Finite Group Databasebeta

← #10588629⋊C3 #13142521

Subgroups — #10588629⋊C3 #13142521

Order 1728.

624 conjugacy classes of subgroups.

The subgroup lattice has 346 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
4C2×(3−1+2⋊C4)⋊C4C212yesmaximal, complemented
53−1+2⋊#95C211yesmaximal, characteristic, complemented
6C3×#4949—31nomaximal
7#10588629C311yesmaximal, 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—
16(3−1+2⋊C4)⋊C4—24no—
17C2×C32×C22⋊C4D311yescharacteristic
18C2×C3×#217—32no—
19C2×C3×C4⋊Dic3—31no—
20C2×C3×Dic3⋊C4—32no—
21C3×C3⋊#95—31no—
22C2×C9×C22⋊C4—21no—
23C2×C9×C22⋊C4C2×C311yescharacteristic
24C2×#3676C2×C312yes—
25C2×C4⋊Dic9C2×C311yescharacteristic, complemented
26C2×Dic9⋊C4C2×C312yescomplemented
27C9⋊#95C2×C311yescharacteristic, complemented
28C23×3−1+2—21no—
29C23×3−1+2—22no—
30C23×3−1+2—21no—
31C23×3−1+2—22no—
32C23×3−1+2Q812yes—
33C23×3−1+2C2311yescharacteristic
34C2×C4×3−1+2—22no—
35C2×C4×3−1+2—21no—
36C2×C4×3−1+2—22no—
37C2×C4×3−1+2D412yes—
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—
43C2×3−1+2⋊C4D414yes—
44C3×#399—91no—
45#4949—31no—
46C24×C32C2×D311yescharacteristic
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C22×C3×Dic3—32no—
50C22×C3×Dic3—32no—
51C24×C9—21no—
52C24×C9C22×C311yescharacteristic
53C32×C22⋊C4—22no—
54C3×#217—64no—
55C3×C4⋊Dic3—62no—
56C3×Dic3⋊C4—64no—
57C22×C4×C9—21no—
58C22×C4×C9—21no—
59C22×C4×C9C22×C311yescharacteristic
60C22×C4×C9C22×C311yescharacteristic
61C22×Dic9C22×C312yes—
62C22×Dic9C22×C312yes—
63C9×C22⋊C4—24no—
64C9×C22⋊C4—22no—
65#3676—24no—
66C4⋊Dic9—22no—
67Dic9⋊C4—24no—
68C22×3−1+2—22no—
69C22×3−1+2—24no—
70C22×3−1+2—42no—
71C22×3−1+2—41no—
72C22×3−1+2—41no—
73C22×3−1+2C2×D412yes—
74C22×3−1+2C2×D411yescharacteristic
75C22×3−1+2C2×Q811yescharacteristic
76C22×3−1+2#4111yescharacteristic
77C22×3−1+2#4112yes—
78C4×3−1+2—22no—
79C4×3−1+2—41no—
803−1+2⋊C4—24no—
813−1+2⋊C4—42no—
82C2×C3×C4⋊C4—92no—
83C2×C3×C4⋊C4—91no—
84C2×C3×C22⋊C4—31no—
85C2×C3×C22⋊C4—92no—
86C2×C3×C22⋊C4C3×D311yescharacteristic
87C2×#217—32no—
88C2×C4⋊Dic3—31no—
89C2×Dic3⋊C4—32no—
90C3×#95—91no—
91C3⋊#95—31no—
92C23×C32—21no—
93C23×C32—22no—
94C23×C32—22no—
95C23×C32—21no—
96C23×C32Dic612yes—
97C23×C32C22×D311yescharacteristic
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4#6712yes—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C23×C9—22no—
109C23×C9—22no—
110C23×C9—22no—
111C23×C9—21no—
112C23×C9—21no—
113C23×C9—22no—
114C23×C9—22no—
115C23×C9—21no—
116C23×C9—42no—
117C23×C9—42no—
118C23×C9C3×Q812yes—
119C23×C9C23×C311yescharacteristic
120C2×C4×C9—22no—
121C2×C4×C9—22no—
122C2×C4×C9—22no—
123C2×C4×C9—21no—
124C2×C4×C9—22no—
125C2×C4×C9—42no—
126C2×C4×C9—42no—
127C2×C4×C9C3×D412yes—
128C2×Dic9—22no—
129C2×Dic9—22no—
130C2×Dic9—22no—
131C2×Dic9—22no—
132C2×Dic9—22no—
133C2×Dic9C3×D414yes—
134#399—91noSylow
135C2×3−1+2—42no—
136C2×3−1+2C22⋊Q812yes—
137C2×3−1+2C22⋊Q811yescharacteristic
138C2×3−1+2#12312yes—
139C2×3−1+2#12311yescharacteristic
140C2×3−1+2#12711yescharacteristic
141C24×C3—31no—
142C24×C3C2×C3×D311yescharacteristic
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—92no—
146C22×C3×C4—92no—
147C22×C3×C4C2×C3×D311yescharacteristic
148C22×C3×C4C2×C3×D311yescharacteristic
149C22×Dic3—32no—
150C22×Dic3—32no—
151C3×C4⋊C4—182no—
152C3×C4⋊C4—184no—
153C3×C22⋊C4—22no—
154C3×C22⋊C4—62no—
155C3×C22⋊C4—184no—
156#217—64no—
157C4⋊Dic3—62no—
158Dic3⋊C4—64no—
159C22×C32—24no—
160C22×C32—22no—
161C22×C32—42no—
162C22×C32—41no—
163C22×C32—41no—
164C22×C32C2×Dic611yescharacteristic
165C22×C32#23511yescharacteristic
166C22×C32D3×D412yes—
167C22×C32#23712yes—
168C22×C32C2×#6711yescharacteristic
169C32×C4—22no—
170C32×C4—41no—
171C3×Dic3—64no—
172C3×Dic3—122no—
173C22×C9—24no—
174C22×C9—24no—
175C22×C9—22no—
176C22×C9—21no—
177C22×C9—21no—
178C22×C9—22no—
179C22×C9—22no—
180C22×C9—21no—
181C22×C9—44no—
182C22×C9—44no—
183C22×C9—42no—
184C22×C9—42no—
185C22×C9—42no—
186C22×C9—41no—
187C22×C9—41no—
188C22×C9C2×C3×D412yes—
189C22×C9C2×C3×D411yescharacteristic
190C22×C9C2×C3×Q811yescharacteristic
191C22×C9C3×#4111yescharacteristic
192C22×C9C3×#4112yes—
193C4×C9—22no—
194C4×C9—42no—
195C4×C9—42no—
196C4×C9—41no—
197Dic9—24no—
198Dic9—42no—
199C2×C4⋊C4—92no—
200C2×C4⋊C4—91no—
201C2×C22⋊C4—92no—
202C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic
203#95—91no—
2043−1+2#39911yescharacteristic, complemented, Sylow
205C23×C3—21no—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—21no—
209C23×C3—32no—
210C23×C3—31no—
211C23×C3—61no—
212C23×C3—62no—
213C23×C3—61no—
214C23×C3—62no—
215C23×C3C3×Dic612yes—
216C23×C3C22×C3×D311yescharacteristic
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—32no—
221C2×C3×C4—62no—
222C2×C3×C4—61no—
223C2×C3×C4—62no—
224C2×C3×C4—94no—
225C2×C3×C4—182no—
226C2×C3×C4—182no—
227C2×C3×C4—182no—
228C2×C3×C4—182no—
229C2×C3×C4—182no—
230C2×C3×C4C3×#6712yes—
231C2×Dic3—34no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×C32—42no—
238C2×C32#87812yes—
239C2×C32#88312yes—
240C2×C32#92411yescharacteristic
241C2×C32#93511yescharacteristic
242C2×C32#94011yescharacteristic
243C2×C9—22no—
244C2×C9—22no—
245C2×C9—21no—
246C2×C9—21no—
247C2×C9—21no—
248C2×C9—44no—
249C2×C9—42no—
250C2×C9C3×C22⋊Q812yes—
251C2×C9C3×C22⋊Q811yescharacteristic
252C2×C9C3×#12312yes—
253C2×C9C3×#12311yescharacteristic
254C2×C9C3×#12711yescharacteristic
255C24C2×AGL(1,ℤ/9)11yescharacteristic
256C22×C4—92no—
257C22×C4—92no—
258C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
259C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
260C4⋊C4—184no—
261C4⋊C4—182no—
262C22⋊C4—22no—
263C22⋊C4—184no—
264C22×C3—24no—
265C22×C3—22no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—32no—
269C22×C3—32no—
270C22×C3—31no—
271C22×C3—42no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—62no—
275C22×C3—64no—
276C22×C3—121no—
277C22×C3—121no—
278C22×C3—122no—
279C22×C3C2×C3×Dic611yescharacteristic
280C22×C3C3×#23511yescharacteristic
281C22×C3C3×D3×D412yes—
282C22×C3C3×#23712yes—
283C22×C3C2×C3×#6711yescharacteristic
284C3×C4—22no—
285C3×C4—41no—
286C3×C4—62no—
287C3×C4—121no—
288C3×C4—184no—
289C3×C4—362no—
290Dic3—64no—
291Dic3—122no—
292C32#494911yescharacteristic
293C9—21no—
294C9C3×#39911yescharacteristic, complemented
295C23—21no—
296C23—22no—
297C23—22no—
298C23—21no—
299C233−1+2⋊Q812yes—
300C23C22×AGL(1,ℤ/9)11yescharacteristic
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—94no—
305C2×C4—182no—
306C2×C4—182no—
307C2×C4—182no—
308C2×C4—182no—
309C2×C4—182no—
310C2×C4#611⋊C312yes—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—32no—
315C2×C3—32no—
316C2×C3—42no—
317C2×C3—122no—
318C2×C3C3×#87812yes—
319C2×C3C3×#88312yes—
320C2×C3C3×#92411yescharacteristic
321C2×C3C3×#93511yescharacteristic
322C2×C3C3×#94011yescharacteristic
323C22—22no—
324C22—24no—
325C22—41no—
326C22—41no—
327C22—42no—
328C22C2×3−1+2⋊Q811yescharacteristic
329C22#3697⋊C311yescharacteristic
330C22D4×AGL(1,ℤ/9)12yes—
331C22#3699⋊C312yes—
332C22C2×#611⋊C311yescharacteristic
333C4—22no—
334C4—41no—
335C4—184no—
336C4—362no—
337C3—31no—
338C3C3×#494911yescharacteristic
339C2—42no—
340C2#63440⋊C312yes—
341C2#63445⋊C312yes—
342C2#63486⋊C311yescharacteristic
343C2#63497⋊C311yescharacteristic
344C2#63502⋊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×#399
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. C2×(3−1+2⋊C4)⋊C4C2semidirect
5. 3−1+2⋊#95C2semidirect
7. #10588629C3semidirect
25. C2×C4⋊Dic9C2×C3semidirect
26. C2×Dic9⋊C4C2×C3semidirect
27. C9⋊#95C2×C3semidirect
204. 3−1+2#399semidirect
294. C9C3×#399semidirect

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
C2×(3−1+2⋊C4)⋊C42
3−1+2⋊#952
C3×#49493
#105886293