Finite Group Databasebeta

← #10588662⋊C3 #13142650

Subgroups — #10588662⋊C3 #13142650

Order 1728.

496 conjugacy classes of subgroups.

The subgroup lattice has 411 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)⋊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×#4982—31nomaximal
9#10588662C311yesmaximal, 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⋊C4×3−1+2—22no—
18C4⋊(3−1+2⋊C4)—22no—
19(3−1+2⋊C4)⋊C4—22no—
20(3−1+2⋊C4)⋊C4—22no—
21C2×C32×C4⋊C4D311yescharacteristic
22C2×C3×C4⋊Dic3—31no—
23C2×C3×Dic3⋊C4—31no—
24C2×C3×Dic3⋊C4—31no—
25C3×C3⋊#95—31no—
26C3×C3⋊#95—31no—
27C3×C3⋊#95—31no—
28C2×C9×C4⋊C4—21no—
29C2×C9×C4⋊C4C2×C311yescharacteristic
30C2×C4⋊Dic9C2×C311yescharacteristic
31C2×Dic9⋊C4C2×C311yescharacteristic
32C2×Dic9⋊C4C2×C311yescharacteristic
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+2D412yes—
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⋊C4—21no—
56C2×3−1+2⋊C4D412yes—
57C2×3−1+2⋊C4Q812yes—
58C3×#398—91no—
59#4982—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—
67C32×C4⋊C4—22no—
68C3×C4⋊Dic3—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
81C9×C4⋊C4—24no—
82C9×C4⋊C4—22no—
83C4⋊Dic9—22no—
84Dic9⋊C4—22no—
85Dic9⋊C4—22no—
86C22×3−1+2C2×D411yescharacteristic
87C22×3−1+2C2×D411yescharacteristic
88C22×3−1+2C2×Q811yescharacteristic
89C22×3−1+2C2×Q811yescharacteristic
90C22×3−1+2#4111yescharacteristic
91C22×3−1+2#4111yescharacteristic
92C22×3−1+2#4111yescharacteristic
93C4×3−1+2—22no—
94C4×3−1+2—22no—
95C4×3−1+2—41no—
963−1+2⋊C4—22no—
973−1+2⋊C4—22no—
983−1+2⋊C4—41no—
993−1+2⋊C4—41no—
100C2×C3×C4⋊C4—31no—
101C2×C3×C4⋊C4—91no—
102C2×C3×C4⋊C4—91no—
103C2×C3×C4⋊C4—91no—
104C2×C3×C4⋊C4C3×D311yescharacteristic
105C2×C4⋊Dic3—31no—
106C2×Dic3⋊C4—31no—
107C2×Dic3⋊C4—31no—
108C3×#95—91no—
109C3×#95—91no—
110C3×#95—91no—
111C3⋊#95—31no—
112C3⋊#95—31no—
113C3⋊#95—31no—
114C23×C32C22×D311yescharacteristic
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×C32×C4#6712yes—
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—
135C2×C3×Dic3—61no—
136C23×C9—21no—
137C23×C9C23×C311yescharacteristic
138C2×C4×C9—22no—
139C2×C4×C9—22no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—22no—
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—21no—
148C2×C4×C9—22no—
149C2×C4×C9—22no—
150C2×C4×C9—41no—
151C2×C4×C9—41no—
152C2×C4×C9—41no—
153C2×C4×C9—41no—
154C2×C4×C9C3×D412yes—
155C2×C4×C9C3×Q812yes—
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×Dic9—21no—
165C2×Dic9—21no—
166C2×Dic9C3×D412yes—
167C2×Dic9C3×Q812yes—
168#398—91noSylow
169C2×3−1+2#12111yescharacteristic
170C2×3−1+2C22⋊Q811yescharacteristic
171C2×3−1+2C22⋊Q811yescharacteristic
172C2×3−1+2#12311yescharacteristic
173C2×3−1+2#12511yescharacteristic
174C2×3−1+2#12511yescharacteristic
175C2×3−1+2C4⋊Q811yescharacteristic
176C22×C3×C4—31no—
177C22×C3×C4—31no—
178C22×C3×C4—31no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×C3×C4—91no—
182C22×C3×C4—91no—
183C22×C3×C4C2×C3×D311yescharacteristic
184C22×C3×C4C2×C3×D311yescharacteristic
185C22×C3×C4C2×C3×D311yescharacteristic
186C22×Dic3—31no—
187C22×Dic3—31no—
188C22×Dic3—31no—
189C22×Dic3—31no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—62no—
192C3×C4⋊C4—182no—
193C3×C4⋊C4—182no—
194C3×C4⋊C4—182no—
195C4⋊Dic3—62no—
196Dic3⋊C4—62no—
197Dic3⋊C4—62no—
198C22×C32C2×Dic611yescharacteristic
199C22×C32#23511yescharacteristic
200C22×C32D3×D411yescharacteristic
201C22×C32#23711yescharacteristic
202C22×C32D3×Q811yescharacteristic
203C22×C32#23911yescharacteristic
204C22×C32C2×#6711yescharacteristic
205C32×C4—22no—
206C32×C4—22no—
207C32×C4—41no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212C22×C9—21no—
213C22×C9—21no—
214C22×C9—21no—
215C22×C9—21no—
216C22×C9—21no—
217C22×C9—21no—
218C22×C9—21no—
219C22×C9C2×C3×D411yescharacteristic
220C22×C9C2×C3×D411yescharacteristic
221C22×C9C2×C3×Q811yescharacteristic
222C22×C9C2×C3×Q811yescharacteristic
223C22×C9C3×#4111yescharacteristic
224C22×C9C3×#4111yescharacteristic
225C22×C9C3×#4111yescharacteristic
226C4×C9—24no—
227C4×C9—22no—
228C4×C9—22no—
229C4×C9—42no—
230C4×C9—42no—
231C4×C9—41no—
232Dic9—22no—
233Dic9—22no—
234Dic9—41no—
235Dic9—41no—
236C2×C4⋊C4—91no—
237C2×C4⋊C4—91no—
238C2×C4⋊C4—91no—
239C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic
240#95—91no—
241#95—91no—
242#95—91no—
2433−1+2#39811yescharacteristic, complemented, Sylow
244C23×C3—31no—
245C23×C3C22×C3×D311yescharacteristic
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—32no—
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—181no—
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×C3×C4C3×#6712yes—
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×Dic3—61no—
288C2×C32#88811yescharacteristic
289C2×C32#88911yescharacteristic
290C2×C32#89011yescharacteristic
291C2×C32#89411yescharacteristic
292C2×C32#92411yescharacteristic
293C2×C32#93911yescharacteristic
294C2×C32#94611yescharacteristic
295C2×C9—21no—
296C2×C9—21no—
297C2×C9—21no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—21no—
302C2×C9C3×#12111yescharacteristic
303C2×C9C3×C22⋊Q811yescharacteristic
304C2×C9C3×C22⋊Q811yescharacteristic
305C2×C9C3×#12311yescharacteristic
306C2×C9C3×#12511yescharacteristic
307C2×C9C3×#12511yescharacteristic
308C2×C9C3×C4⋊Q811yescharacteristic
309C22×C4—91no—
310C22×C4—91no—
311C22×C4—91no—
312C22×C4—91no—
313C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
314C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
315C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
316C4⋊C4—22no—
317C4⋊C4—182no—
318C4⋊C4—182no—
319C4⋊C4—182no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3C2×C3×Dic611yescharacteristic
328C22×C3C3×#23511yescharacteristic
329C22×C3C3×D3×D411yescharacteristic
330C22×C3C3×#23711yescharacteristic
331C22×C3C3×D3×Q811yescharacteristic
332C22×C3C3×#23911yescharacteristic
333C22×C3C2×C3×#6711yescharacteristic
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—41no—
337C3×C4—62no—
338C3×C4—62no—
339C3×C4—121no—
340C3×C4—182no—
341C3×C4—182no—
342C3×C4—361no—
343C3×C4—361no—
344Dic3—62no—
345Dic3—62no—
346Dic3—121no—
347Dic3—121no—
348C32#498211yescharacteristic
349C9—21no—
350C9C3×#39811yescharacteristic, complemented
351C23C22×AGL(1,ℤ/9)11yescharacteristic
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—181no—
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×C4#611⋊C312yes—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3C3×#88811yescharacteristic
381C2×C3C3×#88911yescharacteristic
382C2×C3C3×#89011yescharacteristic
383C2×C3C3×#89411yescharacteristic
384C2×C3C3×#92411yescharacteristic
385C2×C3C3×#93911yescharacteristic
386C2×C3C3×#94611yescharacteristic
387C22C2×3−1+2⋊Q811yescharacteristic
388C22#3697⋊C311yescharacteristic
389C22D4×AGL(1,ℤ/9)11yescharacteristic
390C22#3699⋊C311yescharacteristic
391C22Q8×AGL(1,ℤ/9)11yescharacteristic
392C22#8872411yescharacteristic
393C22C2×#611⋊C311yescharacteristic
394C4—22no—
395C4—22no—
396C4—41no—
397C4—182no—
398C4—182no—
399C4—361no—
400C4—361no—
401C3—31no—
402C3C3×#498211yescharacteristic
403C2#63450⋊C311yescharacteristic
404C2#63451⋊C311yescharacteristic
405C2#63452⋊C311yescharacteristic
406C2#63456⋊C311yescharacteristic
407C2#63486⋊C311yescharacteristic
408C2#63501⋊C311yescharacteristic
409C2#63508⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×3−1+2C2
prime radicalC23×C32C22×D3
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#398
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
9. #10588662C3semidirect
243. 3−1+2#398semidirect
350. C9C3×#398semidirect

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
3−1+2⋊#952
C3×#49823
#105886623