Finite Group Databasebeta

← C4×#88433 #13137399

Subgroups — C4×#88433 #13137399

Order 1728.

429 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C4×#634⋊C3C212yesmaximal
2C2×#88433C211yesmaximal, characteristic
3C4×#6180C211yesmaximal, characteristic
4#11720315C211yesmaximal, characteristic
5#11720313C212yesmaximal
6C4×#3719—32nomaximal
7C2×#634⋊C3C2212yes—
8C42×H(3,𝔽3)C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12#3750⋊C3—22no—
13C4×(C3⋊Dic3)⋊C3—22no—
14#88433C414yesdirect product factor
15C4×#634D312yes—
16C3×C4×Dic6—32no—
17C2×#3719—32no—
18C3×C4×C3⋊C8—32no—
19C3×C4×C3⋊C8—61no—
20#63575—32no—
21#63567—32no—
22#63566—32no—
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×C4×H(3,𝔽3)D411yescharacteristic
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×(C3⋊Dic3)⋊C3—22no—
27C2×(C3⋊Dic3)⋊C3—22no—
28#634⋊C3—22no—
29#634⋊C3C2×C414yes—
30#6180—21no—
31#6180C2×C412yes—
32C4×#216—92no—
33C4×Dic12—91no—
34C2×#634C2×D312yes—
35C2×C3×Dic6—32no—
36C32×C42—21no—
37C32×C42C2×D312yes—
38C4⋊(C3⋊Dic3)C2×D312yes—
39#3750—22no—
40C4×C3⋊Dic3—22no—
41C3×C4⋊Dic3—32no—
42C3×Dic3⋊C4—62no—
43C3×C4×Dic3—62no—
44C2×C3×C3⋊C8—32no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—61no—
47C2×C3×C3⋊C8—61no—
48#3719—38no—
49C22×H(3,𝔽3)C2×D411yescharacteristic
50C4×H(3,𝔽3)—21no—
51C4×H(3,𝔽3)Q1612yescomplemented
52C4×H(3,𝔽3)C22×C411yescharacteristic
53C4×H(3,𝔽3)#4111yescharacteristic
54(C3⋊Dic3)⋊C3—24no—
55(C3⋊Dic3)⋊C3—42no—
56C3×C4×Q8—92no—
57C2×#216—92no—
58C2×Dic12—91no—
59C4×Dic6—32no—
60C4×Dic6—92no—
61C3×C4×C8—91no—
62#835—92no—
63#818—91no—
64#816—92no—
65#810—92no—
66#807—92no—
67C4×C3⋊C8—32no—
68C4×C3⋊C8—61no—
69C2×C32×C4—21no—
70C2×C32×C4—21no—
71C2×C32×C4—21no—
72C2×C32×C4#6712yes—
73C2×C32×C4#6712yes—
74C2×C32×C4C22×D312yes—
75C2×C3⋊Dic3—22no—
76C2×C3⋊Dic3—22no—
77#634—22no—
78#634C4×D314yes—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C3×Dic6—34no—
82C3×Dic6—62no—
83C3×C3⋊C8—34no—
84C3×C3⋊C8—62no—
85C3×C3⋊C8—62no—
86C3×C3⋊C8—62no—
87C4×Q16—271noSylow
88C2×H(3,𝔽3)C4×D411yescharacteristic
89C2×H(3,𝔽3)C2×Q1611yescharacteristic
90C2×H(3,𝔽3)#13511yescharacteristic
91C2×C3×Q8—92no—
92C2×Dic6—32no—
93C2×Dic6—92no—
94C2×C3×C8—91no—
95C2×C3×C8—91no—
96C3×C4⋊C4—92no—
97C3×C4⋊C4—182no—
98C3×C42—32no—
99C3×C42—61no—
100C3×C42—182no—
101C3×C42D3211yescharacteristic
102#216—98no—
103C4⋊Dic3—32no—
104C4⋊Dic3—92no—
105Dic3⋊C4—62no—
106Dic3⋊C4—182no—
107C4×Dic3—62no—
108C4×Dic3—182no—
109C2×C3⋊C8—32no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8—61no—
112C2×C3⋊C8—61no—
113Dic12—94no—
114C22×C32—21no—
115C22×C32C2×#6712yes—
116C32×C4—22no—
117C32×C4—22no—
118C32×C4—22no—
119C32×C4—21no—
120C32×C4—21no—
121C32×C4#21614yescomplemented
122C32×C4C2×C4×D312yes—
123C32×C4#23512yes—
124C3⋊Dic3—24no—
125C3⋊Dic3—42no—
126C3×Dic3—64no—
127C3×Dic3—122no—
128C2×Q16—271no—
129C4×Q8—92no—
130#107—271no—
131#103—272no—
132C4×C8—91no—
133H(3,𝔽3)C4×Q1611yescharacteristic, complemented, Sylow
134C3×Q8—94no—
135C3×Q8—182no—
136C2×C3×C4—32no—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—61no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×C3×C4—182no—
143C2×C3×C4—182no—
144C2×C3×C4#62511yescharacteristic
145C2×C3×C4#62511yescharacteristic
146C2×C3×C4C2×D3211yescharacteristic
147C2×Dic3—62no—
148C2×Dic3—62no—
149C2×Dic3—182no—
150C2×Dic3—182no—
151Dic6—34no—
152Dic6—62no—
153Dic6—94no—
154Dic6—182no—
155C3×C8—92no—
156C3×C8—181no—
157C3⋊C8—34no—
158C3⋊C8—62no—
159C3⋊C8—62no—
160C3⋊C8—62no—
161C2×C32—21no—
162C2×C32—21no—
163C2×C32—21no—
164C2×C32C4×#6712yes—
165C2×C32C2×#21612yes—
166C2×C32#95012yes—
167C2×Q8—92no—
168Q16—274no—
169C2×C8—91no—
170C2×C8—91no—
171C4⋊C4—92no—
172C4⋊C4—182no—
173C42—182no—
174C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
175C22×C3—32no—
176C22×C3—61no—
177C22×C3C2×#62511yescharacteristic
178C3×C4—21no—
179C3×C4—32no—
180C3×C4—32no—
181C3×C4—34no—
182C3×C4—62no—
183C3×C4—62no—
184C3×C4—61no—
185C3×C4—61no—
186C3×C4—62no—
187C3×C4—184no—
188C3×C4—362no—
189C3×C4#371812yes—
190C3×C4#379811yescharacteristic
191C3×C4C4×D3211yescharacteristic
192Dic3—64no—
193Dic3—122no—
194Dic3—184no—
195Dic3—362no—
196C32—21no—
197C32C4×#21612yescomplemented
198Q8—94no—
199Q8—182no—
200C2×C4—182no—
201C2×C4—182no—
202C2×C4#619811yescharacteristic
203C2×C4#619811yescharacteristic
204C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
205C8—92no—
206C8—181no—
207C2×C3—32no—
208C2×C3—32no—
209C2×C3—32no—
210C2×C3—61no—
211C2×C3—61no—
212C2×C3—61no—
213C2×C3#6382211yescharacteristic
214C2×C3C2×#371811yescharacteristic
215C2×C3C4×#62511yescharacteristic
216C22C2×#619811yescharacteristic
217C4—21no—
218C4—184no—
219C4—362no—
220C4#8843312yesdirect product factor
221C4#8865011yescharacteristic
222C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
223C3—32no—
224C3—61no—
225C3C4×#371811yescharacteristic
226C2#1172106011yescharacteristic
227C2C2×#8843311yescharacteristic
228C2C4×#619811yescharacteristic
subgroupG/H
centerC2×C4#6198
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×Q16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
14. #88433C4direct (N × H)
51. C4×H(3,𝔽3)Q16semidirect
121. C32×C4#216semidirect
133. H(3,𝔽3)C4×Q16semidirect
197. C32C4×#216semidirect
220. C4#88433direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#634⋊C32
C2×#884332
C4×#61802
#117203152
#117203132
C4×#37193