Finite Group Databasebeta

← #13143966

Subgroups — #13143966

Order 1728.

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

normal subgroups only

This group has 540 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#11721854C211yesmaximal, characteristic
2C2×#88547C211yesmaximal, characteristic
3C2×#88521C211yesmaximal, characteristic
4#11721743C211yesmaximal, characteristic
5C3×C9×#103C211yesmaximal, characteristic
6#11720537C211yesmaximal, characteristic
7#11720527C211yesmaximal, characteristic
8#10592494—31nomaximal
9#10588475—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×#6226C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15#88547—22no—
16#88534—21no—
17#88534—21no—
18#88534—21no—
19C4×C3⋊Dic9—21no—
20C4×C3⋊Dic9C2211yescharacteristic
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22#88521—22no—
23#64153—31no—
24C2×#3756—31no—
25C2×#3746—31no—
26#64096—31no—
27C32×#103D311yescharacteristic
28#63641—31no—
29#63637—31no—
30#63506—33no—
31C2×#3674—33no—
32C2×Dic36—33no—
33#63449—33no—
34C9×#103D313yes—
35Dic9⋊C8—33no—
36#63368—33no—
37C3×Q8×C9—21no—
38C3×Q8×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9D411yescharacteristic
45C2×C3⋊Dic9D411yescharacteristic
46#6226—21no—
47#6226—21no—
48#6226—21no—
49#6226D412yes—
50C3×C8×C9—21no—
51C3⋊(C9⋊C8)—21no—
52#4795—93no—
53#4795—91no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×#634—31no—
56C2×#634—31no—
57C2×C32×C8C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59#3756—62no—
60#3750—61no—
61#3750—61no—
62#3750—61no—
63C4×C3⋊Dic3—31no—
64C4×C3⋊Dic3—61no—
65C2×C3⋊(C3⋊C8)—31no—
66#3746—62no—
67C2×Q8×C9C2×D313yes—
68C2×Dic18—33no—
69C2×Dic18—33no—
70C2×C8×C9C2×D313yes—
71C9×C4⋊C4C2×D313yes—
72#3674—66no—
73Dic9⋊C4—63no—
74Dic9⋊C4—63no—
75Dic9⋊C4—63no—
76C4×Dic9—33no—
77C4×Dic9—63no—
78C2×C9⋊C8—33no—
79Dic36—66no—
80C22×C3×C9C2×D411yescharacteristic
81C3×C4×C9—41no—
82C3×C4×C9—41no—
83C3×C4×C9C2×D411yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—22no—
86C3⋊Dic9—21no—
87C3⋊Dic9—41no—
88C3⋊Dic9Q1612yescomplemented
89#944—93no—
90#944—91no—
91C2×#216—93no—
92C2×#216—91no—
93C2×Dic12—93no—
94C2×Dic12—91no—
95#887—93no—
96#887—91no—
97C3×#103D913yes—
98C3×#103C3⋊D311yescharacteristic
99Dic3⋊C8—93no—
100Dic3⋊C8—91no—
101#810—93no—
102#810—91no—
103C32×Q8—21no—
104C32×Q8—21no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—61no—
112#634—32no—
113#634—61no—
114#634—61no—
115#634—61no—
116C32×C8—21no—
117C3⋊(C3⋊C8)—61no—
118Q8×C9—23no—
119Q8×C9—23no—
120C2×C4×C9—23no—
121C2×C4×C9—23no—
122C2×C4×C9C22×D313yes—
123C2×Dic9—33no—
124C2×Dic9—33no—
125C2×Dic9—63no—
126C2×Dic9—63no—
127Dic18—36no—
128Dic18—63no—
129Dic18—63no—
130Dic18—63no—
131C8×C9—23no—
132C9⋊C8—63no—
133#462—271noSylow
134C2×C3×C9#12111yescharacteristic
135C2×C3×C9C2×Q1611yescharacteristic
136C2×C3×C9#13711yescharacteristic
137C2×C3×Q8C2×D913yes—
138C2×C3×Q8C2×C3⋊D311yescharacteristic
139C2×Dic6—93no—
140C2×Dic6—91no—
141C2×Dic6—93no—
142C2×Dic6—91no—
143C2×C3×C8C2×D913yes—
144C2×C3×C8C2×C3⋊D311yescharacteristic
145C3×C4⋊C4C2×D913yes—
146C3×C4⋊C4C2×C3⋊D311yescharacteristic
147#216—182no—
148#216—186no—
149Dic3⋊C4—181no—
150Dic3⋊C4—181no—
151Dic3⋊C4—181no—
152Dic3⋊C4—183no—
153Dic3⋊C4—183no—
154Dic3⋊C4—183no—
155C4×Dic3—93no—
156C4×Dic3—91no—
157C4×Dic3—181no—
158C4×Dic3—183no—
159C2×C3⋊C8—93no—
160C2×C3⋊C8—91no—
161Dic12—182no—
162Dic12—186no—
163C22×C32D3×D411yescharacteristic
164C32×C4—41no—
165C32×C4—41no—
166C32×C4#23511yescharacteristic
167C32×C4D3×D411yescharacteristic
168C3⋊Dic3—32no—
169C3⋊Dic3—62no—
170C3⋊Dic3—61no—
171C3⋊Dic3—121no—
172C22×C9D3×D413yes—
173C4×C9—43no—
174C4×C9—43no—
175C4×C9#23513yes—
176C4×C9D3×D413yes—
177Dic9—36no—
178Dic9—66no—
179Dic9—63no—
180Dic9—123no—
181C2×Q16—271no—
182C2×Q16—271no—
183C4⋊Q8—271no—
184C4×Q8—271no—
185C4⋊C8—271no—
186#103—271no—
187#103C3⋊D911yescharacteristic
188C3×C9#46211yescharacteristic, complemented, Sylow
189C3×Q8—23no—
190C3×Q8—23no—
191C3×Q8—21no—
192C3×Q8—21no—
193C2×C3×C4—23no—
194C2×C3×C4—23no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4C22×D913yes—
198C2×C3×C4C22×C3⋊D311yescharacteristic
199C2×Dic3—93no—
200C2×Dic3—91no—
201C2×Dic3—93no—
202C2×Dic3—91no—
203C2×Dic3—183no—
204C2×Dic3—181no—
205C2×Dic3—183no—
206C2×Dic3—181no—
207Dic6—96no—
208Dic6—92no—
209Dic6—183no—
210Dic6—181no—
211Dic6—183no—
212Dic6—183no—
213Dic6—181no—
214Dic6—181no—
215C3×C8—23no—
216C3×C8—21no—
217C3⋊C8—183no—
218C3⋊C8—181no—
219C2×C32#88411yescharacteristic
220C2×C32#91511yescharacteristic
221C2×C32D3×Q1611yescharacteristic
222C2×C9#88413yes—
223C2×C9#91513yes—
224C2×C9D3×Q1613yes—
225C2×Q8—271no—
226C2×Q8—271no—
227C2×Q8C2×C3⋊D911yescharacteristic
228Q16—542no—
229Q16—542no—
230C2×C8—271no—
231C2×C8C2×C3⋊D911yescharacteristic
232C4⋊C4—541no—
233C4⋊C4—541no—
234C4⋊C4—541no—
235C4⋊C4C2×C3⋊D911yescharacteristic
236C42—271no—
237C42—541no—
238C22×C3D4×D913yes—
239C22×C3D4×C3⋊D311yescharacteristic
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—43no—
243C3×C4—43no—
244C3×C4#369713yes—
245C3×C4D4×D913yes—
246C3×C4#382811yescharacteristic
247C3×C4D4×C3⋊D311yescharacteristic
248Dic3—92no—
249Dic3—96no—
250Dic3—182no—
251Dic3—181no—
252Dic3—186no—
253Dic3—183no—
254Dic3—361no—
255Dic3—363no—
256C32#479511yescharacteristic
257C9#479513yescomplemented
258Q8—21no—
259Q8—21no—
260Q8—272no—
261Q8—541no—
262Q8—541no—
263Q8—541no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—271no—
267C2×C4—271no—
268C2×C4—541no—
269C2×C4—541no—
270C2×C4C22×C3⋊D911yescharacteristic
271C8—21no—
272C8—541no—
273C2×C3#6344613yes—
274C2×C3#6347713yes—
275C2×C3Q16×D913yes—
276C2×C3#6409311yescharacteristic
277C2×C3#6412411yescharacteristic
278C2×C3Q16×C3⋊D311yescharacteristic
279C22D4×C3⋊D911yescharacteristic
280C4—41no—
281C4—41no—
282C4—272no—
283C4—542no—
284C4—541no—
285C4—1081no—
286C4#8873611yescharacteristic
287C4D4×C3⋊D911yescharacteristic
288C3#1058847513yescomplemented
289C3#1059249411yescharacteristic
290C2#1172173011yescharacteristic
291C2#1172179711yescharacteristic
292C2Q16×C3⋊D911yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socle, prime radicalC22×C32D3×D4
hypercenter#103C3⋊D9
hypocenterC3×C9#462
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
88. C3⋊Dic9Q16semidirect
188. C3×C9#462semidirect
257. C9#4795semidirect
288. C3#10588475semidirect

Maximal subgroups

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

subgroupindex
#117218542
C2×#885472
C2×#885212
#117217432
C3×C9×#1032
#117205372
#117205272
#105924943
#105884753