Finite Group Databasebeta

← #10588633⋊C3 #13142523

Subgroups — #10588633⋊C3 #13142523

Order 1728.

576 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×3−1+2C211yesmaximal, characteristic
2C2×#3676⋊C3C212yesmaximal
3C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
43−1+2⋊#95C211yesmaximal, characteristic, complemented
53−1+2⋊#95C212yesmaximal, complemented
6C3×#4953—31nomaximal
7#10588633C311yesmaximal, characteristic, complemented
8C24×3−1+2C2211yescharacteristic
9C22×C4×3−1+2C2211yescharacteristic
10C22×C4×3−1+2C2211yescharacteristic
11C22×3−1+2⋊C4C2212yes—
12C22×3−1+2⋊C4C2212yes—
13C22⋊C4×3−1+2—22no—
14#3676⋊C3—24no—
15C4⋊(3−1+2⋊C4)—22no—
16C2×C32×C22⋊C4D311yescharacteristic
17C2×C3×#217—32no—
18C2×C3×C4⋊Dic3—31no—
19C3×C3⋊#95—32no—
20C3×C3⋊#95—31no—
21C2×C9×C22⋊C4—21no—
22C2×C9×C22⋊C4C2×C311yescharacteristic
23C2×#3676C2×C312yes—
24C2×C4⋊Dic9C2×C311yescharacteristic, complemented
25C9⋊#95C2×C311yescharacteristic, complemented
26C9⋊#95C2×C312yescomplemented
27C23×3−1+2—21no—
28C23×3−1+2—21no—
29C23×3−1+2—22no—
30C23×3−1+2—22no—
31C23×3−1+2D412yes—
32C23×3−1+2C2311yescharacteristic
33C2×C4×3−1+2—22no—
34C2×C4×3−1+2—21no—
35C2×C4×3−1+2—22no—
36C2×C4×3−1+2D412yes—
37C2×3−1+2⋊C4—22no—
38C2×3−1+2⋊C4—22no—
39C2×3−1+2⋊C4—22no—
40C2×3−1+2⋊C4—22no—
41C2×3−1+2⋊C4—22no—
42C2×3−1+2⋊C4—22no—
43C3×#397—91no—
44#4953—31no—
45C24×C32C2×D311yescharacteristic
46C22×C32×C4C2×D311yescharacteristic
47C22×C32×C4C2×D311yescharacteristic
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C24×C9—21no—
51C24×C9C22×C311yescharacteristic
52C32×C22⋊C4—22no—
53C3×#217—64no—
54C3×C4⋊Dic3—62no—
55C22×C4×C9—21no—
56C22×C4×C9—21no—
57C22×C4×C9C22×C311yescharacteristic
58C22×C4×C9C22×C311yescharacteristic
59C22×Dic9C22×C312yes—
60C22×Dic9C22×C312yes—
61C9×C22⋊C4—24no—
62C9×C22⋊C4—22no—
63#3676—24no—
64C4⋊Dic9—22no—
65C22×3−1+2—22no—
66C22×3−1+2—24no—
67C22×3−1+2—41no—
68C22×3−1+2—41no—
69C22×3−1+2—42no—
70C22×3−1+2C2×D411yescharacteristic
71C22×3−1+2C2×D411yescharacteristic
72C22×3−1+2#4111yescharacteristic
73C22×3−1+2#4112yes—
74C22×3−1+2#4112yes—
75C4×3−1+2—22no—
76C4×3−1+2—41no—
773−1+2⋊C4—42no—
783−1+2⋊C4—42no—
79C2×C3×C4⋊C4—91no—
80C2×C3×C22⋊C4—31no—
81C2×C3×C22⋊C4—92no—
82C2×C3×C22⋊C4C3×D311yescharacteristic
83C2×#217—32no—
84C2×C4⋊Dic3—31no—
85C3×#95—91no—
86C3×#95—92no—
87C3⋊#95—31no—
88C3⋊#95—32no—
89C23×C32—21no—
90C23×C32—22no—
91C23×C32—22no—
92C23×C32—21no—
93C23×C32D1212yes—
94C23×C32C22×D311yescharacteristic
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4#6712yes—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C23×C9—22no—
106C23×C9—22no—
107C23×C9—21no—
108C23×C9—21no—
109C23×C9—22no—
110C23×C9—22no—
111C23×C9—22no—
112C23×C9—21no—
113C23×C9—42no—
114C23×C9—42no—
115C23×C9C3×D412yes—
116C23×C9C23×C311yescharacteristic
117C2×C4×C9—22no—
118C2×C4×C9—22no—
119C2×C4×C9—21no—
120C2×C4×C9—22no—
121C2×C4×C9—22no—
122C2×C4×C9—42no—
123C2×C4×C9—42no—
124C2×C4×C9C3×D412yes—
125C2×Dic9—22no—
126C2×Dic9—22no—
127C2×Dic9—22no—
128C2×Dic9—22no—
129C2×Dic9—22no—
130C2×Dic9—22no—
131#397—91noSylow
132C2×3−1+2—42no—
133C2×3−1+2#12111yescharacteristic
134C2×3−1+2#12312yes—
135C2×3−1+2#12311yescharacteristic
136C2×3−1+2#12411yescharacteristic
137C2×3−1+2#12612yes—
138C24×C3—31no—
139C24×C3C2×C3×D311yescharacteristic
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—92no—
143C22×C3×C4—92no—
144C22×C3×C4C2×C3×D311yescharacteristic
145C22×C3×C4C2×C3×D311yescharacteristic
146C22×Dic3—32no—
147C22×Dic3—32no—
148C3×C4⋊C4—182no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—62no—
151C3×C22⋊C4—184no—
152#217—64no—
153C4⋊Dic3—62no—
154C22×C32—24no—
155C22×C32—22no—
156C22×C32—42no—
157C22×C32—41no—
158C22×C32—41no—
159C22×C32C2×D1211yescharacteristic
160C22×C32#23511yescharacteristic
161C22×C32#23712yes—
162C22×C32#23712yes—
163C22×C32C2×#6711yescharacteristic
164C32×C4—22no—
165C32×C4—41no—
166C3×Dic3—122no—
167C3×Dic3—122no—
168C22×C9—24no—
169C22×C9—22no—
170C22×C9—24no—
171C22×C9—21no—
172C22×C9—21no—
173C22×C9—22no—
174C22×C9—22no—
175C22×C9—21no—
176C22×C9—44no—
177C22×C9—44no—
178C22×C9—42no—
179C22×C9—42no—
180C22×C9—41no—
181C22×C9—41no—
182C22×C9—42no—
183C22×C9C2×C3×D411yescharacteristic
184C22×C9C2×C3×D411yescharacteristic
185C22×C9C3×#4111yescharacteristic
186C22×C9C3×#4112yes—
187C22×C9C3×#4112yes—
188C4×C9—22no—
189C4×C9—42no—
190C4×C9—42no—
191C4×C9—41no—
192Dic9—42no—
193Dic9—42no—
194C2×C4⋊C4—91no—
195C2×C22⋊C4—92no—
196C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic
197#95—91no—
198#95—92no—
1993−1+2#39711yescharacteristic, complemented, Sylow
200C23×C3—21no—
201C23×C3—22no—
202C23×C3—22no—
203C23×C3—21no—
204C23×C3—32no—
205C23×C3—31no—
206C23×C3—61no—
207C23×C3—61no—
208C23×C3—62no—
209C23×C3—62no—
210C23×C3C3×D1212yes—
211C23×C3C22×C3×D311yescharacteristic
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—21no—
215C2×C3×C4—32no—
216C2×C3×C4—62no—
217C2×C3×C4—61no—
218C2×C3×C4—62no—
219C2×C3×C4—182no—
220C2×C3×C4—182no—
221C2×C3×C4—182no—
222C2×C3×C4—182no—
223C2×C3×C4—182no—
224C2×C3×C4—182no—
225C2×C3×C4C3×#6712yes—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×C32—42no—
233C2×C32#87912yes—
234C2×C32#88612yes—
235C2×C32#93011yescharacteristic
236C2×C32#93511yescharacteristic
237C2×C32#93611yescharacteristic
238C2×C9—22no—
239C2×C9—22no—
240C2×C9—21no—
241C2×C9—21no—
242C2×C9—21no—
243C2×C9—44no—
244C2×C9—42no—
245C2×C9C3×#12111yescharacteristic
246C2×C9C3×#12312yes—
247C2×C9C3×#12311yescharacteristic
248C2×C9C3×#12411yescharacteristic
249C2×C9C3×#12612yes—
250C24C2×AGL(1,ℤ/9)11yescharacteristic
251C22×C4—92no—
252C22×C4—92no—
253C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
254C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
255C4⋊C4—182no—
256C22⋊C4—22no—
257C22⋊C4—184no—
258C22×C3—24no—
259C22×C3—22no—
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—32no—
263C22×C3—32no—
264C22×C3—31no—
265C22×C3—42no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—62no—
269C22×C3—64no—
270C22×C3—121no—
271C22×C3—121no—
272C22×C3—122no—
273C22×C3C2×C3×D1211yescharacteristic
274C22×C3C3×#23511yescharacteristic
275C22×C3C3×#23712yes—
276C22×C3C3×#23712yes—
277C22×C3C2×C3×#6711yescharacteristic
278C3×C4—22no—
279C3×C4—41no—
280C3×C4—62no—
281C3×C4—121no—
282C3×C4—362no—
283C3×C4—362no—
284Dic3—122no—
285Dic3—122no—
286C32#495311yescharacteristic
287C9—21no—
288C9C3×#39711yescharacteristic, complemented
289C23—21no—
290C23—22no—
291C23—22no—
292C23—21no—
293C23C4⋊AGL(1,ℤ/9)12yes—
294C23C22×AGL(1,ℤ/9)11yescharacteristic
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—182no—
299C2×C4—182no—
300C2×C4—182no—
301C2×C4—182no—
302C2×C4—182no—
303C2×C4—182no—
304C2×C4#611⋊C312yes—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—32no—
309C2×C3—32no—
310C2×C3—42no—
311C2×C3—122no—
312C2×C3C3×#87912yes—
313C2×C3C3×#88612yes—
314C2×C3C3×#93011yescharacteristic
315C2×C3C3×#93511yescharacteristic
316C2×C3C3×#93611yescharacteristic
317C22—22no—
318C22—24no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
323C22#3697⋊C311yescharacteristic
324C22#3699⋊C312yes—
325C22#3699⋊C312yes—
326C22C2×#611⋊C311yescharacteristic
327C4—22no—
328C4—41no—
329C4—362no—
330C4—362no—
331C3—31no—
332C3C3×#495311yescharacteristic
333C2—42no—
334C2#63441⋊C312yes—
335C2#63448⋊C312yes—
336C2#63492⋊C311yescharacteristic
337C2#63497⋊C311yescharacteristic
338C2#63498⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C22⋊C4×3−1+2C2
prime radicalC24×C32C2×D3
hypercenterC2×C22⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#397
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊(3−1+2⋊C4)C2semidirect
4. 3−1+2⋊#95C2semidirect
5. 3−1+2⋊#95C2semidirect
7. #10588633C3semidirect
24. C2×C4⋊Dic9C2×C3semidirect
25. C9⋊#95C2×C3semidirect
26. C9⋊#95C2×C3semidirect
199. 3−1+2#397semidirect
288. C9C3×#397semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×3−1+22
C2×#3676⋊C32
C2×C4⋊(3−1+2⋊C4)2
3−1+2⋊#952
C3×#49533
#105886333