Finite Group Databasebeta

← #10588337⋊C3 #13141376

Subgroups — #10588337⋊C3 #13141376

Order 1728.

488 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic
2C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
3C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
4C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
53−1+2×#95C211yesmaximal, characteristic
63−1+2⋊#95C211yesmaximal, characteristic
73−1+2⋊#95C211yesmaximal, characteristic
8C3×#4657—31nomaximal
9#10588337C311yesmaximal, characteristic, complemented
10C22×C4×3−1+2C2211yescharacteristic
11C22×C4×3−1+2C2211yescharacteristic
12C22×C4×3−1+2C2211yescharacteristic
13C22×3−1+2⋊C4C2211yescharacteristic
14C22×3−1+2⋊C4C2211yescharacteristic
15C22×3−1+2⋊C4C2211yescharacteristic
16C22×3−1+2⋊C4C2211yescharacteristic
17C4⋊(3−1+2⋊C4)—22no—
18(3−1+2⋊C4)⋊C4—22no—
19(3−1+2⋊C4)⋊C4—22no—
20(3−1+2⋊C4)⋊C4—22no—
21C2×C3×C4⋊Dic3—31no—
22C2×C3×Dic3⋊C4—31no—
23C2×C3×Dic3⋊C4—31no—
24C2×C3×Dic3⋊C4—31no—
25C32×#95D311yescharacteristic
26C3×C3⋊#95—31no—
27C3×C3⋊#95—31no—
28C2×C4⋊Dic9C2×C311yescharacteristic
29C2×Dic9⋊C4C2×C311yescharacteristic
30C2×Dic9⋊C4C2×C311yescharacteristic
31C2×Dic9⋊C4C2×C311yescharacteristic
32C9×#95—21no—
33C9×#95C2×C311yescharacteristic
34C9⋊#95C2×C311yescharacteristic
35C9⋊#95C2×C311yescharacteristic
36C23×3−1+2C2311yescharacteristic
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2—21no—
41C2×C4×3−1+2—21no—
42C2×C4×3−1+2—21no—
43C2×C4×3−1+2—21no—
44C2×C4×3−1+2—21no—
45C2×C4×3−1+2Q812yes—
46C2×3−1+2⋊C4—21no—
47C2×3−1+2⋊C4—21no—
48C2×3−1+2⋊C4—21no—
49C2×3−1+2⋊C4—21no—
50C2×3−1+2⋊C4—21no—
51C2×3−1+2⋊C4—21no—
52C2×3−1+2⋊C4—21no—
53C2×3−1+2⋊C4—21no—
54C2×3−1+2⋊C4—21no—
55C2×3−1+2⋊C4D412yes—
56C2×3−1+2⋊C4D412yes—
57C2×3−1+2⋊C4Q812yes—
58C3×#398—91no—
59#4657—31no—
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C22×C3×Dic3—31no—
64C22×C3×Dic3—31no—
65C22×C3×Dic3—31no—
66C22×C3×Dic3—31no—
67C3×C4⋊Dic3—62no—
68C3×Dic3⋊C4—62no—
69C3×Dic3⋊C4—62no—
70C3×Dic3⋊C4—62no—
71C22×C4×C9—21no—
72C22×C4×C9—21no—
73C22×C4×C9—21no—
74C22×C4×C9C22×C311yescharacteristic
75C22×C4×C9C22×C311yescharacteristic
76C22×C4×C9C22×C311yescharacteristic
77C22×Dic9C22×C311yescharacteristic
78C22×Dic9C22×C311yescharacteristic
79C22×Dic9C22×C311yescharacteristic
80C22×Dic9C22×C311yescharacteristic
81C4⋊Dic9—22no—
82Dic9⋊C4—22no—
83Dic9⋊C4—22no—
84Dic9⋊C4—22no—
85C22×3−1+2C2×D411yescharacteristic
86C22×3−1+2C2×D411yescharacteristic
87C22×3−1+2C2×Q811yescharacteristic
88C22×3−1+2C2×Q811yescharacteristic
89C22×3−1+2#4111yescharacteristic
90C22×3−1+2#4111yescharacteristic
91C22×3−1+2#4111yescharacteristic
92C4×3−1+2—22no—
93C4×3−1+2—41no—
94C4×3−1+2—41no—
953−1+2⋊C4—22no—
963−1+2⋊C4—22no—
973−1+2⋊C4—22no—
983−1+2⋊C4—41no—
99C2×C3×C4⋊C4—91no—
100C2×C3×C4⋊C4—91no—
101C2×C3×C4⋊C4—91no—
102C2×C3×C4⋊C4—91no—
103C2×C4⋊Dic3—31no—
104C2×Dic3⋊C4—31no—
105C2×Dic3⋊C4—31no—
106C2×Dic3⋊C4—31no—
107C3×#95—31no—
108C3×#95—91no—
109C3×#95—91no—
110C3×#95C3×D311yescharacteristic
111C3⋊#95—31no—
112C3⋊#95—31no—
113C23×C32C22×D311yescharacteristic
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4Dic612yes—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—32no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C23×C9—21no—
136C23×C9C23×C311yescharacteristic
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—22no—
140C2×C4×C9—22no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—22no—
148C2×C4×C9—41no—
149C2×C4×C9—41no—
150C2×C4×C9—41no—
151C2×C4×C9—41no—
152C2×C4×C9—41no—
153C2×C4×C9—41no—
154C2×C4×C9C3×Q812yes—
155C2×Dic9—21no—
156C2×Dic9—21no—
157C2×Dic9—21no—
158C2×Dic9—21no—
159C2×Dic9—21no—
160C2×Dic9—21no—
161C2×Dic9—21no—
162C2×Dic9—21no—
163C2×Dic9—21no—
164C2×Dic9C3×D412yes—
165C2×Dic9C3×D412yes—
166C2×Dic9C3×Q812yes—
167#398—91noSylow
168C2×3−1+2#12111yescharacteristic
169C2×3−1+2C22⋊Q811yescharacteristic
170C2×3−1+2C22⋊Q811yescharacteristic
171C2×3−1+2#12311yescharacteristic
172C2×3−1+2#12511yescharacteristic
173C2×3−1+2#12511yescharacteristic
174C2×3−1+2C4⋊Q811yescharacteristic
175C22×C3×C4—31no—
176C22×C3×C4—31no—
177C22×C3×C4—31no—
178C22×C3×C4—91no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×C3×C4—91no—
182C22×C3×C4C2×C3×D311yescharacteristic
183C22×C3×C4C2×C3×D311yescharacteristic
184C22×C3×C4C2×C3×D311yescharacteristic
185C22×Dic3—31no—
186C22×Dic3—31no—
187C22×Dic3—31no—
188C22×Dic3—31no—
189C3×C4⋊C4—182no—
190C3×C4⋊C4—182no—
191C3×C4⋊C4—182no—
192C3×C4⋊C4—182no—
193C4⋊Dic3—62no—
194Dic3⋊C4—62no—
195Dic3⋊C4—62no—
196Dic3⋊C4—62no—
197C22×C32C2×Dic611yescharacteristic
198C22×C32#23511yescharacteristic
199C22×C32#23511yescharacteristic
200C22×C32D3×D411yescharacteristic
201C22×C32D3×D411yescharacteristic
202C22×C32#23711yescharacteristic
203C22×C32D3×Q811yescharacteristic
204C32×C4—22no—
205C32×C4—41no—
206C32×C4—41no—
207C3×Dic3—62no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C3×Dic3—121no—
211C22×C9—21no—
212C22×C9—21no—
213C22×C9—21no—
214C22×C9—21no—
215C22×C9—21no—
216C22×C9—21no—
217C22×C9—21no—
218C22×C9C2×C3×D411yescharacteristic
219C22×C9C2×C3×D411yescharacteristic
220C22×C9C2×C3×Q811yescharacteristic
221C22×C9C2×C3×Q811yescharacteristic
222C22×C9C3×#4111yescharacteristic
223C22×C9C3×#4111yescharacteristic
224C22×C9C3×#4111yescharacteristic
225C4×C9—22no—
226C4×C9—42no—
227C4×C9—42no—
228C4×C9—42no—
229C4×C9—41no—
230C4×C9—41no—
231Dic9—22no—
232Dic9—22no—
233Dic9—22no—
234Dic9—41no—
235C2×C4⋊C4—91no—
236C2×C4⋊C4—91no—
237C2×C4⋊C4—91no—
238C2×C4⋊C4—91no—
239#95—91no—
240#95—91no—
241#95AGL(1,ℤ/9)11yescharacteristic
2423−1+2#39811yescharacteristic, complemented, Sylow
243C23×C3—31no—
244C23×C3C22×C3×D311yescharacteristic
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—32no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—92no—
263C2×C3×C4—92no—
264C2×C3×C4—92no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4C3×Dic612yes—
275C2×Dic3—32no—
276C2×Dic3—32no—
277C2×Dic3—32no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×C32#87011yescharacteristic
288C2×C32#87811yescharacteristic
289C2×C32#88311yescharacteristic
290C2×C32#88411yescharacteristic
291C2×C32#88811yescharacteristic
292C2×C32#88911yescharacteristic
293C2×C32#89611yescharacteristic
294C2×C9—21no—
295C2×C9—21no—
296C2×C9—21no—
297C2×C9—21no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9C3×#12111yescharacteristic
302C2×C9C3×C22⋊Q811yescharacteristic
303C2×C9C3×C22⋊Q811yescharacteristic
304C2×C9C3×#12311yescharacteristic
305C2×C9C3×#12511yescharacteristic
306C2×C9C3×#12511yescharacteristic
307C2×C9C3×C4⋊Q811yescharacteristic
308C22×C4—91no—
309C22×C4—91no—
310C22×C4—91no—
311C22×C4—91no—
312C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
313C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
314C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
315C4⋊C4—182no—
316C4⋊C4—182no—
317C4⋊C4—182no—
318C4⋊C4—182no—
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3C2×C3×Dic611yescharacteristic
327C22×C3C3×#23511yescharacteristic
328C22×C3C3×#23511yescharacteristic
329C22×C3C3×D3×D411yescharacteristic
330C22×C3C3×D3×D411yescharacteristic
331C22×C3C3×#23711yescharacteristic
332C22×C3C3×D3×Q811yescharacteristic
333C3×C4—22no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—62no—
337C3×C4—121no—
338C3×C4—121no—
339C3×C4—182no—
340C3×C4—182no—
341C3×C4—182no—
342C3×C4—361no—
343Dic3—62no—
344Dic3—62no—
345Dic3—62no—
346Dic3—121no—
347C32#465711yescharacteristic
348C9—21no—
349C9C3×#39811yescharacteristic, complemented
350C23C22×AGL(1,ℤ/9)11yescharacteristic
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—92no—
360C2×C4—92no—
361C2×C4—92no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4—181no—
365C2×C4—181no—
366C2×C4—181no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C43−1+2⋊Q812yes—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3C3×#87011yescharacteristic
380C2×C3C3×#87811yescharacteristic
381C2×C3C3×#88311yescharacteristic
382C2×C3C3×#88411yescharacteristic
383C2×C3C3×#88811yescharacteristic
384C2×C3C3×#88911yescharacteristic
385C2×C3C3×#89611yescharacteristic
386C22C2×3−1+2⋊Q811yescharacteristic
387C22#3697⋊C311yescharacteristic
388C22#3697⋊C311yescharacteristic
389C22D4×AGL(1,ℤ/9)11yescharacteristic
390C22D4×AGL(1,ℤ/9)11yescharacteristic
391C22#3699⋊C311yescharacteristic
392C22Q8×AGL(1,ℤ/9)11yescharacteristic
393C4—22no—
394C4—41no—
395C4—41no—
396C4—182no—
397C4—182no—
398C4—182no—
399C4—361no—
400C3—31no—
401C3C3×#465711yescharacteristic
402C2#63432⋊C311yescharacteristic
403C2#63440⋊C311yescharacteristic
404C2#63445⋊C311yescharacteristic
405C2#63446⋊C311yescharacteristic
406C2#63450⋊C311yescharacteristic
407C2#63451⋊C311yescharacteristic
408C2#63458⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
Fitting3−1+2×#95C2
prime radicalC23×C32C22×D3
hypercenter#95AGL(1,ℤ/9)
hypocenterC9C3×#398
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
9. #10588337C3semidirect
242. 3−1+2#398semidirect
349. C9C3×#398semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
3−1+2×#952
3−1+2⋊#952
C3×#46573
#105883373