Finite Group Databasebeta

← #13147408

Subgroups — #13147408

Order 1728.

565 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 63 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
2C2×#88631C211yesmaximal, characteristic, complemented
3C22×#6188C211yesmaximal, characteristic, complemented
4C22⋊#6188C212yesmaximal, complemented
5#11720784C212yesmaximal, complemented
6C2×D4×H(3,𝔽3)C411yescharacteristic
7C2×#6235C411yescharacteristic
8C2×#6235C411yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11#88744—22no—
12#88744—22no—
13C2×C4×#272C2211yescharacteristic
14C2×#6229C411yescharacteristic
15#88631—22no—
16C2×#6188—22no—
17C2×#6188C2212yes—
18C2×#6188C2212yes—
19C2×C3×#237—62no—
20C23×H(3,𝔽3)C2×C411yescharacteristic
21C23×H(3,𝔽3)C2×C411yescharacteristic
22C22×#272C2×C411yescharacteristic
23D4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)C2×C411yescharacteristic
25#6235—22no—
26#6235—22no—
27C2×#1084—21no—
28C2×#1084—21no—
29C2×#1084C2×C411yescharacteristic
30C2×#1084C2×C411yescharacteristic
31C2×#1084D412yes—
32C2×#1084D412yes—
33C2×#1084C2311yescharacteristic
34C4×#272—21no—
35C4×#272—21no—
36#6229—22no—
37#6188—22no—
38#6188—22no—
39C3×#408—91nomaximal
40C2×C32×D4—22no—
41C2×C3×#67—62no—
42C2×C3×#67—62no—
43C22×C3×Dic3—62no—
44C22×C3×Dic3—62no—
45C3×#237—68no—
46C3×#237—68no—
47C2×C3×C4×D3—62no—
48C2×C3×Dic6—62no—
49C22×H(3,𝔽3)—22no—
50C22×H(3,𝔽3)—21no—
51C22×H(3,𝔽3)—22no—
52C22×H(3,𝔽3)C22⋊C412yes—
53C22×H(3,𝔽3)C22×C411yescharacteristic
54C2×#272—21no—
55C2×#272—22no—
56C4×H(3,𝔽3)—21no—
57#1084—21no—
58#1084C22⋊C412yes—
59#1084C2×D411yescharacteristic
60#1084C2×D411yescharacteristic
61C2×C3×#41—91no—
62C2×#237—62no—
63C2×C3×M16—91no—
64C22×C3×C8—91no—
65C3×C22⋊C8—92no—
66C3×C22⋊C8—92no—
67C23×C32—22no—
68C23×C32—22no—
69C22×C3×D3—62no—
70C32×D4—28no—
71C2×C32×C4—22no—
72C3×#67—68no—
73C3×#67—68no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—62no—
76C2×C3×Dic3—64no—
77C2×C3×Dic3—64no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—64no—
80C2×C3×Dic3—64no—
81C3×C4×D3—64no—
82C3×C4×D3—64no—
83C3×Dic6—68no—
84#408—91noSylow
85C2×H(3,𝔽3)—22no—
86C2×H(3,𝔽3)—41no—
87C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
88C2×H(3,𝔽3)#13112yes—
89#272—41no—
90C3×#41—182no—
91C3×#41—182no—
92C2×C3×Q8—91no—
93C2×C3×D4—62no—
94C2×C3×D4—91no—
95C2×C3×D4—91no—
96C2×C3×D4#15711yescharacteristic
97C22×C3×C4—91no—
98C22×C3×C4—91no—
99C22×C3×C4—91no—
100C2×#67—62no—
101C2×#67—62no—
102C22×Dic3—62no—
103C22×Dic3—62no—
104#237—68no—
105#237—68no—
106C2×C4×D3—62no—
107C2×Dic6—62no—
108C3×M16—182no—
109C2×C3×C8—92no—
110C2×C3×C8—92no—
111C2×C3×C8—182no—
112C22×C32—24no—
113C22×C32—22no—
114C22×C32—24no—
115C22×C32—44no—
116C22×C32—44no—
117C2×C3×D3—64no—
118C2×C3×D3—124no—
119C32×C4—24no—
120C3×Dic3—64no—
121C3×Dic3—62no—
122C3×Dic3—62no—
123C3×Dic3—64no—
124C2×#41—91no—
125C2×M16—91no—
126C22×C8—91no—
127C22⋊C8—92no—
128C22⋊C8—92no—
129H(3,𝔽3)#40811yescharacteristic, complemented, Sylow
130C23×C3—62no—
131C23×C3—62no—
132C23×C3—91no—
133C23×C3C2×#15711yescharacteristic
134C23×C3C2×#15711yescharacteristic
135C22×D3—62no—
136C3×Q8—182no—
137C3×D4—22no—
138C3×D4—68no—
139C3×D4—182no—
140C3×D4—182no—
141C2×C3×C4—62no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—92no—
145C2×C3×C4—92no—
146C2×C3×C4—91no—
147C2×C3×C4—181no—
148C2×C3×C4—181no—
149C2×C3×C4—181no—
150C2×C3×C4—181no—
151C2×C3×C4C2×#15711yescharacteristic
152#67—68no—
153#67—68no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—62no—
157C2×Dic3—64no—
158C2×Dic3—64no—
159C2×Dic3—64no—
160C2×Dic3—64no—
161C4×D3—64no—
162C4×D3—64no—
163Dic6—68no—
164C3×C8—182no—
165C3×C8—182no—
166C2×C32—24no—
167C2×C32—22no—
168C2×C32—44no—
169C2×C32—44no—
170C3×D3—124no—
171#41—182no—
172#41—182no—
173C2×Q8—91no—
174C2×D4—91no—
175C2×D4—91no—
176C2×D4#108811yescharacteristic
177C22×C4—91no—
178C22×C4—91no—
179C22×C4—91no—
180M16—182no—
181C2×C8—92no—
182C2×C8—92no—
183C2×C8—182no—
184C22×C3—22no—
185C22×C3—22no—
186C22×C3—21no—
187C22×C3—64no—
188C22×C3—64no—
189C22×C3—62no—
190C22×C3—124no—
191C22×C3—124no—
192C22×C3—182no—
193C22×C3—181no—
194C22×C3C22⋊#15712yes—
195C22×C3C22×#15711yescharacteristic
196C2×D3—64no—
197C2×D3—124no—
198C3×C4—21no—
199C3×C4—64no—
200C3×C4—91no—
201C3×C4—91no—
202C3×C4—92no—
203C3×C4—181no—
204Dic3—64no—
205Dic3—62no—
206Dic3—62no—
207Dic3—64no—
208C32—22no—
209C23—91no—
210C23C2×#108811yescharacteristic
211C23C2×#108811yescharacteristic
212Q8—182no—
213D4—22no—
214D4—182no—
215D4—182no—
216C2×C4—92no—
217C2×C4—92no—
218C2×C4—91no—
219C2×C4—91no—
220C2×C4—91no—
221C2×C4—181no—
222C2×C4—181no—
223C2×C4—181no—
224C2×C4—181no—
225C2×C4C2×#108811yescharacteristic
226C8—182no—
227C8—182no—
228C2×C3—22no—
229C2×C3—41no—
230C2×C3—64no—
231C2×C3—62no—
232C2×C3—124no—
233C2×C3—124no—
234C2×C3—361no—
235C2×C3#6428712yes—
236C2×C3C2×C22⋊#15711yescharacteristic
237D3—124no—
238C22—22no—
239C22—22no—
240C22—21no—
241C22—182no—
242C22—181no—
243C22C22⋊#108812yes—
244C22C22×#108811yescharacteristic
245C4—21no—
246C4—91no—
247C4—91no—
248C4—92no—
249C4—181no—
250C3—62no—
251C3#1059382211yescharacteristic
252C2—22no—
253C2—41no—
254C2—361no—
255C2#1172244912yes—
256C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×D4×H(3,𝔽3)C4
prime radicalC2×#88744C2
cosocleC2×#1084C23
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#408
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88631C2semidirect
3. C22×#6188C2semidirect
4. C22⋊#6188C2semidirect
5. #11720784C2semidirect
129. H(3,𝔽3)#408semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
C2×#886312
C22×#61882
C22⋊#61882
#117207842
C3×#4089