Finite Group Databasebeta

← C4×#88926 #13166491

Subgroups — C4×#88926 #13166491

Order 1728.

608 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 104 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×#88926C212yesmaximal
2C4×#6289C211yesmaximal, characteristic
3C4×C3⋊#622C211yesmaximal, characteristic, complemented
4C3×C4×#622C211yesmaximal, characteristic, complemented
5#11722724C212yesmaximal, complemented
6C4×#3788—31nomaximal
7C2×C4×C3⋊(C3⋊D3)C411yescharacteristic
8C2×#6289C411yescharacteristic
9C2×#6289C2211yescharacteristic
10C2×C3⋊#622C2212yes—
11C2×C3×#622C2212yes—
12#88926C418yesdirect product factor
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4×Dic3C411yescharacteristic
15Dic3×C3⋊Dic3C411yescharacteristic
16Dic3×C3⋊Dic3C2211yescharacteristic
17C2×#3788—32no—
18C42×C3⋊D3—31no—
19C4×#624—62no—
20#63777—32no—
21C4×#622—31no—
22C4×#622D311yescharacteristic, complemented
23C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
24C2×C33×C4C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
26C4×C3⋊(C3⋊D3)—22no—
27C2×C3×C3⋊Dic3C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3C2×C411yescharacteristic
30C2×C32×Dic3C2×C411yescharacteristic
31#6289—22no—
32#6289C2×C414yes—
33C3⋊#622C2×C414yescomplemented
34C3×#622C2×C414yescomplemented
35C4×ΓL(1,𝔽25)—91nomaximal
36C2×C4×C3⋊D3—31no—
37C2×C4×C3⋊D3—31no—
38C2×C4×C3⋊D3—31no—
39C2×C4×C3⋊D3—62no—
40C2×C4×C3⋊D3—122no—
41C2×#624—62no—
42C2×#624—62no—
43C2×#622—32no—
44C2×#622C2×D312yes—
45#3788—38no—
46C32×C42—31no—
47C4×C3⋊Dic3—31no—
48C4×C3⋊Dic3—31no—
49C4×C3⋊Dic3C2×D311yescharacteristic
50C3×C4×Dic3—22no—
51C3×C4×Dic3—62no—
52Dic32—62no—
53Dic32—62no—
54C22×C33C22×C411yescharacteristic
55C2×C3⋊(C3⋊D3)—21no—
56C2×C3⋊(C3⋊D3)—21no—
57C2×C3⋊(C3⋊D3)C4212yes—
58C33×C4M1612yescomplemented
59C3⋊(C3⋊Dic3)—21no—
60C3×C3⋊Dic3M1612yescomplemented
61C3×C3⋊Dic3C22×C412yes—
62C32×Dic3—21no—
63C32×Dic3C4212yes—
64C2×ΓL(1,𝔽25)—92no—
65C42×D3—91no—
66C42×D3—182no—
67C3×C4×C8—91no—
68#815—92no—
69C4×C3⋊C8—91no—
70C22×C3⋊D3—31no—
71C22×C3⋊D3—62no—
72C22×C3⋊D3—122no—
73C2×C32×C4—22no—
74C2×C32×C4—31no—
75C2×C32×C4—31no—
76C2×C32×C4—42no—
77C2×C32×C4C4×D311yescharacteristic
78C2×C3⋊Dic3—31no—
79C2×C3⋊Dic3—62no—
80C2×C3⋊Dic3—122no—
81C2×C3⋊Dic3C4×D311yescharacteristic
82C2×C3⋊Dic3C22×D311yescharacteristic
83C4×C3⋊D3—34no—
84C4×C3⋊D3—62no—
85C4×C3⋊D3—62no—
86C4×C3⋊D3—68no—
87C4×C3⋊D3—128no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—22no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92#624—68no—
93#624—68no—
94#622—34no—
95#622C4×D314yescomplemented
96C4×M16—271noSylow
97C2×C33C2×C4211yescharacteristic
98C2×C33C2×M1611yescharacteristic
99C2×C33C2×M1611yescharacteristic
100C3⋊(C3⋊D3)—22no—
101C2×C4×D3—91no—
102C2×C4×D3—91no—
103C2×C4×D3—91no—
104C2×C4×D3—182no—
105C2×C4×D3—182no—
106C2×C4×D3—182no—
107C2×C4×D3—362no—
108C2×C3×C8—92no—
109C3×C42—62no—
110C3×C42—91no—
111C4×Dic3—62no—
112C4×Dic3—91no—
113C4×Dic3—91no—
114C4×Dic3—182no—
115C4×Dic3—182no—
116C4×Dic3#15711yescharacteristic
117C2×C3⋊C8—92no—
118ΓL(1,𝔽25)—98no—
119C22×C32—22no—
120C22×C32—42no—
121C22×C32C2×C4×D311yescharacteristic
122C2×C3⋊D3—32no—
123C2×C3⋊D3—64no—
124C2×C3⋊D3—61no—
125C2×C3⋊D3—61no—
126C2×C3⋊D3—64no—
127C2×C3⋊D3—64no—
128C2×C3⋊D3—124no—
129C2×C3⋊D3—124no—
130C2×C3⋊D3—124no—
131C32×C4—24no—
132C32×C4—32no—
133C32×C4—44no—
134C32×C4—61no—
135C32×C4ΓL(1,𝔽25)12yescomplemented
136C3⋊Dic3—61no—
137C3⋊Dic3—64no—
138C3⋊Dic3—124no—
139C3⋊Dic3ΓL(1,𝔽25)12yescomplemented
140C3⋊Dic3C2×C4×D312yes—
141C3×Dic3—24no—
142C3×Dic3—24no—
143C3×Dic3—64no—
144C3×Dic3—64no—
145C2×M16—272no—
146C2×C42—271no—
147#97—272no—
148C4×C8—91no—
149C4×C8—271no—
150C33C4×M1611yescharacteristic, complemented, Sylow
151C22×D3—91no—
152C22×D3—182no—
153C22×D3—362no—
154C2×C3×C4—22no—
155C2×C3×C4—42no—
156C2×C3×C4—62no—
157C2×C3×C4—62no—
158C2×C3×C4—91no—
159C2×C3×C4—91no—
160C2×C3×C4C2×#15711yescharacteristic
161C2×Dic3—62no—
162C2×Dic3—62no—
163C2×Dic3—91no—
164C2×Dic3—182no—
165C2×Dic3—362no—
166C2×Dic3C2×#15711yescharacteristic
167C2×Dic3C2×#15711yescharacteristic
168C4×D3—94no—
169C4×D3—188no—
170C4×D3—182no—
171C4×D3—188no—
172C4×D3—188no—
173C4×D3—182no—
174C4×D3—368no—
175C3×C8—94no—
176C3⋊C8—94no—
177C2×C32—22no—
178C2×C32—22no—
179C2×C32—22no—
180C2×C32—42no—
181C2×C32—42no—
182C2×C32—42no—
183C2×C32C42×D311yescharacteristic
184C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
185C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
186C3⋊D3—62no—
187C3⋊D3—68no—
188C3⋊D3—128no—
189C22×C4—271no—
190C22×C4—271no—
191C22×C4—271no—
192M16—278no—
193C2×C8—92no—
194C2×C8—272no—
195C42—31no—
196C42—91no—
197C42—271no—
198C42—271no—
199C22×C3—22no—
200C22×C3—42no—
201C22×C3C22×#15711yescharacteristic
202C2×D3—92no—
203C2×D3—184no—
204C2×D3—184no—
205C2×D3—184no—
206C2×D3—181no—
207C2×D3—181no—
208C2×D3—364no—
209C2×D3—364no—
210C2×D3—364no—
211C3×C4—24no—
212C3×C4—44no—
213C3×C4—64no—
214C3×C4—64no—
215C3×C4—92no—
216C3×C4—92no—
217C3×C4#378812yescomplemented
218Dic3—21no—
219Dic3—64no—
220Dic3—64no—
221Dic3—184no—
222Dic3—181no—
223Dic3—364no—
224Dic3C4×#15712yes—
225C32—22no—
226C32—42no—
227C32C4×ΓL(1,𝔽25)11yescharacteristic, complemented
228C23—271no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—91no—
232C2×C4—91no—
233C2×C4—274no—
234C2×C4—271no—
235C2×C4—542no—
236C2×C4—542no—
237C2×C4D3×#15711yescharacteristic
238C8—94no—
239C8—274no—
240C2×C3—22no—
241C2×C3—22no—
242C2×C3—22no—
243C2×C3—42no—
244C2×C3—42no—
245C2×C3—42no—
246C2×C3C2×#378811yescharacteristic
247C2×C3#6428311yescharacteristic
248C2×C3C2×C4×#15711yescharacteristic
249D3—188no—
250D3—182no—
251D3—368no—
252C22—272no—
253C22—541no—
254C22—541no—
255C22C2×D3×#15711yescharacteristic
256C4—32no—
257C4—61no—
258C4—92no—
259C4—92no—
260C4—541no—
261C4#8892612yesdirect product factor
262C3—22no—
263C3—42no—
264C3C4×#378811yescharacteristic, complemented
265C2—542no—
266C2#1172425411yescharacteristic
267C2C4×D3×#15711yescharacteristic
268C2C2×#8892611yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×#157
commutatorC2×C33C2×C42
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×C3⋊(C3⋊D3)C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C4×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C4×C3⋊#622C2semidirect
4. C3×C4×#622C2semidirect
5. #11722724C2semidirect
12. #88926C4direct (N × H)
22. C4×#622D3semidirect
33. C3⋊#622C2×C4semidirect
34. C3×#622C2×C4semidirect
58. C33×C4M16semidirect
60. C3×C3⋊Dic3M16semidirect
95. #622C4×D3semidirect
135. C32×C4ΓL(1,𝔽25)semidirect
139. C3⋊Dic3ΓL(1,𝔽25)semidirect
150. C33C4×M16semidirect
217. C3×C4#3788semidirect
227. C32C4×ΓL(1,𝔽25)semidirect
261. C4#88926direct (N × H)
264. C3C4×#3788semidirect

Maximal subgroups

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

subgroupindex
C2×#889262
C4×#62892
C4×C3⋊#6222
C3×C4×#6222
#117227242
C4×#37883
C4×ΓL(1,𝔽25)9