Finite Group Databasebeta

← #10588652⋊C3 #13142647

Subgroups — #10588652⋊C3 #13142647

Order 1728.

556 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×3−1+2C211yesmaximal, characteristic
2C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic
3C2×(3−1+2⋊C4)⋊C4C212yesmaximal
4C2×C4×3−1+2⋊C4C211yesmaximal, characteristic
53−1+2⋊#95C212yesmaximal
6C3×#4972—31nomaximal
7#10588652C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2C2212yes—
9C22×C4×3−1+2C2211yescharacteristic
10C22×3−1+2⋊C4C2212yes—
11C22×3−1+2⋊C4C2211yescharacteristic
12C22×3−1+2⋊C4C2211yescharacteristic
13C4⋊C4×3−1+2—22no—
14C4⋊(3−1+2⋊C4)C414yescomplemented
15(3−1+2⋊C4)⋊C4—24no—
16C4×3−1+2⋊C4—22no—
17C2×C32×C4⋊C4D311yescharacteristic
18C2×C3×C4⋊Dic3—31no—
19C2×C3×Dic3⋊C4—32no—
20C2×C3×C4×Dic3—31no—
21C3×C3⋊#95—32no—
22C2×C9×C4⋊C4—21no—
23C2×C9×C4⋊C4C2×C311yescharacteristic
24C2×C4⋊Dic9C2×C311yescharacteristic
25C2×Dic9⋊C4C2×C312yes—
26C2×C4×Dic9C2×C311yescharacteristic
27C9⋊#95C2×C312yes—
28C23×3−1+2C2311yescharacteristic
29C2×C4×3−1+2—22no—
30C2×C4×3−1+2—22no—
31C2×C4×3−1+2—22no—
32C2×C4×3−1+2C2×C412yes—
33C2×C4×3−1+2D412yes—
34C2×C4×3−1+2Q812yes—
35C2×3−1+2⋊C4—21no—
36C2×3−1+2⋊C4—21no—
37C2×3−1+2⋊C4—22no—
38C2×3−1+2⋊C4—22no—
39C2×3−1+2⋊C4—21no—
40C2×3−1+2⋊C4—21no—
41C2×3−1+2⋊C4C2×C414yes—
42C2×3−1+2⋊C4D412yes—
43C2×3−1+2⋊C4Q812yes—
44C3×#389—91no—
45#4972—31no—
46C22×C32×C4C2×D312yes—
47C22×C32×C4C2×D311yescharacteristic
48C22×C3×Dic3—31no—
49C22×C3×Dic3—31no—
50C22×C3×Dic3—32no—
51C32×C4⋊C4—22no—
52C3×C4⋊Dic3—34no—
53C3×Dic3⋊C4—64no—
54C3×C4×Dic3—62no—
55C22×C4×C9—21no—
56C22×C4×C9—22no—
57C22×C4×C9C22×C312yes—
58C22×C4×C9C22×C311yescharacteristic
59C22×Dic9C22×C312yes—
60C22×Dic9C22×C311yescharacteristic
61C22×Dic9C22×C311yescharacteristic
62C9×C4⋊C4—24no—
63C9×C4⋊C4—22no—
64C4⋊Dic9C3×C414yescomplemented
65Dic9⋊C4—24no—
66C4×Dic9—22no—
67C22×3−1+2C22×C411yescharacteristic
68C22×3−1+2C2×D411yescharacteristic
69C22×3−1+2C2×D411yescharacteristic
70C22×3−1+2C2×Q811yescharacteristic
71C22×3−1+2C2×Q811yescharacteristic
72C22×3−1+2#4111yescharacteristic
73C22×3−1+2#4111yescharacteristic
74C4×3−1+2—42no—
75C4×3−1+2C4⋊C414yescomplemented
763−1+2⋊C4—24no—
773−1+2⋊C4—22no—
783−1+2⋊C4—22no—
79C2×C3×C4⋊C4—31no—
80C2×C3×C4⋊C4—91no—
81C2×C3×C4⋊C4—92no—
82C2×C3×C4⋊C4C3×D311yescharacteristic
83C2×C3×C42—91no—
84C2×C4⋊Dic3—31no—
85C2×Dic3⋊C4—32no—
86C2×C4×Dic3—31no—
87C3×#95—92no—
88C3⋊#95—32no—
89C23×C32C22×D311yescharacteristic
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4Dic612yes—
94C2×C32×C4C4×D312yes—
95C2×C32×C4#6712yes—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—32no—
98C2×C3×Dic3—32no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C23×C9—21no—
106C23×C9C23×C311yescharacteristic
107C2×C4×C9—24no—
108C2×C4×C9—22no—
109C2×C4×C9—22no—
110C2×C4×C9—22no—
111C2×C4×C9—22no—
112C2×C4×C9—22no—
113C2×C4×C9—22no—
114C2×C4×C9—42no—
115C2×C4×C9—42no—
116C2×C4×C9C2×C3×C412yes—
117C2×C4×C9C3×D412yes—
118C2×C4×C9C3×Q812yes—
119C2×Dic9—21no—
120C2×Dic9—21no—
121C2×Dic9—22no—
122C2×Dic9—21no—
123C2×Dic9—22no—
124C2×Dic9—21no—
125C2×Dic9C2×C3×C414yes—
126C2×Dic9C3×D412yes—
127C2×Dic9C3×Q812yes—
128#389—91noSylow
129C2×3−1+2C2×C4⋊C411yescharacteristic
130C2×3−1+2C4×D411yescharacteristic
131C2×3−1+2C4×Q811yescharacteristic
132C2×3−1+2#12111yescharacteristic
133C2×3−1+2C22⋊Q811yescharacteristic
134C2×3−1+2#12511yescharacteristic
135C2×3−1+2C4⋊Q811yescharacteristic
136C22×C3×C4—31no—
137C22×C3×C4—32no—
138C22×C3×C4—91no—
139C22×C3×C4—92no—
140C22×C3×C4—91no—
141C22×C3×C4C2×C3×D312yes—
142C22×C3×C4C2×C3×D311yescharacteristic
143C22×Dic3—31no—
144C22×Dic3—32no—
145C22×Dic3—31no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—62no—
148C3×C4⋊C4—94no—
149C3×C4⋊C4—184no—
150C3×C42—182no—
151C4⋊Dic3—34no—
152Dic3⋊C4—64no—
153C4×Dic3—62no—
154C22×C32C2×Dic611yescharacteristic
155C22×C32C2×C4×D311yescharacteristic
156C22×C32D3×D411yescharacteristic
157C22×C32#23711yescharacteristic
158C22×C32D3×Q811yescharacteristic
159C22×C32#23911yescharacteristic
160C22×C32C2×#6711yescharacteristic
161C32×C4—42no—
162C32×C4Dic3⋊C414yes—
163C3×Dic3—64no—
164C3×Dic3—62no—
165C3×Dic3—62no—
166C22×C9—21no—
167C22×C9—21no—
168C22×C9—21no—
169C22×C9—21no—
170C22×C9—21no—
171C22×C9—21no—
172C22×C9—21no—
173C22×C9C22×C3×C411yescharacteristic
174C22×C9C2×C3×D411yescharacteristic
175C22×C9C2×C3×D411yescharacteristic
176C22×C9C2×C3×Q811yescharacteristic
177C22×C9C2×C3×Q811yescharacteristic
178C22×C9C3×#4111yescharacteristic
179C22×C9C3×#4111yescharacteristic
180C4×C9—24no—
181C4×C9—44no—
182C4×C9—42no—
183C4×C9C3×C4⋊C414yescomplemented
184Dic9—24no—
185Dic9—22no—
186Dic9—22no—
187C2×C4⋊C4—91no—
188C2×C4⋊C4—92no—
189C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic
190C2×C42—91no—
191#95—92no—
1923−1+2#38911yescharacteristic, complemented, Sylow
193C23×C3—31no—
194C23×C3C22×C3×D311yescharacteristic
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—32no—
199C2×C3×C4—32no—
200C2×C3×C4—32no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4—62no—
204C2×C3×C4—94no—
205C2×C3×C4—92no—
206C2×C3×C4—92no—
207C2×C3×C4—181no—
208C2×C3×C4—181no—
209C2×C3×C4—182no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4—182no—
213C2×C3×C4C3×Dic612yes—
214C2×C3×C4C3×C4×D312yes—
215C2×C3×C4C3×#6712yes—
216C2×Dic3—34no—
217C2×Dic3—32no—
218C2×Dic3—32no—
219C2×Dic3—62no—
220C2×Dic3—61no—
221C2×Dic3—62no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×C32#88711yescharacteristic
226C2×C32#88811yescharacteristic
227C2×C32#89011yescharacteristic
228C2×C32#89311yescharacteristic
229C2×C32C2×Dic3⋊C411yescharacteristic
230C2×C32#93811yescharacteristic
231C2×C32#94611yescharacteristic
232C2×C9—21no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9—21no—
238C2×C9—21no—
239C2×C9C2×C3×C4⋊C411yescharacteristic
240C2×C9C3×C4×D411yescharacteristic
241C2×C9C3×C4×Q811yescharacteristic
242C2×C9C3×#12111yescharacteristic
243C2×C9C3×C22⋊Q811yescharacteristic
244C2×C9C3×#12511yescharacteristic
245C2×C9C3×C4⋊Q811yescharacteristic
246C22×C4—91no—
247C22×C4—92no—
248C22×C4—91no—
249C22×C4C2×AGL(1,ℤ/9)12yes—
250C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—94no—
253C4⋊C4—184no—
254C42—182no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—31no—
259C22×C3—31no—
260C22×C3—31no—
261C22×C3—31no—
262C22×C3C2×C3×Dic611yescharacteristic
263C22×C3C2×C3×C4×D311yescharacteristic
264C22×C3C3×D3×D411yescharacteristic
265C22×C3C3×#23711yescharacteristic
266C22×C3C3×D3×Q811yescharacteristic
267C22×C3C3×#23911yescharacteristic
268C22×C3C2×C3×#6711yescharacteristic
269C3×C4—34no—
270C3×C4—42no—
271C3×C4—122no—
272C3×C4—184no—
273C3×C4—182no—
274C3×C4—182no—
275C3×C4C3×Dic3⋊C414yes—
276Dic3—64no—
277Dic3—62no—
278Dic3—62no—
279C32#497211yescharacteristic
280C9—21no—
281C9C3×#38911yescharacteristic, complemented
282C23C22×AGL(1,ℤ/9)11yescharacteristic
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—94no—
287C2×C4—92no—
288C2×C4—92no—
289C2×C4—182no—
290C2×C4—181no—
291C2×C4—182no—
292C2×C4—181no—
293C2×C4—181no—
294C2×C4—181no—
295C2×C43−1+2⋊Q812yes—
296C2×C4C4×AGL(1,ℤ/9)12yes—
297C2×C4#611⋊C312yes—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3C3×#88711yescharacteristic
306C2×C3C3×#88811yescharacteristic
307C2×C3C3×#89011yescharacteristic
308C2×C3C3×#89311yescharacteristic
309C2×C3C2×C3×Dic3⋊C411yescharacteristic
310C2×C3C3×#93811yescharacteristic
311C2×C3C3×#94611yescharacteristic
312C22C2×3−1+2⋊Q811yescharacteristic
313C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
314C22D4×AGL(1,ℤ/9)11yescharacteristic
315C22#3699⋊C311yescharacteristic
316C22Q8×AGL(1,ℤ/9)11yescharacteristic
317C22#8872411yescharacteristic
318C22C2×#611⋊C311yescharacteristic
319C4—42no—
320C4—184no—
321C4—182no—
322C4—182no—
323C4(3−1+2⋊C4)⋊C414yescomplemented
324C3—31no—
325C3C3×#497211yescharacteristic
326C2#63449⋊C311yescharacteristic
327C2#63450⋊C311yescharacteristic
328C2#63452⋊C311yescharacteristic
329C2#63455⋊C311yescharacteristic
330C2C2×(3−1+2⋊C4)⋊C411yescharacteristic
331C2#63500⋊C311yescharacteristic
332C2#63508⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×3−1+2C2
prime radicalC23×C32C22×D3
cosocleC23×C9C23×C3
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#389
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #10588652C3semidirect
14. C4⋊(3−1+2⋊C4)C4semidirect
64. C4⋊Dic9C3×C4semidirect
75. C4×3−1+2C4⋊C4semidirect
183. C4×C9C3×C4⋊C4semidirect
192. 3−1+2#389semidirect
281. C9C3×#389semidirect
323. C4(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
C2×C4⋊C4×3−1+22
C2×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
C2×C4×3−1+2⋊C42
3−1+2⋊#952
C3×#49723
#105886523