Finite Group Databasebeta

← #10588728⋊C3 #13142887

Subgroups — #10588728⋊C3 #13142887

Order 1728.

367 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 96 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
13−1+2×C22⋊Q8C211yesmaximal, characteristic
2C2×#3674⋊C3C211yesmaximal, characteristic, complemented
3#63486⋊C3C211yesmaximal, characteristic
4C2×#3667⋊C3C211yesmaximal, characteristic
5#63395⋊C3C211yesmaximal, characteristic, complemented
6#63368⋊C3C211yesmaximal, characteristic, complemented
7#63367⋊C3C211yesmaximal, characteristic, complemented
8C3×#5052—31nomaximal
9#10588728C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C22×C4×3−1+2C2211yescharacteristic
12C2×3−1+2⋊Q8C2211yescharacteristic
13C4⋊C4×3−1+2—21no—
14C4⋊C4×3−1+2C2211yescharacteristic
15C22⋊C4×3−1+2—21no—
16#3676⋊C3—21no—
17#3674⋊C3—22no—
18C4⋊(3−1+2⋊C4)C2211yescharacteristic
19(3−1+2⋊C4)⋊C4—21no—
20#3667⋊C3—22no—
21C2×3−1+2⋊C8C2211yescharacteristic
22C2×3−1+2⋊C8C2211yescharacteristic
23C32×C22⋊Q8D311yescharacteristic
24C2×C3×#216—31no—
25C3×#924—31no—
26C2×C3×#208—31no—
27C3×#835—31no—
28C3×#810—31no—
29C3×#808—31no—
30C9×C22⋊Q8—21no—
31C9×C22⋊Q8C2×C311yescharacteristic
32C2×#3674C2×C311yescharacteristic, complemented
33#63486C2×C311yescharacteristic
34C2×#3667C2×C311yescharacteristic
35#63395C2×C311yescharacteristic, complemented
36#63368C2×C311yescharacteristic, complemented
37#63367C2×C311yescharacteristic, complemented
38C23×3−1+2D411yescharacteristic
39Q8×3−1+2—21no—
40Q8×3−1+2—21no—
41C2×C4×3−1+2—21no—
42C2×C4×3−1+2—21no—
43C2×C4×3−1+2—21no—
44C2×C4×3−1+2—21no—
45C2×C4×3−1+2D411yescharacteristic
46C2×C4×3−1+2C2311yescharacteristic
47C2×3−1+2⋊C4—21no—
48C2×3−1+2⋊C4—21no—
493−1+2⋊Q8—21no—
503−1+2⋊Q8—21no—
513−1+2⋊C8—21no—
523−1+2⋊C8D412yes—
53C3×#470—91no—
54#5052—31no—
55C2×C32×Q8C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C2×C3×Dic6—31no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4C2×D311yescharacteristic
60C32×C22⋊C4—21no—
61C3×#217—61no—
62C3×#216—62no—
63C3×C4⋊Dic3—31no—
64C3×Dic3⋊C4—61no—
65C3×#208—62no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×Q8×C9—21no—
69C2×Q8×C9C22×C311yescharacteristic
70C22×C4×C9—21no—
71C22×C4×C9C22×C311yescharacteristic
72C2×Dic18C22×C311yescharacteristic
73C9×C4⋊C4—22no—
74C9×C4⋊C4—21no—
75C9×C4⋊C4—21no—
76C9×C4⋊C4C22×C311yescharacteristic
77C9×C22⋊C4—22no—
78C9×C22⋊C4—21no—
79#3676—21no—
80#3674—22no—
81C4⋊Dic9C22×C311yescharacteristic
82Dic9⋊C4—21no—
83#3667—22no—
84C2×C9⋊C8C22×C311yescharacteristic
85C2×C9⋊C8C22×C311yescharacteristic
86C22×3−1+2—21no—
87C22×3−1+2—21no—
88C22×3−1+2—21no—
89C22×3−1+2C2×D411yescharacteristic
90C4×3−1+2—21no—
91C4×3−1+2—41no—
92C4×3−1+2—41no—
93C4×3−1+2C2×D411yescharacteristic
94C4×3−1+2#4111yescharacteristic
953−1+2⋊C4—41no—
963−1+2⋊C4—41no—
97C2×C3×Q16—91no—
98C2×C3×M16—91no—
99C3×C22⋊Q8—31no—
100C3×C22⋊Q8—91no—
101C3×C22⋊Q8C3×D311yescharacteristic
102C2×#216—31no—
103#924—31no—
104C2×#208—31no—
105C3×#106—91no—
106C3×#103—91no—
107C3×#103—91no—
108#835—31no—
109#810—31no—
110#808—31no—
111C23×C32#6711yescharacteristic
112C32×Q8—21no—
113C32×Q8—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4#6711yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C3×Dic6—61no—
123C3×Dic6—61no—
124C23×C9—21no—
125C23×C9C3×D411yescharacteristic
126C3×C3⋊C8—32no—
127C3×C3⋊C8—61no—
128Q8×C9—21no—
129Q8×C9—21no—
130Q8×C9—41no—
131Q8×C9—41no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—22no—
135C2×C4×C9—22no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—41no—
141C2×C4×C9—41no—
142C2×C4×C9C3×D411yescharacteristic
143C2×C4×C9C23×C311yescharacteristic
144C2×Dic9—21no—
145C2×Dic9—21no—
146Dic18—21no—
147Dic18—21no—
148C9⋊C8—21no—
149C9⋊C8C3×D412yes—
150#470—91noSylow
151C2×3−1+2—41no—
152C2×3−1+2#12111yescharacteristic
153C2×3−1+2#13711yescharacteristic
154C2×3−1+2#13711yescharacteristic
155C2×C3×Q8—31no—
156C2×C3×Q8—91no—
157C2×C3×Q8C2×C3×D311yescharacteristic
158C22×C3×C4—31no—
159C22×C3×C4C2×C3×D311yescharacteristic
160C2×Dic6—31no—
161C3×Q16—182no—
162C3×M16—182no—
163C2×C3×C8—91no—
164C2×C3×C8—91no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—181no—
170C3×C4⋊C4C2×C3×D311yescharacteristic
171C3×C22⋊C4—21no—
172C3×C22⋊C4—61no—
173C3×C22⋊C4—181no—
174#217—61no—
175#216—62no—
176C4⋊Dic3—31no—
177Dic3⋊C4—61no—
178#208—62no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32—21no—
184C22×C32C2×#6711yescharacteristic
185C32×C4—21no—
186C32×C4—41no—
187C32×C4—41no—
188C32×C4D3×D411yescharacteristic
189C32×C4#23711yescharacteristic
190C3×Dic3—121no—
191C3×Dic3—121no—
192C22×C9—22no—
193C22×C9—21no—
194C22×C9—21no—
195C22×C9—21no—
196C22×C9—21no—
197C22×C9—41no—
198C22×C9—41no—
199C22×C9C2×C3×D411yescharacteristic
200C4×C9—21no—
201C4×C9—21no—
202C4×C9—21no—
203C4×C9—41no—
204C4×C9—42no—
205C4×C9—42no—
206C4×C9—41no—
207C4×C9—41no—
208C4×C9C2×C3×D411yescharacteristic
209C4×C9C3×#4111yescharacteristic
210Dic9—41no—
211Dic9—41no—
212C2×Q16—91no—
213C2×M16—91no—
214C22⋊Q8—91no—
215C22⋊Q8AGL(1,ℤ/9)11yescharacteristic
216#106—91no—
217#103—91no—
218#103—91no—
2193−1+2#47011yescharacteristic, complemented, Sylow
220C23×C3—31no—
221C23×C3C3×#6711yescharacteristic
222C3×Q8—21no—
223C3×Q8—21no—
224C3×Q8—61no—
225C3×Q8—61no—
226C3×Q8—181no—
227C3×Q8—181no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4C3×#6711yescharacteristic
241C2×C3×C4C22×C3×D311yescharacteristic
242C2×Dic3—61no—
243C2×Dic3—61no—
244Dic6—61no—
245Dic6—61no—
246C3×C8—92no—
247C3×C8—181no—
248C3⋊C8—32no—
249C3⋊C8—61no—
250C2×C32—41no—
251C2×C32#93911yescharacteristic
252C2×C32#94211yescharacteristic
253C2×C32#95111yescharacteristic
254C2×C9—21no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—42no—
258C2×C9—41no—
259C2×C9C3×#12111yescharacteristic
260C2×C9C3×#13711yescharacteristic
261C2×C9C3×#13711yescharacteristic
262C2×Q8—91no—
263C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
264C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
265Q16—182no—
266M16—182no—
267C2×C8—91no—
268C2×C8—91no—
269C4⋊C4—21no—
270C4⋊C4—91no—
271C4⋊C4—181no—
272C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—181no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—31no—
279C22×C3—61no—
280C22×C3—61no—
281C22×C3—61no—
282C22×C3C2×C3×#6711yescharacteristic
283C3×C4—21no—
284C3×C4—31no—
285C3×C4—31no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—61no—
289C3×C4—121no—
290C3×C4—121no—
291C3×C4—361no—
292C3×C4—361no—
293C3×C4C3×D3×D411yescharacteristic
294C3×C4C3×#23711yescharacteristic
295Dic3—121no—
296Dic3—121no—
297C32#505211yescharacteristic
298C9—21no—
299C9C3×#47011yescharacteristic, complemented
300C23#611⋊C311yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—181no—
304Q8—181no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4#611⋊C311yescharacteristic
312C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
313C8—92no—
314C8—181no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—41no—
319C2×C3—121no—
320C2×C3C3×#93911yescharacteristic
321C2×C3C3×#94211yescharacteristic
322C2×C3C3×#95111yescharacteristic
323C22—21no—
324C22—21no—
325C22—21no—
326C22C2×#611⋊C311yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—361no—
331C4—361no—
332C4D4×AGL(1,ℤ/9)11yescharacteristic
333C4#3699⋊C311yescharacteristic
334C3—31no—
335C3C3×#505211yescharacteristic
336C2—41no—
337C2#63501⋊C311yescharacteristic
338C2#63505⋊C311yescharacteristic
339C2#63511⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC23×C32#67
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#470
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#3674⋊C3C2semidirect
5. #63395⋊C3C2semidirect
6. #63368⋊C3C2semidirect
7. #63367⋊C3C2semidirect
9. #10588728C3semidirect
32. C2×#3674C2×C3semidirect
35. #63395C2×C3semidirect
36. #63368C2×C3semidirect
37. #63367C2×C3semidirect
219. 3−1+2#470semidirect
299. C9C3×#470semidirect

Maximal subgroups

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

subgroupindex
3−1+2×C22⋊Q82
C2×#3674⋊C32
#63486⋊C32
C2×#3667⋊C32
#63395⋊C32
#63368⋊C32
#63367⋊C32
C3×#50523
#105887283