46#869C2×D312yes—
47C3×C4×C8—32no—
48#817—64no—
49#808—64no—
50#808—64no—
51C4×C3⋊C8—32no—
52C4×C3⋊C8—61no—
53C4×C3⋊C8C2×D312yes—
54C23×C32—41no—
55C32×D4—42no—
56C32×D4—44no—
57C2×C32×C4C2×D411yescharacteristic
58C2×C32×C4C2×D412yes—
59C2×C3⋊Dic3—41no—
60#634—42no—
61C2×C3×Dic3—42no—
62C3×Dic6—44no—
63C3⋊(C3⋊C8)SD1614yescomplemented
64C3×C3⋊C8SD1618yescomplemented
65#498—92no—
66#498—91no—
67#492—32no—
68#492—91no—
69C82—91no—
70C2×C3×Q8—62no—
71C2×C3×D4—22no—
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—42no—
76C2×Dic6—22no—
77C2×Dic6—62no—
78C2×Dic6—62no—
79C3×SD16—128no—
80C2×C3×C8—34no—
81C3×C4⋊C4—62no—
82C3×C4⋊C4—62no—
83C3×C42—21no—
84C3×C42C22×D312yes—
85#214—48no—
86#214—128no—
87#214—128no—
88C4⋊Dic3—22no—
89C4⋊Dic3—22no—
90C4⋊Dic3—62no—
91C4⋊Dic3—62no—
92C4⋊Dic3—122no—
93C2×C3⋊C8—34no—
94C2×C3⋊C8—62no—
95C2×C3⋊C8#6714yes—
96C22×C32—81no—
97C22×C32—82no—
98C22×C32#12711yescharacteristic
99C32×C4C2×SD1612yes—
100C32×C4C2×SD1614yes—
101C3⋊Dic3—81no—
102C3×Dic3—82no—
103C2×SD16—64no—
104C2×SD16—182no—
105C4⋊Q8—32no—
106C4⋊Q8—91no—
107#127D3211yescharacteristic
108#106—182no—
109#106—184no—
110C4×C8—32no—
111C4×C8—91no—
112C23×C3—42no—
113C23×C3—42no—
114C3×Q8—124no—
115C3×D4—44no—
116C3×D4—44no—
117C3×D4—44no—
118C3×D4—44no—
119C3×D4—84no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—122no—
123C2×C3×C4D3×D412yes—
124C2×C3×C4D3×D412yes—
125C2×C3×C4C2×#6712yes—
126C2×Dic3—42no—
127C2×Dic3—122no—
128C2×Dic3—122no—
129Dic6—44no—
130Dic6—124no—
131Dic6—124no—
132C3×C8—38no—
133C3⋊C8—38no—
134C3⋊C8—64no—
135C3⋊C8#21418yescomplemented
136C2×C32—161no—
137C2×C32#49211yescharacteristic
138C2×C32#49212yes—
139C2×Q8—62no—
140C2×Q8—181no—
141C2×D4—22no—
142C2×D4—21no—
143SD16—128no—
144SD16—364no—
145C2×C8—34no—
146C2×C8—92no—
147C4⋊C4—62no—
148C4⋊C4—62no—
149C4⋊C4—182no—
150C42C2×D3211yescharacteristic
151C22×C3—21no—
152C22×C3—82no—
153C22×C3—84no—
154C22×C3—82no—
155C22×C3—82no—
156C22×C3—82no—
157C22×C3#94012yes—
158C3×C4—24no—
159C3×C4—22no—
160C3×C4—242no—
161C3×C4D3×SD1614yes—
162C3×C4D3×SD1614yes—
163C3×C4C2×#21414yes—
164Dic3—82no—
165Dic3—242no—
166Dic3—242no—
167C32#170511yescharacteristic, complemented, Sylow
168C23—41no—
169Q8—124no—
170Q8—362no—
171D4—44no—
172D4—42no—
173C2×C4—122no—
174C2×C4—361no—
175C2×C4D3×#6712yes—
176C2×C4#381111yescharacteristic
177C8—38no—
178C8—94no—
179C2×C3—22no—
180C2×C3—21no—
181C2×C3—162no—
182C2×C3—162no—
183C2×C3#507612yes—
184C2×C3#517512yes—
185C2×C3#517512yes—
186C22—82no—
187C22—81no—
188C22#6397911yescharacteristic
189C4—242no—
190C4—721no—
191C4D3×#21414yes—
192C4#6393012yes—
193C3—21no—
194C3#7125412yescomplemented
195C2—161no—
196C2#1059149012yes—
197C2#1059149211yescharacteristic
subgroupG/H
centerC22#63979
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
Fitting, prime radicalC32×#127C22
socleC22×C32#127
hypercenter#127D32
hypocenterC32#1705
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
3. #10590309C2semidirect
4. #10590303C2semidirect
5. C3⋊C82C2semidirect
63. C3⋊(C3⋊C8)SD16semidirect
64. C3×C3⋊C8SD16semidirect
135. C3⋊C8#214semidirect
167. C32#1705semidirect
194. C3#71254semidirect

Maximal subgroups

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

subgroupindex
#105927752
C3×#50762
#105903092
#105903032
C3⋊C822
#712543