Finite Group Databasebeta

← #13138474

Subgroups — #13138474

Order 1728.

392 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 89 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#11721038C211yesmaximal, characteristic
2D3×C4⋊Dic9C211yesmaximal, characteristic, complemented
3#11720543C211yesmaximal, characteristic, complemented
4C9×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
5C3×#63377C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic, complemented
7#11720278C211yesmaximal, characteristic, complemented
8#10590677—31nomaximal
9#10588417—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×D3×C9C2211yescharacteristic
12C2×#6195—21no—
13C2×D3×Dic9—21no—
14C2×C3×C8×C9C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88445—21no—
20Dic3⋊Dic9—21no—
21#63909—31no—
22D3×C4⋊Dic3—31no—
23#63648—31no—
24C3×(C2×D3)⋊C8D311yescharacteristic
25C3×#817—31no—
26#63573—31no—
27#63564—31no—
28#63492—31no—
29C2×C4⋊Dic9—31no—
30C9×C22⋊C8—31no—
31#63380—31no—
32#63380—31no—
33#63380—61no—
34#63377—31no—
35#63377D311yescharacteristic, complemented
36C22×C3⋊D9—21no—
37C22×D3×C9D411yescharacteristic
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C4⋊(C3⋊D9)—21no—
41C4⋊(C3⋊D9)—21no—
42C2×C9×Dic3D411yescharacteristic
43C2×C3×Dic9—21no—
44C2×C3×Dic9—21no—
45C4×D3×C9—21no—
46C4×D3×C9—21no—
47#6195—42no—
48D3×Dic9—42no—
49C3×C8×C9—21no—
50C9×C3⋊C8—21no—
51#4863—91no—
52#4735—91no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C3×C4×D3C2×D311yescharacteristic
55C2×#626—61no—
56C2×D3×Dic3—61no—
57C2×C32×C8C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62Dic3⋊Dic3—61no—
63#3722—61no—
64C22×C4×C9—31no—
65C2×#611—61no—
66C22×Dic9—61no—
67C2×D36—31no—
68C2×D36—61no—
69C2×C8×C9—21no—
70C2×C8×C9—31no—
71C2×C8×C9C2×D311yescharacteristic
72(C2×D9)⋊C4—61no—
73C4⋊Dic9—31no—
74C4⋊Dic9—61no—
75C4⋊Dic9—61no—
76C4⋊Dic9C2×D311yescharacteristic
77C4⋊Dic9C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×C3⋊D9—41no—
80C2×C3⋊D9—41no—
81C2×C3⋊D9—41no—
82C2×D3×C9—21no—
83C2×D3×C9—21no—
84C2×D3×C9SD1612yes—
85C3×C4×C9#4111yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C3⋊Dic9—41no—
88C9×Dic3—21no—
89C3×Dic9—41no—
90C3×Dic9—41no—
91#930—91no—
92C2×C4⋊Dic3—91no—
93#895—91no—
94D3×C4⋊C4—91no—
95C3×#106—91no—
96C3×C22⋊C8—31no—
97#821—91no—
98#821—91no—
99#821—91no—
100#821—181no—
101(C2×D3)⋊C8D911yescharacteristic, complemented
102#817—31no—
103#817—91no—
104#809—91no—
105#808—91no—
106C22×C3⋊D3—61no—
107C22×C3×D3D1211yescharacteristic
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—61no—
110C4⋊(C3⋊D3)—61no—
111C4⋊(C3⋊D3)—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3D1211yescharacteristic
115C3×C4×D3—21no—
116C3×C4×D3—21no—
117#626—122no—
118D3×Dic3—122no—
119C23×C9—31no—
120C22×D9—61no—
121C22×D9—121no—
122C32×C8—21no—
123C3×C3⋊C8—21no—
124C2×C4×C9—21no—
125C2×C4×C9—31no—
126C2×C4×C9—61no—
127C2×C4×C9—61no—
128C2×C4×C9C22×D311yescharacteristic
129#611—122no—
130C2×Dic9—21no—
131C2×Dic9—21no—
132C2×Dic9—61no—
133C2×Dic9—121no—
134C2×Dic9—122no—
135D36—62no—
136D36—61no—
137D36—61no—
138D36—121no—
139C8×C9—21no—
140C8×C9—41no—
141C8×C9—61no—
142#481—271noSylow
143C2×C3×C9#12311yescharacteristic
144C2×C3×C9C2×SD1611yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C3⋊D9—81no—
147D3×C9—22no—
148C22×C3×C4—31no—
149C2×#67—181no—
150C22×Dic3—181no—
151C2×D12—91no—
152C2×D12—91no—
153C2×D12—181no—
154C2×D12—181no—
155C2×C4×D3—181no—
156C2×C4×D3C2×D911yescharacteristic
157C2×C3×C8—21no—
158C2×C3×C8—31no—
159C2×C3×C8C2×D911yescharacteristic
160C2×C3×C8D3211yescharacteristic
161C3×C4⋊C4—91no—
162C3×C4⋊C4—91no—
163(C2×D3)⋊C4—181no—
164(C2×D3)⋊C4—181no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—91no—
168C4⋊Dic3—91no—
169C4⋊Dic3—181no—
170C4⋊Dic3—181no—
171Dic3⋊C4—181no—
172C2×C3⋊C8C2×D911yescharacteristic
173C22×C32C2×D1211yescharacteristic
174C2×C3⋊D3—121no—
175C2×C3⋊D3—121no—
176C2×C3⋊D3—121no—
177C2×C3×D3—21no—
178C2×C3×D3—21no—
179C2×C3×D3#20412yes—
180C32×C4#23711yescharacteristic
181C32×C4#23711yescharacteristic
182C3⋊Dic3—121no—
183C3×Dic3—21no—
184C3×Dic3—121no—
185C3×Dic3—121no—
186C22×C9—21no—
187C22×C9—32no—
188C22×C9—61no—
189C22×C9—61no—
190C22×C9D3×D411yescharacteristic
191C2×D9—122no—
192C2×D9—121no—
193C2×D9—121no—
194C2×D9—121no—
195C2×D9—241no—
196C2×D9—241no—
197C4×C9—21no—
198C4×C9—21no—
199C4×C9—61no—
200C4×C9#23511yescharacteristic
201C4×C9#23911yescharacteristic
202Dic9—41no—
203Dic9—41no—
204Dic9—121no—
205Dic9—241no—
206#121—271no—
207C2×C4⋊C4—271no—
208#106—91no—
209#106—271no—
210#102—271no—
211#102—271no—
212C22⋊C8—31no—
213C3×C9#48111yescharacteristic, complemented, Sylow
214C23×C3—31no—
215C22×D3—181no—
216C22×D3—181no—
217C22×D3—361no—
218C22×D3D3611yescharacteristic
219C2×C3×C4—21no—
220C2×C3×C4—31no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4C22×D911yescharacteristic
226C2×C3×C4C2×D3211yescharacteristic
227#67—362no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—181no—
231C2×Dic3—181no—
232C2×Dic3—361no—
233C2×Dic3—362no—
234C2×Dic3D3611yescharacteristic
235D12—182no—
236D12—181no—
237D12—181no—
238D12—181no—
239D12—181no—
240D12—361no—
241D12—362no—
242C4×D3—21no—
243C4×D3—21no—
244C4×D3—362no—
245C3×C8—21no—
246C3×C8—21no—
247C3×C8—41no—
248C3×C8—61no—
249C3⋊C8—21no—
250C2×C32#88611yescharacteristic
251C2×C32C2×#20411yescharacteristic
252C2×C32#90811yescharacteristic
253C3⋊D3—241no—
254C3×D3—22no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9—62no—
259C2×C9#89411yescharacteristic
260C2×C9D3×SD1611yescharacteristic
261C2×C9#91411yescharacteristic
262D9—242no—
263D9—241no—
264C2×D4—271no—
265C2×D4—541no—
266C22×C4—31no—
267C22×C4—541no—
268C2×C8—31no—
269C2×C8D3×D911yescharacteristic
270C4⋊C4—91no—
271C4⋊C4—91no—
272C4⋊C4—271no—
273C4⋊C4—541no—
274C22⋊C4—541no—
275C22×C3—21no—
276C22×C3—32no—
277C22×C3—61no—
278C22×C3—61no—
279C22×C3C2×D3611yescharacteristic
280C22×C3D3×D1211yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—362no—
284C2×D3—361no—
285C2×D3—361no—
286C2×D3—361no—
287C2×D3—361no—
288C2×D3—361no—
289C2×D3—361no—
290C2×D3—721no—
291C2×D3—721no—
292C2×D3#366412yes—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—61no—
296C3×C4—361no—
297C3×C4—361no—
298C3×C4#369911yescharacteristic
299C3×C4#369911yescharacteristic
300C3×C4#379611yescharacteristic
301C3×C4#380411yescharacteristic
302Dic3—21no—
303Dic3—121no—
304Dic3—121no—
305Dic3—361no—
306Dic3—361no—
307Dic3—721no—
308C32#473511yescharacteristic
309C9—21no—
310C9#486311yescharacteristic, complemented
311C23—31no—
312C23—541no—
313D4—541no—
314D4—541no—
315D4—1082no—
316C2×C4—31no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—541no—
322C2×C4—1082no—
323C2×C4C2×D3×D911yescharacteristic
324C8—21no—
325C8—61no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—62no—
330C2×C3#6344811yescharacteristic
331C2×C3C2×#366411yescharacteristic
332C2×C3#6347011yescharacteristic
333C2×C3D3×#20411yescharacteristic
334C2×C3#6379411yescharacteristic
335C2×C3#6389011yescharacteristic
336D3—22no—
337D3—722no—
338D3—721no—
339D3—721no—
340C22—32no—
341C22—61no—
342C22—61no—
343C22—1081no—
344C22—1081no—
345C22—1081no—
346C22D3×D3611yescharacteristic
347C4—61no—
348C4—361no—
349C4—361no—
350C4—1081no—
351C4#8863911yescharacteristic
352C4#8866211yescharacteristic
353C3—21no—
354C3#1058841711yescharacteristic, complemented
355C3#1059067711yescharacteristic
356C2—62no—
357C2—2161no—
358C2D3×#366411yescharacteristic
359C2#1172080411yescharacteristic
360C2#1172102011yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radical#11721038C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#481
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. D3×C4⋊Dic9C2semidirect
3. #11720543C2semidirect
4. C9×(C2×D3)⋊C8C2semidirect
5. C3×#63377C2semidirect
6. #11720284C2semidirect
7. #11720278C2semidirect
35. #63377D3semidirect
101. (C2×D3)⋊C8D9semidirect
213. C3×C9#481semidirect
310. C9#4863semidirect
354. C3#10588417semidirect

Maximal subgroups

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

subgroupindex
#117210382
D3×C4⋊Dic92
#117205432
C9×(C2×D3)⋊C82
C3×#633772
#117202842
#117202782
#105906773
#105884173