Finite Group Databasebeta

← #13140614

Subgroups — #13140614

Order 1728.

580 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 75 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#11721846C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C212yesmaximal, complemented
3#11721144C211yesmaximal, characteristic, complemented
4C2×#88438C212yesmaximal, complemented
5#11720303C211yesmaximal, characteristic, complemented
6#10591483—32nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C2×#6229C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#6198—22no—
11#88552—22no—
12C4×#1084C2211yescharacteristic
13#3753⋊C3—24no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88438—24no—
16C2×#3753D312yescomplemented
17C3×#936—32no—
18C3×#936—61no—
19C2×C3×#213—32no—
20#63911—32no—
21C2×#3717—32no—
22C2×#3715—32no—
23#63554—32no—
24C23×H(3,𝔽3)—21no—
25C22×(C3⋊D3)⋊C3—22no—
26D4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—21no—
29C2×#1084D412yes—
30#6229—22no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32C4⋊((C3⋊D3)⋊C3)—22no—
33#6198—44no—
34(C3⋊(C3⋊C8))⋊C3D414yescomplemented
35#5154—92no—
36#5058—91no—
37C2×C32×D4—21no—
38C2×C32×D4C2×D312yes—
39C2×C4⋊(C3⋊D3)C2×D312yes—
40C2×C3×D12—32no—
41C2×C3×Dic6—32no—
42C2×C3×Dic6—61no—
43C2×#626—64no—
44#3753—24no—
45C2×C3⋊(C3⋊C8)C2×D312yes—
46C3×#217—64no—
47C3×#217—64no—
48C3×#213—64no—
49C3×C4×Dic3—32no—
50C3×C4×Dic3—61no—
51C2×C3×C3⋊C8—32no—
52#3717—64no—
53#3715—64no—
54C22×H(3,𝔽3)—41no—
55C22×H(3,𝔽3)—42no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C2×(C3⋊D3)⋊C3—42no—
58C2×(C3⋊D3)⋊C3—44no—
59C4×H(3,𝔽3)C2×D412yes—
60#1084—22no—
61#1084—41no—
62C2×C3×D8—92no—
63C3×#124—91no—
64C2×#215—92no—
65#940—92no—
66#936—32no—
67#936—61no—
68C2×#214—92no—
69C2×#213—32no—
70C2×#213—92no—
71C2×#204—92no—
72#874—91no—
73#815—92no—
74#803—91no—
75C23×C32—22no—
76C23×C32—22no—
77C22×C3⋊D3—22no—
78C22×C3×D3—62no—
79C32×D4—22no—
80C32×D4—22no—
81C32×D4—42no—
82C2×C32×C4—21no—
83C2×C32×C4C22×D312yes—
84C4⋊(C3⋊D3)—22no—
85C4⋊(C3⋊D3)—22no—
86C2×C3×Dic3—34no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C3×D12—62no—
91C3×D12—62no—
92C3×Dic6—62no—
93C3×Dic6—62no—
94C3×Dic6—122no—
95#626—128no—
96C3⋊(C3⋊C8)#6714yescomplemented
97C3×C3⋊C8—34no—
98#496—271noSylow
99C2×H(3,𝔽3)—81no—
100C2×H(3,𝔽3)#12711yescharacteristic
101C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
102(C3⋊D3)⋊C3—82no—
103C2×C3×Q8—91no—
104C2×C3×D4—32no—
105C2×C3×D4—61no—
106C2×C3×D4—92no—
107C2×C3×D4D3211yescharacteristic
108C2×#67—184no—
109C2×D12—32no—
110C2×D12—92no—
111C2×D12—182no—
112C2×Dic6—32no—
113C2×Dic6—61no—
114C3×D8—184no—
115C2×C3×C8—92no—
116C3×C22⋊C4—182no—
117C3×C42—91no—
118#217—64no—
119#217—64no—
120#215—184no—
121#214—184no—
122#213—64no—
123#213—184no—
124C4×Dic3—32no—
125C4×Dic3—61no—
126C2×C3⋊C8—32no—
127C2×C3⋊C8—92no—
128#204—184no—
129C22×C32—21no—
130C22×C32—42no—
131C22×C32—44no—
132C22×C32—44no—
133C22×C32—42no—
134C22×C32D3×D412yes—
135C2×C3⋊D3—44no—
136C2×C3⋊D3—42no—
137C2×C3×D3—124no—
138C2×C3×D3—122no—
139C32×C4—22no—
140C32×C4D3×D412yes—
141C32×C4C2×#6712yes—
142C3×Dic3—64no—
143C3×Dic3—122no—
144C3×Dic3—122no—
145C3×Dic3—122no—
146C2×SD16—272no—
147C2×D8—92no—
148#127—271no—
149#124—91no—
150#97—271no—
151H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
152C23×C3—21no—
153C23×C3—62no—
154C23×C3—62no—
155C23×C3—182no—
156C22×D3—62no—
157C22×D3—182no—
158C3×Q8—182no—
159C3×D4—22no—
160C3×D4—62no—
161C3×D4—62no—
162C3×D4—122no—
163C3×D4—182no—
164C3×D4—182no—
165C2×C3×C4—32no—
166C2×C3×C4—61no—
167C2×C3×C4—92no—
168C2×C3×C4—181no—
169C2×C3×C4C2×D3211yescharacteristic
170#67—368no—
171C2×Dic3—34no—
172C2×Dic3—62no—
173C2×Dic3—62no—
174C2×Dic3—62no—
175D12—62no—
176D12—62no—
177D12—182no—
178D12—182no—
179D12—364no—
180Dic6—62no—
181Dic6—62no—
182Dic6—122no—
183C3×C8—94no—
184C3⋊C8—34no—
185C3⋊C8—94no—
186C2×C32—22no—
187C2×C32—21no—
188C2×C32—82no—
189C2×C32—82no—
190C2×C32#91114yes—
191C2×C32#94012yes—
192C3⋊D3—82no—
193C3×D3—242no—
194C2×Q8—91no—
195C2×D4—92no—
196C2×D4—542no—
197C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
198SD16—544no—
199D8—184no—
200C2×C8—92no—
201C22⋊C4—182no—
202C42—91no—
203C22×C3—32no—
204C22×C3—42no—
205C22×C3—41no—
206C22×C3—61no—
207C22×C3—122no—
208C22×C3—124no—
209C22×C3—122no—
210C22×C3—124no—
211C22×C3—364no—
212C22×C3—362no—
213C22×C3#381111yescharacteristic
214C2×D3—124no—
215C2×D3—122no—
216C2×D3—362no—
217C2×D3—364no—
218C3×C4—32no—
219C3×C4—32no—
220C3×C4—62no—
221C3×C4—182no—
222C3×C4—361no—
223C3×C4D3×#6712yes—
224Dic3—64no—
225Dic3—122no—
226Dic3—122no—
227Dic3—122no—
228C32—21no—
229C32#515412yescomplemented
230C23—21no—
231C23—182no—
232Q8—182no—
233D4—22no—
234D4—182no—
235D4—182no—
236D4—1084no—
237C2×C4—92no—
238C2×C4—181no—
239C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
240C8—94no—
241C2×C3—32no—
242C2×C3—34no—
243C2×C3—61no—
244C2×C3—62no—
245C2×C3—81no—
246C2×C3—242no—
247C2×C3—242no—
248C2×C3—722no—
249C2×C3#6392712yes—
250C2×C3#6397911yescharacteristic
251D3—242no—
252D3—722no—
253C22—41no—
254C22—42no—
255C22—362no—
256C22—364no—
257C22#8867711yescharacteristic
258C4—182no—
259C4—361no—
260C4#8867612yes—
261C3—32no—
262C3—61no—
263C3#1059148511yescharacteristic
264C2—81no—
265C2—722no—
266C2#1172119112yes—
267C2#1172131711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#496
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721846C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #11721144C2semidirect
4. C2×#88438C2semidirect
5. #11720303C2semidirect
16. C2×#3753D3semidirect
34. (C3⋊(C3⋊C8))⋊C3D4semidirect
96. C3⋊(C3⋊C8)#67semidirect
151. H(3,𝔽3)#496semidirect
229. C32#5154semidirect

Maximal subgroups

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

subgroupindex
#117218462
C2×#3753⋊C32
#117211442
C2×#884382
#117203032
#105914833