Finite Group Databasebeta

← #13147404

Subgroups — #13147404

Order 1728.

757 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 59 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×D4×#272C211yesmaximal, characteristic
2#11722448C211yesmaximal, characteristic, complemented
3#11722445C211yesmaximal, characteristic, complemented
4#11720788C214yesmaximal, complemented
5C23×#272—21no—
6C2×D4×H(3,𝔽3)C411yescharacteristic, complemented
7C2×#6235—21no—
8D4×#272—21no—
9D4×#272—21no—
10D4×#272C412yescomplemented
11D4×#272C2212yes—
12C2×C4⋊#272C411yescharacteristic, complemented
13C2×C4×#272C2211yescharacteristic
14C22⋊#1088—21no—
15#88631—21no—
16C4⋊#1088C2212yescomplemented
17C4×#1088—21no—
18#88627—21no—
19#88626C2212yescomplemented
20C2×C3×D3×D4—62no—
21C23×H(3,𝔽3)—21no—
22C22×#272—21no—
23C22×#272—22no—
24C22×#272—22no—
25C22×#272—44no—
26C22×#272D411yescharacteristic
27C2×#1088—22no—
28D4×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)C2×C412yes—
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31#6235—22no—
32#6235—22no—
33C2×#1084D411yescharacteristic
34C4⋊#272—21no—
35C4⋊#272C2×C412yes—
36C4×#272C2×C411yescharacteristic
37C4×#272D411yescharacteristic
38C4×#272D411yescharacteristic
39C4×#272C2311yescharacteristic
40#6188—22no—
41C3×#418—91nomaximal
42C23×C3×D3—64no—
43C2×C32×D4—22no—
44C2×C3×#67—64no—
45C3×D3×D4—64no—
46C3×D3×D4—64no—
47C3×D3×D4—64no—
48C3×D3×D4—64no—
49C2×C3×D12—62no—
50C2×C3×C4×D3—62no—
51C22×H(3,𝔽3)—22no—
52C22×H(3,𝔽3)—42no—
53C22×H(3,𝔽3)C22⋊C411yescharacteristic
54C2×#272—22no—
55C2×#272—22no—
56C2×#272—44no—
57C2×#272—42no—
58C2×#272—44no—
59C2×#272C22⋊C411yescharacteristic
60C2×#272C2×D411yescharacteristic
61#1088—42no—
62C4×H(3,𝔽3)C22⋊C411yescharacteristic
63C4×H(3,𝔽3)C22×C411yescharacteristic
64#1084C22⋊C411yescharacteristic
65#1084C2×D411yescharacteristic
66C22×C3×D4—91no—
67C2×D3×D4—62no—
68C2×C3×M16—91no—
69C3×#117—91no—
70C3×#102—94no—
71C23×C32—24no—
72C22×C3×D3—62no—
73C22×C3×D3—64no—
74C22×C3×D3—68no—
75C22×C3×D3—68no—
76C22×C3×D3—1216no—
77C32×D4—24no—
78C32×D4—24no—
79C2×C32×C4—22no—
80C3×#67—68no—
81C3×#67—68no—
82C2×C3×Dic3—62no—
83C3×D12—64no—
84C3×D12—64no—
85C3×C4×D3—62no—
86C3×C4×D3—62no—
87C3×C4×D3—62no—
88C3×C4×D3—62no—
89#418—91noSylow
90C2×H(3,𝔽3)—21no—
91C2×H(3,𝔽3)—42no—
92C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
93#272—21no—
94#272—42no—
95#272AGL(1,ℤ/8)12yes—
96C24×C3—181no—
97C23×D3—64no—
98C2×C3×D4—62no—
99C2×C3×D4—92no—
100C2×C3×D4—92no—
101C2×C3×D4—91no—
102C2×C3×D4—181no—
103C2×C3×D4—181no—
104C2×C3×D4—181no—
105C2×C3×D4#15711yescharacteristic
106C22×C3×C4—91no—
107C2×#67—64no—
108D3×D4—64no—
109D3×D4—64no—
110D3×D4—64no—
111D3×D4—64no—
112C2×D12—62no—
113C2×C4×D3—62no—
114C3×M16—181no—
115C3×M16—181no—
116C2×C3×C8—92no—
117C3×C4⋊C4—92no—
118C3×C22⋊C4—181no—
119C3×C42—181no—
120C22×C32—28no—
121C22×C32—22no—
122C22×C32—48no—
123C2×C3×D3—62no—
124C2×C3×D3—62no—
125C2×C3×D3—68no—
126C2×C3×D3—68no—
127C2×C3×D3—1216no—
128C2×C3×D3—1216no—
129C2×C3×D3—128no—
130C32×C4—22no—
131C32×C4—22no—
132C3×Dic3—62no—
133C3×Dic3—62no—
134C22×D4—91no—
135C2×M16—91no—
136#117—91no—
137#102—94no—
138H(3,𝔽3)#41811yescharacteristic, complemented, Sylow
139C23×C3—21no—
140C23×C3—64no—
141C23×C3—91no—
142C23×C3—182no—
143C23×C3—182no—
144C23×C3—181no—
145C23×C3—364no—
146C22×D3—62no—
147C22×D3—64no—
148C22×D3—68no—
149C22×D3—68no—
150C22×D3—1216no—
151C3×D4—21no—
152C3×D4—64no—
153C3×D4—64no—
154C3×D4—92no—
155C3×D4—182no—
156C3×D4—182no—
157C3×D4—181no—
158C3×D4C2×#15712yes—
159C2×C3×C4—62no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—91no—
164C2×C3×C4—91no—
165C2×C3×C4—182no—
166C2×C3×C4C2×#15711yescharacteristic
167#67—68no—
168#67—68no—
169C2×Dic3—62no—
170D12—64no—
171D12—64no—
172C4×D3—62no—
173C4×D3—62no—
174C4×D3—62no—
175C4×D3—62no—
176C3×C8—182no—
177C2×C32—24no—
178C2×C32—22no—
179C2×C32—48no—
180C3×D3—64no—
181C3×D3—64no—
182C3×D3—128no—
183C24—181no—
184C2×D4—92no—
185C2×D4—92no—
186C2×D4—91no—
187C2×D4—181no—
188C2×D4—181no—
189C2×D4—181no—
190C2×D4#108811yescharacteristic, complemented
191C22×C4—91no—
192M16—181no—
193M16—181no—
194C2×C8—92no—
195C4⋊C4—92no—
196C22⋊C4—181no—
197C42—181no—
198C22×C3—22no—
199C22×C3—42no—
200C22×C3—68no—
201C22×C3—62no—
202C22×C3—91no—
203C22×C3—91no—
204C22×C3—128no—
205C22×C3—182no—
206C22×C3—182no—
207C22×C3—362no—
208C22×C3—364no—
209C22×C3—364no—
210C22×C3C22⋊#15711yescharacteristic
211C2×D3—62no—
212C2×D3—62no—
213C2×D3—68no—
214C2×D3—68no—
215C2×D3—1216no—
216C2×D3—1216no—
217C2×D3—128no—
218C3×C4—62no—
219C3×C4—62no—
220C3×C4—91no—
221C3×C4—91no—
222C3×C4—362no—
223C3×C4C22⋊#15711yescharacteristic
224C3×C4C22×#15711yescharacteristic
225Dic3—62no—
226Dic3—62no—
227C32—22no—
228C23—21no—
229C23—91no—
230C23—182no—
231C23—182no—
232C23—181no—
233C23—364no—
234D4—21no—
235D4—92no—
236D4—182no—
237D4—182no—
238D4—181no—
239D4C2×#108812yes—
240C2×C4—91no—
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—91no—
244C2×C4—91no—
245C2×C4—182no—
246C2×C4C2×#108811yescharacteristic
247C8—182no—
248C2×C3—21no—
249C2×C3—42no—
250C2×C3—64no—
251C2×C3—62no—
252C2×C3—92no—
253C2×C3—128no—
254C2×C3—181no—
255C2×C3—362no—
256C2×C3C2×C22⋊#15711yescharacteristic
257D3—64no—
258D3—64no—
259D3—128no—
260C22—22no—
261C22—42no—
262C22—91no—
263C22—91no—
264C22—182no—
265C22—182no—
266C22—362no—
267C22—364no—
268C22—364no—
269C22C22⋊#108811yescharacteristic
270C4—91no—
271C4—91no—
272C4—362no—
273C4C22⋊#108811yescharacteristic
274C4C22×#108811yescharacteristic
275C3—62no—
276C3#1059381411yescharacteristic
277C2—21no—
278C2—42no—
279C2—92no—
280C2—181no—
281C2—362no—
282C2C2×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×D4×#272C2
cosocleC4×#272C23
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#418
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. #11720788C2semidirect
6. C2×D4×H(3,𝔽3)C4semidirect
10. D4×#272C4semidirect
12. C2×C4⋊#272C4semidirect
16. C4⋊#1088C22semidirect
19. #88626C22semidirect
138. H(3,𝔽3)#418semidirect
190. C2×D4#1088semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
#117224482
#117224452
#117207882
C3×#4189