Finite Group Databasebeta

← #10588434⋊C3 #13141707

Subgroups — #10588434⋊C3 #13141707

Order 1728.

331 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 80 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#63505⋊C3C211yesmaximal, characteristic, complemented
2C2×#3675⋊C3C211yesmaximal, characteristic, complemented
3#63470⋊C3C211yesmaximal, characteristic
4#63469⋊C3C211yesmaximal, characteristic, complemented
53−1+2×#101C211yesmaximal, characteristic, complemented
6#63383⋊C3C211yesmaximal, characteristic, complemented
7#63381⋊C3C211yesmaximal, characteristic, complemented
8C3×#4754—31nomaximal
9#10588434C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11#3697⋊C3C2211yescharacteristic
12C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
13M16×3−1+2C2211yescharacteristic
14M16×3−1+2C2211yescharacteristic
15#3675⋊C3—21no—
16#3675⋊C3—21no—
17#3675⋊C3—21no—
18#3674⋊C3—21no—
19#3667⋊C3C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21D72⋊C3—21no—
22#3664⋊C3—21no—
23#3663⋊C3—21no—
24C8×AGL(1,ℤ/9)—21no—
25C3×#942—31no—
26C2×C3×#215—31no—
27C3×#908—31no—
28C3×#907—31no—
29C32×#101D311yescharacteristic
30C3×#823—31no—
31C3×#822—31no—
32#63505C2×C311yescharacteristic, complemented
33C2×#3675C2×C311yescharacteristic, complemented
34#63470C2×C311yescharacteristic
35#63469C2×C311yescharacteristic, complemented
36C9×#101—21no—
37C9×#101C2×C311yescharacteristic, complemented
38#63383C2×C311yescharacteristic, complemented
39#63381C2×C311yescharacteristic, complemented
40C22×AGL(1,ℤ/9)—21no—
41Q8×3−1+2—21no—
42Q8×3−1+2—21no—
43C2×C4×3−1+2—21no—
44C2×C4×3−1+2C2311yescharacteristic
45#611⋊C3—21no—
46C4⋊AGL(1,ℤ/9)—21no—
47C4⋊AGL(1,ℤ/9)—21no—
48C4⋊AGL(1,ℤ/9)D411yescharacteristic
49C4×AGL(1,ℤ/9)—21no—
503−1+2⋊Q8D411yescharacteristic
51C8×3−1+2—21no—
52C8×3−1+2—21no—
533−1+2⋊C8—21no—
543−1+2⋊C8D411yescharacteristic
553−1+2⋊C8D411yescharacteristic
56C3×#471—91no—
57#4754—31no—
58C2×C32×Q8C2×D311yescharacteristic
59C3×#235—31no—
60C2×C3×D12—31no—
61C32×M16C2×D311yescharacteristic
62C32×M16C2×D311yescharacteristic
63C3×#216—61no—
64C3×#215—61no—
65C3×#215—61no—
66C3×#215—61no—
67C3×#208—31no—
68C2×C3×C3⋊C8—31no—
69C3×D24—61no—
70C3×#204—61no—
71C3×ΓL(1,𝔽25)—61no—
72C3×D3×C8—61no—
73C2×Q8×C9—21no—
74C2×Q8×C9C22×C311yescharacteristic
75#3697C22×C311yescharacteristic
76C2×D36C22×C311yescharacteristic
77C9×M16—21no—
78C9×M16—21no—
79C9×M16C22×C311yescharacteristic
80C9×M16C22×C311yescharacteristic
81#3675—21no—
82#3675—21no—
83#3675—21no—
84#3674—21no—
85#3667C22×C311yescharacteristic
86C2×C9⋊C8C22×C311yescharacteristic
87D72—21no—
88#3664—21no—
89#3663—21no—
90C8×D9—21no—
91C22×3−1+2#4111yescharacteristic
92C2×AGL(1,ℤ/9)—21no—
93C2×AGL(1,ℤ/9)—41no—
94C2×AGL(1,ℤ/9)—81no—
95C4×3−1+2—41no—
96C4×3−1+2C2×D411yescharacteristic
97C4×3−1+2C2×D411yescharacteristic
983−1+2⋊C4—21no—
99C3×#137—91no—
100C3×AGL(1,ℤ/8)—91no—
101C2×C3×SD16—91no—
102C3×#131—91no—
103#942—31no—
104C2×#215—31no—
105#908—31no—
106#907—31no—
107C3×#108—91no—
108C3×#101—31no—
109C3×#101C3×D311yescharacteristic
110C3×#100—91no—
111#823—31no—
112#822—31no—
113C22×C3×D3—61no—
114C32×Q8—21no—
115C32×Q8—21no—
116C2×C32×C4—21no—
117C2×C32×C4C22×D311yescharacteristic
118C3×#67—61no—
119C3×D12—31no—
120C3×D12—61no—
121C3×D12—61no—
122C3×C4×D3—61no—
123C3×Dic6—31no—
124C22×D9—21no—
125C32×C8—21no—
126C32×C8—21no—
127C3×C3⋊C8—31no—
128C3×C3⋊C8—31no—
129C3×C3⋊C8—61no—
130Q8×C9—21no—
131Q8×C9—21no—
132Q8×C9—41no—
133Q8×C9—41no—
134C2×C4×C9—22no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9C23×C311yescharacteristic
138#611—21no—
139D36—21no—
140D36—21no—
141D36C3×D411yescharacteristic
142C4×D9—21no—
143Dic18C3×D411yescharacteristic
144C8×C9—21no—
145C8×C9—21no—
146C8×C9—41no—
147C8×C9—41no—
148C9⋊C8—21no—
149C9⋊C8C3×D411yescharacteristic
150C9⋊C8C3×D411yescharacteristic
151#471—91noSylow
152C2×3−1+2—21no—
153C2×3−1+2#12111yescharacteristic
154AGL(1,ℤ/9)—41no—
155AGL(1,ℤ/9)—81no—
156C3×#41—91no—
157C2×C3×Q8—31no—
158C2×C3×Q8C2×C3×D311yescharacteristic
159C2×C3×D4—91no—
160#235—31no—
161C2×D12—31no—
162C3×Q16—181no—
163C3×SD16—181no—
164C3×SD16—181no—
165C3×SD16—181no—
166C3×SD16—181no—
167C3×D8—181no—
168C3×M16—31no—
169C3×M16—31no—
170C3×M16—91no—
171C3×M16—181no—
172C3×M16C2×C3×D311yescharacteristic
173C3×M16C2×C3×D311yescharacteristic
174C2×C3×C8—91no—
175C2×C3×C8—181no—
176#216—61no—
177#215—61no—
178#215—61no—
179#215—61no—
180#208—31no—
181C2×C3⋊C8—31no—
182D24—61no—
183#204—61no—
184ΓL(1,𝔽25)—61no—
185D3×C8—61no—
186C22×C32#23511yescharacteristic
187C2×C3×D3—61no—
188C2×C3×D3—121no—
189C2×C3×D3—241no—
190C32×C4—41no—
191C32×C4D3×D411yescharacteristic
192C32×C4D3×D411yescharacteristic
193C3×Dic3—61no—
194C22×C9—21no—
195C22×C9C3×#4111yescharacteristic
196C2×D9—21no—
197C2×D9—41no—
198C2×D9—81no—
199C4×C9—21no—
200C4×C9—21no—
201C4×C9—42no—
202C4×C9—41no—
203C4×C9C2×C3×D411yescharacteristic
204C4×C9C2×C3×D411yescharacteristic
205Dic9—21no—
206#137—91no—
207AGL(1,ℤ/8)—91no—
208C2×SD16—91no—
209#131—91no—
210#108—91no—
211#101AGL(1,ℤ/9)11yescharacteristic, complemented
212#100—91no—
2133−1+2#47111yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—61no—
216C3×Q8—21no—
217C3×Q8—21no—
218C3×Q8—61no—
219C3×Q8—61no—
220C3×Q8—91no—
221C3×D4—91no—
222C3×D4—181no—
223C3×D4—181no—
224C3×D4—181no—
225C2×C3×C4—21no—
226C2×C3×C4—31no—
227C2×C3×C4—61no—
228C2×C3×C4—181no—
229C2×C3×C4C22×C3×D311yescharacteristic
230#67—61no—
231D12—31no—
232D12—61no—
233D12—61no—
234C4×D3—61no—
235Dic6—31no—
236C3×C8—21no—
237C3×C8—21no—
238C3×C8—61no—
239C3×C8—61no—
240C3×C8—91no—
241C3×C8—91no—
242C3×C8—181no—
243C3⋊C8—31no—
244C3⋊C8—31no—
245C3⋊C8—61no—
246C2×C32—21no—
247C2×C32#88411yescharacteristic
248C3×D3—121no—
249C3×D3—241no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—41no—
253C2×C9C3×#12111yescharacteristic
254D9—41no—
255D9—81no—
256#41—91no—
257C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
258C2×D4—91no—
259Q16—181no—
260SD16—181no—
261SD16—181no—
262SD16—181no—
263SD16—181no—
264D8—181no—
265M16—91no—
266M16—181no—
267M16C2×AGL(1,ℤ/9)11yescharacteristic
268M16C2×AGL(1,ℤ/9)11yescharacteristic
269C2×C8—91no—
270C2×C8—181no—
271C22×C3—31no—
272C22×C3—181no—
273C22×C3—361no—
274C22×C3—721no—
275C22×C3C3×#23511yescharacteristic
276C2×D3—61no—
277C2×D3—121no—
278C2×D3—241no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—41no—
282C3×C4—121no—
283C3×C4—181no—
284C3×C4C3×D3×D411yescharacteristic
285C3×C4C3×D3×D411yescharacteristic
286Dic3—61no—
287C32#475411yescharacteristic
288C9—21no—
289C9C3×#47111yescharacteristic, complemented
290C23—181no—
291Q8—21no—
292Q8—21no—
293Q8—91no—
294D4—91no—
295D4—181no—
296D4—181no—
297D4—181no—
298C2×C4—21no—
299C2×C4—181no—
300C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
301C8—21no—
302C8—21no—
303C8—91no—
304C8—91no—
305C8—181no—
306C2×C3—21no—
307C2×C3—31no—
308C2×C3—61no—
309C2×C3—361no—
310C2×C3—721no—
311C2×C3C3×#88411yescharacteristic
312D3—121no—
313D3—241no—
314C22—181no—
315C22—361no—
316C22—721no—
317C22#3697⋊C311yescharacteristic
318C4—41no—
319C4—181no—
320C4D4×AGL(1,ℤ/9)11yescharacteristic
321C4D4×AGL(1,ℤ/9)11yescharacteristic
322C3—31no—
323C3C3×#475411yescharacteristic
324C2—21no—
325C2—361no—
326C2—721no—
327C2#63446⋊C311yescharacteristic
subgroupG/H
centerC2#63446⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#101C2
socleC2×C3C3×#884
prime radical#63470⋊C3C2
hypercenter#101AGL(1,ℤ/9)
hypocenterC9C3×#471
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
1. #63505⋊C3C2semidirect
2. C2×#3675⋊C3C2semidirect
4. #63469⋊C3C2semidirect
5. 3−1+2×#101C2semidirect
6. #63383⋊C3C2semidirect
7. #63381⋊C3C2semidirect
9. #10588434C3semidirect
32. #63505C2×C3semidirect
33. C2×#3675C2×C3semidirect
35. #63469C2×C3semidirect
37. C9×#101C2×C3semidirect
38. #63383C2×C3semidirect
39. #63381C2×C3semidirect
211. #101AGL(1,ℤ/9)semidirect
213. 3−1+2#471semidirect
289. C9C3×#471semidirect

Maximal subgroups

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

subgroupindex
#63505⋊C32
C2×#3675⋊C32
#63470⋊C32
#63469⋊C32
3−1+2×#1012
#63383⋊C32
#63381⋊C32
C3×#47543
#105884343