Finite Group Databasebeta

← #13137568

Subgroups — #13137568

Order 1728.

378 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 85 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×#63457C211yesmaximal, characteristic
2D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
3#11720526C211yesmaximal, characteristic, complemented
4C9×#808C211yesmaximal, characteristic, complemented
5#11720283C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10590441—31nomaximal
9#10588543—31nomaximal
10C2×C3×D36—21no—
11C2×C3×D36C2211yescharacteristic
12C2×C3×C4×D9C2211yescharacteristic
13C2×Dic3×D9—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C2×C3⋊(C9⋊C8)C2211yescharacteristic
17C3×(C2×D9)⋊C4—21no—
18C9×C4⋊Dic3C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20Dic9⋊Dic3—21no—
21C3×#895—31no—
22D3×C4⋊Dic3—31no—
23#63635—31no—
24C3×#808D311yescharacteristic
25#63563—31no—
26#63562—31no—
27(C2×D3)⋊(C3⋊C8)—31no—
28#63457D311yescharacteristic
29C4⋊C4×D9—31no—
30C9×#106—31no—
31#63380—31no—
32(C2×D9)⋊C8—31no—
33#63369—31no—
34#63367—31no—
35#63367—61no—
36C22×C3×D9—21no—
37C22×C3×D9D411yescharacteristic
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C9×Dic3—21no—
42C2×C3×Dic9D411yescharacteristic
43C3×D36—21no—
44C3×D36—21no—
45C3×D36—42no—
46C3×C4×D9—21no—
47C3×C4×D9—21no—
48Dic3×D9—42no—
49C3⋊(C9⋊C8)—21no—
50C9×C3⋊C8—21no—
51#5034—91no—
52#4863—91no—
53C2×C3×D12—31no—
54C2×C3×D12—61no—
55C2×C3×C4×D3—31no—
56C2×D3×Dic3—61no—
57C32×C4⋊C4C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59C2×C3⋊(C3⋊C8)—31no—
60C3×(C2×D3)⋊C4—61no—
61C3×C4⋊Dic3C2×D311yescharacteristic
62C2×C3×C3⋊C8C2×D311yescharacteristic
63Dic3⋊Dic3—61no—
64C2×D36—21no—
65C2×D36C2×D311yescharacteristic
66C2×C4×D9—61no—
67C2×C4×D9C2×D311yescharacteristic
68C2×C8×C9—31no—
69C9×C4⋊C4—21no—
70C9×C4⋊C4—31no—
71C9×C4⋊C4C2×D311yescharacteristic
72(C2×D9)⋊C4—21no—
73C4⋊Dic9—31no—
74C4⋊Dic9—61no—
75Dic9⋊C4—61no—
76C2×C9⋊C8—31no—
77C2×C9⋊C8—61no—
78C22×C3×C9C2×D411yescharacteristic
79C2×C3×D9—21no—
80C2×C3×D9—21no—
81C2×C3×D9—41no—
82C2×C3×D9—41no—
83C2×C3×D9—41no—
84C2×C3×D9SD1612yes—
85C3×C4×C9—41no—
86C3×C4×C9#4111yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C3⋊Dic9—41no—
89C9×Dic3—41no—
90C3×Dic9—21no—
91C3×#121—91no—
92C2×C4⋊Dic3—91no—
93#895—31no—
94D3×C4⋊C4—91no—
95C3×#106—31no—
96#832—91no—
97#832—91no—
98#830—91no—
99#821—91no—
100(C2×D3)⋊C8—91no—
101#809—91no—
102#808—91no—
103#808—91no—
104#808—181no—
105#808D911yescharacteristic, complemented
106C22×C3×D3—31no—
107C22×C3×D3—61no—
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—61no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—31no—
113C3×D12—61no—
114C3×D12—61no—
115C3×D12—122no—
116C3×C4×D3—61no—
117C3×C4×D3—61no—
118D3×Dic3—122no—
119C22×D9—21no—
120C22×D9#6711yescharacteristic
121C3⋊(C3⋊C8)—61no—
122C3×C3⋊C8—21no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9—41no—
126C2×C4×C9—61no—
127C2×C4×C9C22×D311yescharacteristic
128C2×Dic9—61no—
129C2×Dic9—121no—
130C2×Dic9#6711yescharacteristic
131D36—21no—
132D36—21no—
133D36—42no—
134C4×D9—21no—
135C4×D9—21no—
136C4×D9—122no—
137C8×C9—61no—
138C9⋊C8—62no—
139C9⋊C8—61no—
140#481—271noSylow
141C2×C3×C9#12311yescharacteristic
142C2×C3×C9C2×SD1611yescharacteristic
143C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
144C3×D9—22no—
145C3×D9—81no—
146C2×C3×D4—91no—
147C2×C3×D4—181no—
148C22×C3×C4—91no—
149C22×Dic3—181no—
150C2×D12—31no—
151C2×D12—61no—
152C2×C4×D3—31no—
153C2×C4×D3—181no—
154C2×C3×C8—31no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4C2×D911yescharacteristic
158C3×C4⋊C4D3211yescharacteristic
159C3×C22⋊C4—181no—
160(C2×D3)⋊C4—61no—
161C4⋊Dic3—91no—
162C4⋊Dic3—91no—
163C4⋊Dic3—181no—
164C4⋊Dic3—181no—
165C4⋊Dic3C2×D911yescharacteristic
166Dic3⋊C4—181no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—181no—
170C2×C3⋊C8C2×D911yescharacteristic
171C22×C32D3×D411yescharacteristic
172C2×C3×D3—32no—
173C2×C3×D3—61no—
174C2×C3×D3—61no—
175C2×C3×D3—121no—
176C2×C3×D3—121no—
177C2×C3×D3—121no—
178C32×C4—41no—
179C32×C4#23511yescharacteristic
180C32×C4#23911yescharacteristic
181C3⋊Dic3—121no—
182C3×Dic3—41no—
183C3×Dic3—61no—
184C22×C9—21no—
185C22×C9C2×#6711yescharacteristic
186C2×D9—21no—
187C2×D9—21no—
188C2×D9—41no—
189C2×D9—41no—
190C2×D9—41no—
191C2×D9#21412yes—
192C4×C9—21no—
193C4×C9—21no—
194C4×C9—41no—
195C4×C9—81no—
196C4×C9—121no—
197C4×C9#23711yescharacteristic
198C4×C9#23711yescharacteristic
199Dic9—21no—
200Dic9—121no—
201Dic9—241no—
202#121—91no—
203C2×C4⋊C4—271no—
204#106—31no—
205#106—271no—
206#102—271no—
207#102—271no—
208C22⋊C8—271no—
209C3×C9#48111yescharacteristic, complemented, Sylow
210C23×C3—91no—
211C23×C3—181no—
212C22×D3—31no—
213C22×D3—61no—
214C3×D4—181no—
215C3×D4—181no—
216C3×D4—362no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—41no—
221C2×C3×C4—61no—
222C2×C3×C4—91no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4C22×D911yescharacteristic
226C2×C3×C4C2×D3211yescharacteristic
227C2×Dic3—21no—
228C2×Dic3—31no—
229C2×Dic3—181no—
230C2×Dic3—181no—
231C2×Dic3—361no—
232C2×Dic3—362no—
233D12—61no—
234D12—61no—
235D12—122no—
236C4×D3—61no—
237C4×D3—61no—
238C4×D3—362no—
239C3×C8—61no—
240C3⋊C8—21no—
241C3⋊C8—182no—
242C3⋊C8—181no—
243C3⋊C8—181no—
244C2×C32#89411yescharacteristic
245C2×C32D3×SD1611yescharacteristic
246C2×C32#91411yescharacteristic
247C3×D3—62no—
248C3×D3—241no—
249C2×C9—21no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9C2×#21411yescharacteristic
253C2×C9#93511yescharacteristic
254C2×C9#94911yescharacteristic
255D9—22no—
256D9—81no—
257C2×D4—91no—
258C2×D4—181no—
259C22×C4—91no—
260C22×C4—541no—
261C2×C8—31no—
262C2×C8—271no—
263C4⋊C4—31no—
264C4⋊C4—271no—
265C4⋊C4—541no—
266C4⋊C4D3×D911yescharacteristic
267C22⋊C4—181no—
268C22×C3—21no—
269C22×C3—92no—
270C22×C3—181no—
271C22×C3—181no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3—361no—
275C22×C3D4×D911yescharacteristic
276C22×C3D3×#6711yescharacteristic
277C2×D3—32no—
278C2×D3—61no—
279C2×D3—61no—
280C2×D3—121no—
281C2×D3—121no—
282C2×D3—121no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—81no—
288C3×C4—121no—
289C3×C4—181no—
290C3×C4#369711yescharacteristic
291C3×C4#370111yescharacteristic
292C3×C4#379511yescharacteristic
293C3×C4#379611yescharacteristic
294Dic3—41no—
295Dic3—61no—
296Dic3—361no—
297Dic3—361no—
298Dic3—721no—
299C32#486311yescharacteristic
300C9—21no—
301C9#503411yescharacteristic, complemented
302C23—91no—
303C23—181no—
304D4—181no—
305D4—181no—
306D4—362no—
307C2×C4—21no—
308C2×C4—61no—
309C2×C4—91no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4—541no—
313C2×C4—1082no—
314C2×C4C2×D3×D911yescharacteristic
315C8—61no—
316C8—541no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—182no—
321C2×C3—721no—
322C2×C3#6345611yescharacteristic
323C2×C3SD16×D911yescharacteristic
324C2×C3#6347611yescharacteristic
325C2×C3#6388411yescharacteristic
326C2×C3D3×#21411yescharacteristic
327C2×C3#6393911yescharacteristic
328D3—62no—
329D3—241no—
330C22—92no—
331C22—181no—
332C22—181no—
333C22—361no—
334C22—361no—
335C22—361no—
336C22D9×#6711yescharacteristic
337C4—41no—
338C4—121no—
339C4—181no—
340C4—1081no—
341C4#8863411yescharacteristic
342C4#8864111yescharacteristic
343C3—21no—
344C3#1058854311yescharacteristic, complemented
345C3#1059044111yescharacteristic
346C2—182no—
347C2—721no—
348C2#1172098111yescharacteristic
349C2D9×#21411yescharacteristic
350C2#1172121111yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC3×#63457C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#481
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
2. D9×C4⋊Dic3C2semidirect
3. #11720526C2semidirect
4. C9×#808C2semidirect
5. #11720283C2semidirect
6. (C2×D9)⋊(C3⋊C8)C2semidirect
7. #11720266C2semidirect
105. #808D9semidirect
209. C3×C9#481semidirect
301. C9#5034semidirect
344. C3#10588543semidirect

Maximal subgroups

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

subgroupindex
C3×#634572
D9×C4⋊Dic32
#117205262
C9×#8082
#117202832
(C2×D9)⋊(C3⋊C8)2
#117202662
#105904413
#105885433