Finite Group Databasebeta

← #10588498⋊C3 #13141932

Subgroups — #10588498⋊C3 #13141932

Order 1728.

381 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 94 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#63506⋊C3C211yesmaximal, characteristic, complemented
2C2×#3675⋊C3C211yesmaximal, characteristic
3C2×#3664⋊C3C211yesmaximal, characteristic
4#63455⋊C3C211yesmaximal, characteristic
53−1+2×#103C211yesmaximal, characteristic, complemented
6(3−1+2⋊C4)⋊C8C211yesmaximal, characteristic, complemented
7#63368⋊C3C211yesmaximal, characteristic, complemented
8C3×#4818—31nomaximal
9#10588498C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C2×C4×AGL(1,ℤ/9)—21no—
13C2×3−1+2⋊Q8C2211yescharacteristic
14C2×C8×3−1+2C2211yescharacteristic
15C4⋊C4×3−1+2C2211yescharacteristic
16#3675⋊C3—22no—
17(C2×AGL(1,ℤ/9))⋊C4—21no—
18(3−1+2⋊C4)⋊C4—21no—
19(3−1+2⋊C4)⋊C4—21no—
20C4×3−1+2⋊C4C2211yescharacteristic
21C2×3−1+2⋊C8C2211yescharacteristic
22#3664⋊C3—22no—
23C3×#944—31no—
24C2×C3×#215—31no—
25C2×C3×#204—31no—
26C3×#893—31no—
27C32×#103D311yescharacteristic
28C3×Dic3⋊C8—31no—
29C3×#810—31no—
30#63506C2×C311yescharacteristic, complemented
31C2×#3675C2×C311yescharacteristic
32C2×#3664C2×C311yescharacteristic
33#63455C2×C311yescharacteristic
34C9×#103—21no—
35C9×#103C2×C311yescharacteristic, complemented
36Dic9⋊C8C2×C311yescharacteristic, complemented
37#63368C2×C311yescharacteristic, complemented
38C22×AGL(1,ℤ/9)—21no—
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+2C2311yescharacteristic
44C2×3−1+2⋊C4—21no—
45C2×3−1+2⋊C4D411yescharacteristic
46C2×3−1+2⋊C4D411yescharacteristic
47C4⋊AGL(1,ℤ/9)—21no—
48C4⋊AGL(1,ℤ/9)D412yes—
49C4×AGL(1,ℤ/9)—42no—
503−1+2⋊Q8—21no—
513−1+2⋊Q8—21no—
52C8×3−1+2—21no—
533−1+2⋊C8—21no—
54C3×#461—91no—
55#4818—31no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×C3×D12—31no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6—31no—
60C2×C32×C8C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62C3×#215—62no—
63C3×(C2×D3)⋊C4—61no—
64C3×Dic3⋊C4—61no—
65C3×Dic3⋊C4—61no—
66C3×C4×Dic3—31no—
67C2×C3×C3⋊C8—31no—
68C3×#204—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9C22×C311yescharacteristic
71C2×D36C22×C311yescharacteristic
72C2×C4×D9—21no—
73C2×Dic18C22×C311yescharacteristic
74C2×C8×C9—21no—
75C2×C8×C9C22×C311yescharacteristic
76C9×C4⋊C4—21no—
77C9×C4⋊C4C22×C311yescharacteristic
78#3675—22no—
79(C2×D9)⋊C4—21no—
80Dic9⋊C4—21no—
81Dic9⋊C4—21no—
82C4×Dic9C22×C311yescharacteristic
83C2×C9⋊C8C22×C311yescharacteristic
84#3664—22no—
85C22×3−1+2C2×D411yescharacteristic
86C2×AGL(1,ℤ/9)—22no—
87C2×AGL(1,ℤ/9)—41no—
88C2×AGL(1,ℤ/9)—41no—
89C4×3−1+2—41no—
90C4×3−1+2—41no—
91C4×3−1+2C2×D411yescharacteristic
92C4×3−1+2#4111yescharacteristic
933−1+2⋊C4—21no—
943−1+2⋊C4—41no—
953−1+2⋊C4SD1612yescomplemented
96C2×C3×SD16—91no—
97C2×C3×SD16—91no—
98C3×C4⋊Q8—91no—
99C3×C4×D4—91no—
100#944—31no—
101C2×#215—31no—
102C2×#204—31no—
103#893—31no—
104C3×C4⋊C8—91no—
105C3×#103—31no—
106C3×#103—91no—
107C3×#103C3×D311yescharacteristic
108Dic3⋊C8—31no—
109#810—31no—
110C22×C3×D3—61no—
111C32×Q8—21no—
112C32×Q8—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—61no—
119C3×D12—32no—
120C3×D12—61no—
121C3×C4×D3—122no—
122C3×Dic6—61no—
123C3×Dic6—61no—
124C22×D9—21no—
125C32×C8—21no—
126C3×C3⋊C8—61no—
127Q8×C9—22no—
128Q8×C9—21no—
129Q8×C9—21no—
130Q8×C9—41no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—41no—
135C2×C4×C9—41no—
136C2×C4×C9C23×C311yescharacteristic
137C2×Dic9—21no—
138C2×Dic9C3×D411yescharacteristic
139C2×Dic9C3×D411yescharacteristic
140D36—21no—
141D36C3×D412yes—
142C4×D9—42no—
143Dic18—21no—
144Dic18—21no—
145C8×C9—21no—
146C8×C9—41no—
147C9⋊C8—21no—
148#461—91noSylow
149C2×3−1+2#12111yescharacteristic
150C2×3−1+2C2×SD1611yescharacteristic
151C2×3−1+2#13711yescharacteristic
152AGL(1,ℤ/9)—42no—
153C2×C3×Q8—31no—
154C2×C3×Q8—91no—
155C2×C3×Q8C2×C3×D311yescharacteristic
156C2×C3×D4—91no—
157C22×C3×C4—181no—
158C2×D12—31no—
159C2×C4×D3—61no—
160C2×Dic6—31no—
161C3×SD16—182no—
162C3×SD16—182no—
163C2×C3×C8—31no—
164C2×C3×C8—91no—
165C2×C3×C8C2×C3×D311yescharacteristic
166C3×C4⋊C4—31no—
167C3×C4⋊C4—181no—
168C3×C4⋊C4—181no—
169C3×C4⋊C4C2×C3×D311yescharacteristic
170C3×C22⋊C4—181no—
171C3×C42—91no—
172#215—62no—
173(C2×D3)⋊C4—61no—
174Dic3⋊C4—61no—
175Dic3⋊C4—61no—
176C4×Dic3—31no—
177C2×C3⋊C8—31no—
178#204—62no—
179C22×C32D3×D411yescharacteristic
180C2×C3×D3—62no—
181C2×C3×D3—121no—
182C2×C3×D3—121no—
183C32×C4—41no—
184C32×C4—41no—
185C32×C4#23511yescharacteristic
186C32×C4D3×D411yescharacteristic
187C3×Dic3—32no—
188C3×Dic3—61no—
189C3×Dic3—121no—
190C22×C9—21no—
191C22×C9C2×C3×D411yescharacteristic
192C2×D9—22no—
193C2×D9—41no—
194C2×D9—41no—
195C4×C9—21no—
196C4×C9—21no—
197C4×C9—42no—
198C4×C9—41no—
199C4×C9—41no—
200C4×C9—81no—
201C4×C9C2×C3×D411yescharacteristic
202C4×C9C3×#4111yescharacteristic
203Dic9—21no—
204Dic9—41no—
205Dic9C3×SD1612yescomplemented
206C2×SD16—91no—
207C2×SD16—91no—
208C4⋊Q8—91no—
209C4×D4—91no—
210C4⋊C8—91no—
211#103—91no—
212#103AGL(1,ℤ/9)11yescharacteristic, complemented
2133−1+2#46111yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—61no—
216C3×Q8—21no—
217C3×Q8—21no—
218C3×Q8—61no—
219C3×Q8—61no—
220C3×Q8—181no—
221C3×Q8—181no—
222C3×D4—92no—
223C3×D4—181no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—31no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—181no—
232C2×C3×C4—362no—
233C2×C3×C4C22×C3×D311yescharacteristic
234C2×Dic3—31no—
235C2×Dic3—31no—
236C2×Dic3—61no—
237D12—32no—
238D12—61no—
239C4×D3—122no—
240Dic6—61no—
241Dic6—61no—
242C3×C8—21no—
243C3×C8—61no—
244C3×C8—181no—
245C3⋊C8—61no—
246C2×C32#88411yescharacteristic
247C2×C32D3×SD1611yescharacteristic
248C2×C32#91811yescharacteristic
249C3×D3—122no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—21no—
253C2×C9C3×#12111yescharacteristic
254C2×C9C2×C3×SD1611yescharacteristic
255C2×C9C3×#13711yescharacteristic
256D9—42no—
257C2×Q8—91no—
258C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
259C2×D4—91no—
260C22×C4—181no—
261SD16—182no—
262SD16—182no—
263C2×C8—91no—
264C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
265C4⋊C4—181no—
266C4⋊C4—181no—
267C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
268C22⋊C4—181no—
269C42—91no—
270C22×C3—31no—
271C22×C3—182no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3C3×D3×D411yescharacteristic
275C2×D3—62no—
276C2×D3—121no—
277C2×D3—121no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—92no—
283C3×C4—121no—
284C3×C4—121no—
285C3×C4—181no—
286C3×C4—361no—
287C3×C4C3×#23511yescharacteristic
288C3×C4C3×D3×D411yescharacteristic
289Dic3—32no—
290Dic3—61no—
291Dic3—121no—
292C32#481811yescharacteristic
293C9—21no—
294C9C3×#46111yescharacteristic, complemented
295C23—181no—
296Q8—21no—
297Q8—21no—
298Q8—181no—
299Q8—181no—
300D4—92no—
301D4—181no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—91no—
305C2×C4—91no—
306C2×C4—181no—
307C2×C4—362no—
308C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
309C8—21no—
310C8—181no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—362no—
315C2×C3C3×#88411yescharacteristic
316C2×C3C3×D3×SD1611yescharacteristic
317C2×C3C3×#91811yescharacteristic
318D3—122no—
319C22—182no—
320C22—361no—
321C22—361no—
322C22D4×AGL(1,ℤ/9)11yescharacteristic
323C4—41no—
324C4—41no—
325C4—92no—
326C4—181no—
327C4—361no—
328C4#3697⋊C311yescharacteristic
329C4D4×AGL(1,ℤ/9)11yescharacteristic
330C3—31no—
331C3C3×#481811yescharacteristic
332C2—362no—
333C2#63446⋊C311yescharacteristic
334C2SD16×AGL(1,ℤ/9)11yescharacteristic
335C2#63480⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#103C2
socleC22×C3C3×D3×D4
prime radicalC2×C4⋊AGL(1,ℤ/9)C22
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×#461
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. #63506⋊C3C2semidirect
5. 3−1+2×#103C2semidirect
6. (3−1+2⋊C4)⋊C8C2semidirect
7. #63368⋊C3C2semidirect
9. #10588498C3semidirect
30. #63506C2×C3semidirect
35. C9×#103C2×C3semidirect
36. Dic9⋊C8C2×C3semidirect
37. #63368C2×C3semidirect
95. 3−1+2⋊C4SD16semidirect
205. Dic9C3×SD16semidirect
212. #103AGL(1,ℤ/9)semidirect
213. 3−1+2#461semidirect
294. C9C3×#461semidirect

Maximal subgroups

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

subgroupindex
#63506⋊C32
C2×#3675⋊C32
C2×#3664⋊C32
#63455⋊C32
3−1+2×#1032
(3−1+2⋊C4)⋊C82
#63368⋊C32
C3×#48183
#105884983