Finite Group Databasebeta

← #13146565

Subgroups — #13146565

Order 1728.

385 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 47 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
1#11721152C211yesmaximal, characteristic
2#11721147C211yesmaximal, characteristic
3#11720787C211yesmaximal, characteristic
4#11720707C212yesmaximal
5#11720707C212yesmaximal
6C22×#1088C2212yes—
7C2×C4×#272C2211yescharacteristic
8C2×#6199C2211yescharacteristic
9C2×#6199C2212yes—
10C2×#6199C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)—21no—
12#3750⋊C3—21no—
13#88449—21no—
14#88448—21no—
15D3×C4⋊Dic3—61no—
16D3×Dic3⋊C4—61no—
17C22×#272C2311yescharacteristic
18C2×#1088—22no—
19C2×#1088—22no—
20C2×#1088—22no—
21C2×C4×H(3,𝔽3)D411yescharacteristic
22C2×#1084D411yescharacteristic
23C4×#272—22no—
24C2×(C3⋊Dic3)⋊C3—21no—
25C2×(C3⋊Dic3)⋊C3—21no—
26C2×(C3⋊Dic3)⋊C3—22no—
27#6199—21no—
28#6199—22no—
29#6199—24no—
30#6199Q812yes—
31#4659—91nomaximal
32C2×C3×C4×D3—61no—
33C2×C3×C4×D3—61no—
34C2×D3×Dic3—61no—
35C2×D3×Dic3—61no—
36C2×D3×Dic3—62no—
37C4⋊(C3⋊Dic3)—21no—
38#3750—21no—
39C3×C4⋊Dic3—61no—
40C3×Dic3⋊C4—61no—
41#3724—61no—
42Dic3⋊Dic3—61no—
43Dic3⋊Dic3—62no—
44C22×H(3,𝔽3)C2×D411yescharacteristic
45C2×#272C2×Q811yescharacteristic
46C2×#272#4112yes—
47C2×#272#4112yes—
48C2×#272#4111yescharacteristic
49#1088—42no—
50C4×H(3,𝔽3)—21no—
51#1084—21no—
52(C3⋊Dic3)⋊C3—22no—
53(C3⋊Dic3)⋊C3—41no—
54(C3⋊Dic3)⋊C3—42no—
55C2×C4⋊Dic3—91no—
56C2×Dic3⋊C4—91no—
57D3×C4⋊C4—181no—
58D3×C4⋊C4—181no—
59C3×#95—91no—
60C3⋊#95—92no—
61C3⋊#95—92no—
62C22×C3×D3—61no—
63C22×C3×D3—61no—
64C2×C32×C4—21no—
65C2×C32×C4—21no—
66C2×C3⋊Dic3—21no—
67C2×C3⋊Dic3—21no—
68C2×C3⋊Dic3—22no—
69C2×C3×Dic3—61no—
70C2×C3×Dic3—61no—
71C2×C3×Dic3—61no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—61no—
74C3×C4×D3—64no—
75C3×C4×D3—122no—
76D3×Dic3—62no—
77D3×Dic3—62no—
78D3×Dic3—124no—
79D3×Dic3—122no—
80#400—271noSylow
81C2×H(3,𝔽3)C22⋊Q811yescharacteristic
82C2×H(3,𝔽3)#12311yescharacteristic
83C2×H(3,𝔽3)#12411yescharacteristic
84#272#12512yes—
85#272#12612yes—
86C22×C3×C4—91no—
87C22×C3×C4—92no—
88C22×Dic3—91no—
89C22×Dic3—91no—
90C22×Dic3—92no—
91C2×C4×D3—61no—
92C2×C4×D3—61no—
93C2×C4×D3—181no—
94C2×C4×D3—181no—
95C2×C4×D3—182no—
96C3×C4⋊C4—181no—
97C3×C4⋊C4—181no—
98C4⋊Dic3—61no—
99C4⋊Dic3—181no—
100C4⋊Dic3—181no—
101C4⋊Dic3—181no—
102Dic3⋊C4—61no—
103Dic3⋊C4—181no—
104Dic3⋊C4—181no—
105Dic3⋊C4—182no—
106Dic3⋊C4—181no—
107C22×C32—21no—
108C22×C32—21no—
109C2×C3×D3—61no—
110C2×C3×D3—61no—
111C2×C3×D3—61no—
112C2×C3×D3—62no—
113C2×C3×D3—62no—
114C2×C3×D3—62no—
115C2×C3×D3—61no—
116C2×C3×D3—62no—
117C32×C4—22no—
118C32×C4—41no—
119C3⋊Dic3—22no—
120C3⋊Dic3—42no—
121C3⋊Dic3—41no—
122C3×Dic3—62no—
123C3×Dic3—62no—
124C3×Dic3—121no—
125C3×Dic3—122no—
126C3×Dic3—121no—
127C2×C4⋊C4—271no—
128C2×C4⋊C4—271no—
129#95—91no—
130#95—272no—
131#95—272no—
132H(3,𝔽3)#40011yescharacteristic, complemented, Sylow
133C23×C3—91no—
134C22×D3—61no—
135C22×D3—61no—
136C2×C3×C4—61no—
137C2×C3×C4—61no—
138C2×C3×C4—91no—
139C2×C3×C4—182no—
140C2×C3×C4—181no—
141C2×C3×C4—182no—
142C2×C3×C4—181no—
143C2×C3×C4—182no—
144C2×C3×C4—182no—
145C2×C3×C4—182no—
146C2×C3×C4GO+(4,𝔽2)11yescharacteristic
147C2×Dic3—61no—
148C2×Dic3—61no—
149C2×Dic3—61no—
150C2×Dic3—61no—
151C2×Dic3—62no—
152C2×Dic3—92no—
153C2×Dic3—181no—
154C2×Dic3—181no—
155C2×Dic3—182no—
156C2×Dic3—181no—
157C2×Dic3—184no—
158C2×Dic3—182no—
159C4×D3—64no—
160C4×D3—122no—
161C4×D3—182no—
162C4×D3—182no—
163C4×D3—362no—
164C4×D3—364no—
165C2×C32—21no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32—21no—
169C2×C32—21no—
170C2×C32—21no—
171C3×D3—62no—
172C3×D3—62no—
173C3×D3—62no—
174C3×D3—62no—
175C22×C4—91no—
176C22×C4—92no—
177C22×C4—271no—
178C22×C4—271no—
179C22×C4—272no—
180C4⋊C4—181no—
181C4⋊C4—181no—
182C4⋊C4—541no—
183C4⋊C4—541no—
184C22×C3—61no—
185C22×C3—61no—
186C22×C3—91no—
187C22×C3—91no—
188C22×C3—92no—
189C22×C3—92no—
190C22×C3C2×GO+(4,𝔽2)11yescharacteristic
191C2×D3—62no—
192C2×D3—62no—
193C2×D3—61no—
194C2×D3—61no—
195C2×D3—62no—
196C2×D3—61no—
197C2×D3—61no—
198C2×D3—62no—
199C3×C4—21no—
200C3×C4—62no—
201C3×C4—121no—
202C3×C4—182no—
203C3×C4—181no—
204C3×C4—362no—
205C3×C4—361no—
206C3×C4—362no—
207Dic3—62no—
208Dic3—62no—
209Dic3—121no—
210Dic3—122no—
211Dic3—121no—
212Dic3—182no—
213Dic3—362no—
214Dic3—361no—
215C32—21no—
216C32—21no—
217C23—91no—
218C2×C4—91no—
219C2×C4—182no—
220C2×C4—181no—
221C2×C4—182no—
222C2×C4—182no—
223C2×C4—182no—
224C2×C4—182no—
225C2×C4—181no—
226C2×C4—272no—
227C2×C4—541no—
228C2×C4—544no—
229C2×C4—542no—
230C2×C4#1088⋊C211yescharacteristic
231C2×C3—61no—
232C2×C3—61no—
233C2×C3—61no—
234C2×C3—61no—
235C2×C3—61no—
236C2×C3—61no—
237C2×C3—92no—
238C2×C3—92no—
239C2×C3D32⋊Q811yescharacteristic
240C2×C3#6422111yescharacteristic
241C2×C3#6423311yescharacteristic
242D3—62no—
243D3—62no—
244D3—62no—
245D3—62no—
246C22—91no—
247C22—91no—
248C22—92no—
249C22—92no—
250C22C2×#1088⋊C211yescharacteristic
251C4—21no—
252C4—182no—
253C4—181no—
254C4—362no—
255C4—362no—
256C4—361no—
257C3—61no—
258C3—61no—
259C3#1059334311yescharacteristic
260C2—92no—
261C2—92no—
262C2(((C3⋊D3)⋊C3)⋊C2)⋊Q811yescharacteristic
263C2#1172230211yescharacteristic
264C2#1172231411yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, prime radical, cosocleC22×#272C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#400
LCSG ▷ C22×#272 ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
132. H(3,𝔽3)#400semidirect

Maximal subgroups

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

subgroupindex
#117211522
#117211472
#117207872
#117207072
#46599