Finite Group Databasebeta

← #13143970

Subgroups — #13143970

Order 1728.

560 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 135 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#11721857C211yesmaximal, characteristic
2C2×#88547C211yesmaximal, characteristic, complemented
3C2×#88524C211yesmaximal, characteristic
4#11721743C211yesmaximal, characteristic, complemented
5C3×C9×#103C211yesmaximal, characteristic, complemented
6#11720537C211yesmaximal, characteristic, complemented
7#11720528C211yesmaximal, characteristic
8#10592497—31nomaximal
9#10588478—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×#6226C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15#88547—22no—
16#88536—21no—
17#88536—21no—
18#88534—21no—
19C4×C3⋊Dic9—21no—
20C4×C3⋊Dic9C2211yescharacteristic
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22#88524—22no—
23#64156—31no—
24C2×#3756—31no—
25C2×#3744—31no—
26#64096—31no—
27C32×#103D311yescharacteristic
28#63641—31no—
29#63636—31no—
30#63509—33no—
31C2×#3674—33no—
32C2×#3664—33no—
33#63449—33no—
34C9×#103D313yescomplemented
35Dic9⋊C8—33no—
36#63369—33no—
37C22×C3⋊D9—21no—
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⋊Dic9D411yescharacteristic
45C2×C3⋊Dic9D411yescharacteristic
46C4⋊(C3⋊D9)—21no—
47C4⋊(C3⋊D9)—21no—
48#6226—21no—
49#6226D412yes—
50C3×C8×C9—21no—
51C3⋊(C9⋊C8)—21no—
52#4798—93no—
53#4798—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—
62#3750—61no—
63C4×C3⋊Dic3—31no—
64C4×C3⋊Dic3—61no—
65C2×C3⋊(C3⋊C8)—31no—
66#3744—62no—
67C2×Q8×C9C2×D313yes—
68C2×D36—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—
75Dic9⋊C4—63no—
76C4×Dic9—33no—
77C4×Dic9—63no—
78C2×C9⋊C8—33no—
79#3664—66no—
80C22×C3×C9C2×D411yescharacteristic
81C2×C3⋊D9—41no—
82C2×C3⋊D9—41no—
83C2×C3⋊D9—41no—
84C3×C4×C9—41no—
85C3×C4×C9—41no—
86C3×C4×C9C2×D411yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C3⋊Dic9—22no—
89C3⋊Dic9—21no—
90C3⋊Dic9—21no—
91#947—93no—
92#947—91no—
93C2×#216—93no—
94C2×#216—91no—
95C2×#204—93no—
96C2×#204—91no—
97#887—93no—
98#887—91no—
99C3×#103D913yescomplemented
100C3×#103C3⋊D311yescharacteristic
101Dic3⋊C8—93no—
102Dic3⋊C8—91no—
103#809—93no—
104#809—91no—
105C22×C3⋊D3—61no—
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—31no—
113C2×C3⋊Dic3—61no—
114C4⋊(C3⋊D3)—61no—
115C4⋊(C3⋊D3)—61no—
116#634—32no—
117#634—61no—
118C22×D9—63no—
119C32×C8—21no—
120C3⋊(C3⋊C8)—61no—
121Q8×C9—23no—
122Q8×C9—23no—
123C2×C4×C9—23no—
124C2×C4×C9—23no—
125C2×C4×C9C22×D313yes—
126C2×Dic9—33no—
127C2×Dic9—33no—
128C2×Dic9—63no—
129D36—63no—
130D36—63no—
131Dic18—36no—
132Dic18—63no—
133C8×C9—23no—
134C9⋊C8—63no—
135#464—271noSylow
136C2×C3×C9#12111yescharacteristic
137C2×C3×C9#13511yescharacteristic
138C2×C3×C9#13711yescharacteristic
139C3⋊D9—81no—
140C2×C3×Q8C2×D913yes—
141C2×C3×Q8C2×C3⋊D311yescharacteristic
142C2×D12—93no—
143C2×D12—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—
156Dic3⋊C4—181no—
157Dic3⋊C4—183no—
158C4×Dic3—93no—
159C4×Dic3—91no—
160C4×Dic3—181no—
161C4×Dic3—183no—
162C2×C3⋊C8—93no—
163C2×C3⋊C8—91no—
164#204—182no—
165#204—186no—
166C22×C32D3×D411yescharacteristic
167C2×C3⋊D3—121no—
168C2×C3⋊D3—121no—
169C2×C3⋊D3—121no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4#23511yescharacteristic
173C32×C4D3×D411yescharacteristic
174C3⋊Dic3—62no—
175C3⋊Dic3—61no—
176C3⋊Dic3—61no—
177C22×C9D3×D413yes—
178C2×D9—123no—
179C2×D9—123no—
180C2×D9—123no—
181C4×C9—43no—
182C4×C9—43no—
183C4×C9#23513yes—
184C4×C9D3×D413yes—
185Dic9—66no—
186Dic9—63no—
187Dic9—63no—
188C2×Q16—271no—
189C2×SD16—271no—
190#124—271no—
191C4×Q8—271no—
192C4⋊C8—271no—
193#103C3⋊D911yescharacteristic, complemented
194#102—271no—
195C3×C9#46411yescharacteristic, complemented, Sylow
196C22×D3—183no—
197C22×D3—181no—
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—93no—
210C2×Dic3—91no—
211C2×Dic3—91no—
212C2×Dic3—183no—
213C2×Dic3—181no—
214D12—183no—
215D12—183no—
216D12—181no—
217D12—181no—
218Dic6—96no—
219Dic6—92no—
220Dic6—183no—
221Dic6—181no—
222C3×C8—23no—
223C3×C8—21no—
224C3⋊C8—183no—
225C3⋊C8—181no—
226C2×C32#88411yescharacteristic
227C2×C32#91611yescharacteristic
228C2×C32#91811yescharacteristic
229C3⋊D3—241no—
230C2×C9#88413yes—
231C2×C9#91613yes—
232C2×C9#91813yes—
233D9—243no—
234C2×Q8—271no—
235C2×Q8C2×C3⋊D911yescharacteristic
236C2×D4—271no—
237Q16—542no—
238SD16—542no—
239C2×C8—271no—
240C2×C8C2×C3⋊D911yescharacteristic
241C4⋊C4—541no—
242C4⋊C4C2×C3⋊D911yescharacteristic
243C22⋊C4—541no—
244C22⋊C4—541no—
245C42—271no—
246C42—541no—
247C22×C3D4×D913yes—
248C22×C3D4×C3⋊D311yescharacteristic
249C2×D3—361no—
250C2×D3—361no—
251C2×D3—361no—
252C2×D3—363no—
253C2×D3—363no—
254C2×D3—363no—
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—182no—
264Dic3—181no—
265Dic3—181no—
266Dic3—186no—
267Dic3—183no—
268Dic3—183no—
269C32#479811yescharacteristic
270C9#479813yescomplemented
271C23—541no—
272Q8—21no—
273Q8—21no—
274Q8—272no—
275Q8—541no—
276D4—541no—
277D4—541no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—271no—
281C2×C4—271no—
282C2×C4—541no—
283C2×C4C22×C3⋊D911yescharacteristic
284C8—21no—
285C8—541no—
286C2×C3#6344613yes—
287C2×C3#6347813yes—
288C2×C3#6348013yes—
289C2×C3#6409311yescharacteristic
290C2×C3#6412511yescharacteristic
291C2×C3#6412711yescharacteristic
292D3—723no—
293D3—721no—
294C22—1081no—
295C22—1081no—
296C22—1081no—
297C22D4×C3⋊D911yescharacteristic
298C4—41no—
299C4—41no—
300C4—542no—
301C4—541no—
302C4—541no—
303C4#8873611yescharacteristic
304C4D4×C3⋊D911yescharacteristic
305C3#1058847813yescomplemented
306C3#1059249711yescharacteristic
307C2—2161no—
308C2#1172173011yescharacteristic
309C2#1172179811yescharacteristic
310C2#1172180411yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenter#103C3⋊D9
hypocenterC3×C9#464
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
4. #11721743C2semidirect
5. C3×C9×#103C2semidirect
6. #11720537C2semidirect
34. C9×#103D3semidirect
99. C3×#103D9semidirect
193. #103C3⋊D9semidirect
195. C3×C9#464semidirect
270. C9#4798semidirect
305. C3#10588478semidirect

Maximal subgroups

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

subgroupindex
#117218572
C2×#885472
C2×#885242
#117217432
C3×C9×#1032
#117205372
#117205282
#105924973
#105884783