Finite Group Databasebeta

← #10588671⋊C3 #13142662

Subgroups — #10588671⋊C3 #13142662

Order 1728.

760 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic
2C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
3C2×(C2×AGL(1,ℤ/9))⋊C4C212yesmaximal, complemented
4C2×(C2×AGL(1,ℤ/9))⋊C4C212yesmaximal, complemented
53−1+2⋊#95C211yesmaximal, characteristic, complemented
6C3×#4991—31nomaximal
7#10588671C311yesmaximal, characteristic, complemented
8C23×AGL(1,ℤ/9)C2212yes—
9C22×C4×3−1+2C2211yescharacteristic
10C22×C4×3−1+2C2211yescharacteristic
11C22×C4×3−1+2C2211yescharacteristic
12C22×3−1+2⋊C4C2212yes—
13C2×C4⋊AGL(1,ℤ/9)—24no—
14C2×C4⋊AGL(1,ℤ/9)—22no—
15C4⋊C4×3−1+2—22no—
16(C2×AGL(1,ℤ/9))⋊C4—24no—
17(C2×AGL(1,ℤ/9))⋊C4—24no—
18C22×C3×D12—31no—
19C2×C32×C4⋊C4D311yescharacteristic
20C2×C3×(C2×D3)⋊C4—32no—
21C2×C3×(C2×D3)⋊C4—32no—
22C22×D36C2×C311yescharacteristic
23C3×C3⋊#95—31no—
24C2×C9×C4⋊C4—21no—
25C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
26C2×(C2×D9)⋊C4C2×C312yescomplemented
27C2×(C2×D9)⋊C4C2×C312yescomplemented
28C9⋊#95C2×C311yescharacteristic, complemented
29C23×3−1+2C2311yescharacteristic
30C22×AGL(1,ℤ/9)—22no—
31C22×AGL(1,ℤ/9)—22no—
32C22×AGL(1,ℤ/9)—22no—
33C22×AGL(1,ℤ/9)—22no—
34C22×AGL(1,ℤ/9)—22no—
35C22×AGL(1,ℤ/9)—22no—
36C22×AGL(1,ℤ/9)D414yes—
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2—22no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2—21no—
41C2×C4×3−1+2—22no—
42C2×C4×3−1+2D412yes—
43C2×C4×3−1+2D412yescomplemented
44C2×3−1+2⋊C4—22no—
45C2×3−1+2⋊C4—22no—
46C2×3−1+2⋊C4—22no—
47C4⋊AGL(1,ℤ/9)—28no—
48C3×#394—91no—
49#4991—31no—
50C23×C3×D3—32no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C2×D311yescharacteristic
54C22×C3×Dic3—32no—
55C2×C3×D12—64no—
56C2×C3×D12—62no—
57C23×D9C22×C312yes—
58C32×C4⋊C4—22no—
59C3×(C2×D3)⋊C4—64no—
60C3×(C2×D3)⋊C4—64no—
61C22×C4×C9—21no—
62C22×C4×C9—21no—
63C22×C4×C9—21no—
64C22×C4×C9C22×C311yescharacteristic
65C22×C4×C9C22×C311yescharacteristic
66C22×C4×C9C22×C311yescharacteristic
67C22×Dic9C22×C312yes—
68C2×D36—24no—
69C2×D36—22no—
70C9×C4⋊C4—24no—
71C9×C4⋊C4—22no—
72(C2×D9)⋊C4—24no—
73(C2×D9)⋊C4—24no—
74C22×3−1+2C2×D412yes—
75C22×3−1+2C2×D411yescharacteristic
76C22×3−1+2C2×D411yescharacteristic
77C22×3−1+2#4111yescharacteristic
78C22×3−1+2#4112yes—
79C2×AGL(1,ℤ/9)—24no—
80C2×AGL(1,ℤ/9)—24no—
81C2×AGL(1,ℤ/9)—24no—
82C2×AGL(1,ℤ/9)—42no—
83C2×AGL(1,ℤ/9)—42no—
84C2×AGL(1,ℤ/9)—42no—
85C2×AGL(1,ℤ/9)—42no—
86C4×3−1+2—22no—
87C4×3−1+2—22no—
88C4×3−1+2—41no—
893−1+2⋊C4—42no—
90C22×C3×D4—91no—
91C22×D12—31no—
92C2×C3×C4⋊C4—31no—
93C2×C3×C4⋊C4C3×D311yescharacteristic
94C2×C3×C22⋊C4—92no—
95C2×C3×C22⋊C4—92no—
96C2×(C2×D3)⋊C4—32no—
97C2×(C2×D3)⋊C4—32no—
98C3×#95—91no—
99C3⋊#95—31no—
100C23×C32C22×D311yescharacteristic
101C22×C3×D3—34no—
102C22×C3×D3—62no—
103C22×C3×D3—62no—
104C22×C3×D3—62no—
105C22×C3×D3—62no—
106C22×C3×D3—62no—
107C22×C3×D3—62no—
108C2×C32×C4—21no—
109C2×C32×C4—22no—
110C2×C32×C4—21no—
111C2×C32×C4—22no—
112C2×C32×C4—21no—
113C2×C32×C4D1212yes—
114C2×C32×C4#6712yes—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C3×D12—68no—
119C23×C9—21no—
120C23×C9C23×C311yescharacteristic
121C22×D9—22no—
122C22×D9—22no—
123C22×D9—22no—
124C22×D9—22no—
125C22×D9—22no—
126C22×D9—22no—
127C22×D9C3×D414yes—
128C2×C4×C9—22no—
129C2×C4×C9—22no—
130C2×C4×C9—21no—
131C2×C4×C9—22no—
132C2×C4×C9—21no—
133C2×C4×C9—22no—
134C2×C4×C9—22no—
135C2×C4×C9—21no—
136C2×C4×C9—22no—
137C2×C4×C9—22no—
138C2×C4×C9—42no—
139C2×C4×C9—42no—
140C2×C4×C9C3×D412yes—
141C2×C4×C9C3×D412yescomplemented
142C2×Dic9—22no—
143C2×Dic9—22no—
144C2×Dic9—22no—
145D36—28no—
146#394—91noSylow
147C2×3−1+2#12011yescharacteristic
148C2×3−1+2#12112yes—
149C2×3−1+2#12111yescharacteristic
150C2×3−1+2#12312yes—
151C2×3−1+2#12411yescharacteristic
152AGL(1,ℤ/9)—44no—
153C24×C3—92no—
154C23×D3—32no—
155C2×C3×D4—182no—
156C2×C3×D4—184no—
157C22×C3×C4—31no—
158C22×C3×C4—31no—
159C22×C3×C4—31no—
160C22×C3×C4—92no—
161C22×C3×C4C2×C3×D311yescharacteristic
162C22×C3×C4C2×C3×D311yescharacteristic
163C22×C3×C4C2×C3×D311yescharacteristic
164C22×Dic3—32no—
165C2×D12—64no—
166C2×D12—62no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—62no—
169C3×C22⋊C4—184no—
170C3×C22⋊C4—184no—
171(C2×D3)⋊C4—64no—
172(C2×D3)⋊C4—64no—
173C22×C32C2×D1211yescharacteristic
174C22×C32#23511yescharacteristic
175C22×C32D3×D412yes—
176C22×C32#23912yes—
177C22×C32C2×#6711yescharacteristic
178C2×C3×D3—64no—
179C2×C3×D3—64no—
180C2×C3×D3—64no—
181C2×C3×D3—122no—
182C2×C3×D3—122no—
183C2×C3×D3—122no—
184C2×C3×D3—122no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—41no—
188C3×Dic3—122no—
189C22×C9—21no—
190C22×C9—21no—
191C22×C9—22no—
192C22×C9—22no—
193C22×C9—21no—
194C22×C9C2×C3×D411yescharacteristic
195C22×C9C2×C3×D412yes—
196C22×C9C2×C3×D411yescharacteristic
197C22×C9C3×#4111yescharacteristic
198C22×C9C3×#4112yes—
199C2×D9—24no—
200C2×D9—24no—
201C2×D9—24no—
202C2×D9—42no—
203C2×D9—42no—
204C2×D9—42no—
205C2×D9—42no—
206C4×C9—24no—
207C4×C9—22no—
208C4×C9—22no—
209C4×C9—42no—
210C4×C9—42no—
211C4×C9—41no—
212Dic9—42no—
213C22×D4—91no—
214C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
215C2×C22⋊C4—92no—
216C2×C22⋊C4—92no—
217#95—91no—
2183−1+2#39411yescharacteristic, complemented, Sylow
219C23×C3—31no—
220C23×C3—94no—
221C23×C3—182no—
222C23×C3—182no—
223C23×C3—182no—
224C23×C3—182no—
225C23×C3—182no—
226C23×C3—182no—
227C23×C3C22×C3×D311yescharacteristic
228C22×D3—34no—
229C22×D3—62no—
230C22×D3—62no—
231C22×D3—62no—
232C22×D3—62no—
233C22×D3—62no—
234C22×D3—62no—
235C3×D4—188no—
236C2×C3×C4—21no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—62no—
246C2×C3×C4—62no—
247C2×C3×C4—61no—
248C2×C3×C4—182no—
249C2×C3×C4—182no—
250C2×C3×C4—182no—
251C2×C3×C4C3×D1212yes—
252C2×C3×C4C3×#6712yes—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256D12—68no—
257C2×C32#89412yes—
258C2×C32#89512yes—
259C2×C32#93011yescharacteristic
260C2×C32#93711yescharacteristic
261C2×C32#94711yescharacteristic
262C3×D3—124no—
263C2×C9—22no—
264C2×C9—22no—
265C2×C9—21no—
266C2×C9—21no—
267C2×C9—21no—
268C2×C9C3×#12011yescharacteristic
269C2×C9C3×#12112yes—
270C2×C9C3×#12111yescharacteristic
271C2×C9C3×#12312yes—
272C2×C9C3×#12411yescharacteristic
273D9—44no—
274C24—92no—
275C2×D4—184no—
276C2×D4—182no—
277C22×C4—92no—
278C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
279C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
280C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
281C4⋊C4—22no—
282C22⋊C4—184no—
283C22⋊C4—184no—
284C22×C3—31no—
285C22×C3—31no—
286C22×C3—32no—
287C22×C3—32no—
288C22×C3—31no—
289C22×C3—184no—
290C22×C3—184no—
291C22×C3—184no—
292C22×C3—362no—
293C22×C3—362no—
294C22×C3—362no—
295C22×C3—362no—
296C22×C3C2×C3×D1211yescharacteristic
297C22×C3C3×#23511yescharacteristic
298C22×C3C3×D3×D412yes—
299C22×C3C3×#23912yes—
300C22×C3C2×C3×#6711yescharacteristic
301C2×D3—64no—
302C2×D3—64no—
303C2×D3—64no—
304C2×D3—122no—
305C2×D3—122no—
306C2×D3—122no—
307C2×D3—122no—
308C3×C4—22no—
309C3×C4—22no—
310C3×C4—41no—
311C3×C4—62no—
312C3×C4—62no—
313C3×C4—121no—
314C3×C4—362no—
315Dic3—122no—
316C32#499111yescharacteristic
317C9—21no—
318C9C3×#39411yescharacteristic, complemented
319C23—94no—
320C23—182no—
321C23—182no—
322C23—182no—
323C23—182no—
324C23—182no—
325C23—182no—
326C23C22×AGL(1,ℤ/9)11yescharacteristic
327D4—188no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—182no—
334C2×C4—182no—
335C2×C4—182no—
336C2×C4C4⋊AGL(1,ℤ/9)12yescomplemented
337C2×C4#611⋊C312yes—
338C2×C3—32no—
339C2×C3—32no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—364no—
344C2×C3C3×#89412yes—
345C2×C3C3×#89512yes—
346C2×C3C3×#93011yescharacteristic
347C2×C3C3×#93711yescharacteristic
348C2×C3C3×#94711yescharacteristic
349D3—124no—
350C22—184no—
351C22—184no—
352C22—184no—
353C22—362no—
354C22—362no—
355C22—362no—
356C22—362no—
357C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
358C22#3697⋊C311yescharacteristic
359C22D4×AGL(1,ℤ/9)12yes—
360C22#8872412yes—
361C22C2×#611⋊C311yescharacteristic
362C4—22no—
363C4—22no—
364C4—41no—
365C4—362no—
366C3—31no—
367C3C3×#499111yescharacteristic
368C2—364no—
369C2#63456⋊C312yes—
370C2#63457⋊C312yes—
371C2#63492⋊C311yescharacteristic
372C2#63499⋊C311yescharacteristic
373C2#63509⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×3−1+2C2
prime radicalC22×C4⋊AGL(1,ℤ/9)C2
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#394
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×3−1+2C2semidirect
3. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
4. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
5. 3−1+2⋊#95C2semidirect
7. #10588671C3semidirect
25. C2×C9×C4⋊C4C2×C3semidirect
26. C2×(C2×D9)⋊C4C2×C3semidirect
27. C2×(C2×D9)⋊C4C2×C3semidirect
28. C9⋊#95C2×C3semidirect
43. C2×C4×3−1+2D4semidirect
141. C2×C4×C9C3×D4semidirect
214. C2×C4⋊C4AGL(1,ℤ/9)semidirect
218. 3−1+2#394semidirect
318. C9C3×#394semidirect
336. C2×C4C4⋊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
C22×C4⋊AGL(1,ℤ/9)2
C2×C4⋊C4×3−1+22
C2×(C2×AGL(1,ℤ/9))⋊C42
3−1+2⋊#952
C3×#49913
#105886713