Finite Group Databasebeta

← #10588732⋊C3 #13142889

Subgroups — #10588732⋊C3 #13142889

Order 1728.

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

normal subgroups only

This group has 397 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, complemented
2C2×#3675⋊C3C211yesmaximal, characteristic, complemented
3#63492⋊C3C211yesmaximal, characteristic
4C2×#3667⋊C3C211yesmaximal, characteristic, complemented
5#63395⋊C3C211yesmaximal, characteristic, complemented
6#63369⋊C3C211yesmaximal, characteristic, complemented
7#63366⋊C3C211yesmaximal, characteristic, complemented
8C3×#5049—31nomaximal
9#10588732C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C22×C4×3−1+2C2211yescharacteristic
12C2×#611⋊C3—21no—
13C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
14C4⋊C4×3−1+2—21no—
15C4⋊C4×3−1+2C2211yescharacteristic
16C22⋊C4×3−1+2—21no—
17#3675⋊C3—22no—
18(C2×AGL(1,ℤ/9))⋊C4—21no—
19C4⋊(3−1+2⋊C4)C2211yescharacteristic
20#3667⋊C3—22no—
21C2×3−1+2⋊C8C2211yescharacteristic
22C2×3−1+2⋊C8C2211yescharacteristic
23C32×C22⋊Q8D311yescharacteristic
24C2×C3×#215—31no—
25C3×#930—31no—
26C2×C3×#208—31no—
27C3×#835—31no—
28C3×#809—31no—
29C3×#807—31no—
30C9×C22⋊Q8—21no—
31C9×C22⋊Q8C2×C311yescharacteristic, complemented
32C2×#3675C2×C311yescharacteristic, complemented
33#63492C2×C311yescharacteristic
34C2×#3667C2×C311yescharacteristic, complemented
35#63395C2×C311yescharacteristic, complemented
36#63369C2×C311yescharacteristic, complemented
37#63366C2×C311yescharacteristic, complemented
38C23×3−1+2D411yescharacteristic
39C22×AGL(1,ℤ/9)—21no—
40Q8×3−1+2—21no—
41Q8×3−1+2—21no—
42C2×C4×3−1+2—21no—
43C2×C4×3−1+2—21no—
44C2×C4×3−1+2—21no—
45C2×C4×3−1+2—21no—
46C2×C4×3−1+2D411yescharacteristic
47C2×C4×3−1+2C2311yescharacteristic
48#611⋊C3—42no—
49C2×3−1+2⋊C4—21no—
50C4⋊AGL(1,ℤ/9)—21no—
51C4⋊AGL(1,ℤ/9)—21no—
523−1+2⋊C8—21no—
533−1+2⋊C8D412yescomplemented
54C3×#468—91no—
55#5049—31no—
56C2×C32×Q8C2×D311yescharacteristic
57C22×C32×C4C2×D311yescharacteristic
58C2×C3×#67—61no—
59C2×C3×D12—31no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C22⋊C4—21no—
63C3×#215—62no—
64C3×(C2×D3)⋊C4—61no—
65C3×C4⋊Dic3—31no—
66C3×#208—62no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×Q8×C9—21no—
70C2×Q8×C9C22×C311yescharacteristic
71C22×C4×C9—21no—
72C22×C4×C9C22×C311yescharacteristic
73C2×#611—21no—
74C2×D36C22×C311yescharacteristic
75C9×C4⋊C4—22no—
76C9×C4⋊C4—21no—
77C9×C4⋊C4—21no—
78C9×C4⋊C4C22×C311yescharacteristic
79C9×C22⋊C4—22no—
80C9×C22⋊C4—21no—
81#3675—22no—
82(C2×D9)⋊C4—21no—
83C4⋊Dic9C22×C311yescharacteristic
84#3667—22no—
85C2×C9⋊C8C22×C311yescharacteristic
86C2×C9⋊C8C22×C311yescharacteristic
87C22×3−1+2—21no—
88C22×3−1+2—21no—
89C22×3−1+2—21no—
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—41no—
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×M16—91no—
102C3×C22⋊Q8—31no—
103C3×C22⋊Q8C3×D311yescharacteristic
104C3×#121—91no—
105C2×#215—31no—
106#930—31no—
107C2×#208—31no—
108C3×#107—91no—
109C3×#103—91no—
110C3×#102—91no—
111#835—31no—
112#809—31no—
113#807—31no—
114C23×C32#6711yescharacteristic
115C22×C3×D3—61no—
116C32×Q8—21no—
117C32×Q8—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4#6711yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C3×#67—122no—
125C2×C3×Dic3—61no—
126C3×D12—61no—
127C3×D12—61no—
128C23×C9—21no—
129C23×C9C3×D411yescharacteristic
130C22×D9—21no—
131C3×C3⋊C8—32no—
132C3×C3⋊C8—61no—
133Q8×C9—21no—
134Q8×C9—21no—
135Q8×C9—41no—
136Q8×C9—41no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—22no—
140C2×C4×C9—22no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9—41no—
146C2×C4×C9—41no—
147C2×C4×C9C3×D411yescharacteristic
148C2×C4×C9C23×C311yescharacteristic
149#611—42no—
150C2×Dic9—21no—
151D36—21no—
152D36—21no—
153C9⋊C8—21no—
154C9⋊C8C3×D412yescomplemented
155#468—91noSylow
156C2×3−1+2—41no—
157C2×3−1+2#12111yescharacteristic
158C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
159C2×3−1+2#13711yescharacteristic
160AGL(1,ℤ/9)—81no—
161C2×C3×Q8—31no—
162C2×C3×Q8C2×C3×D311yescharacteristic
163C2×C3×D4—91no—
164C2×C3×D4—181no—
165C22×C3×C4—31no—
166C22×C3×C4C2×C3×D311yescharacteristic
167C2×#67—61no—
168C2×D12—31no—
169C3×SD16—182no—
170C3×M16—182no—
171C2×C3×C8—91no—
172C2×C3×C8—91no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—31no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4—91no—
177C3×C4⋊C4C2×C3×D311yescharacteristic
178C3×C22⋊C4—21no—
179C3×C22⋊C4—61no—
180C3×C22⋊C4—181no—
181#215—62no—
182(C2×D3)⋊C4—61no—
183C4⋊Dic3—31no—
184#208—62no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—31no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—21no—
190C22×C32C2×#6711yescharacteristic
191C2×C3×D3—121no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C32×C4—21no—
195C32×C4—41no—
196C32×C4—41no—
197C32×C4D3×D411yescharacteristic
198C32×C4#23711yescharacteristic
199C3×Dic3—121no—
200C22×C9—22no—
201C22×C9—21no—
202C22×C9—21no—
203C22×C9—21no—
204C22×C9—21no—
205C22×C9—41no—
206C22×C9—41no—
207C22×C9C2×C3×D411yescharacteristic
208C2×D9—41no—
209C2×D9—41no—
210C2×D9—41no—
211C4×C9—21no—
212C4×C9—21no—
213C4×C9—21no—
214C4×C9—41no—
215C4×C9—42no—
216C4×C9—42no—
217C4×C9—41no—
218C4×C9—41no—
219C4×C9C2×C3×D411yescharacteristic
220C4×C9C3×#4111yescharacteristic
221Dic9—41no—
222C2×SD16—91no—
223C2×M16—91no—
224C22⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
225#121—91no—
226#107—91no—
227#103—91no—
228#102—91no—
2293−1+2#46811yescharacteristic, complemented, Sylow
230C23×C3—31no—
231C23×C3—181no—
232C23×C3C3×#6711yescharacteristic
233C22×D3—61no—
234C3×Q8—21no—
235C3×Q8—21no—
236C3×Q8—61no—
237C3×Q8—61no—
238C3×D4—181no—
239C3×D4—181no—
240C3×D4—362no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—181no—
252C2×C3×C4C3×#6711yescharacteristic
253C2×C3×C4C22×C3×D311yescharacteristic
254#67—122no—
255C2×Dic3—61no—
256D12—61no—
257D12—61no—
258C3×C8—92no—
259C3×C8—181no—
260C3⋊C8—32no—
261C3⋊C8—61no—
262C2×C32—41no—
263C2×C32#93911yescharacteristic
264C2×C32#94211yescharacteristic
265C2×C32#94911yescharacteristic
266C3×D3—241no—
267C2×C9—21no—
268C2×C9—21no—
269C2×C9—21no—
270C2×C9—42no—
271C2×C9—41no—
272C2×C9C3×#12111yescharacteristic
273C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
274C2×C9C3×#13711yescharacteristic
275D9—81no—
276C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
277C2×D4—91no—
278C2×D4—181no—
279C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
280SD16—182no—
281M16—182no—
282C2×C8—91no—
283C2×C8—91no—
284C4⋊C4—21no—
285C4⋊C4—91no—
286C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—181no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—31no—
293C22×C3—61no—
294C22×C3—61no—
295C22×C3—61no—
296C22×C3—361no—
297C22×C3—361no—
298C22×C3—361no—
299C22×C3C2×C3×#6711yescharacteristic
300C2×D3—121no—
301C2×D3—121no—
302C2×D3—121no—
303C3×C4—21no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—61no—
309C3×C4—121no—
310C3×C4—121no—
311C3×C4—361no—
312C3×C4C3×D3×D411yescharacteristic
313C3×C4C3×#23711yescharacteristic
314Dic3—121no—
315C32#504911yescharacteristic
316C9—21no—
317C9C3×#46811yescharacteristic, complemented
318C23—181no—
319C23#611⋊C311yescharacteristic
320Q8—21no—
321Q8—21no—
322D4—181no—
323D4—181no—
324D4—362no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—181no—
330C2×C4#611⋊C311yescharacteristic
331C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
332C8—92no—
333C8—181no—
334C2×C3—31no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—41no—
338C2×C3—121no—
339C2×C3—721no—
340C2×C3C3×#93911yescharacteristic
341C2×C3C3×#94211yescharacteristic
342C2×C3C3×#94911yescharacteristic
343D3—241no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—361no—
348C22—361no—
349C22—361no—
350C22C2×#611⋊C311yescharacteristic
351C4—21no—
352C4—41no—
353C4—41no—
354C4—361no—
355C4D4×AGL(1,ℤ/9)11yescharacteristic
356C4#3699⋊C311yescharacteristic
357C3—31no—
358C3C3×#504911yescharacteristic
359C2—41no—
360C2—721no—
361C2#63501⋊C311yescharacteristic
362C2#63505⋊C311yescharacteristic
363C2#63512⋊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 radical#63492⋊C3C2
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#468
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
1. 3−1+2×C22⋊Q8C2semidirect
2. C2×#3675⋊C3C2semidirect
4. C2×#3667⋊C3C2semidirect
5. #63395⋊C3C2semidirect
6. #63369⋊C3C2semidirect
7. #63366⋊C3C2semidirect
9. #10588732C3semidirect
31. C9×C22⋊Q8C2×C3semidirect
32. C2×#3675C2×C3semidirect
34. C2×#3667C2×C3semidirect
35. #63395C2×C3semidirect
36. #63369C2×C3semidirect
37. #63366C2×C3semidirect
53. 3−1+2⋊C8D4semidirect
154. C9⋊C8C3×D4semidirect
224. C22⋊Q8AGL(1,ℤ/9)semidirect
229. 3−1+2#468semidirect
317. C9C3×#468semidirect

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×#3675⋊C32
#63492⋊C32
C2×#3667⋊C32
#63395⋊C32
#63369⋊C32
#63366⋊C32
C3×#50493
#105887323