Finite Group Databasebeta

← #10588626⋊C3 #13142516

Subgroups — #10588626⋊C3 #13142516

Order 1728.

768 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×3−1+2⋊C4C211yesmaximal, characteristic
2C2×C22⋊C4×3−1+2C211yesmaximal, characteristic
3C2×#3676⋊C3C211yesmaximal, characteristic
4C2×(3−1+2⋊C4)⋊C4C212yesmaximal, complemented
53−1+2⋊#95C212yesmaximal, complemented
6C3×#4946—31nomaximal
7#10588626C311yesmaximal, characteristic, complemented
8C24×3−1+2C2211yescharacteristic
9C22×C4×3−1+2C2211yescharacteristic
10C22×C4×3−1+2C2211yescharacteristic
11C22×3−1+2⋊C4—22no—
12C22×3−1+2⋊C4—24no—
13C22×3−1+2⋊C4C2212yes—
14C22×3−1+2⋊C4C2212yes—
15C22⋊C4×3−1+2—22no—
16#3676⋊C3C414yescomplemented
17(3−1+2⋊C4)⋊C4—24no—
18C23×C3×Dic3—31no—
19C2×C32×C22⋊C4D311yescharacteristic
20C2×C3×#217—31no—
21C2×C3×Dic3⋊C4—32no—
22C23×Dic9C2×C311yescharacteristic
23C3×C3⋊#95—32no—
24C2×C9×C22⋊C4—21no—
25C2×C9×C22⋊C4C2×C311yescharacteristic
26C2×#3676C2×C311yescharacteristic
27C2×Dic9⋊C4C2×C312yescomplemented
28C9⋊#95C2×C312yescomplemented
29C23×3−1+2—22no—
30C23×3−1+2—22no—
31C23×3−1+2C2×C412yes—
32C23×3−1+2D412yes—
33C23×3−1+2Q812yes—
34C23×3−1+2C2311yescharacteristic
35C2×C4×3−1+2—21no—
36C2×C4×3−1+2—21no—
37C2×C4×3−1+2—22no—
38C2×C4×3−1+2—22no—
39C2×3−1+2⋊C4—28no—
40C2×3−1+2⋊C4—22no—
41C2×3−1+2⋊C4—22no—
42C2×3−1+2⋊C4—22no—
43C2×3−1+2⋊C4—22no—
44C2×3−1+2⋊C4C2×C414yescomplemented
45C2×3−1+2⋊C4D414yes—
46C3×#385—91no—
47#4946—31no—
48C24×C32C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C22×C32×C4C2×D311yescharacteristic
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—64no—
54C22×C3×Dic3—62no—
55C24×C9—21no—
56C24×C9C22×C311yescharacteristic
57C32×C22⋊C4—22no—
58C3×#217—34no—
59C3×Dic3⋊C4—64no—
60C22×C4×C9—21no—
61C22×C4×C9—21no—
62C22×C4×C9C22×C311yescharacteristic
63C22×C4×C9C22×C311yescharacteristic
64C22×Dic9—22no—
65C22×Dic9—24no—
66C22×Dic9C22×C312yes—
67C22×Dic9C22×C312yes—
68C9×C22⋊C4—24no—
69C9×C22⋊C4—22no—
70#3676C3×C414yescomplemented
71Dic9⋊C4—24no—
72C22×3−1+2—24no—
73C22×3−1+2—22no—
74C22×3−1+2—24no—
75C22×3−1+2—22no—
76C22×3−1+2C4⋊C414yescomplemented
77C22×3−1+2C22×C411yescharacteristic
78C22×3−1+2C2×D411yescharacteristic
79C22×3−1+2C2×D412yes—
80C22×3−1+2C2×Q811yescharacteristic
81C22×3−1+2#4112yes—
82C4×3−1+2—41no—
83C4×3−1+2—41no—
843−1+2⋊C4—24no—
853−1+2⋊C4—24no—
86C23×C3×C4—91no—
87C23×Dic3—31no—
88C2×C3×C4⋊C4—92no—
89C2×C3×C22⋊C4—31no—
90C2×C3×C22⋊C4—91no—
91C2×C3×C22⋊C4C3×D311yescharacteristic
92C2×#217—31no—
93C2×Dic3⋊C4—32no—
94C3×#95—92no—
95C3⋊#95—32no—
96C23×C32—22no—
97C23×C32—22no—
98C23×C32Dic612yes—
99C23×C32C4×D312yes—
100C23×C32#6712yes—
101C23×C32C22×D311yescharacteristic
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—68no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C23×C9—22no—
114C23×C9—22no—
115C23×C9—22no—
116C23×C9—22no—
117C23×C9—22no—
118C23×C9—21no—
119C23×C9—42no—
120C23×C9—42no—
121C23×C9C2×C3×C412yes—
122C23×C9C3×D412yes—
123C23×C9C3×Q812yes—
124C23×C9C23×C311yescharacteristic
125C2×C4×C9—22no—
126C2×C4×C9—22no—
127C2×C4×C9—22no—
128C2×C4×C9—21no—
129C2×C4×C9—21no—
130C2×C4×C9—22no—
131C2×C4×C9—42no—
132C2×C4×C9—42no—
133C2×Dic9—28no—
134C2×Dic9—22no—
135C2×Dic9—22no—
136C2×Dic9—22no—
137C2×Dic9—22no—
138C2×Dic9C2×C3×C414yescomplemented
139C2×Dic9C3×D414yes—
140#385—91noSylow
141C2×3−1+2—24no—
142C2×3−1+2C2×C4⋊C411yescharacteristic
143C2×3−1+2C4×D412yes—
144C2×3−1+2#12011yescharacteristic
145C2×3−1+2C22⋊Q812yes—
146C2×3−1+2#12311yescharacteristic
147C24×C3—31no—
148C24×C3C2×C3×D311yescharacteristic
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—92no—
152C22×C3×C4—92no—
153C22×C3×C4—184no—
154C22×C3×C4—182no—
155C22×C3×C4C2×C3×D311yescharacteristic
156C22×C3×C4C2×C3×D311yescharacteristic
157C22×Dic3—32no—
158C22×Dic3—32no—
159C22×Dic3—64no—
160C22×Dic3—62no—
161C3×C4⋊C4—184no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—62no—
164C3×C22⋊C4—94no—
165#217—34no—
166Dic3⋊C4—64no—
167C22×C32—24no—
168C22×C32—24no—
169C22×C32—22no—
170C22×C32—22no—
171C22×C32Dic3⋊C414yes—
172C22×C32C2×Dic611yescharacteristic
173C22×C32C2×C4×D311yescharacteristic
174C22×C32D3×D412yes—
175C22×C32#23712yes—
176C22×C32C2×#6711yescharacteristic
177C32×C4—41no—
178C32×C4—41no—
179C3×Dic3—64no—
180C3×Dic3—64no—
181C22×C9—24no—
182C22×C9—22no—
183C22×C9—24no—
184C22×C9—22no—
185C22×C9—24no—
186C22×C9—21no—
187C22×C9—21no—
188C22×C9—22no—
189C22×C9—22no—
190C22×C9—21no—
191C22×C9—44no—
192C22×C9—44no—
193C22×C9—42no—
194C22×C9—42no—
195C22×C9C3×C4⋊C414yescomplemented
196C22×C9C22×C3×C411yescharacteristic
197C22×C9C2×C3×D412yes—
198C22×C9C2×C3×D411yescharacteristic
199C22×C9C2×C3×Q811yescharacteristic
200C22×C9C3×#4112yes—
201C4×C9—42no—
202C4×C9—42no—
203C4×C9—41no—
204C4×C9—41no—
205Dic9—24no—
206Dic9—24no—
207C23×C4—91no—
208C2×C4⋊C4—92no—
209C2×C22⋊C4—91no—
210C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic
211#95—92no—
2123−1+2#38511yescharacteristic, complemented, Sylow
213C23×C3—22no—
214C23×C3—22no—
215C23×C3—32no—
216C23×C3—32no—
217C23×C3—32no—
218C23×C3—31no—
219C23×C3—62no—
220C23×C3—62no—
221C23×C3C3×Dic612yes—
222C23×C3C3×C4×D312yes—
223C23×C3C3×#6712yes—
224C23×C3C22×C3×D311yescharacteristic
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—94no—
234C2×C3×C4—94no—
235C2×C3×C4—182no—
236C2×C3×C4—182no—
237C2×C3×C4—182no—
238C2×C3×C4—182no—
239C2×C3×C4—188no—
240C2×Dic3—34no—
241C2×Dic3—34no—
242C2×Dic3—68no—
243C2×Dic3—62no—
244C2×Dic3—62no—
245C2×Dic3—62no—
246C2×Dic3—62no—
247C2×C32—24no—
248C2×C32#87812yes—
249C2×C32#67⋊C412yes—
250C2×C32C2×Dic3⋊C411yescharacteristic
251C2×C32#93511yescharacteristic
252C2×C32#93711yescharacteristic
253C2×C9—24no—
254C2×C9—22no—
255C2×C9—22no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9—21no—
259C2×C9—44no—
260C2×C9C2×C3×C4⋊C411yescharacteristic
261C2×C9C3×C4×D412yes—
262C2×C9C3×#12011yescharacteristic
263C2×C9C3×C22⋊Q812yes—
264C2×C9C3×#12311yescharacteristic
265C24C2×AGL(1,ℤ/9)11yescharacteristic
266C22×C4—92no—
267C22×C4—92no—
268C22×C4—184no—
269C22×C4—182no—
270C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
271C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
272C4⋊C4—184no—
273C22⋊C4—22no—
274C22⋊C4—94no—
275C22×C3—24no—
276C22×C3—24no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—31no—
280C22×C3—31no—
281C22×C3—32no—
282C22×C3—32no—
283C22×C3—31no—
284C22×C3—34no—
285C22×C3—62no—
286C22×C3—62no—
287C22×C3—64no—
288C22×C3—64no—
289C22×C3C3×Dic3⋊C414yes—
290C22×C3C2×C3×Dic611yescharacteristic
291C22×C3C2×C3×C4×D311yescharacteristic
292C22×C3C3×D3×D412yes—
293C22×C3C3×#23712yes—
294C22×C3C2×C3×#6711yescharacteristic
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—121no—
298C3×C4—121no—
299C3×C4—184no—
300C3×C4—184no—
301Dic3—64no—
302Dic3—64no—
303C32#494611yescharacteristic
304C9—21no—
305C9C3×#38511yescharacteristic, complemented
306C23—22no—
307C23—22no—
308C233−1+2⋊Q812yes—
309C23C4×AGL(1,ℤ/9)12yes—
310C23#611⋊C312yes—
311C23C22×AGL(1,ℤ/9)11yescharacteristic
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—94no—
317C2×C4—94no—
318C2×C4—188no—
319C2×C4—182no—
320C2×C4—182no—
321C2×C4—182no—
322C2×C4—182no—
323C2×C3—24no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—32no—
328C2×C3—32no—
329C2×C3—64no—
330C2×C3C3×#87812yes—
331C2×C3C3×#67⋊C412yes—
332C2×C3C2×C3×Dic3⋊C411yescharacteristic
333C2×C3C3×#93511yescharacteristic
334C2×C3C3×#93711yescharacteristic
335C22—22no—
336C22—22no—
337C22—24no—
338C22—24no—
339C22(3−1+2⋊C4)⋊C414yescomplemented
340C22C2×3−1+2⋊Q811yescharacteristic
341C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
342C22D4×AGL(1,ℤ/9)12yes—
343C22#3699⋊C312yes—
344C22C2×#611⋊C311yescharacteristic
345C4—41no—
346C4—41no—
347C4—184no—
348C4—184no—
349C3—31no—
350C3C3×#494611yescharacteristic
351C2—24no—
352C2#63440⋊C312yes—
353C2(#611⋊C3)⋊C412yes—
354C2C2×(3−1+2⋊C4)⋊C411yescharacteristic
355C2#63497⋊C311yescharacteristic
356C2#63499⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingC2×C22⋊C4×3−1+2C2
prime radicalC24×C32C2×D3
cosocleC23×C9C23×C3
hypercenterC2×C22⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#385
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×(3−1+2⋊C4)⋊C4C2semidirect
5. 3−1+2⋊#95C2semidirect
7. #10588626C3semidirect
16. #3676⋊C3C4semidirect
27. C2×Dic9⋊C4C2×C3semidirect
28. C9⋊#95C2×C3semidirect
44. C2×3−1+2⋊C4C2×C4semidirect
70. #3676C3×C4semidirect
76. C22×3−1+2C4⋊C4semidirect
138. C2×Dic9C2×C3×C4semidirect
195. C22×C9C3×C4⋊C4semidirect
212. 3−1+2#385semidirect
305. C9C3×#385semidirect
339. C22(3−1+2⋊C4)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×3−1+2⋊C42
C2×C22⋊C4×3−1+22
C2×#3676⋊C32
C2×(3−1+2⋊C4)⋊C42
3−1+2⋊#952
C3×#49463
#105886263