Finite Group Databasebeta

← #13143926

Subgroups — #13143926

Order 1728.

660 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 161 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
1D4×C3⋊Dic9C211yesmaximal, characteristic, complemented
2C2×#88544C211yesmaximal, characteristic
3#11721749C211yesmaximal, characteristic, complemented
4C3×C9×#102C211yesmaximal, characteristic, complemented
5#11720543C211yesmaximal, characteristic, complemented
6#11720538C211yesmaximal, characteristic, complemented
7#11720526C211yesmaximal, characteristic, complemented
8#10592484—31nomaximal
9#10588465—33nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3⋊Dic9—21no—
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×C4×C3⋊D9—21no—
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16#88551—21no—
17#88544C414yescomplemented
18#88536—21no—
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C4×C3⋊Dic9C2211yescharacteristic
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22D4×C3⋊Dic3—31no—
23C2×#3753—31no—
24#64102—31no—
25C32×#102D311yescharacteristic
26#63648—31no—
27#63642—31no—
28#63635—31no—
29D4×Dic9—33no—
30C2×#3673—33no—
31#63455—33no—
32C9×#102D313yescomplemented
33#63380—33no—
34#63375—33no—
35#63367—33no—
36C23×C3×C9—21no—
37C22×C3⋊D9—21no—
38C3×D4×C9—21no—
39C3×D4×C9C2×C412yescomplemented
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9—42no—
44C2×C3⋊Dic9D411yescharacteristic
45C2×C3⋊Dic9D411yescharacteristic
46C4⋊(C3⋊D9)—21no—
47C4⋊(C3⋊D9)C2×C412yescomplemented
48C4×C3⋊D9—42no—
49C3×C8×C9—21no—
50C3⋊(C9⋊C8)C2×C412yescomplemented
51#4785—93no—
52#4785—91no—
53C2×C32×D4C2×D311yescharacteristic
54C22×C3⋊Dic3—61no—
55C2×C4⋊(C3⋊D3)—31no—
56C2×C4×C3⋊D3—61no—
57C2×C32×C8C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59#3757—61no—
60#3753—34no—
61#3752—61no—
62C4⋊(C3⋊Dic3)—31no—
63C4×C3⋊Dic3—31no—
64C2×C3⋊(C3⋊C8)—31no—
65C2×D4×C9C2×D313yes—
66C22×Dic9—63no—
67C2×D36—33no—
68C2×C4×D9—63no—
69C2×C8×C9C2×D313yes—
70C9×C4⋊C4C2×D313yes—
71#3676—63no—
72#3673—312no—
73(C2×D9)⋊C4—63no—
74C4⋊Dic9—33no—
75C4×Dic9—33no—
76C2×C9⋊C8—33no—
77C22×C3×C9—22no—
78C22×C3×C9—41no—
79C22×C3×C9—41no—
80C22×C3×C9C2×D411yescharacteristic
81C2×C3⋊D9—22no—
82C2×C3⋊D9—41no—
83C2×C3⋊D9—41no—
84C3×C4×C9—41no—
85C3×C4×C9C22×C411yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C3⋊Dic9—21no—
88C3⋊Dic9—21no—
89C3⋊Dic9—41no—
90D4×Dic3—93no—
91D4×Dic3—91no—
92C2×#213—93no—
93C2×#213—91no—
94#893—93no—
95#893—91no—
96C3×#102D913yescomplemented
97C3×#102C3⋊D311yescharacteristic
98#821—93no—
99#821—91no—
100#815—93no—
101#815—91no—
102#808—93no—
103#808—91no—
104C23×C32—21no—
105C22×C3⋊D3—61no—
106C32×D4—21no—
107C32×D4C4×D312yes—
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—122no—
114C4⋊(C3⋊D3)—32no—
115C4⋊(C3⋊D3)—61no—
116C4×C3⋊D3—122no—
117C23×C9—23no—
118C22×D9—63no—
119C32×C8—21no—
120C3⋊(C3⋊C8)—32no—
121D4×C9—23no—
122D4×C9C4×D316yescomplemented
123C2×C4×C9—23no—
124C2×C4×C9C22×D313yes—
125C2×Dic9—33no—
126C2×Dic9—33no—
127C2×Dic9—63no—
128C2×Dic9—126no—
129D36—36no—
130D36—63no—
131C4×D9—126no—
132C8×C9—23no—
133C9⋊C8—36no—
134#442—271noSylow
135C2×C3×C9—42no—
136C2×C3×C9C4×D411yescharacteristic
137C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
138C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
139C3⋊D9—42no—
140C2×C3×D4C2×D913yes—
141C2×C3×D4C2×C3⋊D311yescharacteristic
142C22×Dic3—181no—
143C22×Dic3—183no—
144C2×D12—93no—
145C2×D12—91no—
146C2×C4×D3—181no—
147C2×C4×D3—183no—
148C2×C3×C8C2×D913yes—
149C2×C3×C8C2×C3⋊D311yescharacteristic
150C3×C4⋊C4C2×D913yes—
151C3×C4⋊C4C2×C3⋊D311yescharacteristic
152#217—181no—
153#217—183no—
154#213—912no—
155#213—94no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—183no—
158C4⋊Dic3—93no—
159C4⋊Dic3—91no—
160C4×Dic3—93no—
161C4×Dic3—91no—
162C2×C3⋊C8—93no—
163C2×C3⋊C8—91no—
164C22×C32—22no—
165C22×C32—41no—
166C22×C32—41no—
167C22×C32D3×D411yescharacteristic
168C2×C3⋊D3—62no—
169C2×C3⋊D3—121no—
170C2×C3⋊D3—121no—
171C32×C4—41no—
172C32×C4C2×C4×D311yescharacteristic
173C32×C4#23711yescharacteristic
174C3⋊Dic3—61no—
175C3⋊Dic3—61no—
176C3⋊Dic3—121no—
177C22×C9—26no—
178C22×C9—43no—
179C22×C9—43no—
180C22×C9D3×D413yes—
181C2×D9—66no—
182C2×D9—123no—
183C2×D9—123no—
184C4×C9—43no—
185C4×C9C2×C4×D313yes—
186C4×C9#23713yes—
187Dic9—63no—
188Dic9—63no—
189Dic9—123no—
190C2×D8—271no—
191C4×D4—271no—
192C4×D4—271no—
193#106—271no—
194#102—271no—
195#102C3⋊D911yescharacteristic, complemented
196#97—271no—
197C3×C9#44211yescharacteristic, complemented, Sylow
198C23×C3—23no—
199C23×C3—21no—
200C22×D3—183no—
201C22×D3—181no—
202C3×D4—23no—
203C3×D4—21no—
204C3×D4C4×D916yescomplemented
205C3×D4C4×C3⋊D312yes—
206C2×C3×C4—23no—
207C2×C3×C4—21no—
208C2×C3×C4C22×D913yes—
209C2×C3×C4C22×C3⋊D311yescharacteristic
210C2×Dic3—93no—
211C2×Dic3—93no—
212C2×Dic3—91no—
213C2×Dic3—91no—
214C2×Dic3—183no—
215C2×Dic3—181no—
216C2×Dic3—362no—
217C2×Dic3—366no—
218D12—96no—
219D12—92no—
220D12—183no—
221D12—181no—
222C4×D3—362no—
223C4×D3—366no—
224C3×C8—23no—
225C3×C8—21no—
226C3⋊C8—96no—
227C3⋊C8—92no—
228C2×C32—42no—
229C2×C32#67⋊C411yescharacteristic
230C2×C32#91111yescharacteristic
231C2×C32#91411yescharacteristic
232C3⋊D3—122no—
233C2×C9—46no—
234C2×C9#67⋊C413yes—
235C2×C9#91113yes—
236C2×C9#91413yes—
237D9—126no—
238C2×D4—271no—
239C2×D4C2×C3⋊D911yescharacteristic
240C22×C4—541no—
241C22×C4—541no—
242D8—274no—
243C2×C8—271no—
244C2×C8C2×C3⋊D911yescharacteristic
245C4⋊C4—271no—
246C4⋊C4C2×C3⋊D911yescharacteristic
247C22⋊C4—541no—
248C22⋊C4—541no—
249C42—271no—
250C22×C3—22no—
251C22×C3—26no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—43no—
255C22×C3—43no—
256C22×C3D4×D913yes—
257C22×C3D4×C3⋊D311yescharacteristic
258C2×D3—182no—
259C2×D3—186no—
260C2×D3—361no—
261C2×D3—361no—
262C2×D3—363no—
263C2×D3—363no—
264C3×C4—41no—
265C3×C4—43no—
266C3×C4C2×C4×D913yes—
267C3×C4#369913yes—
268C3×C4C2×C4×C3⋊D311yescharacteristic
269C3×C4#383011yescharacteristic
270Dic3—181no—
271Dic3—181no—
272Dic3—183no—
273Dic3—183no—
274Dic3—361no—
275Dic3—363no—
276C32#478511yescharacteristic
277C9#478513yescomplemented
278C23—21no—
279C23—541no—
280D4—21no—
281D4—272no—
282D4—541no—
283D4C4×C3⋊D912yescomplemented
284C2×C4—21no—
285C2×C4—271no—
286C2×C4—271no—
287C2×C4—541no—
288C2×C4—1082no—
289C2×C4—1082no—
290C2×C4C22×C3⋊D911yescharacteristic
291C8—21no—
292C8—272no—
293C2×C3—46no—
294C2×C3—42no—
295C2×C3#611⋊C413yes—
296C2×C3#6347313yes—
297C2×C3#6347613yes—
298C2×C3#6409011yescharacteristic
299C2×C3#6412011yescharacteristic
300C2×C3#6412311yescharacteristic
301D3—366no—
302D3—362no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—542no—
307C22—1081no—
308C22—1081no—
309C22D4×C3⋊D911yescharacteristic
310C4—41no—
311C4—541no—
312C4—541no—
313C4—1081no—
314C4C2×C4×C3⋊D911yescharacteristic
315C4#8874211yescharacteristic
316C3#1058846513yescomplemented
317C3#1059248411yescharacteristic
318C2—42no—
319C2—1082no—
320C2#1172172711yescharacteristic
321C2#1172179011yescharacteristic
322C2#1172179611yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radicalC2×#88544C2
cosocleC2×C3×C4×C9C23
hypercenter#102C3⋊D9
hypocenterC3×C9#442
LCSG ▷ 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
1. D4×C3⋊Dic9C2semidirect
3. #11721749C2semidirect
4. C3×C9×#102C2semidirect
5. #11720543C2semidirect
6. #11720538C2semidirect
7. #11720526C2semidirect
17. #88544C4semidirect
32. C9×#102D3semidirect
39. C3×D4×C9C2×C4semidirect
47. C4⋊(C3⋊D9)C2×C4semidirect
50. C3⋊(C9⋊C8)C2×C4semidirect
96. C3×#102D9semidirect
122. D4×C9C4×D3semidirect
195. #102C3⋊D9semidirect
197. C3×C9#442semidirect
204. C3×D4C4×D9semidirect
277. C9#4785semidirect
283. D4C4×C3⋊D9semidirect
316. C3#10588465semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊Dic92
C2×#885442
#117217492
C3×C9×#1022
#117205432
#117205382
#117205262
#105924843
#105884653