Finite Group Databasebeta

← #13147184

Subgroups — #13147184

Order 1728.

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

normal subgroups only

This group has 296 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
1C2×C9×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×D9×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×Dic9C211yesmaximal, characteristic, complemented
4#5831⋊C3—31nomaximal
5#10589933C311yesmaximal, characteristic, complemented
6C9×#41⋊C3—21no—
7C2×C9×SL(2,𝔽3)C2211yescharacteristic
8D9×SL(2,𝔽3)—21no—
9C3×(C2×D9)⋊C4—41nomaximal
10C2×C3×#41⋊C3D311yescharacteristic
11C2×D3×SL(2,𝔽3)—31no—
12Dic3×SL(2,𝔽3)—31no—
13C2×C9×#41C2×C311yescharacteristic, complemented
14C2×Q8×D9C2×C311yescharacteristic
15C2×#41⋊C9—21no—
16Q8×Dic9C2×C311yescharacteristic, complemented
17#63500—31no—
18#63498—31no—
19C4×#611—31no—
20#63486—31no—
21C22×C3×D9—41no—
22C2×C3×C4×C9—41no—
23C2×C3×Dic9—41no—
24C9×SL(2,𝔽3)D411yescharacteristic, complemented
25#5831—31no—
26#548⋊C3—91no—
27C3×#41⋊C3—21no—
28C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
29D3×SL(2,𝔽3)—61no—
30C3×(C2×D3)⋊C4—121no—
31C9×#41—21no—
32C9×#41—61no—
33C2×Q8×C9C22×C311yescharacteristic
34C2×D4×C9—31no—
35C22×C4×C9—31no—
36C2×#611—31no—
37Q8×D9—21no—
38Q8×D9—61no—
39C2×C4×D9—31no—
40C2×Dic18—31no—
41#41⋊C9—22no—
42C2×Q8⋊C9—21no—
43#3676—32no—
44#3676—31no—
45(C2×D9)⋊C4A411yescharacteristic
46C4⋊Dic9—31no—
47Dic9⋊C4—31no—
48C4×Dic9—31no—
49C22×C3×C9—41no—
50C2×C3×D9—42no—
51C2×C3×D9—81no—
52C2×C3×D9—81no—
53C3×C4×C9—81no—
54C3×Dic9—81no—
55C2×C3×#41C3×D311yescharacteristic
56C2×D3×Q8—31no—
57C2×#41⋊C3—21no—
58C2×#41⋊C3D911yescharacteristic, complemented
59C22×SL(2,𝔽3)—91no—
60Q8×Dic3—31no—
61#938—91no—
62#936—91no—
63C4×#67—91no—
64#924—91no—
65C4×SL(2,𝔽3)—91no—
66C22×C3×D3—121no—
67C2×C32×C4—41no—
68C2×C3×Dic3—121no—
69C3×SL(2,𝔽3)#6711yescharacteristic
70C23×C9—31no—
71C22×D9C2×A411yescharacteristic
72Q8×C9—31no—
73Q8×C9C3×D411yescharacteristic
74D4×C9—61no—
75D4×C9—61no—
76C2×C4×C9—31no—
77C2×C4×C9—61no—
78C2×C4×C9—61no—
79C2×C4×C9—81no—
80C2×C4×C9C2×A411yescharacteristic
81#611—62no—
82C2×Dic9—31no—
83C2×Dic9—31no—
84C2×Dic9C2×A411yescharacteristic
85C4×D9—61no—
86C4×D9—61no—
87Dic18—61no—
88Dic18—61no—
89Q8⋊C9—21no—
90#548—91noSylow
91C2×C3×C9—41no—
92C2×C3×C9—41no—
93C2×C3×C9—41no—
94C3×D9—82no—
95C3×#41—21no—
96C3×#41—61no—
97C2×C3×Q8C2×C3×D311yescharacteristic
98C2×C3×D4—31no—
99C22×C3×C4—31no—
100C2×#67—91no—
101D3×Q8—61no—
102D3×Q8—181no—
103C2×C4×D3—91no—
104C2×Dic6—91no—
105#41⋊C3—22no—
106#41⋊C3—21no—
107C2×SL(2,𝔽3)—21no—
108C2×SL(2,𝔽3)—181no—
109C2×SL(2,𝔽3)C2×D911yescharacteristic
110C3×C22⋊C4—361no—
111#217—91no—
112#217—92no—
113(C2×D3)⋊C4—31no—
114C4⋊Dic3—91no—
115Dic3⋊C4—91no—
116C4×Dic3—91no—
117C22×C32—41no—
118C2×C3×D3—122no—
119C2×C3×D3—241no—
120C2×C3×D3—241no—
121C32×C4—81no—
122C3×Dic3—241no—
123C22×C9—61no—
124C22×C9—61no—
125C22×C9—61no—
126C22×C9—81no—
127C22×C9C22×A411yescharacteristic
128C2×D9—21no—
129C2×D9—21no—
130C2×D9#41⋊C312yescomplemented
131C4×C9—21no—
132C4×C9—31no—
133C4×C9—31no—
134C4×C9—82no—
135Dic9—21no—
136Dic9—61no—
137Dic9—61no—
138C2×#41C3×D911yescharacteristic, complemented
139C22×Q8—91no—
140#124—271no—
141C22⋊Q8—271no—
142#121—271no—
143C4×Q8—91no—
144C4×D4—271no—
145C3×C9—41noSylow
146C23×C3—31no—
147C23×C3—361no—
148C22×D3—31no—
149C3×Q8—31no—
150C3×Q8C3×#6711yescharacteristic
151C3×D4—61no—
152C3×D4—61no—
153C2×C3×C4—31no—
154C2×C3×C4—41no—
155C2×C3×C4—61no—
156C2×C3×C4—61no—
157C2×C3×C4—81no—
158C2×C3×C4—361no—
159C2×C3×C4D3×A411yescharacteristic
160#67—182no—
161C2×Dic3—31no—
162C2×Dic3—91no—
163C2×Dic3—91no—
164C4×D3—181no—
165C4×D3—181no—
166Dic6—181no—
167Dic6—181no—
168SL(2,𝔽3)—21no—
169SL(2,𝔽3)#61111yescharacteristic, complemented
170C2×C32—41no—
171C2×C32—41no—
172C2×C32—41no—
173C3×D3—242no—
174C2×C9—81no—
175C2×C9—81no—
176C2×C9—81no—
177C2×C9—121no—
178C2×C9D4×A411yescharacteristic
179C2×C9C2×#41⋊C311yescharacteristic
180C2×C92−1+4⋊C311yescharacteristic
181D9—22no—
182#41—21no—
183#41—61no—
184C2×Q8—181no—
185C2×Q8—271no—
186C2×Q8—541no—
187C2×Q8C2×C3×D911yescharacteristic
188C2×D4—31no—
189C2×D4—271no—
190C22×C4—31no—
191C22×C4—271no—
192C4⋊C4—271no—
193C4⋊C4—271no—
194C22⋊C4—91no—
195C22⋊C4—271no—
196C22⋊C4—272no—
197C42—271no—
198C22×C3—41no—
199C22×C3—61no—
200C22×C3—61no—
201C22×C3—61no—
202C22×C3—81no—
203C22×C3—362no—
204C22×C3—721no—
205C22×C3—721no—
206C22×C3C2×D3×A411yescharacteristic
207C2×D3—32no—
208C2×D3—61no—
209C2×D3—61no—
210C3×C4—21no—
211C3×C4—31no—
212C3×C4—31no—
213C3×C4—81no—
214C3×C4—82no—
215C3×C4—721no—
216Dic3—61no—
217Dic3—181no—
218Dic3—181no—
219C32—41no—
220C9—81no—
221C9#548⋊C311yescharacteristic, complemented
222C23—31no—
223C23—91no—
224Q8—31no—
225Q8—541no—
226Q8—541no—
227Q8C3×#61111yescharacteristic
228D4—61no—
229D4—61no—
230D4—542no—
231C2×C4—31no—
232C2×C4—61no—
233C2×C4—61no—
234C2×C4—91no—
235C2×C4—271no—
236C2×C4—271no—
237C2×C4—541no—
238C2×C4—541no—
239C2×C4A4×D911yescharacteristic
240C2×C3—41no—
241C2×C3—41no—
242C2×C3—41no—
243C2×C3—81no—
244C2×C3—81no—
245C2×C3—81no—
246C2×C3—121no—
247C2×C3—722no—
248C2×C3D3×#41⋊C311yescharacteristic
249C2×C3#1010⋊C311yescharacteristic
250C2×C3A4×#6711yescharacteristic
251D3—62no—
252C22—61no—
253C22—61no—
254C22—61no—
255C22—92no—
256C22—181no—
257C22—181no—
258C22C2×A4×D911yescharacteristic
259C4—21no—
260C4—31no—
261C4—31no—
262C4—181no—
263C4—541no—
264C4—541no—
265C3—41no—
266C3—81no—
267C3#5831⋊C311yescharacteristic
268C2—121no—
269C2—182no—
270C2D9×#41⋊C311yescharacteristic
271C2#1172241611yescharacteristic
272C2A4×#61111yescharacteristic
subgroupG/H
centerC22C2×A4×D9
commutator, cosocleC2×Q8×C9C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC2×C9×#41C2×C3
hypercenterC2×C4A4×D9
hypocenterQ8×C9C3×D4
LCSG ▷ C2×Q8×C9 ▷ Q8×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C9×#41⋊C3C2semidirect
2. C2×D9×SL(2,𝔽3)C2semidirect
3. SL(2,𝔽3)×Dic9C2semidirect
5. #10589933C3semidirect
13. C2×C9×#41C2×C3semidirect
16. Q8×Dic9C2×C3semidirect
24. C9×SL(2,𝔽3)D4semidirect
58. C2×#41⋊C3D9semidirect
130. C2×D9#41⋊C3semidirect
138. C2×#41C3×D9semidirect
169. SL(2,𝔽3)#611semidirect
221. C9#548⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#41⋊C32
C2×D9×SL(2,𝔽3)2
SL(2,𝔽3)×Dic92
#5831⋊C33
#105899333
C3×(C2×D9)⋊C44