Finite Group Databasebeta

← #13143984

Subgroups — #13143984

Order 1728.

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

normal subgroups only

This group has 570 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#11721856C211yesmaximal, characteristic
2C2×#88547C211yesmaximal, characteristic, complemented
3C2×C8×C3⋊D9C211yesmaximal, characteristic
4#11721753C211yesmaximal, characteristic
5C3×C9×#103C211yesmaximal, characteristic, complemented
6#11720539C211yesmaximal, characteristic, complemented
7#11720525C211yesmaximal, characteristic, complemented
8#10592512—31nomaximal
9#10588493—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×#6226C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15#88547—22no—
16#88536—21no—
17#88536—21no—
18C4⋊(C3⋊Dic9)—21no—
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20#88534—21no—
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22C8×C3⋊D9—22no—
23#64155—31no—
24C2×#3756—31no—
25C2×C8×C3⋊D3—31no—
26#64106—31no—
27C32×#103D311yescharacteristic
28#63643—31no—
29#63634—31no—
30#63508—33no—
31C2×#3674—33no—
32C2×C8×D9—33no—
33#63459—33no—
34C9×#103D313yescomplemented
35#63376—33no—
36#63366—33no—
37C22×C3⋊D9D411yescharacteristic
38C3×Q8×C9—21no—
39C3×Q8×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9D411yescharacteristic
46C4×C3⋊D9—21no—
47C4×C3⋊D9—21no—
48#6226—21no—
49#6226—21no—
50C3×C8×C9—21no—
51C3⋊(C9⋊C8)D412yes—
52#4813—93no—
53#4813—91no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C4×C3⋊D3—31no—
56C2×#634—31no—
57C2×C32×C8C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59#3756—62no—
60#3752—61no—
61#3752—61no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—61no—
64#3750—61no—
65C2×C3⋊(C3⋊C8)—31no—
66C8×C3⋊D3—62no—
67C2×Q8×C9C2×D313yes—
68C2×C4×D9—33no—
69C2×Dic18—33no—
70C2×C8×C9C2×D313yes—
71C9×C4⋊C4C2×D313yes—
72#3674—66no—
73(C2×D9)⋊C4—63no—
74(C2×D9)⋊C4—63no—
75C4⋊Dic9—33no—
76C4⋊Dic9—63no—
77Dic9⋊C4—63no—
78C2×C9⋊C8—33no—
79C8×D9—66no—
80C22×C3×C9C2×D411yescharacteristic
81C2×C3⋊D9—21no—
82C2×C3⋊D9—21no—
83C2×C3⋊D9Q1612yescomplemented
84C3×C4×C9—41no—
85C3×C4×C9—41no—
86C3×C4×C9C2×D411yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C3⋊Dic9—21no—
89C3⋊Dic9—41no—
90C3⋊Dic9—41no—
91#946—93no—
92#946—91no—
93C2×#216—93no—
94C2×#216—91no—
95C2×D3×C8—93no—
96C2×D3×C8—91no—
97#897—93no—
98#897—91no—
99C3×#103D913yescomplemented
100C3×#103C3⋊D311yescharacteristic
101#816—93no—
102#816—91no—
103#807—93no—
104#807—91no—
105C22×C3⋊D3—31no—
106C32×Q8—21no—
107C32×Q8—21no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—61no—
114C4×C3⋊D3—61no—
115C4×C3⋊D3—61no—
116#634—61no—
117#634—61no—
118C22×D9—33no—
119C32×C8—21no—
120C3⋊(C3⋊C8)—32no—
121Q8×C9—23no—
122Q8×C9—23no—
123C2×C4×C9—23no—
124C2×C4×C9—23no—
125C2×C4×C9C22×D313yes—
126C2×Dic9—33no—
127C2×Dic9—63no—
128C2×Dic9—63no—
129C4×D9—63no—
130C4×D9—63no—
131Dic18—63no—
132Dic18—63no—
133C8×C9—23no—
134C9⋊C8—36no—
135#467—271noSylow
136C2×C3×C9#12111yescharacteristic
137C2×C3×C9C2×Q1611yescharacteristic
138C2×C3×C9#13511yescharacteristic
139C3⋊D9—22no—
140C2×C3×Q8C2×D913yes—
141C2×C3×Q8C2×C3⋊D311yescharacteristic
142C2×C4×D3—93no—
143C2×C4×D3—91no—
144C2×Dic6—93no—
145C2×Dic6—91no—
146C2×C3×C8C2×D913yes—
147C2×C3×C8C2×C3⋊D311yescharacteristic
148C3×C4⋊C4C2×D913yes—
149C3×C4⋊C4C2×C3⋊D311yescharacteristic
150#216—182no—
151#216—186no—
152(C2×D3)⋊C4—181no—
153(C2×D3)⋊C4—181no—
154(C2×D3)⋊C4—183no—
155(C2×D3)⋊C4—183no—
156C4⋊Dic3—93no—
157C4⋊Dic3—91no—
158C4⋊Dic3—181no—
159C4⋊Dic3—183no—
160Dic3⋊C4—181no—
161Dic3⋊C4—183no—
162C2×C3⋊C8—93no—
163C2×C3⋊C8—91no—
164D3×C8—182no—
165D3×C8—186no—
166C22×C32D3×D411yescharacteristic
167C2×C3⋊D3—32no—
168C2×C3⋊D3—61no—
169C2×C3⋊D3—61no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4#23511yescharacteristic
173C32×C4D3×D411yescharacteristic
174C3⋊Dic3—61no—
175C3⋊Dic3—121no—
176C3⋊Dic3—121no—
177C22×C9D3×D413yes—
178C2×D9—36no—
179C2×D9—63no—
180C2×D9—63no—
181C4×C9—43no—
182C4×C9—43no—
183C4×C9#23513yes—
184C4×C9D3×D413yes—
185Dic9—63no—
186Dic9—123no—
187Dic9—123no—
188C2×Q16—271no—
189C22×C8—271no—
190C22⋊Q8—271no—
191C22⋊Q8—271no—
192#107—271no—
193#103—271no—
194#103C3⋊D911yescharacteristic, complemented
195C3×C9#46711yescharacteristic, complemented, Sylow
196C22×D3—93no—
197C22×D3—91no—
198C3×Q8—23no—
199C3×Q8—23no—
200C3×Q8—21no—
201C3×Q8—21no—
202C2×C3×C4—23no—
203C2×C3×C4—23no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4C22×D913yes—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208C2×Dic3—93no—
209C2×Dic3—91no—
210C2×Dic3—183no—
211C2×Dic3—183no—
212C2×Dic3—181no—
213C2×Dic3—181no—
214C4×D3—183no—
215C4×D3—183no—
216C4×D3—181no—
217C4×D3—181no—
218Dic6—183no—
219Dic6—183no—
220Dic6—181no—
221Dic6—181no—
222C3×C8—23no—
223C3×C8—21no—
224C3⋊C8—96no—
225C3⋊C8—92no—
226C2×C32#88411yescharacteristic
227C2×C32#91611yescharacteristic
228C2×C32D3×Q1611yescharacteristic
229C3⋊D3—62no—
230C2×C9#88413yes—
231C2×C9#91613yes—
232C2×C9D3×Q1613yes—
233D9—66no—
234C2×Q8—271no—
235C2×Q8C2×C3⋊D911yescharacteristic
236C22×C4—271no—
237Q16—542no—
238C2×C8—271no—
239C2×C8—542no—
240C2×C8C2×C3⋊D911yescharacteristic
241C4⋊C4—271no—
242C4⋊C4—541no—
243C4⋊C4—541no—
244C4⋊C4C2×C3⋊D911yescharacteristic
245C22⋊C4—541no—
246C22⋊C4—541no—
247C22×C3D4×D913yes—
248C22×C3D4×C3⋊D311yescharacteristic
249C2×D3—92no—
250C2×D3—96no—
251C2×D3—181no—
252C2×D3—181no—
253C2×D3—183no—
254C2×D3—183no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—43no—
258C3×C4—43no—
259C3×C4#369713yes—
260C3×C4D4×D913yes—
261C3×C4#382811yescharacteristic
262C3×C4D4×C3⋊D311yescharacteristic
263Dic3—181no—
264Dic3—183no—
265Dic3—361no—
266Dic3—361no—
267Dic3—363no—
268Dic3—363no—
269C32#481311yescharacteristic
270C9#481313yescomplemented
271C23—271no—
272Q8—21no—
273Q8—21no—
274Q8—541no—
275Q8—541no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—271no—
279C2×C4—541no—
280C2×C4—541no—
281C2×C4—541no—
282C2×C4—541no—
283C2×C4C22×C3⋊D911yescharacteristic
284C8—21no—
285C8—272no—
286C2×C3#6344613yes—
287C2×C3#6347813yes—
288C2×C3Q16×D913yes—
289C2×C3#6409311yescharacteristic
290C2×C3#6412511yescharacteristic
291C2×C3Q16×C3⋊D311yescharacteristic
292D3—182no—
293D3—186no—
294C22—272no—
295C22—541no—
296C22—541no—
297C22D4×C3⋊D911yescharacteristic
298C4—41no—
299C4—41no—
300C4—541no—
301C4—1081no—
302C4—1081no—
303C4#8873611yescharacteristic
304C4D4×C3⋊D911yescharacteristic
305C3#1058849313yescomplemented
306C3#1059251211yescharacteristic
307C2—542no—
308C2#1172173011yescharacteristic
309C2#1172179811yescharacteristic
310C2Q16×C3⋊D911yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenter#103C3⋊D9
hypocenterC3×C9#467
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×#88547C2semidirect
5. C3×C9×#103C2semidirect
6. #11720539C2semidirect
7. #11720525C2semidirect
34. C9×#103D3semidirect
83. C2×C3⋊D9Q16semidirect
99. C3×#103D9semidirect
194. #103C3⋊D9semidirect
195. C3×C9#467semidirect
270. C9#4813semidirect
305. C3#10588493semidirect

Maximal subgroups

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

subgroupindex
#117218562
C2×#885472
C2×C8×C3⋊D92
#117217532
C3×C9×#1032
#117205392
#117205252
#105925123
#105884933