Finite Group Databasebeta

← #10588711⋊C3 #13142811

Subgroups — #10588711⋊C3 #13142811

Order 1728.

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

normal subgroups only

This group has 371 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×Q8×3−1+2C211yesmaximal, characteristic, complemented
2C2×#3675⋊C3C211yesmaximal, characteristic
3C2×#3674⋊C3C211yesmaximal, characteristic, complemented
4#63437⋊C3C211yesmaximal, characteristic
5#63369⋊C3C211yesmaximal, characteristic
6#63368⋊C3C211yesmaximal, characteristic, complemented
7C4⋊(3−1+2⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5028—31nomaximal
9#10588711C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C2×3−1+2⋊Q8C2211yescharacteristic
13C4⋊C4×3−1+2—21no—
14C4⋊C4×3−1+2C2211yescharacteristic
15C42×3−1+2—21no—
16C42×3−1+2C2211yescharacteristic
17#3675⋊C3—22no—
18#3674⋊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×Q8D311yescharacteristic
24C2×C3×#216—31no—
25C2×C3×#215—31no—
26C3×#875—31no—
27C3×#810—31no—
28C3×#809—31no—
29C3×C4⋊(C3⋊C8)—31no—
30C4×Q8×C9—21no—
31C4×Q8×C9C2×C311yescharacteristic, complemented
32C2×#3675C2×C311yescharacteristic
33C2×#3674C2×C311yescharacteristic, complemented
34#63437C2×C311yescharacteristic
35#63369C2×C311yescharacteristic
36#63368C2×C311yescharacteristic, complemented
37C4⋊(C9⋊C8)C2×C311yescharacteristic, complemented
38C22×AGL(1,ℤ/9)—21no—
39Q8×3−1+2—21no—
40Q8×3−1+2D412yes—
41C2×C4×3−1+2—21no—
42C2×C4×3−1+2—21no—
43C2×C4×3−1+2D411yescharacteristic
44C2×C4×3−1+2D411yescharacteristic
45C2×C4×3−1+2C2311yescharacteristic
46C2×3−1+2⋊C4—21no—
47C4⋊AGL(1,ℤ/9)—21no—
48C4⋊AGL(1,ℤ/9)—21no—
493−1+2⋊Q8—21no—
503−1+2⋊Q8—21no—
513−1+2⋊C8—21no—
523−1+2⋊C8—21no—
53C3×#464—91no—
54#5028—31no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C3×D12—31no—
57C2×C3×Dic6—31no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4C2×D311yescharacteristic
60C32×C42—21no—
61C32×C42C2×D311yescharacteristic
62C3×#216—62no—
63C3×#215—62no—
64C3×(C2×D3)⋊C4—61no—
65C3×(C2×D3)⋊C4—61no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×Q8×C9—21no—
69C2×Q8×C9C22×C311yescharacteristic
70C2×D36C22×C311yescharacteristic
71C2×Dic18C22×C311yescharacteristic
72C9×C4⋊C4—22no—
73C9×C4⋊C4—21no—
74C9×C4⋊C4—21no—
75C9×C4⋊C4C22×C311yescharacteristic
76C42×C9—22no—
77C42×C9—21no—
78C42×C9—21no—
79C42×C9C22×C311yescharacteristic
80#3675—22no—
81#3674—22no—
82(C2×D9)⋊C4—21no—
83(C2×D9)⋊C4—21no—
84C2×C9⋊C8C22×C311yescharacteristic
85C2×C9⋊C8C22×C311yescharacteristic
86C22×3−1+2C2×D411yescharacteristic
87C2×AGL(1,ℤ/9)—41no—
88C2×AGL(1,ℤ/9)—41no—
89C2×AGL(1,ℤ/9)—41no—
90C4×3−1+2—22no—
91C4×3−1+2—21no—
92C4×3−1+2—21no—
93C4×3−1+2—41no—
94C4×3−1+2C2×D411yescharacteristic
95C4×3−1+2#4111yescharacteristic
963−1+2⋊C4—41no—
97C2×C3×Q16—91no—
98C2×C3×SD16—91no—
99C3×#124—91no—
100C3×C4×Q8—31no—
101C3×C4×Q8C3×D311yescharacteristic
102C2×#216—31no—
103C2×#215—31no—
104#875—31no—
105C3×C4⋊C8—91no—
106C3×#103—91no—
107C3×#102—91no—
108#810—31no—
109#809—31no—
110C4⋊(C3⋊C8)—31no—
111C22×C3×D3—61no—
112C32×Q8—21no—
113C32×Q8D1212yes—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4#6711yescharacteristic
117C2×C32×C4#6711yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C3×Dic3—61no—
120C3×D12—61no—
121C3×D12—61no—
122C3×Dic6—61no—
123C3×Dic6—61no—
124C22×D9—21no—
125C3×C3⋊C8—61no—
126C3×C3⋊C8—61no—
127Q8×C9—22no—
128Q8×C9—21no—
129Q8×C9—22no—
130Q8×C9C3×D412yes—
131C2×C4×C9—22no—
132C2×C4×C9—21no—
133C2×C4×C9—22no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9C3×D411yescharacteristic
139C2×C4×C9C3×D411yescharacteristic
140C2×C4×C9C23×C311yescharacteristic
141C2×Dic9—21no—
142D36—21no—
143D36—21no—
144Dic18—21no—
145Dic18—21no—
146C9⋊C8—21no—
147C9⋊C8—21no—
148#464—91noSylow
149C2×3−1+2#12111yescharacteristic
150C2×3−1+2#13511yescharacteristic
151C2×3−1+2#13711yescharacteristic
152AGL(1,ℤ/9)—81no—
153C2×C3×Q8—31no—
154C2×C3×Q8—91no—
155C2×C3×Q8C2×C3×D311yescharacteristic
156C2×C3×D4—91no—
157C2×D12—31no—
158C2×Dic6—31no—
159C3×Q16—182no—
160C3×SD16—182no—
161C2×C3×C8—91no—
162C2×C3×C8—91no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4C2×C3×D311yescharacteristic
167C3×C22⋊C4—181no—
168C3×C22⋊C4—181no—
169C3×C42—21no—
170C3×C42—31no—
171C3×C42—61no—
172C3×C42C2×C3×D311yescharacteristic
173#216—62no—
174#215—62no—
175(C2×D3)⋊C4—61no—
176(C2×D3)⋊C4—61no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8—31no—
179C22×C32C2×#6711yescharacteristic
180C2×C3×D3—121no—
181C2×C3×D3—121no—
182C2×C3×D3—121no—
183C32×C4—22no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—41no—
187C32×C4C2×D1211yescharacteristic
188C32×C4#23511yescharacteristic
189C3×Dic3—121no—
190C22×C9—21no—
191C22×C9C2×C3×D411yescharacteristic
192C2×D9—41no—
193C2×D9—41no—
194C2×D9—41no—
195C4×C9—22no—
196C4×C9—24no—
197C4×C9—22no—
198C4×C9—21no—
199C4×C9—21no—
200C4×C9—21no—
201C4×C9—21no—
202C4×C9—41no—
203C4×C9—42no—
204C4×C9—41no—
205C4×C9C2×C3×D411yescharacteristic
206C4×C9C3×#4111yescharacteristic
207Dic9—41no—
208C2×Q16—91no—
209C2×SD16—91no—
210#124—91no—
211C4×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
212C4⋊C8—91no—
213#103—91no—
214#102—91no—
2153−1+2#46411yescharacteristic, complemented, Sylow
216C23×C3—181no—
217C22×D3—61no—
218C3×Q8—21no—
219C3×Q8—32no—
220C3×Q8—61no—
221C3×Q8—181no—
222C3×Q8—181no—
223C3×Q8C3×D1212yes—
224C3×D4—181no—
225C3×D4—181no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—181no—
234C2×C3×C4C3×#6711yescharacteristic
235C2×C3×C4C3×#6711yescharacteristic
236C2×C3×C4C22×C3×D311yescharacteristic
237C2×Dic3—61no—
238D12—61no—
239D12—61no—
240Dic6—61no—
241Dic6—61no—
242C3×C8—181no—
243C3×C8—181no—
244C3⋊C8—61no—
245C3⋊C8—61no—
246C2×C32#93011yescharacteristic
247C2×C32#94211yescharacteristic
248C2×C32#95011yescharacteristic
249C3×D3—241no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—21no—
253C2×C9C3×#12111yescharacteristic
254C2×C9C3×#13511yescharacteristic
255C2×C9C3×#13711yescharacteristic
256D9—81no—
257C2×Q8—91no—
258C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
259C2×D4—91no—
260Q16—182no—
261SD16—182no—
262C2×C8—91no—
263C2×C8—91no—
264C4⋊C4—21no—
265C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
266C22⋊C4—181no—
267C22⋊C4—181no—
268C42—21no—
269C42C2×AGL(1,ℤ/9)11yescharacteristic
270C22×C3—31no—
271C22×C3—361no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3C2×C3×#6711yescharacteristic
275C2×D3—121no—
276C2×D3—121no—
277C2×D3—121no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—41no—
284C3×C4—61no—
285C3×C4—61no—
286C3×C4—62no—
287C3×C4—121no—
288C3×C4—361no—
289C3×C4C2×C3×D1211yescharacteristic
290C3×C4C3×#23511yescharacteristic
291Dic3—121no—
292C32#502811yescharacteristic
293C9—21no—
294C9C3×#46411yescharacteristic, complemented
295C23—181no—
296Q8—21no—
297Q8—181no—
298Q8—181no—
299Q8C4⋊AGL(1,ℤ/9)12yes—
300D4—181no—
301D4—181no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—181no—
305C2×C4#611⋊C311yescharacteristic
306C2×C4#611⋊C311yescharacteristic
307C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
308C8—181no—
309C8—181no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—721no—
314C2×C3C3×#93011yescharacteristic
315C2×C3C3×#94211yescharacteristic
316C2×C3C3×#95011yescharacteristic
317D3—241no—
318C22—361no—
319C22—361no—
320C22—361no—
321C22C2×#611⋊C311yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—361no—
327C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
328C4#3697⋊C311yescharacteristic
329C3—31no—
330C3C3×#502811yescharacteristic
331C2—721no—
332C2#63492⋊C311yescharacteristic
333C2#63505⋊C311yescharacteristic
334C2#63513⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×Q8×3−1+2C2
socleC22×C3C2×C3×#67
prime radicalC2×C4⋊AGL(1,ℤ/9)C22
hypercenterC4×Q8AGL(1,ℤ/9)
hypocenterC9C3×#464
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×3−1+2C2semidirect
3. C2×#3674⋊C3C2semidirect
6. #63368⋊C3C2semidirect
7. C4⋊(3−1+2⋊C8)C2semidirect
9. #10588711C3semidirect
31. C4×Q8×C9C2×C3semidirect
33. C2×#3674C2×C3semidirect
36. #63368C2×C3semidirect
37. C4⋊(C9⋊C8)C2×C3semidirect
211. C4×Q8AGL(1,ℤ/9)semidirect
215. 3−1+2#464semidirect
294. C9C3×#464semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×3−1+22
C2×#3675⋊C32
C2×#3674⋊C32
#63437⋊C32
#63369⋊C32
#63368⋊C32
C4⋊(3−1+2⋊C8)2
C3×#50283
#105887113