Finite Group Databasebeta

← #10588627⋊C3 #13142517

Subgroups — #10588627⋊C3 #13142517

Order 1728.

672 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×3−1+2⋊Q8C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×3−1+2C211yesmaximal, characteristic
3C2×#3676⋊C3C212yesmaximal
43−1+2⋊#95C211yesmaximal, characteristic, complemented
53−1+2⋊#95C212yesmaximal, complemented
6C3×#4947—31nomaximal
7#10588627C311yesmaximal, 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—
13C2×3−1+2⋊Q8—22no—
14C2×3−1+2⋊Q8—24no—
15C22⋊C4×3−1+2—22no—
16#3676⋊C3—24no—
17C22×C3×Dic6—31no—
18C2×C32×C22⋊C4D311yescharacteristic
19C2×C3×#217—32no—
20C22×Dic18C2×C311yescharacteristic, complemented
21C3×C3⋊#95—32no—
22C3×C3⋊#95—31no—
23C2×C9×C22⋊C4—21no—
24C2×C9×C22⋊C4C2×C311yescharacteristic
25C2×#3676C2×C312yes—
26C9⋊#95C2×C311yescharacteristic, complemented
27C9⋊#95C2×C312yescomplemented
28C23×3−1+2—21no—
29C23×3−1+2—22no—
30C23×3−1+2—21no—
31C23×3−1+2—22no—
32C23×3−1+2Q812yescomplemented
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—
443−1+2⋊Q8—28no—
45C3×#393—91no—
46#4947—31no—
47C24×C32C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C2×C3×Dic6—64no—
53C2×C3×Dic6—62no—
54C24×C9—21no—
55C24×C9C22×C311yescharacteristic
56C32×C22⋊C4—22no—
57C3×#217—64no—
58C22×C4×C9—21no—
59C22×C4×C9—21no—
60C22×C4×C9C22×C311yescharacteristic
61C22×C4×C9C22×C311yescharacteristic
62C22×Dic9C22×C312yes—
63C22×Dic9C22×C312yes—
64C2×Dic18—22no—
65C2×Dic18—24no—
66C9×C22⋊C4—24no—
67C9×C22⋊C4—22no—
68#3676—24no—
69C22×3−1+2—22no—
70C22×3−1+2—24no—
71C22×3−1+2—41no—
72C22×3−1+2—42no—
73C22×3−1+2—41no—
74C22×3−1+2C2×D412yes—
75C22×3−1+2C2×D411yescharacteristic
76C22×3−1+2C2×Q811yescharacteristic
77C22×3−1+2#4111yescharacteristic
78C22×3−1+2#4112yes—
79C4×3−1+2—22no—
80C4×3−1+2—41no—
813−1+2⋊C4—24no—
823−1+2⋊C4—42no—
83C22×C3×Q8—91no—
84C22×Dic6—31no—
85C2×C3×C22⋊C4—31no—
86C2×C3×C22⋊C4—92no—
87C2×C3×C22⋊C4C3×D311yescharacteristic
88C2×#217—32no—
89C3×#95—91no—
90C3×#95—92no—
91C3⋊#95—32no—
92C3⋊#95—31no—
93C23×C32—21no—
94C23×C32—22no—
95C23×C32—22no—
96C23×C32—21no—
97C23×C32Dic612yes—
98C23×C32C22×D311yescharacteristic
99C2×C32×C4—22no—
100C2×C32×C4—22no—
101C2×C32×C4—21no—
102C2×C32×C4#6712yes—
103C2×C3×Dic3—34no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C3×Dic6—68no—
110C23×C9—22no—
111C23×C9—22no—
112C23×C9—22no—
113C23×C9—21no—
114C23×C9—21no—
115C23×C9—22no—
116C23×C9—22no—
117C23×C9—21no—
118C23×C9—42no—
119C23×C9—42no—
120C23×C9C3×Q812yescomplemented
121C23×C9C23×C311yescharacteristic
122C2×C4×C9—22no—
123C2×C4×C9—22no—
124C2×C4×C9—21no—
125C2×C4×C9—22no—
126C2×C4×C9—22no—
127C2×C4×C9—42no—
128C2×C4×C9—42no—
129C2×C4×C9C3×D412yes—
130C2×Dic9—22no—
131C2×Dic9—22no—
132C2×Dic9—22no—
133C2×Dic9—22no—
134C2×Dic9—22no—
135C2×Dic9C3×D414yes—
136Dic18—28no—
137#393—91noSylow
138C2×3−1+2—42no—
139C2×3−1+2#12011yescharacteristic
140C2×3−1+2C22⋊Q812yes—
141C2×3−1+2C22⋊Q811yescharacteristic
142C2×3−1+2#12412yes—
143C2×3−1+2#12411yescharacteristic
144C24×C3—31no—
145C24×C3C2×C3×D311yescharacteristic
146C2×C3×Q8—182no—
147C2×C3×Q8—184no—
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—92no—
151C22×C3×C4—92no—
152C22×C3×C4C2×C3×D311yescharacteristic
153C22×C3×C4C2×C3×D311yescharacteristic
154C22×Dic3—32no—
155C22×Dic3—32no—
156C2×Dic6—64no—
157C2×Dic6—62no—
158C3×C22⋊C4—22no—
159C3×C22⋊C4—62no—
160C3×C22⋊C4—184no—
161#217—64no—
162C22×C32—24no—
163C22×C32—22no—
164C22×C32—42no—
165C22×C32—41no—
166C22×C32—41no—
167C22×C32C2×Dic611yescharacteristic
168C22×C32#23511yescharacteristic
169C22×C32D3×D412yes—
170C22×C32#23712yes—
171C22×C32C2×#6711yescharacteristic
172C32×C4—22no—
173C32×C4—41no—
174C3×Dic3—64no—
175C3×Dic3—122no—
176C22×C9—24no—
177C22×C9—24no—
178C22×C9—22no—
179C22×C9—21no—
180C22×C9—21no—
181C22×C9—22no—
182C22×C9—22no—
183C22×C9—21no—
184C22×C9—44no—
185C22×C9—44no—
186C22×C9—42no—
187C22×C9—42no—
188C22×C9—41no—
189C22×C9—42no—
190C22×C9—41no—
191C22×C9C2×C3×D412yes—
192C22×C9C2×C3×D411yescharacteristic
193C22×C9C2×C3×Q811yescharacteristic
194C22×C9C3×#4111yescharacteristic
195C22×C9C3×#4112yes—
196C4×C9—22no—
197C4×C9—42no—
198C4×C9—42no—
199C4×C9—41no—
200Dic9—24no—
201Dic9—42no—
202C22×Q8—91no—
203C2×C22⋊C4—92no—
204C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic
205#95—92no—
206#95—91no—
2073−1+2#39311yescharacteristic, complemented, Sylow
208C23×C3—21no—
209C23×C3—22no—
210C23×C3—22no—
211C23×C3—21no—
212C23×C3—32no—
213C23×C3—31no—
214C23×C3—61no—
215C23×C3—62no—
216C23×C3—61no—
217C23×C3—62no—
218C23×C3C3×Dic612yes—
219C23×C3C22×C3×D311yescharacteristic
220C3×Q8—188no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—21no—
224C2×C3×C4—32no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—62no—
228C2×C3×C4—94no—
229C2×C3×C4—182no—
230C2×C3×C4—182no—
231C2×C3×C4—182no—
232C2×C3×C4—182no—
233C2×C3×C4—182no—
234C2×C3×C4C3×#6712yes—
235C2×Dic3—34no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241Dic6—68no—
242C2×C32—42no—
243C2×C32#87812yes—
244C2×C32#88512yes—
245C2×C32#92411yescharacteristic
246C2×C32#93611yescharacteristic
247C2×C32#93711yescharacteristic
248C2×C9—22no—
249C2×C9—22no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—21no—
253C2×C9—44no—
254C2×C9—42no—
255C2×C9C3×#12011yescharacteristic
256C2×C9C3×C22⋊Q812yes—
257C2×C9C3×C22⋊Q811yescharacteristic
258C2×C9C3×#12412yes—
259C2×C9C3×#12411yescharacteristic
260C24C2×AGL(1,ℤ/9)11yescharacteristic
261C2×Q8—184no—
262C2×Q8—182no—
263C22×C4—92no—
264C22×C4—92no—
265C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
266C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
267C22⋊C4—22no—
268C22⋊C4—184no—
269C22×C3—24no—
270C22×C3—22no—
271C22×C3—31no—
272C22×C3—31no—
273C22×C3—32no—
274C22×C3—32no—
275C22×C3—31no—
276C22×C3—42no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—62no—
280C22×C3—64no—
281C22×C3—121no—
282C22×C3—121no—
283C22×C3—122no—
284C22×C3C2×C3×Dic611yescharacteristic
285C22×C3C3×#23511yescharacteristic
286C22×C3C3×D3×D412yes—
287C22×C3C3×#23712yes—
288C22×C3C2×C3×#6711yescharacteristic
289C3×C4—22no—
290C3×C4—41no—
291C3×C4—62no—
292C3×C4—121no—
293C3×C4—184no—
294C3×C4—362no—
295Dic3—64no—
296Dic3—122no—
297C32#494711yescharacteristic
298C9—21no—
299C9C3×#39311yescharacteristic, complemented
300C23—21no—
301C23—22no—
302C23—22no—
303C23—21no—
304C233−1+2⋊Q812yescomplemented
305C23C22×AGL(1,ℤ/9)11yescharacteristic
306Q8—188no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—94no—
311C2×C4—182no—
312C2×C4—182no—
313C2×C4—182no—
314C2×C4—182no—
315C2×C4—182no—
316C2×C4#611⋊C312yes—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—32no—
321C2×C3—32no—
322C2×C3—42no—
323C2×C3—122no—
324C2×C3C3×#87812yes—
325C2×C3C3×#88512yes—
326C2×C3C3×#92411yescharacteristic
327C2×C3C3×#93611yescharacteristic
328C2×C3C3×#93711yescharacteristic
329C22—22no—
330C22—24no—
331C22—41no—
332C22—41no—
333C22—42no—
334C22C2×3−1+2⋊Q811yescharacteristic
335C22#3697⋊C311yescharacteristic
336C22D4×AGL(1,ℤ/9)12yes—
337C22#3699⋊C312yes—
338C22C2×#611⋊C311yescharacteristic
339C4—22no—
340C4—41no—
341C4—184no—
342C4—362no—
343C3—31no—
344C3C3×#494711yescharacteristic
345C2—42no—
346C2#63440⋊C312yes—
347C2#63447⋊C312yes—
348C2#63486⋊C311yescharacteristic
349C2#63498⋊C311yescharacteristic
350C2#63499⋊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×#393
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×3−1+2⋊Q8C2semidirect
4. 3−1+2⋊#95C2semidirect
5. 3−1+2⋊#95C2semidirect
7. #10588627C3semidirect
20. C22×Dic18C2×C3semidirect
26. C9⋊#95C2×C3semidirect
27. C9⋊#95C2×C3semidirect
32. C23×3−1+2Q8semidirect
120. C23×C9C3×Q8semidirect
207. 3−1+2#393semidirect
299. C9C3×#393semidirect
304. C233−1+2⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×3−1+2⋊Q82
C2×C22⋊C4×3−1+22
C2×#3676⋊C32
3−1+2⋊#952
C3×#49473
#105886273