Finite Group Databasebeta

← #13147398

Subgroups — #13147398

Order 1728.

823 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 823 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C22×C4×#272C211yesmaximal, characteristic
2C2×C22⋊#1088C212yesmaximal
3C2×C4⋊#1088C212yesmaximal, complemented
4#11720787C212yesmaximal, complemented
5C23×#272C2211yescharacteristic
6C22×#1088C2214yes—
7C22×C4×H(3,𝔽3)C411yescharacteristic, complemented
8C22×#1084C411yescharacteristic, complemented
9C2×C4×#272—24no—
10C2×C4×#272C411yescharacteristic, complemented
11C2×C4×#272C411yescharacteristic, complemented
12C2×C4×#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C22⋊#1088—24no—
15C4⋊#1088—24no—
16C22×C3×C4×D3—62no—
17C23×H(3,𝔽3)C2×C411yescharacteristic
18C22×#272—22no—
19C22×#272—22no—
20C22×#272C2×C411yescharacteristic
21C22×#272D412yes—
22C22×#272Q812yes—
23C22×#272C2311yescharacteristic
24C2×#1088—24no—
25C2×#1088—24no—
26C2×#1088—24no—
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)C2×C411yescharacteristic
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×#1084—22no—
31C2×#1084C2×C411yescharacteristic
32C2×#1084C2×C411yescharacteristic
33C4×#272—22no—
34C4×#272—22no—
35C4×#272—22no—
36C4×#272D414yes—
37C3×#380—91nomaximal
38C23×C3×D3—62no—
39C22×C32×C4—22no—
40C22×C3×Dic3—62no—
41C2×C3×C4×D3—62no—
42C2×C3×C4×D3—62no—
43C2×C3×C4×D3—62no—
44C2×C3×C4×D3—62no—
45C2×C3×C4×D3—616no—
46C22×H(3,𝔽3)—21no—
47C22×H(3,𝔽3)—21no—
48C22×H(3,𝔽3)C4⋊C412yescomplemented
49C22×H(3,𝔽3)C22×C411yescharacteristic
50C2×#272—21no—
51C2×#272—21no—
52C2×#272—24no—
53C2×#272—24no—
54C2×#272C4⋊C412yescomplemented
55C2×#272C2×D411yescharacteristic
56C2×#272C2×D412yes—
57C2×#272C2×Q811yescharacteristic
58C2×#272#4112yes—
59#1088—44no—
60C4×H(3,𝔽3)—21no—
61C4×H(3,𝔽3)C22⋊C412yescomplemented
62#1084—21no—
63#1084C22⋊C412yescomplemented
64C23×C3×C4—91no—
65C22×C4×D3—62no—
66C2×C3×C4⋊C4—92no—
67C2×C3×C22⋊C4—92no—
68C3×#95—92no—
69C23×C32—22no—
70C22×C3×D3—62no—
71C22×C3×D3—64no—
72C22×C3×D3—64no—
73C22×C3×D3—62no—
74C22×C3×D3—68no—
75C22×C3×D3—68no—
76C2×C32×C4—28no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—68no—
82C3×C4×D3—68no—
83C3×C4×D3—68no—
84C3×C4×D3—68no—
85C3×C4×D3—68no—
86#380—91noSylow
87C2×H(3,𝔽3)—22no—
88C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
89C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
90C2×H(3,𝔽3)#11711yescharacteristic
91#272—22no—
92#272#12112yes—
93#272C22⋊Q812yes—
94C24×C3—91no—
95C23×D3—62no—
96C22×C3×C4—62no—
97C22×C3×C4—91no—
98C22×C3×C4—91no—
99C22×C3×C4—91no—
100C22×C3×C4—91no—
101C22×C3×C4—91no—
102C22×C3×C4—94no—
103C22×C3×C4—184no—
104C22×C3×C4#15711yescharacteristic
105C22×Dic3—62no—
106C2×C4×D3—62no—
107C2×C4×D3—62no—
108C2×C4×D3—62no—
109C2×C4×D3—62no—
110C2×C4×D3—616no—
111C3×C4⋊C4—184no—
112C3×C22⋊C4—184no—
113C22×C32—24no—
114C22×C32—24no—
115C22×C32—22no—
116C22×C32—24no—
117C2×C3×D3—64no—
118C2×C3×D3—64no—
119C2×C3×D3—62no—
120C2×C3×D3—62no—
121C2×C3×D3—64no—
122C2×C3×D3—64no—
123C2×C3×D3—616no—
124C2×C3×D3—616no—
125C2×C3×D3—64no—
126C32×C4—24no—
127C32×C4—24no—
128C3×Dic3—64no—
129C3×Dic3—64no—
130C23×C4—91no—
131C2×C4⋊C4—92no—
132C2×C22⋊C4—92no—
133#95—92no—
134H(3,𝔽3)#38011yescharacteristic, complemented, Sylow
135C23×C3—62no—
136C23×C3—91no—
137C23×C3—91no—
138C23×C3—92no—
139C23×C3—92no—
140C23×C3—182no—
141C23×C3—182no—
142C23×C3C2×#15711yescharacteristic
143C22×D3—62no—
144C22×D3—62no—
145C22×D3—64no—
146C22×D3—64no—
147C22×D3—68no—
148C22×D3—68no—
149C2×C3×C4—22no—
150C2×C3×C4—68no—
151C2×C3×C4—62no—
152C2×C3×C4—62no—
153C2×C3×C4—91no—
154C2×C3×C4—94no—
155C2×C3×C4—91no—
156C2×C3×C4—182no—
157C2×C3×C4—182no—
158C2×C3×C4—182no—
159C2×C3×C4—182no—
160C2×C3×C4—184no—
161C2×C3×C4—184no—
162C2×C3×C4—184no—
163C2×C3×C4C2×#15711yescharacteristic
164C2×C3×C4C2×#15711yescharacteristic
165C2×Dic3—62no—
166C2×Dic3—62no—
167C2×Dic3—68no—
168C4×D3—68no—
169C4×D3—68no—
170C4×D3—68no—
171C4×D3—68no—
172C2×C32—28no—
173C2×C32—22no—
174C2×C32—22no—
175C2×C32—22no—
176C3×D3—64no—
177C3×D3—64no—
178C3×D3—68no—
179C24—91no—
180C22×C4—91no—
181C22×C4—91no—
182C22×C4—91no—
183C22×C4—91no—
184C22×C4—91no—
185C22×C4—94no—
186C22×C4—184no—
187C22×C4#108811yescharacteristic, complemented
188C4⋊C4—184no—
189C22⋊C4—184no—
190C22×C3—21no—
191C22×C3—21no—
192C22×C3—64no—
193C22×C3—64no—
194C22×C3—64no—
195C22×C3—62no—
196C22×C3—91no—
197C22×C3—91no—
198C22×C3—92no—
199C22×C3—92no—
200C22×C3—92no—
201C22×C3—184no—
202C22×C3—184no—
203C22×C3—181no—
204C22×C3—181no—
205C22×C3C4⋊#15712yes—
206C22×C3C22×#15711yescharacteristic
207C2×D3—64no—
208C2×D3—62no—
209C2×D3—62no—
210C2×D3—64no—
211C2×D3—64no—
212C2×D3—64no—
213C2×D3—616no—
214C2×D3—616no—
215C2×D3—64no—
216C3×C4—21no—
217C3×C4—64no—
218C3×C4—64no—
219C3×C4—92no—
220C3×C4—181no—
221C3×C4—364no—
222C3×C4C22⋊#15712yes—
223Dic3—64no—
224Dic3—64no—
225C32—22no—
226C23—91no—
227C23—92no—
228C23—92no—
229C23—91no—
230C23—182no—
231C23—182no—
232C23C2×#108811yescharacteristic
233C2×C4—22no—
234C2×C4—91no—
235C2×C4—94no—
236C2×C4—91no—
237C2×C4—182no—
238C2×C4—182no—
239C2×C4—182no—
240C2×C4—182no—
241C2×C4—184no—
242C2×C4—184no—
243C2×C4—184no—
244C2×C4C2×#108811yescharacteristic
245C2×C4C2×#108811yescharacteristic
246C2×C3—22no—
247C2×C3—68no—
248C2×C3—62no—
249C2×C3—62no—
250C2×C3—62no—
251C2×C3—92no—
252C2×C3—92no—
253C2×C3—182no—
254C2×C3C2×C4⋊#15711yescharacteristic
255C2×C3#6428611yescharacteristic
256C2×C3C2×C22⋊#15711yescharacteristic
257D3—64no—
258D3—64no—
259D3—68no—
260C22—21no—
261C22—21no—
262C22—91no—
263C22—91no—
264C22—92no—
265C22—92no—
266C22—92no—
267C22—184no—
268C22—184no—
269C22—181no—
270C22—181no—
271C22C4⋊#108812yescomplemented
272C22C22×#108811yescharacteristic
273C4—21no—
274C4—92no—
275C4—181no—
276C4—364no—
277C4C22⋊#108812yescomplemented
278C3—62no—
279C3#1059381211yescharacteristic
280C2—22no—
281C2—92no—
282C2—92no—
283C2—182no—
284C2C2×C4⋊#108811yescharacteristic
285C2#1172244811yescharacteristic
286C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×#272C22
cosocleC22×#272C23
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)#380
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊#1088C2semidirect
4. #11720787C2semidirect
7. C22×C4×H(3,𝔽3)C4semidirect
8. C22×#1084C4semidirect
10. C2×C4×#272C4semidirect
11. C2×C4×#272C4semidirect
48. C22×H(3,𝔽3)C4⋊C4semidirect
54. C2×#272C4⋊C4semidirect
61. C4×H(3,𝔽3)C22⋊C4semidirect
63. #1084C22⋊C4semidirect
134. H(3,𝔽3)#380semidirect
187. C22×C4#1088semidirect
271. C22C4⋊#1088semidirect
277. C4C22⋊#1088semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×C22⋊#10882
C2×C4⋊#10882
#117207872
C3×#3809