Finite Group Databasebeta

← C4⋊#88615 #13147100

Subgroups — C4⋊#88615 #13147100

Order 1728.

283 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 51 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×#88615C212yesmaximal, complemented
2C4×C9×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊(#41⋊C3))—31nomaximal
4#10589679C311yesmaximal, characteristic, complemented
5C2×C9×SL(2,𝔽3)C2211yescharacteristic
6#88615—22no—
7C3×C4⋊Dic9—41nomaximal
8C2×C3⋊(#41⋊C3)—32no—
9C3×C4×SL(2,𝔽3)D311yescharacteristic
10C2×#3701C2×C312yes—
11C4×Q8×C9C2×C311yescharacteristic
12#63457—32no—
13#63436—31no—
14C4×D36—31no—
15C4×Q8⋊C9—21no—
16C2×C3×C4×C9—41no—
17C2×C3×Dic9—42no—
18C9×SL(2,𝔽3)D411yescharacteristic, complemented
19#5577—31no—
20C4⋊(#41⋊C3)—91no—
21C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
22C3⋊(#41⋊C3)—62no—
23C3×C4⋊Dic3—121no—
24C2×Q8×C9C22×C311yescharacteristic
25#3701—22no—
26#3701—62no—
27C2×D36—32no—
28C2×D36—31no—
29C2×D36—32no—
30C2×C4×D9—32no—
31C2×Q8⋊C9—21no—
32C9×C4⋊C4—31no—
33C42×C9—31no—
34(C2×D9)⋊C4—32no—
35C4⋊Dic9A411yescharacteristic
36C22×C3×C9—41no—
37C3×C4×C9—42no—
38C3×Dic9—82no—
39C2×#239—32no—
40C2×#41⋊C3—92no—
41C3×C4×Q8C3×D311yescharacteristic
42#895—92no—
43#874—91no—
44C4×D12—91no—
45C4×SL(2,𝔽3)—21no—
46C4×SL(2,𝔽3)D911yescharacteristic, complemented
47C2×C32×C4—41no—
48C2×C3×Dic3—122no—
49C3×SL(2,𝔽3)D1211yescharacteristic
50C22×D9—32no—
51Q8×C9—31no—
52Q8×C9C3×D411yescharacteristic
53C2×C4×C9—31no—
54C2×C4×C9—31no—
55C2×C4×C9—81no—
56C2×C4×C9C2×A411yescharacteristic
57C2×Dic9C2×A412yes—
58D36—62no—
59D36—62no—
60D36—62no—
61D36—122no—
62C4×D9—62no—
63C4×D9—62no—
64Q8⋊C9—21no—
65#550—91noSylow
66C2×C3×C9—41no—
67C2×C3×C9—41no—
68C2×C3×C9—41no—
69C2×C3×Q8C2×C3×D311yescharacteristic
70#239—62no—
71#239—182no—
72C2×D12—92no—
73C2×D12—91no—
74C2×D12—92no—
75C2×C4×D3—92no—
76#41⋊C3—182no—
77C2×SL(2,𝔽3)—21no—
78C2×SL(2,𝔽3)C2×D911yescharacteristic
79C3×C4⋊C4—31no—
80C3×C4⋊C4—361no—
81C3×C42—31no—
82(C2×D3)⋊C4—92no—
83C4⋊Dic3—31no—
84C22×C32—41no—
85C32×C4—42no—
86C3×Dic3—242no—
87C22×C9—81no—
88C22×C9C22×A411yescharacteristic
89C2×D9—62no—
90C2×D9—62no—
91C2×D9—62no—
92C4×C9—31no—
93C4×C9—31no—
94C4×C9—61no—
95C4×C9—82no—
96C4×C9#41⋊C312yescomplemented
97Dic9—22no—
98C2×#41—92no—
99#127—271no—
100#121—272no—
101C4×Q8C3×D911yescharacteristic
102C4×D4—271no—
103C3×C9—41noSylow
104C22×D3—92no—
105C3×Q8—31no—
106C3×Q8C3×D1211yescharacteristic
107C2×C3×C4—31no—
108C2×C3×C4—31no—
109C2×C3×C4—41no—
110C2×C3×C4—81no—
111C2×C3×C4—362no—
112C2×C3×C4D3×A411yescharacteristic
113C2×Dic3—32no—
114D12—182no—
115D12—182no—
116D12—182no—
117D12—362no—
118C4×D3—182no—
119C4×D3—182no—
120SL(2,𝔽3)—21no—
121SL(2,𝔽3)D3611yescharacteristic, complemented
122C2×C32—41no—
123C2×C32—41no—
124C2×C32—41no—
125C2×C9—81no—
126C2×C9—81no—
127C2×C9—81no—
128C2×C9D4×A411yescharacteristic
129C2×C9C2×#41⋊C311yescharacteristic
130C2×C9#99411yescharacteristic
131D9—122no—
132#41—182no—
133#41—542no—
134C2×Q8C2×C3×D911yescharacteristic
135C2×D4—272no—
136C2×D4—272no—
137C2×D4—271no—
138C22×C4—272no—
139C4⋊C4—31no—
140C4⋊C4—91no—
141C22⋊C4—272no—
142C42—31no—
143C22×C3—41no—
144C22×C3—81no—
145C22×C3C2×D3×A411yescharacteristic
146C2×D3—182no—
147C2×D3—182no—
148C2×D3—182no—
149C3×C4—31no—
150C3×C4—31no—
151C3×C4—42no—
152C3×C4—61no—
153C3×C4—82no—
154C3×C4—722no—
155C3×C4C3⋊(#41⋊C3)12yes—
156Dic3—62no—
157C32—41no—
158C9—81no—
159C9C4⋊(#41⋊C3)11yescharacteristic, complemented
160C23—272no—
161Q8—31no—
162Q8C3×D3611yescharacteristic
163D4—542no—
164D4—542no—
165D4—542no—
166D4—1082no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—92no—
170C2×C4—542no—
171C2×C4—542no—
172C2×C4A4×D911yescharacteristic
173C2×C3—41no—
174C2×C3—41no—
175C2×C3—41no—
176C2×C3—81no—
177C2×C3—81no—
178C2×C3—81no—
179C2×C3A4×D1211yescharacteristic
180C2×C3C2×C3⋊(#41⋊C3)11yescharacteristic
181C2×C3C3⋊#99411yescharacteristic
182D3—362no—
183C22—542no—
184C22—542no—
185C22—542no—
186C22C2×A4×D911yescharacteristic
187C4—31no—
188C4—31no—
189C4—61no—
190C4—182no—
191C4#8861512yescomplemented
192C3—41no—
193C3—81no—
194C3C4⋊(C3⋊(#41⋊C3))11yescharacteristic
195C2—1082no—
196C2A4×D3611yescharacteristic
197C2C2×#8861511yescharacteristic
198C2#1172241211yescharacteristic
subgroupG/H
centerC22C2×A4×D9
commutator, cosocleC2×Q8×C9C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC4×Q8×C9C2×C3
hypercenterC2×C4A4×D9
hypocenterQ8×C9C3×D4
LCSG ▷ C2×Q8×C9 ▷ Q8×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88615C2semidirect
2. C4×C9×SL(2,𝔽3)C2semidirect
4. #10589679C3semidirect
18. C9×SL(2,𝔽3)D4semidirect
46. C4×SL(2,𝔽3)D9semidirect
96. C4×C9#41⋊C3semidirect
121. SL(2,𝔽3)D36semidirect
159. C9C4⋊(#41⋊C3)semidirect
191. C4#88615semidirect

Maximal subgroups

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

subgroupindex
C2×#886152
C4×C9×SL(2,𝔽3)2
C4⋊(C3⋊(#41⋊C3))3
#105896793
C3×C4⋊Dic94