Finite Group Databasebeta

← #13140595

Subgroups — #13140595

Order 1728.

605 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721850C211yesmaximal, characteristic
2C2×#3754⋊C3C212yesmaximal, complemented
3#11721099C211yesmaximal, characteristic, complemented
4C2×#88437C212yesmaximal, complemented
5#11720302C211yesmaximal, characteristic, complemented
6#10591490—32nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C2×#6235—22no—
9C2×C4⋊#272C2211yescharacteristic
10C2×#634⋊C3C2212yes—
11C4×#1084C2211yescharacteristic
12#3754⋊C3—24no—
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14#88449—22no—
15#88437—24no—
16C2×#3754D312yescomplemented
17C3×#940—32no—
18C3×#940—61no—
19C2×C3×#214—32no—
20#63866—32no—
21C2×#3716—32no—
22C2×#3715—32no—
23Dic3×C3⋊C8—32no—
24C23×H(3,𝔽3)—21no—
25C22×#272—21no—
26D4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28#6235—44no—
29C2×#1084D412yes—
30C4⋊#272—22no—
31C2×(C3⋊Dic3)⋊C3—22no—
32#634⋊C3—22no—
33#634⋊C3—22no—
34(C3⋊(C3⋊C8))⋊C3D414yescomplemented
35#5175—92no—
36#5076—91no—
37C2×C32×D4—21no—
38C2×C32×D4C2×D312yes—
39C2×#634C2×D312yes—
40C2×C3×#67—64no—
41C2×C3×#67—64no—
42C2×C3×D12—32no—
43C2×C3×D12—61no—
44C2×C3×Dic6—32no—
45#3754—24no—
46C2×C3⋊(C3⋊C8)C2×D312yes—
47C3×#214—64no—
48C3×C4×Dic3—32no—
49C3×C4×Dic3—61no—
50C2×C3×C3⋊C8—32no—
51Dic3⋊Dic3—64no—
52#3716—64no—
53#3715—64no—
54C22×H(3,𝔽3)—41no—
55C22×H(3,𝔽3)—42no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C2×#272—41no—
58C2×#272—42no—
59C4×H(3,𝔽3)C2×D412yes—
60#1084SD1614yescomplemented
61(C3⋊Dic3)⋊C3—42no—
62C2×C3×SD16—92no—
63C3×#127—91no—
64#944—92no—
65C2×#215—92no—
66#940—32no—
67#940—61no—
68C2×#214—32no—
69C2×#214—92no—
70C2×#214—92no—
71C2×#204—92no—
72#869—91no—
73C8×Dic3—92no—
74C4×C3⋊C8—91no—
75C23×C32—22no—
76C23×C32—22no—
77C22×C3×D3—62no—
78C22×C3×D3—62no—
79C32×D4—22no—
80C32×D4—22no—
81C32×D4—42no—
82C2×C32×C4—21no—
83C2×C32×C4C22×D312yes—
84C2×C3⋊Dic3—22no—
85#634—22no—
86#634—22no—
87C3×#67—128no—
88C3×#67—128no—
89C2×C3×Dic3—34no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C3×D12—62no—
93C3×D12—62no—
94C3×D12—122no—
95C3×Dic6—62no—
96C3×Dic6—62no—
97C3⋊(C3⋊C8)#6714yescomplemented
98C3×C3⋊C8—34no—
99#492—271noSylow
100C2×H(3,𝔽3)—81no—
101C2×H(3,𝔽3)#12711yescharacteristic
102C2×H(3,𝔽3)C2×SD1612yes—
103#272—81no—
104C2×C3×Q8—92no—
105C2×C3×D4—32no—
106C2×C3×D4—61no—
107C2×C3×D4—91no—
108C2×C3×D4—182no—
109C2×C3×D4D3211yescharacteristic
110C2×#67—64no—
111C2×#67—64no—
112C2×D12—32no—
113C2×D12—61no—
114C2×Dic6—32no—
115C2×Dic6—92no—
116C3×SD16—184no—
117C2×C3×C8—92no—
118C3×C42—91no—
119#215—184no—
120#214—64no—
121#214—184no—
122#214—184no—
123C4⋊Dic3—182no—
124Dic3⋊C4—184no—
125C4×Dic3—32no—
126C4×Dic3—61no—
127C2×C3⋊C8—32no—
128C2×C3⋊C8—92no—
129#204—184no—
130C22×C32—21no—
131C22×C32—44no—
132C22×C32—42no—
133C22×C32—44no—
134C22×C32—42no—
135C22×C32D3×D412yes—
136C2×C3×D3—122no—
137C2×C3×D3—124no—
138C2×C3×D3—124no—
139C2×C3×D3—122no—
140C32×C4—22no—
141C32×C4D3×D412yes—
142C32×C4C2×#6712yes—
143C3⋊Dic3—42no—
144C3×Dic3—38no—
145C3×Dic3—64no—
146C3×Dic3—122no—
147C2×SD16—92no—
148C2×SD16—272no—
149C4⋊Q8—271no—
150#127—91no—
151C4×C8—271no—
152H(3,𝔽3)#49211yescharacteristic, complemented, Sylow
153C23×C3—21no—
154C23×C3—62no—
155C23×C3—62no—
156C23×C3—181no—
157C22×D3—62no—
158C22×D3—62no—
159C3×Q8—182no—
160C3×Q8—182no—
161C3×D4—22no—
162C3×D4—62no—
163C3×D4—62no—
164C3×D4—122no—
165C3×D4—182no—
166C3×D4—364no—
167C2×C3×C4—32no—
168C2×C3×C4—61no—
169C2×C3×C4—92no—
170C2×C3×C4—182no—
171C2×C3×C4C2×D3211yescharacteristic
172#67—128no—
173#67—128no—
174C2×Dic3—34no—
175C2×Dic3—62no—
176C2×Dic3—62no—
177C2×Dic3—182no—
178D12—62no—
179D12—62no—
180D12—122no—
181Dic6—62no—
182Dic6—62no—
183Dic6—182no—
184Dic6—182no—
185C3×C8—94no—
186C3⋊C8—34no—
187C3⋊C8—94no—
188C2×C32—22no—
189C2×C32—21no—
190C2×C32—82no—
191C2×C32—82no—
192C2×C32D3×SD1614yes—
193C2×C32#94012yes—
194C3×D3—242no—
195C3×D3—242no—
196C2×Q8—92no—
197C2×D4—91no—
198C2×D4—182no—
199C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
200SD16—184no—
201SD16—544no—
202C2×C8—92no—
203C4⋊C4—542no—
204C42—91no—
205C22×C3—32no—
206C22×C3—42no—
207C22×C3—41no—
208C22×C3—61no—
209C22×C3—124no—
210C22×C3—122no—
211C22×C3—124no—
212C22×C3—122no—
213C22×C3—362no—
214C22×C3—361no—
215C22×C3#381111yescharacteristic
216C2×D3—122no—
217C2×D3—124no—
218C2×D3—122no—
219C2×D3—124no—
220C3×C4—32no—
221C3×C4—32no—
222C3×C4—62no—
223C3×C4—94no—
224C3×C4—362no—
225C3×C4D3×#6712yes—
226Dic3—38no—
227Dic3—64no—
228Dic3—122no—
229Dic3—362no—
230C32—21no—
231C32#517512yescomplemented
232C23—21no—
233C23—181no—
234Q8—182no—
235Q8—182no—
236D4—22no—
237D4—182no—
238D4—364no—
239C2×C4—92no—
240C2×C4—182no—
241C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
242C8—94no—
243C2×C3—32no—
244C2×C3—34no—
245C2×C3—61no—
246C2×C3—62no—
247C2×C3—81no—
248C2×C3—242no—
249C2×C3—242no—
250C2×C3—721no—
251C2×C3#6393012yes—
252C2×C3#6397911yescharacteristic
253D3—242no—
254D3—242no—
255C22—41no—
256C22—42no—
257C22—361no—
258C22—362no—
259C22#8867711yescharacteristic
260C4—94no—
261C4—362no—
262C4#8867612yes—
263C3—32no—
264C3—61no—
265C3#1059149211yescharacteristic
266C2—81no—
267C2—721no—
268C2#1172118812yes—
269C2#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
prime radical#11721850C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#492
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
2. C2×#3754⋊C3C2semidirect
3. #11721099C2semidirect
4. C2×#88437C2semidirect
5. #11720302C2semidirect
16. C2×#3754D3semidirect
34. (C3⋊(C3⋊C8))⋊C3D4semidirect
60. #1084SD16semidirect
97. C3⋊(C3⋊C8)#67semidirect
152. H(3,𝔽3)#492semidirect
231. C32#5175semidirect

Maximal subgroups

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

subgroupindex
#117218502
C2×#3754⋊C32
#117210992
C2×#884372
#117203022
#105914903