Finite Group Databasebeta

← #13143898

Subgroups — #13143898

Order 1728.

590 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 137 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#11721840C211yesmaximal, characteristic
2#11721744C211yesmaximal, characteristic, complemented
3C3×C9×#102C211yesmaximal, characteristic, complemented
4#11720576C211yesmaximal, characteristic, complemented
5#11720543C211yesmaximal, characteristic, complemented
6C8×C3⋊Dic9C211yesmaximal, characteristic, complemented
7#11720528C211yesmaximal, characteristic, complemented
8#10592459—31nomaximal
9#10588440—33nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×#6233—21no—
13C2×C4⋊(C3⋊D9)C2211yescharacteristic
14C2×#6226—21no—
15C2×C3×C8×C9C2211yescharacteristic
16C3×C9×C4⋊C4C2211yescharacteristic
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18#88534—21no—
19C4×C3⋊Dic9C2211yescharacteristic
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#64149—31no—
22#64097—31no—
23C32×#102D311yescharacteristic
24#63659—31no—
25#63648—31no—
26C8×C3⋊Dic3—31no—
27#63636—31no—
28#63502—33no—
29#63450—33no—
30C9×#102D313yescomplemented
31#63392—33no—
32#63380—33no—
33C8×Dic9—33no—
34#63369—33no—
35C23×C3×C9—21no—
36C22×C3⋊D9—21no—
37C3×D4×C9—21no—
38C3×D4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41#6233—42no—
42#6233—42no—
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9D411yescharacteristic
45C2×C3⋊Dic9D411yescharacteristic
46C4⋊(C3⋊D9)—21no—
47C4⋊(C3⋊D9)—21no—
48#6226—42no—
49C3×C8×C9—21no—
50C3⋊(C9⋊C8)—21no—
51#4760—93no—
52#4760—91no—
53C2×C32×D4C2×D311yescharacteristic
54C2×#638—61no—
55C2×#638—61no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×#634—61no—
58C2×C32×C8C2×D311yescharacteristic
59C32×C4⋊C4C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)—31no—
61#3750—61no—
62C4×C3⋊Dic3—31no—
63C2×C3⋊(C3⋊C8)—31no—
64C2×D4×C9C2×D313yes—
65C2×#611—63no—
66C2×#611—63no—
67C2×D36—33no—
68C2×Dic18—63no—
69C2×C8×C9C2×D313yes—
70C9×C4⋊C4C2×D313yes—
71C4⋊Dic9—33no—
72Dic9⋊C4—63no—
73C4×Dic9—33no—
74C2×C9⋊C8—33no—
75C22×C3×C9—41no—
76C22×C3×C9—41no—
77C22×C3×C9—41no—
78C22×C3×C9C2×D411yescharacteristic
79C2×C3⋊D9—41no—
80C2×C3⋊D9—41no—
81C2×C3⋊D9—41no—
82C3×C4×C9—41no—
83C3×C4×C9#4111yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—41no—
86C3⋊Dic9D812yes—
87C3⋊Dic9SD1612yes—
88#940—93no—
89#940—91no—
90#888—93no—
91#888—91no—
92C3×#102D913yescomplemented
93C3×#102C3⋊D311yescharacteristic
94#832—93no—
95#832—91no—
96#821—93no—
97#821—91no—
98C8×Dic3—93no—
99C8×Dic3—91no—
100#809—93no—
101#809—91no—
102C23×C32—21no—
103C22×C3⋊D3—61no—
104C32×D4—21no—
105C32×D4—21no—
106C2×C32×C4—21no—
107C2×C32×C4C22×D311yescharacteristic
108#638—122no—
109#638—122no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—61no—
113C4⋊(C3⋊D3)—61no—
114C4⋊(C3⋊D3)—61no—
115#634—122no—
116C23×C9—23no—
117C22×D9—63no—
118C32×C8—21no—
119C3⋊(C3⋊C8)—61no—
120D4×C9—23no—
121D4×C9—23no—
122C2×C4×C9—23no—
123C2×C4×C9C22×D313yes—
124#611—126no—
125#611—126no—
126C2×Dic9—33no—
127C2×Dic9—33no—
128C2×Dic9—63no—
129D36—63no—
130D36—63no—
131Dic18—126no—
132C8×C9—23no—
133C9⋊C8—63no—
134#486—271noSylow
135C2×C3×C9—81no—
136C2×C3×C9#12411yescharacteristic
137C2×C3×C9C2×D811yescharacteristic
138C2×C3×C9C2×SD1611yescharacteristic
139C3⋊D9—81no—
140C2×C3×D4C2×D913yes—
141C2×C3×D4C2×C3⋊D311yescharacteristic
142C2×#67—181no—
143C2×#67—181no—
144C2×#67—183no—
145C2×#67—183no—
146C2×D12—93no—
147C2×D12—91no—
148C2×Dic6—181no—
149C2×Dic6—183no—
150C2×C3×C8C2×D913yes—
151C2×C3×C8C2×C3⋊D311yescharacteristic
152C3×C4⋊C4C2×D913yes—
153C3×C4⋊C4C2×C3⋊D311yescharacteristic
154C4⋊Dic3—93no—
155C4⋊Dic3—91no—
156Dic3⋊C4—181no—
157Dic3⋊C4—183no—
158C4×Dic3—93no—
159C4×Dic3—91no—
160C2×C3⋊C8—93no—
161C2×C3⋊C8—91no—
162C22×C32—41no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32D3×D411yescharacteristic
166C2×C3⋊D3—121no—
167C2×C3⋊D3—121no—
168C2×C3⋊D3—121no—
169C32×C4—41no—
170C32×C4#23511yescharacteristic
171C32×C4#23711yescharacteristic
172C3⋊Dic3—32no—
173C3⋊Dic3—32no—
174C3⋊Dic3—121no—
175C22×C9—43no—
176C22×C9—43no—
177C22×C9—43no—
178C22×C9D3×D413yes—
179C2×D9—123no—
180C2×D9—123no—
181C2×D9—123no—
182C4×C9—43no—
183C4×C9#23513yes—
184C4×C9#23713yes—
185Dic9—36no—
186Dic9—36no—
187Dic9—123no—
188C4⋊Q8—271no—
189#127—271no—
190#102—271no—
191#102—271no—
192#102—271no—
193#102C3⋊D911yescharacteristic, complemented
194C4×C8—271no—
195C3×C9#48611yescharacteristic, complemented, Sylow
196C23×C3—23no—
197C23×C3—21no—
198C22×D3—183no—
199C22×D3—181no—
200C3×D4—23no—
201C3×D4—23no—
202C3×D4—21no—
203C3×D4—21no—
204C2×C3×C4—23no—
205C2×C3×C4—21no—
206C2×C3×C4C22×D913yes—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208#67—362no—
209#67—362no—
210#67—366no—
211#67—366no—
212C2×Dic3—93no—
213C2×Dic3—93no—
214C2×Dic3—91no—
215C2×Dic3—91no—
216C2×Dic3—183no—
217C2×Dic3—181no—
218D12—183no—
219D12—183no—
220D12—181no—
221D12—181no—
222Dic6—362no—
223Dic6—366no—
224C3×C8—23no—
225C3×C8—21no—
226C3⋊C8—183no—
227C3⋊C8—181no—
228C2×C32—81no—
229C2×C32#88511yescharacteristic
230C2×C32D3×D811yescharacteristic
231C2×C32D3×SD1611yescharacteristic
232C3⋊D3—241no—
233C2×C9—83no—
234C2×C9#88513yes—
235C2×C9D3×D813yes—
236C2×C9D3×SD1613yes—
237D9—243no—
238C2×Q8—541no—
239C2×D4—271no—
240C2×D4—541no—
241C2×D4—541no—
242C2×D4C2×C3⋊D911yescharacteristic
243C2×C8—271no—
244C2×C8C2×C3⋊D911yescharacteristic
245C4⋊C4—271no—
246C4⋊C4—541no—
247C4⋊C4C2×C3⋊D911yescharacteristic
248C42—271no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—43no—
253C22×C3—43no—
254C22×C3—43no—
255C22×C3D4×D913yes—
256C22×C3D4×C3⋊D311yescharacteristic
257C2×D3—361no—
258C2×D3—361no—
259C2×D3—361no—
260C2×D3—363no—
261C2×D3—363no—
262C2×D3—363no—
263C3×C4—41no—
264C3×C4—43no—
265C3×C4#369713yes—
266C3×C4#369913yes—
267C3×C4#382811yescharacteristic
268C3×C4#383011yescharacteristic
269Dic3—92no—
270Dic3—92no—
271Dic3—96no—
272Dic3—96no—
273Dic3—361no—
274Dic3—363no—
275C32#476011yescharacteristic
276C9#476013yescomplemented
277C23—21no—
278C23—541no—
279Q8—1082no—
280D4—21no—
281D4—21no—
282D4—541no—
283D4—541no—
284D4—1082no—
285D4—1082no—
286C2×C4—21no—
287C2×C4—271no—
288C2×C4—271no—
289C2×C4—541no—
290C2×C4C22×C3⋊D911yescharacteristic
291C8—21no—
292C8—541no—
293C2×C3—83no—
294C2×C3—81no—
295C2×C3#6344713yes—
296C2×C3D8×D913yes—
297C2×C3SD16×D913yes—
298C2×C3#6409411yescharacteristic
299C2×C3D8×C3⋊D311yescharacteristic
300C2×C3SD16×C3⋊D311yescharacteristic
301D3—723no—
302D3—721no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—1081no—
307C22—1081no—
308C22—1081no—
309C22D4×C3⋊D911yescharacteristic
310C4—41no—
311C4—272no—
312C4—272no—
313C4—1081no—
314C4#8873611yescharacteristic
315C4#8874211yescharacteristic
316C3#1058844013yescomplemented
317C3#1059245911yescharacteristic
318C2—81no—
319C2—2161no—
320C2#1172173111yescharacteristic
321C2D8×C3⋊D911yescharacteristic
322C2SD16×C3⋊D911yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radical#11721840C2
hypercenter#102C3⋊D9
hypocenterC3×C9#486
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. #11721744C2semidirect
3. C3×C9×#102C2semidirect
4. #11720576C2semidirect
5. #11720543C2semidirect
6. C8×C3⋊Dic9C2semidirect
7. #11720528C2semidirect
30. C9×#102D3semidirect
92. C3×#102D9semidirect
193. #102C3⋊D9semidirect
195. C3×C9#486semidirect
276. C9#4760semidirect
316. C3#10588440semidirect

Maximal subgroups

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

subgroupindex
#117218402
#117217442
C3×C9×#1022
#117205762
#117205432
C8×C3⋊Dic92
#117205282
#105924593
#105884403