Finite Group Databasebeta

← #13147378

Subgroups — #13147378

Order 1728.

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

normal subgroups only

This group has 565 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×#88740C211yesmaximal, characteristic
2#11722448C212yesmaximal, complemented
3#11720792C212yesmaximal, complemented
4#11720780C212yesmaximal, complemented
5C22×C4×H(3,𝔽3)C411yescharacteristic, complemented
6C2×#6235C411yescharacteristic, complemented
7C2×#6235C2211yescharacteristic
8#88740—22no—
9#88740—22no—
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272C411yescharacteristic, complemented
12C2×C4×#272C2211yescharacteristic
13C2×#6229C411yescharacteristic, complemented
14C22⋊#1088C2214yes—
15C4⋊#1088—22no—
16C4×#1088—22no—
17C2×C3×#235—62no—
18C23×H(3,𝔽3)C2×C411yescharacteristic
19C22×#272C2×C411yescharacteristic
20C22×#272C2311yescharacteristic
21C2×#1088—24no—
22C2×C4×H(3,𝔽3)—21no—
23C2×C4×H(3,𝔽3)—21no—
24C2×C4×H(3,𝔽3)C2×C411yescharacteristic
25C2×C4×H(3,𝔽3)C2×C411yescharacteristic
26#6235—22no—
27#6235—22no—
28C2×#1084C2×C411yescharacteristic
29C2×#1084C2×C411yescharacteristic
30C4⋊#272—22no—
31C4×#272—21no—
32C4×#272—21no—
33C4×#272D412yes—
34C4×#272D412yes—
35#6229—22no—
36C3×#410—91nomaximal
37C22×C32×C4—22no—
38C2×C3×#67—62no—
39C2×C3×#67—62no—
40C3×#235—68no—
41C3×#235—68no—
42C2×C3×D12—62no—
43C2×C3×C4×D3—62no—
44C2×C3×C4×D3—62no—
45C2×C3×Dic6—62no—
46C22×H(3,𝔽3)—21no—
47C22×H(3,𝔽3)—41no—
48C22×H(3,𝔽3)C22×C411yescharacteristic
49C2×#272—21no—
50C2×#272—41no—
51C2×#272—41no—
52C2×#272C2×D412yes—
53#1088—44no—
54C4×H(3,𝔽3)—21no—
55C4×H(3,𝔽3)C22⋊C411yescharacteristic
56C4×H(3,𝔽3)C22⋊C411yescharacteristic
57#1084—21no—
58#1084C22⋊C412yes—
59C2×C3×#41—91no—
60C2×#235—62no—
61C3×#117—92no—
62C3×#99—92no—
63C3×#99—92no—
64C23×C32—22no—
65C22×C3×D3—62no—
66C22×C3×D3—62no—
67C2×C32×C4—24no—
68C2×C32×C4—24no—
69C2×C32×C4—22no—
70C2×C32×C4—22no—
71C3×#67—68no—
72C3×#67—68no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—62no—
75C3×D12—68no—
76C3×C4×D3—64no—
77C3×C4×D3—64no—
78C3×C4×D3—64no—
79C3×C4×D3—64no—
80C3×Dic6—68no—
81#410—91noSylow
82C2×H(3,𝔽3)—21no—
83C2×H(3,𝔽3)—41no—
84C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
85#272—22no—
86#272—41no—
87C3×#41—182no—
88C3×#41—182no—
89C2×C3×Q8—91no—
90C2×C3×D4—91no—
91C2×C3×D4—91no—
92C2×C3×D4—91no—
93C22×C3×C4—62no—
94C22×C3×C4—91no—
95C22×C3×C4—91no—
96C22×C3×C4#15711yescharacteristic
97C2×#67—62no—
98C2×#67—62no—
99#235—68no—
100#235—68no—
101C2×D12—62no—
102C2×C4×D3—62no—
103C2×C4×D3—62no—
104C2×Dic6—62no—
105C3×C4⋊C4—182no—
106C3×C22⋊C4—94no—
107C3×C42—182no—
108C22×C32—24no—
109C22×C32—22no—
110C22×C32—44no—
111C2×C3×D3—64no—
112C2×C3×D3—64no—
113C2×C3×D3—124no—
114C2×C3×D3—124no—
115C32×C4—24no—
116C32×C4—22no—
117C32×C4—22no—
118C3×Dic3—64no—
119C3×Dic3—64no—
120C2×#41—91no—
121#117—92no—
122#99—92no—
123#99—92no—
124H(3,𝔽3)#41011yescharacteristic, complemented, Sylow
125C23×C3—62no—
126C23×C3—91no—
127C23×C3—91no—
128C23×C3C2×#15711yescharacteristic
129C22×D3—62no—
130C22×D3—62no—
131C3×Q8—182no—
132C3×D4—182no—
133C3×D4—182no—
134C3×D4—182no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—64no—
138C2×C3×C4—64no—
139C2×C3×C4—62no—
140C2×C3×C4—62no—
141C2×C3×C4—91no—
142C2×C3×C4—92no—
143C2×C3×C4—92no—
144C2×C3×C4—91no—
145C2×C3×C4—181no—
146C2×C3×C4—181no—
147C2×C3×C4—184no—
148C2×C3×C4C2×#15711yescharacteristic
149C2×C3×C4C2×#15711yescharacteristic
150#67—68no—
151#67—68no—
152C2×Dic3—62no—
153C2×Dic3—62no—
154D12—68no—
155C4×D3—64no—
156C4×D3—64no—
157C4×D3—64no—
158C4×D3—64no—
159Dic6—68no—
160C2×C32—24no—
161C2×C32—22no—
162C2×C32—44no—
163C3×D3—124no—
164C3×D3—124no—
165#41—182no—
166#41—182no—
167C2×Q8—91no—
168C2×D4—91no—
169C2×D4—91no—
170C2×D4—91no—
171C22×C4—91no—
172C22×C4—91no—
173C22×C4#108811yescharacteristic, complemented
174C4⋊C4—182no—
175C22⋊C4—94no—
176C42—182no—
177C22×C3—21no—
178C22×C3—41no—
179C22×C3—64no—
180C22×C3—62no—
181C22×C3—92no—
182C22×C3—124no—
183C22×C3—181no—
184C22×C3—361no—
185C22×C3—361no—
186C22×C3C22×#15711yescharacteristic
187C2×D3—64no—
188C2×D3—64no—
189C2×D3—124no—
190C2×D3—124no—
191C3×C4—21no—
192C3×C4—64no—
193C3×C4—62no—
194C3×C4—62no—
195C3×C4—92no—
196C3×C4—181no—
197C3×C4—364no—
198C3×C4C22⋊#15711yescharacteristic
199C3×C4C22⋊#15711yescharacteristic
200Dic3—64no—
201Dic3—64no—
202C32—22no—
203C23—91no—
204C23—91no—
205C23C2×#108811yescharacteristic
206Q8—182no—
207D4—182no—
208D4—182no—
209D4—182no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—91no—
213C2×C4—92no—
214C2×C4—92no—
215C2×C4—91no—
216C2×C4—181no—
217C2×C4—181no—
218C2×C4—184no—
219C2×C4C2×#108811yescharacteristic
220C2×C4C2×#108811yescharacteristic
221C2×C3—21no—
222C2×C3—41no—
223C2×C3—64no—
224C2×C3—62no—
225C2×C3—124no—
226C2×C3—182no—
227C2×C3—361no—
228C2×C3C2×C22⋊#15711yescharacteristic
229D3—124no—
230D3—124no—
231C22—21no—
232C22—41no—
233C22—92no—
234C22—181no—
235C22—361no—
236C22—361no—
237C22C22×#108811yescharacteristic
238C4—21no—
239C4—92no—
240C4—181no—
241C4—364no—
242C4C22⋊#108811yescharacteristic
243C4C22⋊#108811yescharacteristic
244C3—62no—
245C3#1059379211yescharacteristic
246C2—21no—
247C2—41no—
248C2—182no—
249C2—361no—
250C2C2×C22⋊#108811yescharacteristic
subgroupG/H
centerC3×C4C22⋊#157
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC22×C3C22×#157
FittingC22×C4×H(3,𝔽3)C4
socleC2×C3C2×C22⋊#157
prime radicalC2×#88740C2
cosocleC22×#272C23
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)#410
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. #11722448C2semidirect
3. #11720792C2semidirect
4. #11720780C2semidirect
5. C22×C4×H(3,𝔽3)C4semidirect
6. C2×#6235C4semidirect
11. C2×C4×#272C4semidirect
13. C2×#6229C4semidirect
124. H(3,𝔽3)#410semidirect
173. C22×C4#1088semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
#117224482
#117207922
#117207802
C3×#4109