Finite Group Databasebeta

← #13147394

Subgroups — #13147394

Order 1728.

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

normal subgroups only

This group has 553 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#11722445C212yesmaximal, complemented
3#11720794C212yesmaximal, complemented
4#11720786C212yesmaximal, complemented
5C22×C4×H(3,𝔽3)C411yescharacteristic
6C2×#6235C411yescharacteristic
7C2×#6235C2211yescharacteristic
8#88740—22no—
9#88740—22no—
10C2×C4⋊#272C411yescharacteristic
11C2×C4×#272C411yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×#6229C2211yescharacteristic
14#88631C2214yes—
15#88627—22no—
16#88626—22no—
17C2×C3×#235—62no—
18C23×H(3,𝔽3)C2×C411yescharacteristic
19C22×#272C2×C411yescharacteristic
20C22×#272C2×C411yescharacteristic
21C2×C4×H(3,𝔽3)—21no—
22C2×C4×H(3,𝔽3)—21no—
23C2×C4×H(3,𝔽3)C2×C411yescharacteristic
24C2×C4×H(3,𝔽3)C2×C411yescharacteristic
25#6235—22no—
26#6235—22no—
27C2×#1084C2×C411yescharacteristic
28C2×#1084C2311yescharacteristic
29C4⋊#272—22no—
30C4×#272—21no—
31C4×#272—21no—
32C4×#272D412yes—
33C4×#272D412yes—
34#6229—22no—
35#6188—24no—
36C3×#413—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×#272C22⋊C412yes—
53C4×H(3,𝔽3)—21no—
54C4×H(3,𝔽3)C22⋊C411yescharacteristic
55C4×H(3,𝔽3)C22⋊C411yescharacteristic
56#1084—21no—
57#1084C2×D412yes—
58C2×C3×#41—91no—
59C2×#235—62no—
60C2×C3×M16—92no—
61C3×#101—92no—
62C3×#100—92no—
63C23×C32—22no—
64C22×C3×D3—62no—
65C22×C3×D3—62no—
66C2×C32×C4—24no—
67C2×C32×C4—24no—
68C2×C32×C4—22no—
69C2×C32×C4—22no—
70C3×#67—68no—
71C3×#67—68no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—62no—
74C3×D12—68no—
75C3×C4×D3—64no—
76C3×C4×D3—64no—
77C3×C4×D3—64no—
78C3×C4×D3—64no—
79C3×Dic6—68no—
80#413—91noSylow
81C2×H(3,𝔽3)—21no—
82C2×H(3,𝔽3)—41no—
83C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
84#272—22no—
85#272—41no—
86C3×#41—182no—
87C3×#41—182no—
88C2×C3×Q8—91no—
89C2×C3×D4—91no—
90C2×C3×D4—91no—
91C2×C3×D4—91no—
92C22×C3×C4—62no—
93C22×C3×C4—91no—
94C22×C3×C4—91no—
95C22×C3×C4#15711yescharacteristic
96C2×#67—62no—
97C2×#67—62no—
98#235—68no—
99#235—68no—
100C2×D12—62no—
101C2×C4×D3—62no—
102C2×C4×D3—62no—
103C2×Dic6—62no—
104C3×M16—94no—
105C3×M16—182no—
106C2×C3×C8—182no—
107C22×C32—24no—
108C22×C32—22no—
109C22×C32—44no—
110C2×C3×D3—64no—
111C2×C3×D3—64no—
112C2×C3×D3—124no—
113C2×C3×D3—124no—
114C32×C4—24no—
115C32×C4—22no—
116C32×C4—22no—
117C3×Dic3—64no—
118C3×Dic3—64no—
119C2×#41—91no—
120C2×M16—92no—
121#101—92no—
122#100—92no—
123H(3,𝔽3)#41311yescharacteristic, complemented, Sylow
124C23×C3—62no—
125C23×C3—91no—
126C23×C3—91no—
127C23×C3C2×#15711yescharacteristic
128C22×D3—62no—
129C22×D3—62no—
130C3×Q8—182no—
131C3×D4—182no—
132C3×D4—182no—
133C3×D4—182no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—64no—
137C2×C3×C4—64no—
138C2×C3×C4—62no—
139C2×C3×C4—62no—
140C2×C3×C4—91no—
141C2×C3×C4—92no—
142C2×C3×C4—92no—
143C2×C3×C4—91no—
144C2×C3×C4—181no—
145C2×C3×C4—181no—
146C2×C3×C4C2×#15711yescharacteristic
147C2×C3×C4C2×#15711yescharacteristic
148#67—68no—
149#67—68no—
150C2×Dic3—62no—
151C2×Dic3—62no—
152D12—68no—
153C4×D3—64no—
154C4×D3—64no—
155C4×D3—64no—
156C4×D3—64no—
157Dic6—68no—
158C3×C8—184no—
159C2×C32—24no—
160C2×C32—22no—
161C2×C32—44no—
162C3×D3—124no—
163C3×D3—124no—
164#41—182no—
165#41—182no—
166C2×Q8—91no—
167C2×D4—91no—
168C2×D4—91no—
169C2×D4—91no—
170C22×C4—91no—
171C22×C4—91no—
172C22×C4#108811yescharacteristic
173M16—94no—
174M16—182no—
175C2×C8—182no—
176C22×C3—21no—
177C22×C3—41no—
178C22×C3—64no—
179C22×C3—62no—
180C22×C3—92no—
181C22×C3—124no—
182C22×C3—181no—
183C22×C3—361no—
184C22×C3—361no—
185C22×C3C22×#15711yescharacteristic
186C2×D3—64no—
187C2×D3—64no—
188C2×D3—124no—
189C2×D3—124no—
190C3×C4—21no—
191C3×C4—64no—
192C3×C4—62no—
193C3×C4—62no—
194C3×C4—92no—
195C3×C4—181no—
196C3×C4C22⋊#15711yescharacteristic
197C3×C4C22⋊#15711yescharacteristic
198Dic3—64no—
199Dic3—64no—
200C32—22no—
201C23—91no—
202C23—91no—
203C23C2×#108811yescharacteristic
204Q8—182no—
205D4—182no—
206D4—182no—
207D4—182no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—91no—
211C2×C4—92no—
212C2×C4—92no—
213C2×C4—91no—
214C2×C4—181no—
215C2×C4—181no—
216C2×C4C2×#108811yescharacteristic
217C2×C4C2×#108811yescharacteristic
218C8—184no—
219C2×C3—21no—
220C2×C3—41no—
221C2×C3—64no—
222C2×C3—62no—
223C2×C3—124no—
224C2×C3—182no—
225C2×C3—361no—
226C2×C3C2×C22⋊#15711yescharacteristic
227D3—124no—
228D3—124no—
229C22—21no—
230C22—41no—
231C22—92no—
232C22—181no—
233C22—361no—
234C22—361no—
235C22C22×#108811yescharacteristic
236C4—21no—
237C4—92no—
238C4—181no—
239C4C22⋊#108811yescharacteristic
240C4C22⋊#108811yescharacteristic
241C3—62no—
242C3#1059380811yescharacteristic
243C2—21no—
244C2—41no—
245C2—182no—
246C2—361no—
247C2C2×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
cosocleC2×#1084C23
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)#413
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. #11722445C2semidirect
3. #11720794C2semidirect
4. #11720786C2semidirect
123. H(3,𝔽3)#413semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
#117224452
#117207942
#117207862
C3×#4139