Finite Group Databasebeta

← #10588569⋊C3 #13142250

Subgroups — #10588569⋊C3 #13142250

Order 1728.

378 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C2×#3664⋊C3C211yesmaximal, characteristic
2#63455⋊C3C211yesmaximal, characteristic
3#63449⋊C3C211yesmaximal, characteristic, complemented
43−1+2×#107C211yesmaximal, characteristic, complemented
5#63375⋊C3C211yesmaximal, characteristic, complemented
6#63369⋊C3C211yesmaximal, characteristic
7#63368⋊C3C211yesmaximal, characteristic, complemented
8C3×#4889—31nomaximal
9#10588569C311yesmaximal, characteristic, complemented
10C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
11C2×C4×AGL(1,ℤ/9)—21no—
12C2×3−1+2⋊Q8C2211yescharacteristic
13C2×C8×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15C4⋊C4×3−1+2C2211yescharacteristic
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17(3−1+2⋊C4)⋊C4—21no—
18C4×3−1+2⋊C4—21no—
19C4×3−1+2⋊C4C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21#3664⋊C3C414yescomplemented
22C2×C3×#204—31no—
23C3×#893—31no—
24C3×#887—31no—
25C32×#107D311yescharacteristic
26C3×#815—31no—
27C3×#810—31no—
28C3×#809—31no—
29C2×#3664C2×C311yescharacteristic
30#63455C2×C311yescharacteristic
31#63449C2×C311yescharacteristic, complemented
32C9×#107—21no—
33C9×#107C2×C311yescharacteristic, complemented
34#63375C2×C311yescharacteristic, complemented
35#63369C2×C311yescharacteristic
36#63368C2×C311yescharacteristic, complemented
37C22×AGL(1,ℤ/9)—21no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2C2311yescharacteristic
41C2×3−1+2⋊C4—21no—
42C2×3−1+2⋊C4D411yescharacteristic
43C2×3−1+2⋊C4D411yescharacteristic
44C4⋊AGL(1,ℤ/9)—21no—
45C4⋊AGL(1,ℤ/9)C2×C412yes—
46C4×AGL(1,ℤ/9)—42no—
473−1+2⋊Q8—21no—
483−1+2⋊Q8C2×C412yescomplemented
49C8×3−1+2C2×C412yescomplemented
503−1+2⋊C8—21no—
51C3×#440—91no—
52#4889—31no—
53C2×C3×D12—31no—
54C2×C3×C4×D3—61no—
55C2×C3×Dic6—31no—
56C2×C32×C8C2×D311yescharacteristic
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59C3×(C2×D3)⋊C4—61no—
60C3×Dic3⋊C4—61no—
61C3×C4×Dic3—31no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—31no—
64C3×#204—34no—
65C2×D36C22×C311yescharacteristic
66C2×C4×D9—21no—
67C2×Dic18C22×C311yescharacteristic
68C2×C8×C9—21no—
69C2×C8×C9C22×C311yescharacteristic
70C9×C4⋊C4—21no—
71C9×C4⋊C4—21no—
72C9×C4⋊C4C22×C311yescharacteristic
73C9×C4⋊C4C22×C311yescharacteristic
74(C2×D9)⋊C4—21no—
75Dic9⋊C4—21no—
76C4×Dic9—21no—
77C4×Dic9C22×C311yescharacteristic
78C2×C9⋊C8C22×C311yescharacteristic
79#3664C3×C414yescomplemented
80C22×3−1+2C2×D411yescharacteristic
81C2×AGL(1,ℤ/9)—22no—
82C2×AGL(1,ℤ/9)—41no—
83C2×AGL(1,ℤ/9)—41no—
84C4×3−1+2—41no—
85C4×3−1+2—41no—
86C4×3−1+2C22×C411yescharacteristic
87C4×3−1+2#4111yescharacteristic
883−1+2⋊C4—22no—
893−1+2⋊C4—21no—
903−1+2⋊C4—21no—
91C2×C3×SD16—91no—
92C3×C4×Q8—91no—
93C3×C4×D4—91no—
94C2×#204—31no—
95#893—31no—
96#887—31no—
97C3×#107—31no—
98C3×#107C3×D311yescharacteristic
99C3×#103—91no—
100C3×#102—91no—
101C3×#97—91no—
102#815—31no—
103#810—31no—
104#809—31no—
105C22×C3×D3—61no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3×Dic3—31no—
110C2×C3×Dic3—31no—
111C2×C3×Dic3—61no—
112C3×D12—32no—
113C3×D12—61no—
114C3×C4×D3—122no—
115C3×Dic6—32no—
116C3×Dic6—61no—
117C22×D9—21no—
118C32×C8C4×D312yes—
119C3×C3⋊C8—61no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—41no—
124C2×C4×C9—41no—
125C2×C4×C9C23×C311yescharacteristic
126C2×Dic9—21no—
127C2×Dic9C3×D411yescharacteristic
128C2×Dic9C3×D411yescharacteristic
129D36—21no—
130D36C2×C3×C412yes—
131C4×D9—42no—
132Dic18—21no—
133Dic18C2×C3×C412yescomplemented
134C8×C9—22no—
135C8×C9C2×C3×C412yescomplemented
136C9⋊C8—21no—
137#440—91noSylow
138C2×3−1+2C4×D411yescharacteristic
139C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
140C2×3−1+2#13711yescharacteristic
141AGL(1,ℤ/9)—42no—
142C2×C3×Q8—91no—
143C2×C3×D4—91no—
144C22×C3×C4—181no—
145C2×D12—31no—
146C2×C4×D3—61no—
147C2×Dic6—31no—
148C3×SD16—94no—
149C2×C3×C8—31no—
150C2×C3×C8—91no—
151C2×C3×C8C2×C3×D311yescharacteristic
152C3×C4⋊C4—31no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—181no—
155C3×C4⋊C4C2×C3×D311yescharacteristic
156C3×C4⋊C4C2×C3×D311yescharacteristic
157C3×C22⋊C4—181no—
158C3×C42—91no—
159C3×C42—181no—
160(C2×D3)⋊C4—61no—
161Dic3⋊C4—61no—
162C4×Dic3—31no—
163C4×Dic3—61no—
164C2×C3⋊C8—31no—
165#204—34no—
166C22×C32D3×D411yescharacteristic
167C2×C3×D3—62no—
168C2×C3×D3—121no—
169C2×C3×D3—121no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4C2×C4×D311yescharacteristic
173C32×C4#23911yescharacteristic
174C3×Dic3—62no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C22×C9—21no—
178C22×C9C2×C3×D411yescharacteristic
179C2×D9—22no—
180C2×D9—41no—
181C2×D9—41no—
182C4×C9—21no—
183C4×C9—21no—
184C4×C9—41no—
185C4×C9—41no—
186C4×C9—81no—
187C4×C9—81no—
188C4×C9C22×C3×C411yescharacteristic
189C4×C9C3×#4111yescharacteristic
190Dic9—22no—
191Dic9—21no—
192Dic9—21no—
193C2×SD16—91no—
194C4×Q8—91no—
195C4×D4—91no—
196#107AGL(1,ℤ/9)11yescharacteristic, complemented
197#103—91no—
198#102—91no—
199#97—91no—
2003−1+2#44011yescharacteristic, complemented, Sylow
201C23×C3—181no—
202C22×D3—61no—
203C3×Q8—92no—
204C3×Q8—181no—
205C3×D4—92no—
206C3×D4—181no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—31no—
210C2×C3×C4—61no—
211C2×C3×C4—61no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—181no—
215C2×C3×C4—362no—
216C2×C3×C4C22×C3×D311yescharacteristic
217C2×Dic3—31no—
218C2×Dic3—31no—
219C2×Dic3—61no—
220D12—32no—
221D12—61no—
222C4×D3—122no—
223Dic6—32no—
224Dic6—61no—
225C3×C8—32no—
226C3×C8—181no—
227C3×C8C3×C4×D312yes—
228C3⋊C8—61no—
229C2×C32#89311yescharacteristic
230C2×C32#91111yescharacteristic
231C2×C32#91811yescharacteristic
232C3×D3—122no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9—21no—
236C2×C9C3×C4×D411yescharacteristic
237C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
238C2×C9C3×#13711yescharacteristic
239D9—42no—
240C2×Q8—91no—
241C2×D4—91no—
242C22×C4—181no—
243SD16—94no—
244C2×C8—91no—
245C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
246C4⋊C4—181no—
247C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
248C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
249C22⋊C4—181no—
250C42—91no—
251C42—181no—
252C22×C3—31no—
253C22×C3—182no—
254C22×C3—361no—
255C22×C3—361no—
256C22×C3C3×D3×D411yescharacteristic
257C2×D3—62no—
258C2×D3—121no—
259C2×D3—121no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—121no—
265C3×C4—121no—
266C3×C4—181no—
267C3×C4—181no—
268C3×C4—182no—
269C3×C4C2×C3×C4×D311yescharacteristic
270C3×C4C3×#23911yescharacteristic
271Dic3—62no—
272Dic3—61no—
273Dic3—61no—
274C32#488911yescharacteristic
275C9—21no—
276C9C3×#44011yescharacteristic, complemented
277C23—181no—
278Q8—92no—
279Q8—181no—
280D4—92no—
281D4—181no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—181no—
287C2×C4—362no—
288C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
289C8—181no—
290C8C4×AGL(1,ℤ/9)12yescomplemented
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—362no—
295C2×C3C3×#89311yescharacteristic
296C2×C3C3×#91111yescharacteristic
297C2×C3C3×#91811yescharacteristic
298D3—122no—
299C22—182no—
300C22—361no—
301C22—361no—
302C22D4×AGL(1,ℤ/9)11yescharacteristic
303C4—41no—
304C4—41no—
305C4—182no—
306C4—181no—
307C4—181no—
308C4C2×C4×AGL(1,ℤ/9)11yescharacteristic
309C4#8872411yescharacteristic
310C3—31no—
311C3C3×#488911yescharacteristic
312C2—362no—
313C2#63455⋊C311yescharacteristic
314C2#63473⋊C311yescharacteristic
315C2#63480⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
prime radicalC2×C4⋊AGL(1,ℤ/9)C22
cosocleC2×C4×C9C23×C3
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C3×#440
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
3. #63449⋊C3C2semidirect
4. 3−1+2×#107C2semidirect
5. #63375⋊C3C2semidirect
7. #63368⋊C3C2semidirect
9. #10588569C3semidirect
21. #3664⋊C3C4semidirect
31. #63449C2×C3semidirect
33. C9×#107C2×C3semidirect
34. #63375C2×C3semidirect
36. #63368C2×C3semidirect
48. 3−1+2⋊Q8C2×C4semidirect
49. C8×3−1+2C2×C4semidirect
79. #3664C3×C4semidirect
133. Dic18C2×C3×C4semidirect
135. C8×C9C2×C3×C4semidirect
196. #107AGL(1,ℤ/9)semidirect
200. 3−1+2#440semidirect
276. C9C3×#440semidirect
290. C8C4×AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C2×#3664⋊C32
#63455⋊C32
#63449⋊C32
3−1+2×#1072
#63375⋊C32
#63369⋊C32
#63368⋊C32
C3×#48893
#105885693