Finite Group Databasebeta

← #13147400

Subgroups — #13147400

Order 1728.

559 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 57 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×#88744C211yesmaximal, characteristic
2#11722448C211yesmaximal, characteristic, complemented
3#11722445C211yesmaximal, characteristic, complemented
4#11720789C214yesmaximal, complemented
5C2×D4×H(3,𝔽3)C411yescharacteristic, complemented
6C2×#6235—21no—
7C22×#1084—21no—
8#88744—21no—
9#88744—21no—
10#88744C412yescomplemented
11#88744C2212yes—
12C2×C4×#272C2211yescharacteristic
13C2×#6229C411yescharacteristic, complemented
14C22⋊#1088—21no—
15#88631—21no—
16C4⋊#1088—21no—
17C4×#1088C2212yescomplemented
18#88627C2212yescomplemented
19#88626—21no—
20C2×C3×#237—62no—
21C23×H(3,𝔽3)—21no—
22C22×#272D411yescharacteristic
23C2×#1088—22no—
24D4×H(3,𝔽3)—21no—
25D4×H(3,𝔽3)C2×C412yes—
26C2×C4×H(3,𝔽3)C2×C411yescharacteristic
27#6235—22no—
28#6235—22no—
29C2×#1084—21no—
30C2×#1084—22no—
31C2×#1084—22no—
32C2×#1084D411yescharacteristic
33C4×#272C2×C411yescharacteristic
34C4×#272D411yescharacteristic
35C4×#272D411yescharacteristic
36C4×#272C2311yescharacteristic
37#6229—21no—
38#6229C2×C412yes—
39#6188—22no—
40C3×#421—91nomaximal
41C2×C32×D4—22no—
42C2×C3×#67—64no—
43C22×C3×Dic3—64no—
44C3×#237—64no—
45C3×#237—64no—
46C3×#237—64no—
47C3×#237—64no—
48C2×C3×C4×D3—62no—
49C2×C3×Dic6—62no—
50C22×H(3,𝔽3)—22no—
51C22×H(3,𝔽3)—42no—
52C22×H(3,𝔽3)C22⋊C411yescharacteristic
53C2×#272—41no—
54C2×#272C22⋊C411yescharacteristic
55C2×#272C2×D411yescharacteristic
56#1088—42no—
57C4×H(3,𝔽3)C22⋊C411yescharacteristic
58C4×H(3,𝔽3)C22×C411yescharacteristic
59#1084—22no—
60#1084C22⋊C411yescharacteristic
61#1084C2×D411yescharacteristic
62C2×C3×#41—91no—
63C2×#237—62no—
64C2×C3×M16—91no—
65C3×#117—91no—
66C3×#104—94no—
67C23×C32—24no—
68C22×C3×D3—62no—
69C32×D4—24no—
70C32×D4—24no—
71C2×C32×C4—22no—
72C3×#67—68no—
73C3×#67—68no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—64no—
76C2×C3×Dic3—68no—
77C2×C3×Dic3—68no—
78C3×C4×D3—62no—
79C3×C4×D3—62no—
80C3×C4×D3—62no—
81C3×C4×D3—62no—
82C3×Dic6—64no—
83C3×Dic6—64no—
84#421—91noSylow
85C2×H(3,𝔽3)—21no—
86C2×H(3,𝔽3)—42no—
87C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
88#272—21no—
89#272—21no—
90C3×#41—92no—
91C3×#41—92no—
92C3×#41—181no—
93C3×#41—181no—
94C2×C3×Q8—91no—
95C2×C3×D4—62no—
96C2×C3×D4—181no—
97C2×C3×D4#15711yescharacteristic
98C22×C3×C4—91no—
99C22×C3×C4—181no—
100C2×#67—64no—
101C22×Dic3—64no—
102#237—64no—
103#237—64no—
104#237—64no—
105#237—64no—
106C2×C4×D3—62no—
107C2×Dic6—62no—
108C3×M16—92no—
109C3×M16—181no—
110C2×C3×C8—181no—
111C3×C4⋊C4—181no—
112C3×C22⋊C4—181no—
113C3×C42—92no—
114C22×C32—28no—
115C22×C32—22no—
116C22×C32—48no—
117C2×C3×D3—62no—
118C2×C3×D3—62no—
119C2×C3×D3—124no—
120C32×C4—22no—
121C32×C4—22no—
122C3×Dic3—62no—
123C3×Dic3—62no—
124C3×Dic3—68no—
125C2×#41—91no—
126C2×M16—91no—
127#117—91no—
128#104—94no—
129H(3,𝔽3)#42111yescharacteristic, complemented, Sylow
130C23×C3—21no—
131C23×C3—64no—
132C23×C3—91no—
133C22×D3—62no—
134C3×Q8—92no—
135C3×Q8—181no—
136C3×D4—21no—
137C3×D4—64no—
138C3×D4—64no—
139C3×D4—182no—
140C3×D4—182no—
141C3×D4C2×#15712yes—
142C2×C3×C4—62no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—182no—
149C2×C3×C4—182no—
150C2×C3×C4—181no—
151C2×C3×C4—182no—
152C2×C3×C4C2×#15711yescharacteristic
153#67—68no—
154#67—68no—
155C2×Dic3—62no—
156C2×Dic3—64no—
157C2×Dic3—68no—
158C2×Dic3—68no—
159C4×D3—62no—
160C4×D3—62no—
161C4×D3—62no—
162C4×D3—62no—
163Dic6—64no—
164Dic6—64no—
165C3×C8—182no—
166C2×C32—24no—
167C2×C32—22no—
168C2×C32—48no—
169C3×D3—122no—
170C3×D3—122no—
171#41—92no—
172#41—92no—
173#41—181no—
174#41—181no—
175C2×Q8—91no—
176C2×D4—181no—
177C2×D4#108811yescharacteristic, complemented
178C22×C4—91no—
179C22×C4—181no—
180M16—92no—
181M16—181no—
182C2×C8—181no—
183C4⋊C4—181no—
184C22⋊C4—181no—
185C42—92no—
186C22×C3—22no—
187C22×C3—42no—
188C22×C3—68no—
189C22×C3—62no—
190C22×C3—91no—
191C22×C3—91no—
192C22×C3—128no—
193C22×C3—361no—
194C22×C3C22⋊#15711yescharacteristic
195C2×D3—62no—
196C2×D3—62no—
197C2×D3—124no—
198C3×C4—62no—
199C3×C4—62no—
200C3×C4—91no—
201C3×C4—91no—
202C3×C4—182no—
203C3×C4—362no—
204C3×C4C22⋊#15711yescharacteristic
205C3×C4C22×#15711yescharacteristic
206Dic3—62no—
207Dic3—62no—
208Dic3—68no—
209C32—22no—
210C23—21no—
211C23—91no—
212Q8—92no—
213Q8—181no—
214D4—21no—
215D4—182no—
216D4—182no—
217D4C2×#108812yes—
218C2×C4—91no—
219C2×C4—91no—
220C2×C4—91no—
221C2×C4—91no—
222C2×C4—91no—
223C2×C4—182no—
224C2×C4—182no—
225C2×C4—181no—
226C2×C4—182no—
227C2×C4C2×#108811yescharacteristic
228C8—182no—
229C2×C3—21no—
230C2×C3—42no—
231C2×C3—64no—
232C2×C3—62no—
233C2×C3—128no—
234C2×C3—181no—
235C2×C3—181no—
236C2×C3C2×C22⋊#15711yescharacteristic
237D3—122no—
238D3—122no—
239C22—22no—
240C22—42no—
241C22—91no—
242C22—91no—
243C22—361no—
244C22C22⋊#108811yescharacteristic
245C4—91no—
246C4—91no—
247C4—182no—
248C4—362no—
249C4C22⋊#108811yescharacteristic
250C4C22×#108811yescharacteristic
251C3—62no—
252C3#1059381611yescharacteristic
253C2—21no—
254C2—42no—
255C2—181no—
256C2—181no—
257C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC2×C3C2×C22⋊#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingC2×D4×H(3,𝔽3)C4
prime radicalC2×#88744C2
cosocleC4×#272C23
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#421
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. #11722448C2semidirect
3. #11722445C2semidirect
4. #11720789C2semidirect
5. C2×D4×H(3,𝔽3)C4semidirect
10. #88744C4semidirect
13. C2×#6229C4semidirect
17. C4×#1088C22semidirect
18. #88627C22semidirect
129. H(3,𝔽3)#421semidirect
177. C2×D4#1088semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
#117224482
#117224452
#117207892
C3×#4219