Finite Group Databasebeta

← #10588699⋊C3 #13142763

Subgroups — #10588699⋊C3 #13142763

Order 1728.

419 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C4×D4×3−1+2C211yesmaximal, characteristic, complemented
2C2×#3673⋊C3C211yesmaximal, characteristic
3C2×#3672⋊C3C211yesmaximal, characteristic, complemented
4#63437⋊C3C211yesmaximal, characteristic, complemented
5#63369⋊C3C211yesmaximal, characteristic, complemented
6#63368⋊C3C211yesmaximal, characteristic, complemented
7C4⋊(3−1+2⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5016—31nomaximal
9#10588699C311yesmaximal, characteristic, complemented
10C2×D4×3−1+2C2211yescharacteristic
11C22×C4×3−1+2—21no—
12C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
13C2×3−1+2⋊Q8C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15C22⋊C4×3−1+2—21no—
16C42×3−1+2C2211yescharacteristic
17#3673⋊C3—22no—
18#3672⋊C3—22no—
19(C2×AGL(1,ℤ/9))⋊C4—21no—
20(C2×AGL(1,ℤ/9))⋊C4—21no—
21C2×3−1+2⋊C8C2211yescharacteristic
22C2×3−1+2⋊C8C2211yescharacteristic
23C32×C4×D4D311yescharacteristic
24C2×C3×#214—31no—
25C2×C3×#213—31no—
26C3×#875—31no—
27C3×#810—31no—
28C3×#809—31no—
29C3×C4⋊(C3⋊C8)—31no—
30C4×D4×C9—21no—
31C4×D4×C9C2×C311yescharacteristic, complemented
32C2×#3673C2×C311yescharacteristic
33C2×#3672C2×C311yescharacteristic, complemented
34#63437C2×C311yescharacteristic, complemented
35#63369C2×C311yescharacteristic, complemented
36#63368C2×C311yescharacteristic, complemented
37C4⋊(C9⋊C8)C2×C311yescharacteristic, complemented
38C23×3−1+2—21no—
39C22×AGL(1,ℤ/9)—21no—
40D4×3−1+2—21no—
41D4×3−1+2D412yes—
42C2×C4×3−1+2—21no—
43C2×C4×3−1+2—42no—
44C2×C4×3−1+2D411yescharacteristic
45C2×C4×3−1+2D411yescharacteristic
46C2×C4×3−1+2C2311yescharacteristic
47C2×3−1+2⋊C4—21no—
48C4⋊AGL(1,ℤ/9)—21no—
49C4⋊AGL(1,ℤ/9)—21no—
503−1+2⋊Q8—21no—
513−1+2⋊Q8—21no—
523−1+2⋊C8—21no—
533−1+2⋊C8—21no—
54C3×#463—91no—
55#5016—31no—
56C2×C32×D4C2×D311yescharacteristic
57C22×C32×C4—21no—
58C2×C3×D12—31no—
59C2×C3×Dic6—31no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C22⋊C4—21no—
62C32×C42C2×D311yescharacteristic
63C3×#214—62no—
64C3×#213—62no—
65C3×(C2×D3)⋊C4—61no—
66C3×(C2×D3)⋊C4—61no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×D4×C9—21no—
70C2×D4×C9C22×C311yescharacteristic
71C22×C4×C9—22no—
72C22×C4×C9—21no—
73C2×D36C22×C311yescharacteristic
74C2×Dic18C22×C311yescharacteristic
75C9×C4⋊C4—21no—
76C9×C4⋊C4C22×C311yescharacteristic
77C9×C22⋊C4—22no—
78C9×C22⋊C4—21no—
79C42×C9—21no—
80C42×C9C22×C311yescharacteristic
81#3673—22no—
82#3672—22no—
83(C2×D9)⋊C4—21no—
84(C2×D9)⋊C4—21no—
85C2×C9⋊C8C22×C311yescharacteristic
86C2×C9⋊C8C22×C311yescharacteristic
87C22×3−1+2—22no—
88C22×3−1+2—41no—
89C22×3−1+2—41no—
90C22×3−1+2C2×D411yescharacteristic
91C2×AGL(1,ℤ/9)—41no—
92C2×AGL(1,ℤ/9)—41no—
93C2×AGL(1,ℤ/9)—41no—
94C4×3−1+2—21no—
95C4×3−1+2—21no—
96C4×3−1+2—41no—
97C4×3−1+2C2×D411yescharacteristic
98C4×3−1+2#4111yescharacteristic
993−1+2⋊C4—41no—
100C2×C3×SD16—91no—
101C2×C3×D8—91no—
102C3×#124—91no—
103C3×C4×D4—31no—
104C3×C4×D4C3×D311yescharacteristic
105C2×#214—31no—
106C2×#213—31no—
107#875—31no—
108C3×C4⋊C8—91no—
109C3×#103—91no—
110C3×#102—91no—
111#810—31no—
112#809—31no—
113C4⋊(C3⋊C8)—31no—
114C23×C32—21no—
115C22×C3×D3—61no—
116C32×D4—21no—
117C32×D4D1212yes—
118C2×C32×C4—21no—
119C2×C32×C4—42no—
120C2×C32×C4#6711yescharacteristic
121C2×C32×C4#6711yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C3×Dic3—61no—
124C3×D12—61no—
125C3×D12—61no—
126C3×Dic6—61no—
127C3×Dic6—61no—
128C23×C9—22no—
129C23×C9—21no—
130C22×D9—21no—
131C3×C3⋊C8—61no—
132C3×C3⋊C8—61no—
133D4×C9—22no—
134D4×C9—21no—
135D4×C9—22no—
136D4×C9C3×D412yes—
137C2×C4×C9—22no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—44no—
143C2×C4×C9—42no—
144C2×C4×C9C3×D411yescharacteristic
145C2×C4×C9C3×D411yescharacteristic
146C2×C4×C9C23×C311yescharacteristic
147C2×Dic9—21no—
148D36—21no—
149D36—21no—
150Dic18—21no—
151Dic18—21no—
152C9⋊C8—21no—
153C9⋊C8—21no—
154#463—91noSylow
155C2×3−1+2—42no—
156C2×3−1+2#12111yescharacteristic
157C2×3−1+2#13511yescharacteristic
158C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
159AGL(1,ℤ/9)—81no—
160C2×C3×Q8—91no—
161C2×C3×D4—31no—
162C2×C3×D4—91no—
163C2×C3×D4C2×C3×D311yescharacteristic
164C22×C3×C4—21no—
165C22×C3×C4—61no—
166C2×D12—31no—
167C2×Dic6—31no—
168C3×SD16—182no—
169C3×D8—182no—
170C2×C3×C8—91no—
171C2×C3×C8—91no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4C2×C3×D311yescharacteristic
174C3×C22⋊C4—21no—
175C3×C22⋊C4—61no—
176C3×C22⋊C4—181no—
177C3×C22⋊C4—181no—
178C3×C42—31no—
179C3×C42C2×C3×D311yescharacteristic
180#214—62no—
181#213—62no—
182(C2×D3)⋊C4—61no—
183(C2×D3)⋊C4—61no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—31no—
186C22×C32—22no—
187C22×C32—41no—
188C22×C32—41no—
189C22×C32C2×#6711yescharacteristic
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C2×C3×D3—121no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—41no—
196C32×C4C2×D1211yescharacteristic
197C32×C4#23511yescharacteristic
198C3×Dic3—121no—
199C22×C9—24no—
200C22×C9—22no—
201C22×C9—21no—
202C22×C9—42no—
203C22×C9—42no—
204C22×C9—41no—
205C22×C9—41no—
206C22×C9C2×C3×D411yescharacteristic
207C2×D9—41no—
208C2×D9—41no—
209C2×D9—41no—
210C4×C9—22no—
211C4×C9—21no—
212C4×C9—21no—
213C4×C9—21no—
214C4×C9—21no—
215C4×C9—41no—
216C4×C9—42no—
217C4×C9—41no—
218C4×C9C2×C3×D411yescharacteristic
219C4×C9C3×#4111yescharacteristic
220Dic9—41no—
221C2×SD16—91no—
222C2×D8—91no—
223#124—91no—
224C4×D4AGL(1,ℤ/9)11yescharacteristic, complemented
225C4⋊C8—91no—
226#103—91no—
227#102—91no—
2283−1+2#46311yescharacteristic, complemented, Sylow
229C23×C3—21no—
230C23×C3—61no—
231C23×C3—181no—
232C22×D3—61no—
233C3×Q8—181no—
234C3×Q8—181no—
235C3×D4—21no—
236C3×D4—32no—
237C3×D4—61no—
238C3×D4—181no—
239C3×D4—181no—
240C3×D4C3×D1212yes—
241C2×C3×C4—21no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—42no—
246C2×C3×C4—61no—
247C2×C3×C4—122no—
248C2×C3×C4—181no—
249C2×C3×C4C3×#6711yescharacteristic
250C2×C3×C4C3×#6711yescharacteristic
251C2×C3×C4C22×C3×D311yescharacteristic
252C2×Dic3—61no—
253D12—61no—
254D12—61no—
255Dic6—61no—
256Dic6—61no—
257C3×C8—181no—
258C3×C8—181no—
259C3⋊C8—61no—
260C3⋊C8—61no—
261C2×C32—42no—
262C2×C32#93011yescharacteristic
263C2×C32#93211yescharacteristic
264C2×C32#95011yescharacteristic
265C3×D3—241no—
266C2×C9—21no—
267C2×C9—21no—
268C2×C9—21no—
269C2×C9—44no—
270C2×C9—42no—
271C2×C9C3×#12111yescharacteristic
272C2×C9C3×#13511yescharacteristic
273C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
274D9—81no—
275C2×Q8—91no—
276C2×D4—91no—
277C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
278C22×C4—21no—
279SD16—182no—
280D8—182no—
281C2×C8—91no—
282C2×C8—91no—
283C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—181no—
286C22⋊C4—181no—
287C42C2×AGL(1,ℤ/9)11yescharacteristic
288C22×C3—22no—
289C22×C3—31no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—62no—
293C22×C3—121no—
294C22×C3—121no—
295C22×C3—361no—
296C22×C3—361no—
297C22×C3—361no—
298C22×C3C2×C3×#6711yescharacteristic
299C2×D3—121no—
300C2×D3—121no—
301C2×D3—121no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—41no—
307C3×C4—61no—
308C3×C4—61no—
309C3×C4—121no—
310C3×C4—361no—
311C3×C4C2×C3×D1211yescharacteristic
312C3×C4C3×#23511yescharacteristic
313Dic3—121no—
314C32#501611yescharacteristic
315C9—21no—
316C9C3×#46311yescharacteristic, complemented
317C23—21no—
318C23—181no—
319Q8—181no—
320Q8—181no—
321D4—21no—
322D4—181no—
323D4—181no—
324D4C4⋊AGL(1,ℤ/9)12yes—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—181no—
328C2×C4#611⋊C311yescharacteristic
329C2×C4#611⋊C311yescharacteristic
330C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
331C8—181no—
332C8—181no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—31no—
336C2×C3—42no—
337C2×C3—122no—
338C2×C3—721no—
339C2×C3C3×#93011yescharacteristic
340C2×C3C3×#93211yescharacteristic
341C2×C3C3×#95011yescharacteristic
342D3—241no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—361no—
347C22—361no—
348C22—361no—
349C22C2×#611⋊C311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—361no—
354C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
355C4#3697⋊C311yescharacteristic
356C3—31no—
357C3C3×#501611yescharacteristic
358C2—42no—
359C2—721no—
360C2#63492⋊C311yescharacteristic
361C2#63495⋊C311yescharacteristic
362C2#63513⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×D4×3−1+2C2
socleC22×C3C2×C3×#67
prime radicalC2×#3673⋊C3C2
hypercenterC4×D4AGL(1,ℤ/9)
hypocenterC9C3×#463
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×D4×3−1+2C2semidirect
3. C2×#3672⋊C3C2semidirect
4. #63437⋊C3C2semidirect
5. #63369⋊C3C2semidirect
6. #63368⋊C3C2semidirect
7. C4⋊(3−1+2⋊C8)C2semidirect
9. #10588699C3semidirect
31. C4×D4×C9C2×C3semidirect
33. C2×#3672C2×C3semidirect
34. #63437C2×C3semidirect
35. #63369C2×C3semidirect
36. #63368C2×C3semidirect
37. C4⋊(C9⋊C8)C2×C3semidirect
224. C4×D4AGL(1,ℤ/9)semidirect
228. 3−1+2#463semidirect
316. C9C3×#463semidirect

Maximal subgroups

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

subgroupindex
C4×D4×3−1+22
C2×#3673⋊C32
C2×#3672⋊C32
#63437⋊C32
#63369⋊C32
#63368⋊C32
C4⋊(3−1+2⋊C8)2
C3×#50163
#105886993