Finite Group Databasebeta

← #10588478⋊C3 #13141942

Subgroups — #10588478⋊C3 #13141942

Order 1728.

357 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 90 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#63509⋊C3C211yesmaximal, characteristic
2C2×#3674⋊C3C211yesmaximal, characteristic, complemented
3C2×#3664⋊C3C211yesmaximal, characteristic
4#63449⋊C3C211yesmaximal, characteristic, complemented
53−1+2×#103C211yesmaximal, characteristic, complemented
6(3−1+2⋊C4)⋊C8C211yesmaximal, characteristic, complemented
7#63369⋊C3C211yesmaximal, characteristic
8C3×#4798—31nomaximal
9#10588478C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C2×3−1+2⋊Q8C2211yescharacteristic
13C2×C8×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15#3674⋊C3—22no—
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17(C2×AGL(1,ℤ/9))⋊C4—21no—
18(3−1+2⋊C4)⋊C4—21no—
19C4×3−1+2⋊C4—21no—
20C4×3−1+2⋊C4C2211yescharacteristic
21C2×3−1+2⋊C8C2211yescharacteristic
22#3664⋊C3—22no—
23C3×#947—31no—
24C2×C3×#216—31no—
25C2×C3×#204—31no—
26C3×#887—31no—
27C32×#103D311yescharacteristic
28C3×Dic3⋊C8—31no—
29C3×#809—31no—
30#63509C2×C311yescharacteristic
31C2×#3674C2×C311yescharacteristic, complemented
32C2×#3664C2×C311yescharacteristic
33#63449C2×C311yescharacteristic, complemented
34C9×#103—21no—
35C9×#103C2×C311yescharacteristic, complemented
36Dic9⋊C8C2×C311yescharacteristic, complemented
37#63369C2×C311yescharacteristic
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)—21no—
493−1+2⋊Q8—21no—
503−1+2⋊Q8D412yes—
51C8×3−1+2—21no—
523−1+2⋊C8—21no—
53C3×#464—91no—
54#4798—31no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C3×D12—31no—
57C2×C3×Dic6—31no—
58C2×C32×C8C2×D311yescharacteristic
59C32×C4⋊C4C2×D311yescharacteristic
60C3×#216—62no—
61C3×(C2×D3)⋊C4—61no—
62C3×(C2×D3)⋊C4—61no—
63C3×Dic3⋊C4—61no—
64C3×C4×Dic3—31no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—31no—
67C3×#204—62no—
68C2×Q8×C9—21no—
69C2×Q8×C9C22×C311yescharacteristic
70C2×D36C22×C311yescharacteristic
71C2×Dic18C22×C311yescharacteristic
72C2×C8×C9—21no—
73C2×C8×C9C22×C311yescharacteristic
74C9×C4⋊C4—21no—
75C9×C4⋊C4C22×C311yescharacteristic
76#3674—22no—
77(C2×D9)⋊C4—21no—
78(C2×D9)⋊C4—21no—
79Dic9⋊C4—21no—
80C4×Dic9—21no—
81C4×Dic9C22×C311yescharacteristic
82C2×C9⋊C8C22×C311yescharacteristic
83#3664—22no—
84C22×3−1+2C2×D411yescharacteristic
85C2×AGL(1,ℤ/9)—41no—
86C2×AGL(1,ℤ/9)—41no—
87C2×AGL(1,ℤ/9)—41no—
88C4×3−1+2—41no—
89C4×3−1+2—41no—
90C4×3−1+2C2×D411yescharacteristic
91C4×3−1+2#4111yescharacteristic
923−1+2⋊C4—22no—
933−1+2⋊C4—21no—
943−1+2⋊C4—21no—
95C2×C3×Q16—91no—
96C2×C3×SD16—91no—
97C3×#124—91no—
98C3×C4×Q8—91no—
99#947—31no—
100C2×#216—31no—
101C2×#204—31no—
102#887—31no—
103C3×C4⋊C8—91no—
104C3×#103—31no—
105C3×#103C3×D311yescharacteristic
106C3×#102—91no—
107Dic3⋊C8—31no—
108#809—31no—
109C22×C3×D3—61no—
110C32×Q8—21no—
111C32×Q8—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3×Dic3—31no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—61no—
118C3×D12—61no—
119C3×D12—61no—
120C3×Dic6—32no—
121C3×Dic6—61no—
122C22×D9—21no—
123C32×C8—21no—
124C3×C3⋊C8—61no—
125Q8×C9—22no—
126Q8×C9—21no—
127Q8×C9—21no—
128Q8×C9—41no—
129C2×C4×C9—21no—
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9—41no—
133C2×C4×C9—41no—
134C2×C4×C9C23×C311yescharacteristic
135C2×Dic9—21no—
136C2×Dic9C3×D411yescharacteristic
137C2×Dic9C3×D411yescharacteristic
138D36—21no—
139D36—21no—
140Dic18—21no—
141Dic18C3×D412yes—
142C8×C9—21no—
143C8×C9—41no—
144C9⋊C8—21no—
145#464—91noSylow
146C2×3−1+2#12111yescharacteristic
147C2×3−1+2#13511yescharacteristic
148C2×3−1+2#13711yescharacteristic
149AGL(1,ℤ/9)—81no—
150C2×C3×Q8—31no—
151C2×C3×Q8—91no—
152C2×C3×Q8C2×C3×D311yescharacteristic
153C2×C3×D4—91no—
154C2×D12—31no—
155C2×Dic6—31no—
156C3×Q16—182no—
157C3×SD16—182no—
158C2×C3×C8—31no—
159C2×C3×C8—91no—
160C2×C3×C8C2×C3×D311yescharacteristic
161C3×C4⋊C4—31no—
162C3×C4⋊C4—181no—
163C3×C4⋊C4C2×C3×D311yescharacteristic
164C3×C22⋊C4—181no—
165C3×C22⋊C4—181no—
166C3×C42—91no—
167C3×C42—181no—
168#216—62no—
169(C2×D3)⋊C4—61no—
170(C2×D3)⋊C4—61no—
171Dic3⋊C4—61no—
172C4×Dic3—31no—
173C4×Dic3—61no—
174C2×C3⋊C8—31no—
175#204—62no—
176C22×C32D3×D411yescharacteristic
177C2×C3×D3—121no—
178C2×C3×D3—121no—
179C2×C3×D3—121no—
180C32×C4—41no—
181C32×C4—41no—
182C32×C4#23511yescharacteristic
183C32×C4D3×D411yescharacteristic
184C3×Dic3—62no—
185C3×Dic3—61no—
186C3×Dic3—61no—
187C22×C9—21no—
188C22×C9C2×C3×D411yescharacteristic
189C2×D9—41no—
190C2×D9—41no—
191C2×D9—41no—
192C4×C9—21no—
193C4×C9—21no—
194C4×C9—42no—
195C4×C9—41no—
196C4×C9—41no—
197C4×C9—81no—
198C4×C9C2×C3×D411yescharacteristic
199C4×C9C3×#4111yescharacteristic
200Dic9—22no—
201Dic9—21no—
202Dic9—21no—
203C2×Q16—91no—
204C2×SD16—91no—
205#124—91no—
206C4×Q8—91no—
207C4⋊C8—91no—
208#103AGL(1,ℤ/9)11yescharacteristic, complemented
209#102—91no—
2103−1+2#46411yescharacteristic, complemented, Sylow
211C23×C3—181no—
212C22×D3—61no—
213C3×Q8—21no—
214C3×Q8—21no—
215C3×Q8—61no—
216C3×Q8—61no—
217C3×Q8—92no—
218C3×Q8—181no—
219C3×D4—181no—
220C3×D4—181no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—31no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—91no—
227C2×C3×C4—91no—
228C2×C3×C4—181no—
229C2×C3×C4C22×C3×D311yescharacteristic
230C2×Dic3—31no—
231C2×Dic3—31no—
232C2×Dic3—61no—
233D12—61no—
234D12—61no—
235Dic6—32no—
236Dic6—61no—
237C3×C8—21no—
238C3×C8—61no—
239C3×C8—181no—
240C3⋊C8—61no—
241C2×C32#88411yescharacteristic
242C2×C32#91611yescharacteristic
243C2×C32#91811yescharacteristic
244C3×D3—241no—
245C2×C9—21no—
246C2×C9—21no—
247C2×C9—21no—
248C2×C9C3×#12111yescharacteristic
249C2×C9C3×#13511yescharacteristic
250C2×C9C3×#13711yescharacteristic
251D9—81no—
252C2×Q8—91no—
253C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
254C2×D4—91no—
255Q16—182no—
256SD16—182no—
257C2×C8—91no—
258C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
259C4⋊C4—181no—
260C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
261C22⋊C4—181no—
262C22⋊C4—181no—
263C42—91no—
264C42—181no—
265C22×C3—31no—
266C22×C3—361no—
267C22×C3—361no—
268C22×C3—361no—
269C22×C3C3×D3×D411yescharacteristic
270C2×D3—121no—
271C2×D3—121no—
272C2×D3—121no—
273C3×C4—31no—
274C3×C4—31no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—121no—
278C3×C4—121no—
279C3×C4—181no—
280C3×C4—181no—
281C3×C4—182no—
282C3×C4C3×#23511yescharacteristic
283C3×C4C3×D3×D411yescharacteristic
284Dic3—62no—
285Dic3—61no—
286Dic3—61no—
287C32#479811yescharacteristic
288C9—21no—
289C9C3×#46411yescharacteristic, complemented
290C23—181no—
291Q8—21no—
292Q8—21no—
293Q8—92no—
294Q8—181no—
295D4—181no—
296D4—181no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—181no—
302C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
303C8—21no—
304C8—181no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—721no—
309C2×C3C3×#88411yescharacteristic
310C2×C3C3×#91611yescharacteristic
311C2×C3C3×#91811yescharacteristic
312D3—241no—
313C22—361no—
314C22—361no—
315C22—361no—
316C22D4×AGL(1,ℤ/9)11yescharacteristic
317C4—41no—
318C4—41no—
319C4—181no—
320C4—181no—
321C4—182no—
322C4#3697⋊C311yescharacteristic
323C4D4×AGL(1,ℤ/9)11yescharacteristic
324C3—31no—
325C3C3×#479811yescharacteristic
326C2—721no—
327C2#63446⋊C311yescharacteristic
328C2#63478⋊C311yescharacteristic
329C2#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×#464
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
2. C2×#3674⋊C3C2semidirect
4. #63449⋊C3C2semidirect
5. 3−1+2×#103C2semidirect
6. (3−1+2⋊C4)⋊C8C2semidirect
9. #10588478C3semidirect
31. C2×#3674C2×C3semidirect
33. #63449C2×C3semidirect
35. C9×#103C2×C3semidirect
36. Dic9⋊C8C2×C3semidirect
208. #103AGL(1,ℤ/9)semidirect
210. 3−1+2#464semidirect
289. C9C3×#464semidirect

Maximal subgroups

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

subgroupindex
#63509⋊C32
C2×#3674⋊C32
C2×#3664⋊C32
#63449⋊C32
3−1+2×#1032
(3−1+2⋊C4)⋊C82
#63369⋊C32
C3×#47983
#105884783