Finite Group Databasebeta

← #11792756

Subgroups — #11792756

Order 1152.

1026 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 191 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
1C22×C32×Q16C211yesmaximal, characteristic, complemented
2C2×#64151C212yesmaximal
3C2×#63663C212yesmaximal, complemented
4#10592809C211yesmaximal, characteristic
5#10589418C211yesmaximal, characteristic, complemented
6#72074—34nomaximal
7C22×C32×Q8C2212yes—
8C2×#3828C2211yescharacteristic
9C2×C32×Q16—24no—
10C2×C32×Q16—22no—
11C22×C32×C8C2211yescharacteristic
12C2×#3756—22no—
13#64151—24no—
14#64151—24no—
15C2×#3755—22no—
16C2×#3748C2211yescharacteristic
17C2×#3748C2212yes—
18#63663—24no—
19#63647—22no—
20C22×C3×Q16D314yescomplemented
21C2×#942—38no—
22C2×#836—38no—
23#5110—34no—
24#4552—34no—
25C2×C32×Q8—22no—
26C2×C32×Q8—22no—
27C2×C32×Q8—22no—
28C2×C32×Q8—22no—
29C2×C32×Q8—22no—
30C2×C32×Q8D414yes—
31C22×C32×C4—22no—
32C22×C32×C4C2311yescharacteristic
33C2×#638—21no—
34#3828—22no—
35#3828—22no—
36C2×C4⋊(C3⋊D3)D411yescharacteristic
37C2×C4×C3⋊D3—21no—
38C2×#634D411yescharacteristic
39C32×Q16—28no—
40C2×C32×C8—22no—
41C2×C32×C8D412yes—
42#3756—28no—
43#3755—28no—
44#3748—22no—
45#3748—22no—
46#3748D414yes—
47C2×C3⋊(C3⋊C8)—22no—
48C2×C3⋊(C3⋊C8)—21no—
49#2020—91noSylow
50C22×C3×Q8C2×D318yes—
51C2×#235—34no—
52C2×C3×Q16—216no—
53C2×C3×Q16—28no—
54C22×C3×C8C2×D314yes—
55C2×#216—68no—
56#942—616no—
57#942—616no—
58C2×#215—68no—
59C2×#208—38no—
60C2×#208—34no—
61#836—616no—
62(C2×D3)⋊C8—68no—
63C23×C32#4111yescharacteristic
64C22×C3⋊D3—21no—
65C32×Q8—24no—
66C32×Q8—24no—
67C32×Q8—42no—
68C32×Q8—42no—
69C2×C32×C4—24no—
70C2×C32×C4—42no—
71C2×C32×C4—42no—
72C2×C32×C4C2×D411yescharacteristic
73C2×C32×C4C2×D412yes—
74C2×C32×C4C2×D412yes—
75C2×C32×C4C2×D411yescharacteristic
76#638—42no—
77C2×C3⋊Dic3—21no—
78C4⋊(C3⋊D3)—22no—
79C4×C3⋊D3—41no—
80C4×C3⋊D3—41no—
81#634—22no—
82C32×C8—22no—
83C3⋊(C3⋊C8)—24no—
84C3⋊(C3⋊C8)—41no—
85C2×#137—92no—
86C22×Q16C3⋊D311yescharacteristic, complemented
87C2×#101—92no—
88#408—91no—
89#341—91no—
90C2×C3×Q8—28no—
91C2×C3×Q8—28no—
92C2×C3×Q8—28no—
93C2×C3×Q8—28no—
94C2×C3×Q8—28no—
95C2×C3×Q8#67116yes—
96C22×C3×C4—28no—
97C22×C3×C4C22×D314yes—
98C2×#67—64no—
99#235—68no—
100#235—68no—
101C2×D12—34no—
102C2×C4×D3—64no—
103C2×Dic6—34no—
104C3×Q16—232no—
105C2×C3×C8—28no—
106C2×C3×C8#6718yes—
107#216—632no—
108#215—632no—
109#208—316no—
110#208—68no—
111#208—68no—
112C2×C3⋊C8—68no—
113C2×C3⋊C8—64no—
114C22×C32—22no—
115C22×C32#12111yescharacteristic
116C22×C32#12112yes—
117C2×C3⋊D3—41no—
118C2×C3⋊D3—42no—
119C32×C4—44no—
120C32×C4#12012yes—
121C32×C4#12011yescharacteristic
122C32×C4#12711yescharacteristic
123C3⋊Dic3—41no—
124C2×#41—91no—
125C22×Q8C2×C3⋊D312yes—
126#137—184no—
127#137—184no—
128C2×Q16—24no—
129C2×Q16—22no—
130C2×Q16—182no—
131C2×SD16—182no—
132C2×M16—92no—
133C2×M16—91no—
134C22×C8C2×C3⋊D311yescharacteristic
135#101—184no—
136C22⋊C8—182no—
137C23×C3#23714yes—
138C22×D3—64no—
139C3×Q8—216no—
140C3×Q8—216no—
141C3×Q8—48no—
142C3×Q8—48no—
143C2×C3×C4—216no—
144C2×C3×C4—48no—
145C2×C3×C4—48no—
146C2×C3×C4D3×D414yes—
147C2×C3×C4D3×D418yes—
148C2×C3×C4C2×#6718yes—
149C2×C3×C4C2×#6714yes—
150#67—128no—
151C2×Dic3—64no—
152D12—68no—
153C4×D3—124no—
154C4×D3—124no—
155Dic6—68no—
156C3×C8—28no—
157C3⋊C8—616no—
158C3⋊C8—124no—
159C2×C32—22no—
160C2×C32#39211yescharacteristic
161C2×C32#47312yes—
162C3⋊D3—81no—
163#41—182no—
164#41—182no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—91no—
171C2×Q8#63814yes—
172C2×D4—91no—
173C2×D4—181no—
174C22×C4—22no—
175C22×C4—181no—
176C22×C4C22×C3⋊D311yescharacteristic
177Q16—28no—
178Q16—188no—
179SD16—188no—
180M16—94no—
181M16—182no—
182M16—182no—
183C2×C8—22no—
184C2×C8—182no—
185C2×C8—181no—
186C2×C8#63812yes—
187C22×C3—28no—
188C22×C3#93814yes—
189C22×C3#93918yes—
190C2×D3—128no—
191C2×D3—124no—
192C3×C4—416no—
193C3×C4#93718yes—
194C3×C4#94014yes—
195C3×C4#95314yes—
196Dic3—124no—
197C32#202011yescharacteristic, complemented, Sylow
198C23—181no—
199C23#383011yescharacteristic
200Q8—24no—
201Q8—24no—
202Q8—42no—
203Q8—42no—
204Q8—182no—
205D4—182no—
206D4—362no—
207C2×C4—24no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—181no—
211C2×C4—361no—
212C2×C4—361no—
213C2×C4D4×C3⋊D311yescharacteristic
214C2×C4D4×C3⋊D312yes—
215C2×C4C2×#63812yes—
216C2×C4C2×#63811yescharacteristic
217C8—22no—
218C8—184no—
219C8—361no—
220C2×C3—28no—
221C2×C3#519218yes—
222C2×C3#522314yes—
223D3—244no—
224C22—22no—
225C22—362no—
226C22—361no—
227C22#6414711yescharacteristic
228C22#6414812yes—
229C4—44no—
230C4—361no—
231C4#6414612yes—
232C4#6414911yescharacteristic
233C4#6416211yescharacteristic
234C3#7207414yescomplemented
235C2—22no—
236C2—721no—
237C2#1059289112yes—
238C2#1059292211yescharacteristic
subgroupG/H
centerC22#64147
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC22×C32×Q16C2
socleC22×C32#121
prime radicalC2×#3828C22
hypercenterC22×Q16C3⋊D3
hypocenterC32#2020
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C22×C32×Q16C2semidirect
3. C2×#63663C2semidirect
5. #10589418C2semidirect
20. C22×C3×Q16D3semidirect
86. C22×Q16C3⋊D3semidirect
197. C32#2020semidirect
234. C3#72074semidirect

Maximal subgroups

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

subgroupindex
C22×C32×Q162
C2×#641512
C2×#636632
#105928092
#105894182
#720743