Finite Group Databasebeta

← #10588630⋊C3 #13142518

Subgroups — #10588630⋊C3 #13142518

Order 1728.

768 conjugacy classes of subgroups.

The subgroup lattice has 355 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×C4⋊(3−1+2⋊C4)C212yesmaximal, complemented
53−1+2⋊#95C212yesmaximal, complemented
6C3×#4950—31nomaximal
7#10588630C311yesmaximal, characteristic, complemented
8C24×3−1+2C2211yescharacteristic
9C22×C4×3−1+2C2212yes—
10C22×3−1+2⋊C4—22no—
11C22×3−1+2⋊C4—24no—
12C22×3−1+2⋊C4C2211yescharacteristic
13C22×3−1+2⋊C4C2211yescharacteristic
14C22×3−1+2⋊C4C2212yes—
15C22⋊C4×3−1+2C414yescomplemented
16#3676⋊C3—22no—
17C4⋊(3−1+2⋊C4)—24no—
18C23×C3×Dic3—31no—
19C2×C32×C22⋊C4D311yescharacteristic
20C2×C3×#217—31no—
21C2×C3×C4⋊Dic3—32no—
22C23×Dic9C2×C311yescharacteristic
23C3×C3⋊#95—32no—
24C2×C9×C22⋊C4—21no—
25C2×C9×C22⋊C4C2×C311yescharacteristic
26C2×#3676C2×C311yescharacteristic
27C2×C4⋊Dic9C2×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—22no—
36C2×C4×3−1+2—22no—
37C2×C4×3−1+2C2×C414yescomplemented
38C2×3−1+2⋊C4—28no—
39C2×3−1+2⋊C4—22no—
40C2×3−1+2⋊C4—22no—
41C2×3−1+2⋊C4—22no—
42C2×3−1+2⋊C4—21no—
43C2×3−1+2⋊C4—21no—
44C2×3−1+2⋊C4—22no—
45C2×3−1+2⋊C4D414yes—
46C3×#385—91no—
47#4950—31no—
48C24×C32C2×D311yescharacteristic
49C22×C32×C4C2×D312yes—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—31no—
52C22×C3×Dic3—31no—
53C22×C3×Dic3—64no—
54C22×C3×Dic3—62no—
55C24×C9—21no—
56C24×C9C22×C311yescharacteristic
57C32×C22⋊C4Dic314yes—
58C3×#217—62no—
59C3×C4⋊Dic3—64no—
60C22×C4×C9—22no—
61C22×C4×C9C22×C312yes—
62C22×Dic9—22no—
63C22×Dic9—24no—
64C22×Dic9C22×C311yescharacteristic
65C22×Dic9C22×C311yescharacteristic
66C22×Dic9C22×C312yes—
67C9×C22⋊C4—24no—
68C9×C22⋊C4C3×C414yescomplemented
69#3676—22no—
70C4⋊Dic9—24no—
71C22×3−1+2—24no—
72C22×3−1+2—22no—
73C22×3−1+2—24no—
74C22×3−1+2—22no—
75C22×3−1+2C4⋊C414yescomplemented
76C22×3−1+2C22×C411yescharacteristic
77C22×3−1+2C2×D411yescharacteristic
78C22×3−1+2C2×D412yes—
79C22×3−1+2C2×Q811yescharacteristic
80C22×3−1+2#4112yes—
81C4×3−1+2—24no—
823−1+2⋊C4—24no—
833−1+2⋊C4—41no—
843−1+2⋊C4—41no—
85C23×C3×C4—91no—
86C23×Dic3—31no—
87C2×C3×C4⋊C4—92no—
88C2×C3×C22⋊C4—31no—
89C2×C3×C22⋊C4—91no—
90C2×C3×C22⋊C4C3×D311yescharacteristic
91C2×#217—31no—
92C2×C4⋊Dic3—32no—
93C3×#95—92no—
94C3⋊#95—32no—
95C23×C32—22no—
96C23×C32—22no—
97C23×C32Dic612yes—
98C23×C32D1212yes—
99C23×C32C2×Dic312yes—
100C23×C32C22×D311yescharacteristic
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4C2×Dic314yes—
104C2×C3×Dic3—34no—
105C2×C3×Dic3—68no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C23×C9—22no—
113C23×C9—22no—
114C23×C9—22no—
115C23×C9—22no—
116C23×C9—22no—
117C23×C9—21no—
118C23×C9—42no—
119C23×C9—42no—
120C23×C9C2×C3×C412yes—
121C23×C9C3×D412yes—
122C23×C9C3×Q812yes—
123C23×C9C23×C311yescharacteristic
124C2×C4×C9—22no—
125C2×C4×C9—22no—
126C2×C4×C9—24no—
127C2×C4×C9—42no—
128C2×C4×C9—42no—
129C2×C4×C9C2×C3×C414yescomplemented
130C2×Dic9—28no—
131C2×Dic9—22no—
132C2×Dic9—22no—
133C2×Dic9—22no—
134C2×Dic9—21no—
135C2×Dic9—21no—
136C2×Dic9—22no—
137C2×Dic9C3×D414yes—
138#385—91noSylow
139C2×3−1+2—24no—
140C2×3−1+2C2×C4⋊C411yescharacteristic
141C2×3−1+2C4×D412yes—
142C2×3−1+2#12011yescharacteristic
143C2×3−1+2C22⋊Q812yes—
144C2×3−1+2#12311yescharacteristic
145C24×C3—31no—
146C24×C3C2×C3×D311yescharacteristic
147C22×C3×C4—32no—
148C22×C3×C4—92no—
149C22×C3×C4—91no—
150C22×C3×C4—91no—
151C22×C3×C4—184no—
152C22×C3×C4—182no—
153C22×C3×C4C2×C3×D312yes—
154C22×Dic3—32no—
155C22×Dic3—31no—
156C22×Dic3—31no—
157C22×Dic3—64no—
158C22×Dic3—62no—
159C3×C4⋊C4—184no—
160C3×C22⋊C4—34no—
161C3×C22⋊C4—182no—
162C3×C22⋊C4C3×Dic314yes—
163#217—62no—
164C4⋊Dic3—64no—
165C22×C32—24no—
166C22×C32—22no—
167C22×C32—22no—
168C22×C32—24no—
169C22×C32C4⋊Dic314yes—
170C22×C32C2×Dic611yescharacteristic
171C22×C32C2×D1211yescharacteristic
172C22×C32D3×D412yes—
173C22×C32#23712yes—
174C22×C32C22×Dic311yescharacteristic
175C32×C4—24no—
176C3×Dic3—64no—
177C3×Dic3—121no—
178C3×Dic3—121no—
179C22×C9—24no—
180C22×C9—22no—
181C22×C9—24no—
182C22×C9—22no—
183C22×C9—24no—
184C22×C9—21no—
185C22×C9—21no—
186C22×C9—22no—
187C22×C9—22no—
188C22×C9—21no—
189C22×C9—44no—
190C22×C9—42no—
191C22×C9—42no—
192C22×C9—44no—
193C22×C9C3×C4⋊C414yescomplemented
194C22×C9C22×C3×C411yescharacteristic
195C22×C9C2×C3×D411yescharacteristic
196C22×C9C2×C3×D412yes—
197C22×C9C2×C3×Q811yescharacteristic
198C22×C9C3×#4112yes—
199C4×C9—24no—
200C4×C9—44no—
201Dic9—24no—
202Dic9—41no—
203Dic9—41no—
204C23×C4—91no—
205C2×C4⋊C4—92no—
206C2×C22⋊C4—91no—
207C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic
208#95—92no—
2093−1+2#38511yescharacteristic, complemented, Sylow
210C23×C3—22no—
211C23×C3—22no—
212C23×C3—32no—
213C23×C3—32no—
214C23×C3—32no—
215C23×C3—31no—
216C23×C3—62no—
217C23×C3—62no—
218C23×C3C3×Dic612yes—
219C23×C3C3×D1212yes—
220C23×C3C2×C3×Dic312yes—
221C23×C3C22×C3×D311yescharacteristic
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—34no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—94no—
228C2×C3×C4—182no—
229C2×C3×C4—182no—
230C2×C3×C4—188no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4—182no—
234C2×C3×C4—182no—
235C2×C3×C4C2×C3×Dic314yes—
236C2×Dic3—34no—
237C2×Dic3—68no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244C2×C32—24no—
245C2×C32#87812yes—
246C2×C32#88211yescharacteristic
247C2×C32#88611yescharacteristic
248C2×C32C2×C4⋊Dic311yescharacteristic
249C2×C32D4×Dic312yes—
250C2×C9—24no—
251C2×C9—22no—
252C2×C9—21no—
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—22no—
256C2×C9—44no—
257C2×C9C2×C3×C4⋊C411yescharacteristic
258C2×C9C3×C4×D412yes—
259C2×C9C3×#12011yescharacteristic
260C2×C9C3×C22⋊Q812yes—
261C2×C9C3×#12311yescharacteristic
262C24C2×AGL(1,ℤ/9)11yescharacteristic
263C22×C4—92no—
264C22×C4—91no—
265C22×C4—91no—
266C22×C4—184no—
267C22×C4—182no—
268C22×C4C2×AGL(1,ℤ/9)12yes—
269C4⋊C4—184no—
270C22⋊C4—182no—
271C22⋊C43−1+2⋊C414yescomplemented
272C22×C3—24no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—24no—
276C22×C3—31no—
277C22×C3—31no—
278C22×C3—32no—
279C22×C3—32no—
280C22×C3—31no—
281C22×C3—34no—
282C22×C3—62no—
283C22×C3—64no—
284C22×C3—64no—
285C22×C3—62no—
286C22×C3C3×C4⋊Dic314yes—
287C22×C3C2×C3×Dic611yescharacteristic
288C22×C3C2×C3×D1211yescharacteristic
289C22×C3C3×D3×D412yes—
290C22×C3C3×#23712yes—
291C22×C3C22×C3×Dic311yescharacteristic
292C3×C4—24no—
293C3×C4—64no—
294C3×C4—184no—
295C3×C4—361no—
296C3×C4—361no—
297Dic3—64no—
298Dic3—121no—
299Dic3—121no—
300C32#495011yescharacteristic
301C9—21no—
302C9C3×#38511yescharacteristic, complemented
303C23—22no—
304C23—22no—
305C233−1+2⋊Q812yes—
306C23C4⋊AGL(1,ℤ/9)12yes—
307C23C2×3−1+2⋊C412yes—
308C23C22×AGL(1,ℤ/9)11yescharacteristic
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—94no—
312C2×C4—188no—
313C2×C4—182no—
314C2×C4—182no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C4C2×3−1+2⋊C414yescomplemented
320C2×C3—24no—
321C2×C3—31no—
322C2×C3—32no—
323C2×C3—32no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—64no—
327C2×C3C3×#87812yes—
328C2×C3C3×#88211yescharacteristic
329C2×C3C3×#88611yescharacteristic
330C2×C3C2×C3×C4⋊Dic311yescharacteristic
331C2×C3C3×D4×Dic312yes—
332C22—24no—
333C22—24no—
334C22—22no—
335C22—22no—
336C22C4⋊(3−1+2⋊C4)14yescomplemented
337C22C2×3−1+2⋊Q811yescharacteristic
338C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
339C22D4×AGL(1,ℤ/9)12yes—
340C22#3699⋊C312yes—
341C22C22×3−1+2⋊C411yescharacteristic
342C4—24no—
343C4—184no—
344C4—361no—
345C4—361no—
346C3—31no—
347C3C3×#495011yescharacteristic
348C2—24no—
349C2#63440⋊C312yes—
350C2#63444⋊C311yescharacteristic
351C2#63448⋊C311yescharacteristic
352C2C2×C4⋊(3−1+2⋊C4)11yescharacteristic
353C2D4×3−1+2⋊C412yes—
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×C4⋊(3−1+2⋊C4)C2semidirect
5. 3−1+2⋊#95C2semidirect
7. #10588630C3semidirect
15. C22⋊C4×3−1+2C4semidirect
27. C2×C4⋊Dic9C2×C3semidirect
28. C9⋊#95C2×C3semidirect
37. C2×C4×3−1+2C2×C4semidirect
68. C9×C22⋊C4C3×C4semidirect
75. C22×3−1+2C4⋊C4semidirect
129. C2×C4×C9C2×C3×C4semidirect
193. C22×C9C3×C4⋊C4semidirect
209. 3−1+2#385semidirect
271. C22⋊C43−1+2⋊C4semidirect
302. C9C3×#385semidirect
319. C2×C4C2×3−1+2⋊C4semidirect
336. C22C4⋊(3−1+2⋊C4)semidirect

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×C4⋊(3−1+2⋊C4)2
3−1+2⋊#952
C3×#49503
#105886303