Finite Group Databasebeta

← #13137595

Subgroups — #13137595

Order 1728.

368 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C3×#63454C211yesmaximal, characteristic, complemented
2#11721004C211yesmaximal, characteristic
3#11720528C211yesmaximal, characteristic, complemented
4C9×#808C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720263C211yesmaximal, characteristic, complemented
8#10590425—31nomaximal
9#10588546—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×#6192—21no—
13C3×C9×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×(C2×D9)⋊C4—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C3×C4×Dic9—21no—
19C2×C9×C3⋊C8C2211yescharacteristic
20#88445—21no—
21C3×#892—31no—
22#63909—31no—
23#63636—31no—
24C3×#808D311yescharacteristic
25#63576—31no—
26#63564—31no—
27(C2×D3)⋊(C3⋊C8)—31no—
28#63457—31no—
29#63454D311yescharacteristic, complemented
30C9×#106—31no—
31#63380—31no—
32(C2×D9)⋊C8—31no—
33#63369—31no—
34#63369—61no—
35#63366—31no—
36C22×C3⋊D9—21no—
37C22×C3×D9D411yescharacteristic
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C4⋊(C3⋊D9)—21no—
41C4⋊(C3⋊D9)—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9D411yescharacteristic
45C3×C4×D9—21no—
46C3×C4×D9—21no—
47#6192—42no—
48C3⋊(C9⋊C8)—21no—
49C9×C3⋊C8—21no—
50#5003—91no—
51#4866—91no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C3×C4×D3—31no—
54C2×#626—61no—
55C32×C4⋊C4C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)—31no—
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3C2×D311yescharacteristic
60C3×C4×Dic3—61no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62#3722—61no—
63C2×D36—31no—
64C2×D36—61no—
65C2×D36—61no—
66C2×C4×D9C2×D311yescharacteristic
67C2×C8×C9—31no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4—31no—
70C9×C4⋊C4C2×D311yescharacteristic
71(C2×D9)⋊C4—21no—
72(C2×D9)⋊C4—61no—
73C4⋊Dic9C2×D311yescharacteristic
74C4×Dic9—21no—
75C2×C9⋊C8—31no—
76C2×C9⋊C8—61no—
77C22×C3×C9C2×D411yescharacteristic
78C2×C3⋊D9—41no—
79C2×C3⋊D9—41no—
80C2×C3⋊D9—41no—
81C2×C3×D9—21no—
82C2×C3×D9—21no—
83C2×C3×D9—21no—
84C3×C4×C9—41no—
85C3×C4×C9#4111yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C9×Dic3—41no—
88C3×Dic9—21no—
89C3×Dic9—41no—
90C3×#117—91no—
91#930—91no—
92#895—91no—
93#892—31no—
94C3×#106—31no—
95#830—91no—
96#821—91no—
97(C2×D3)⋊C8—91no—
98#809—91no—
99#809—91no—
100#809—91no—
101#809—181no—
102#808D911yescharacteristic, complemented
103#807—91no—
104#807—91no—
105C22×C3⋊D3—61no—
106C22×C3×D3—31no—
107C2×C32×C4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C4⋊(C3⋊D3)—61no—
110C4⋊(C3⋊D3)—61no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—31no—
113C2×C3×Dic3—61no—
114C3×C4×D3—61no—
115C3×C4×D3—61no—
116#626—122no—
117C22×D9—61no—
118C22×D9—121no—
119C22×D9#6711yescharacteristic
120C3⋊(C3⋊C8)—61no—
121C3×C3⋊C8—21no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—41no—
125C2×C4×C9—61no—
126C2×C4×C9C22×D311yescharacteristic
127C2×Dic9—21no—
128C2×Dic9#6711yescharacteristic
129D36—62no—
130D36—61no—
131D36—61no—
132D36—121no—
133D36—122no—
134C4×D9—21no—
135C4×D9—21no—
136C8×C9—61no—
137C9⋊C8—61no—
138C9⋊C8—121no—
139#482—271noSylow
140C2×C3×C9#12311yescharacteristic
141C2×C3×C9#13511yescharacteristic
142C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
143C3⋊D9—81no—
144C3×D9—41no—
145C22×C3×C4—91no—
146C2×#67—181no—
147C2×D12—91no—
148C2×D12—91no—
149C2×D12—181no—
150C2×D12—181no—
151C2×C4×D3—31no—
152C2×C3×C8—31no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4C2×D911yescharacteristic
157C3×C4⋊C4D3211yescharacteristic
158C3×C22⋊C4—181no—
159C3×C42—181no—
160(C2×D3)⋊C4—61no—
161(C2×D3)⋊C4—181no—
162(C2×D3)⋊C4—181no—
163C4⋊Dic3—31no—
164C4⋊Dic3C2×D911yescharacteristic
165C4×Dic3—61no—
166C2×C3⋊C8—91no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—181no—
169C2×C3⋊C8C2×D911yescharacteristic
170C22×C32D3×D411yescharacteristic
171C2×C3⋊D3—121no—
172C2×C3⋊D3—121no—
173C2×C3⋊D3—121no—
174C2×C3×D3—61no—
175C2×C3×D3—61no—
176C2×C3×D3—61no—
177C32×C4—41no—
178C32×C4#23511yescharacteristic
179C32×C4#23911yescharacteristic
180C3×Dic3—41no—
181C3×Dic3—61no—
182C3×Dic3—121no—
183C22×C9—21no—
184C22×C9C2×#6711yescharacteristic
185C2×D9—21no—
186C2×D9—21no—
187C2×D9—21no—
188C2×D9—122no—
189C2×D9—121no—
190C2×D9—121no—
191C2×D9—121no—
192C2×D9—241no—
193C2×D9—241no—
194C4×C9—21no—
195C4×C9—21no—
196C4×C9—41no—
197C4×C9—81no—
198C4×C9—121no—
199C4×C9#23511yescharacteristic
200C4×C9#23511yescharacteristic
201Dic9—21no—
202Dic9—41no—
203#121—271no—
204#117—91no—
205#107—271no—
206#106—31no—
207#102—271no—
208#102—271no—
209C22⋊C8—271no—
210C3×C9#48211yescharacteristic, complemented, Sylow
211C23×C3—91no—
212C22×D3—31no—
213C22×D3—181no—
214C22×D3—181no—
215C22×D3—361no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—41no—
220C2×C3×C4—61no—
221C2×C3×C4—91no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4C22×D911yescharacteristic
226C2×C3×C4C2×D3211yescharacteristic
227#67—362no—
228C2×Dic3—21no—
229C2×Dic3—31no—
230C2×Dic3—61no—
231D12—182no—
232D12—181no—
233D12—181no—
234D12—181no—
235D12—181no—
236D12—361no—
237D12—362no—
238C4×D3—61no—
239C4×D3—61no—
240C3×C8—61no—
241C3⋊C8—21no—
242C3⋊C8—181no—
243C3⋊C8—181no—
244C3⋊C8—361no—
245C2×C32#89411yescharacteristic
246C2×C32#91411yescharacteristic
247C2×C32#91611yescharacteristic
248C3⋊D3—241no—
249C3×D3—121no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—21no—
253C2×C9#92911yescharacteristic
254C2×C9#94911yescharacteristic
255C2×C9#95011yescharacteristic
256D9—41no—
257D9—242no—
258D9—241no—
259C2×D4—271no—
260C2×D4—541no—
261C22×C4—91no—
262C2×C8—31no—
263C2×C8—271no—
264C4⋊C4—31no—
265C4⋊C4—91no—
266C4⋊C4D3×D911yescharacteristic
267C22⋊C4—181no—
268C22⋊C4—541no—
269C42—181no—
270C22×C3—21no—
271C22×C3—181no—
272C22×C3—181no—
273C22×C3—181no—
274C22×C3D4×D911yescharacteristic
275C22×C3D3×#6711yescharacteristic
276C2×D3—61no—
277C2×D3—61no—
278C2×D3—61no—
279C2×D3—362no—
280C2×D3—361no—
281C2×D3—361no—
282C2×D3—361no—
283C2×D3—361no—
284C2×D3—361no—
285C2×D3—361no—
286C2×D3—721no—
287C2×D3—721no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—81no—
293C3×C4—121no—
294C3×C4—181no—
295C3×C4—361no—
296C3×C4#369711yescharacteristic
297C3×C4#370111yescharacteristic
298C3×C4#379811yescharacteristic
299C3×C4#379911yescharacteristic
300Dic3—41no—
301Dic3—61no—
302Dic3—121no—
303C32#486611yescharacteristic
304C9—21no—
305C9#500311yescharacteristic, complemented
306C23—91no—
307C23—541no—
308D4—541no—
309D4—541no—
310D4—1082no—
311C2×C4—21no—
312C2×C4—61no—
313C2×C4—91no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4C2×D3×D911yescharacteristic
318C8—61no—
319C8—541no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—361no—
324C2×C3#6345611yescharacteristic
325C2×C3#6347611yescharacteristic
326C2×C3#6347811yescharacteristic
327C2×C3#6385011yescharacteristic
328C2×C3#6393511yescharacteristic
329C2×C3#6393911yescharacteristic
330D3—121no—
331D3—722no—
332D3—721no—
333D3—721no—
334C22—181no—
335C22—181no—
336C22—181no—
337C22—1081no—
338C22—1081no—
339C22—1081no—
340C22D9×#6711yescharacteristic
341C4—41no—
342C4—121no—
343C4—181no—
344C4—361no—
345C4#8863511yescharacteristic
346C4#8864011yescharacteristic
347C3—21no—
348C3#1058854611yescharacteristic, complemented
349C3#1059042511yescharacteristic
350C2—361no—
351C2—2161no—
352C2#1172093811yescharacteristic
353C2#1172118511yescharacteristic
354C2#1172121111yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721004C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#482
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#63454C2semidirect
3. #11720528C2semidirect
4. C9×#808C2semidirect
5. #11720284C2semidirect
6. (C2×D9)⋊(C3⋊C8)C2semidirect
7. #11720263C2semidirect
29. #63454D3semidirect
102. #808D9semidirect
210. C3×C9#482semidirect
305. C9#5003semidirect
348. C3#10588546semidirect

Maximal subgroups

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

subgroupindex
C3×#634542
#117210042
#117205282
C9×#8082
#117202842
(C2×D9)⋊(C3⋊C8)2
#117202632
#105904253
#105885463