Finite Group Databasebeta

← #13147332

Subgroups — #13147332

Order 1728.

615 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C23×#1088C211yesmaximal, characteristic
2C2×C22⋊#1088C211yesmaximal, characteristic
3C2×C4⋊#1088C212yesmaximal, complemented
4C22⋊C4×#272C211yesmaximal, characteristic
5#11720787C212yesmaximal, complemented
6C23×#272C2211yescharacteristic
7C22×#1088—22no—
8C22×#1088—24no—
9C22×#1088C2211yescharacteristic
10C22×#1088C2211yescharacteristic
11C22×#1088C2212yes—
12C2×C4×#272C2212yes—
13C22⋊#1088—22no—
14C4⋊#1088—24no—
15C22⋊C4×H(3,𝔽3)C411yescharacteristic, complemented
16#88552C411yescharacteristic, complemented
17#88542C412yescomplemented
18C3×D3×C22⋊C4—62no—
19C23×H(3,𝔽3)C2×C411yescharacteristic
20C22×#272—22no—
21C22×#272—22no—
22C22×#272C2×C411yescharacteristic
23C22×#272D412yes—
24C22×#272Q812yes—
25C22×#272C2311yescharacteristic
26C2×#1088—28no—
27C2×#1088—22no—
28C2×#1088—22no—
29C2×#1088—22no—
30C2×#1088—21no—
31C2×#1088—21no—
32C2×#1088—22no—
33C2×#1088D414yes—
34C2×C4×H(3,𝔽3)C2×C412yescomplemented
35C2×#1084C2×C412yescomplemented
36C4×#272—22no—
37C4×#272—22no—
38C3×#385—91nomaximal
39C23×C3×D3—62no—
40C2×C3×C4×D3—64no—
41C32×C22⋊C4—22no—
42C3×#217—62no—
43C3×(C2×D3)⋊C4—64no—
44C22×H(3,𝔽3)—22no—
45C22×H(3,𝔽3)C4⋊C412yescomplemented
46C22×H(3,𝔽3)C22×C411yescharacteristic
47C2×#272—22no—
48C2×#272—22no—
49C2×#272—24no—
50C2×#272—22no—
51C2×#272C4⋊C412yescomplemented
52C2×#272C2×D412yes—
53C2×#272C2×D411yescharacteristic
54C2×#272C2×Q811yescharacteristic
55C2×#272#4112yes—
56#1088—24no—
57#1088—41no—
58#1088—41no—
59C4×H(3,𝔽3)—22no—
60#1084—22no—
61C23×C3×C4—91no—
62C2×C3×C4⋊C4—92no—
63C2×C3×C22⋊C4—91no—
64C2×C3×C22⋊C4—91no—
65D3×C22⋊C4—62no—
66C3×#95—92no—
67C23×C32—22no—
68C22×C3×D3—62no—
69C22×C3×D3—62no—
70C22×C3×D3—64no—
71C22×C3×D3—64no—
72C22×C3×D3—124no—
73C22×C3×D3—124no—
74C2×C32×C4—24no—
75C2×C3×Dic3—64no—
76C3×C4×D3—124no—
77C3×C4×D3—124no—
78#385—91noSylow
79C2×H(3,𝔽3)—22no—
80C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
81C2×H(3,𝔽3)C4×D412yes—
82#272—22no—
83#272#12011yescharacteristic
84#272C22⋊Q812yes—
85#272#12311yescharacteristic
86C24×C3—91no—
87C23×D3—62no—
88C22×C3×C4—92no—
89C22×C3×C4—92no—
90C22×C3×C4—91no—
91C22×C3×C4—91no—
92C22×C3×C4—184no—
93C22×C3×C4—182no—
94C2×C4×D3—64no—
95C3×C4⋊C4—184no—
96C3×C22⋊C4—62no—
97C3×C22⋊C4—91no—
98C3×C22⋊C4—92no—
99C3×C22⋊C4—182no—
100C3×C22⋊C4#15711yescharacteristic
101#217—62no—
102(C2×D3)⋊C4—64no—
103C22×C32—22no—
104C22×C32—24no—
105C22×C32—44no—
106C2×C3×D3—64no—
107C2×C3×D3—62no—
108C2×C3×D3—62no—
109C2×C3×D3—64no—
110C2×C3×D3—64no—
111C2×C3×D3—124no—
112C2×C3×D3—128no—
113C2×C3×D3—124no—
114C2×C3×D3—124no—
115C32×C4—44no—
116C3×Dic3—124no—
117C23×C4—91no—
118C2×C4⋊C4—92no—
119C2×C22⋊C4—91no—
120C2×C22⋊C4—91no—
121#95—92no—
122H(3,𝔽3)#38511yescharacteristic, complemented, Sylow
123C23×C3—62no—
124C23×C3—91no—
125C23×C3—91no—
126C23×C3—92no—
127C23×C3—92no—
128C23×C3—182no—
129C23×C3—182no—
130C23×C3C2×#15711yescharacteristic
131C22×D3—62no—
132C22×D3—62no—
133C22×D3—64no—
134C22×D3—64no—
135C22×D3—124no—
136C22×D3—124no—
137C2×C3×C4—64no—
138C2×C3×C4—94no—
139C2×C3×C4—92no—
140C2×C3×C4—182no—
141C2×C3×C4—182no—
142C2×C3×C4—182no—
143C2×C3×C4—182no—
144C2×C3×C4—188no—
145C2×C3×C4—181no—
146C2×C3×C4—181no—
147C2×C3×C4—182no—
148C2×C3×C4—182no—
149C2×C3×C4C2×#15712yes—
150C2×Dic3—64no—
151C4×D3—124no—
152C4×D3—124no—
153C2×C32—22no—
154C2×C32—24no—
155C2×C32—44no—
156C3×D3—62no—
157C3×D3—64no—
158C3×D3—62no—
159C3×D3—124no—
160C24—91no—
161C22×C4—92no—
162C22×C4—92no—
163C22×C4—91no—
164C22×C4—91no—
165C22×C4—184no—
166C22×C4—182no—
167C4⋊C4—184no—
168C22⋊C4—91no—
169C22⋊C4—92no—
170C22⋊C4—182no—
171C22⋊C4#108811yescharacteristic, complemented
172C22×C3—22no—
173C22×C3—64no—
174C22×C3—62no—
175C22×C3—91no—
176C22×C3—91no—
177C22×C3—92no—
178C22×C3—92no—
179C22×C3—92no—
180C22×C3—124no—
181C22×C3—182no—
182C22×C3—182no—
183C22×C3—184no—
184C22×C3—182no—
185C22×C3C4⋊#15712yes—
186C22×C3C22×#15711yescharacteristic
187C2×D3—64no—
188C2×D3—64no—
189C2×D3—62no—
190C2×D3—62no—
191C2×D3—64no—
192C2×D3—124no—
193C2×D3—124no—
194C2×D3—128no—
195C2×D3—124no—
196C3×C4—22no—
197C3×C4—124no—
198C3×C4—184no—
199C3×C4—182no—
200C3×C4—361no—
201C3×C4—361no—
202Dic3—124no—
203C32—22no—
204C23—91no—
205C23—92no—
206C23—92no—
207C23—91no—
208C23—182no—
209C23—182no—
210C23C2×#108811yescharacteristic
211C2×C4—94no—
212C2×C4—92no—
213C2×C4—182no—
214C2×C4—182no—
215C2×C4—182no—
216C2×C4—182no—
217C2×C4—188no—
218C2×C4—181no—
219C2×C4—181no—
220C2×C4—182no—
221C2×C4—182no—
222C2×C4C2×#108812yescomplemented
223C2×C3—22no—
224C2×C3—62no—
225C2×C3—64no—
226C2×C3—91no—
227C2×C3—91no—
228C2×C3—92no—
229C2×C3—124no—
230C2×C3—182no—
231C2×C3C2×C4⋊#15711yescharacteristic
232C2×C3D4×#15712yes—
233D3—62no—
234D3—62no—
235D3—64no—
236D3—124no—
237C22—22no—
238C22—91no—
239C22—91no—
240C22—92no—
241C22—92no—
242C22—92no—
243C22—182no—
244C22—182no—
245C22—184no—
246C22—182no—
247C22C4⋊#108812yescomplemented
248C22C22×#108811yescharacteristic
249C4—22no—
250C4—184no—
251C4—182no—
252C4—361no—
253C4—361no—
254C3—62no—
255C3#1059374611yescharacteristic
256C2—22no—
257C2—91no—
258C2—91no—
259C2—92no—
260C2—182no—
261C2C2×C4⋊#108811yescharacteristic
262C2D4×#108812yes—
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC22⋊C4×H(3,𝔽3)C4
prime radicalC23×#272C22
cosocleC22×#272C23
hypercenterC3×C22⋊C4#157
hypocenterH(3,𝔽3)#385
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊#1088C2semidirect
5. #11720787C2semidirect
15. C22⋊C4×H(3,𝔽3)C4semidirect
16. #88552C4semidirect
17. #88542C4semidirect
34. C2×C4×H(3,𝔽3)C2×C4semidirect
35. C2×#1084C2×C4semidirect
45. C22×H(3,𝔽3)C4⋊C4semidirect
51. C2×#272C4⋊C4semidirect
122. H(3,𝔽3)#385semidirect
171. C22⋊C4#1088semidirect
222. C2×C4C2×#1088semidirect
247. C22C4⋊#1088semidirect

Maximal subgroups

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

subgroupindex
C23×#10882
C2×C22⋊#10882
C2×C4⋊#10882
C22⋊C4×#2722
#117207872
C3×#3859