Finite Group Databasebeta

← #10588656⋊C3 #13142644

Subgroups — #10588656⋊C3 #13142644

Order 1728.

556 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×3−1+2C211yesmaximal, characteristic
2C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
3C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
4C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
5C2×C4×3−1+2⋊C4C211yesmaximal, characteristic
63−1+2⋊#95C211yesmaximal, characteristic
73−1+2⋊#95C211yesmaximal, characteristic
8C3×#4976—31nomaximal
9#10588656C311yesmaximal, 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—
18(3−1+2⋊C4)⋊C4—22no—
19(3−1+2⋊C4)⋊C4—22no—
20(3−1+2⋊C4)⋊C4C414yescomplemented
21C4×3−1+2⋊C4—22no—
22C2×C32×C4⋊C4D311yescharacteristic
23C2×C3×Dic3⋊C4—31no—
24C2×C3×Dic3⋊C4—31no—
25C2×C3×Dic3⋊C4—31no—
26C2×C3×C4×Dic3—31no—
27C3×C3⋊#95—31no—
28C3×C3⋊#95—31no—
29C2×C9×C4⋊C4—21no—
30C2×C9×C4⋊C4C2×C311yescharacteristic
31C2×Dic9⋊C4C2×C311yescharacteristic
32C2×Dic9⋊C4C2×C311yescharacteristic
33C2×Dic9⋊C4C2×C311yescharacteristic
34C2×C4×Dic9C2×C311yescharacteristic
35C9⋊#95C2×C311yescharacteristic
36C9⋊#95C2×C311yescharacteristic
37C23×3−1+2C2311yescharacteristic
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+2C2×C412yes—
46C2×C4×3−1+2D412yes—
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⋊C4C2×C412yes—
55C2×3−1+2⋊C4C2×C412yes—
56C2×3−1+2⋊C4D412yes—
57C2×3−1+2⋊C4Q812yes—
58C2×3−1+2⋊C4Q812yes—
59C3×#389—91no—
60#4976—31no—
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C3×Dic3—31no—
65C22×C3×Dic3—31no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—31no—
68C32×C4⋊C4—22no—
69C3×Dic3⋊C4—34no—
70C3×Dic3⋊C4—62no—
71C3×Dic3⋊C4—62no—
72C3×C4×Dic3—62no—
73C22×C4×C9—21no—
74C22×C4×C9—21no—
75C22×C4×C9—21no—
76C22×C4×C9C22×C311yescharacteristic
77C22×C4×C9C22×C311yescharacteristic
78C22×C4×C9C22×C311yescharacteristic
79C22×Dic9C22×C311yescharacteristic
80C22×Dic9C22×C311yescharacteristic
81C22×Dic9C22×C311yescharacteristic
82C22×Dic9C22×C311yescharacteristic
83C9×C4⋊C4—24no—
84C9×C4⋊C4—22no—
85Dic9⋊C4—22no—
86Dic9⋊C4—22no—
87Dic9⋊C4C3×C414yescomplemented
88C4×Dic9—22no—
89C22×3−1+2C22×C411yescharacteristic
90C22×3−1+2C2×D411yescharacteristic
91C22×3−1+2C2×D411yescharacteristic
92C22×3−1+2C2×Q811yescharacteristic
93C22×3−1+2C2×Q811yescharacteristic
94C22×3−1+2#4111yescharacteristic
95C22×3−1+2#4111yescharacteristic
96C4×3−1+2—22no—
97C4×3−1+2—22no—
98C4×3−1+2—41no—
993−1+2⋊C4—22no—
1003−1+2⋊C4—22no—
1013−1+2⋊C4—41no—
1023−1+2⋊C4C4⋊C414yescomplemented
103C2×C3×C4⋊C4—31no—
104C2×C3×C4⋊C4—91no—
105C2×C3×C4⋊C4—91no—
106C2×C3×C4⋊C4—91no—
107C2×C3×C4⋊C4C3×D311yescharacteristic
108C2×C3×C42—91no—
109C2×Dic3⋊C4—31no—
110C2×Dic3⋊C4—31no—
111C2×Dic3⋊C4—31no—
112C2×C4×Dic3—31no—
113C3×#95—91no—
114C3×#95—91no—
115C3⋊#95—31no—
116C3⋊#95—31no—
117C23×C32C22×D311yescharacteristic
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4C4×D312yes—
126C2×C32×C4#6712yes—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C23×C9—21no—
140C23×C9C23×C311yescharacteristic
141C2×C4×C9—22no—
142C2×C4×C9—22no—
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—22no—
148C2×C4×C9—21no—
149C2×C4×C9—21no—
150C2×C4×C9—21no—
151C2×C4×C9—22no—
152C2×C4×C9—22no—
153C2×C4×C9—41no—
154C2×C4×C9—41no—
155C2×C4×C9—41no—
156C2×C4×C9—41no—
157C2×C4×C9C2×C3×C412yes—
158C2×C4×C9C3×D412yes—
159C2×Dic9—21no—
160C2×Dic9—21no—
161C2×Dic9—21no—
162C2×Dic9—21no—
163C2×Dic9—21no—
164C2×Dic9—21no—
165C2×Dic9—21no—
166C2×Dic9C2×C3×C412yes—
167C2×Dic9C2×C3×C412yes—
168C2×Dic9C3×D412yes—
169C2×Dic9C3×Q812yes—
170C2×Dic9C3×Q812yes—
171#389—91noSylow
172C2×3−1+2C2×C4⋊C411yescharacteristic
173C2×3−1+2C4×D411yescharacteristic
174C2×3−1+2C4×Q811yescharacteristic
175C2×3−1+2#12111yescharacteristic
176C2×3−1+2C22⋊Q811yescharacteristic
177C2×3−1+2#12511yescharacteristic
178C2×3−1+2C4⋊Q811yescharacteristic
179C22×C3×C4—31no—
180C22×C3×C4—31no—
181C22×C3×C4—31no—
182C22×C3×C4—91no—
183C22×C3×C4—91no—
184C22×C3×C4—91no—
185C22×C3×C4—91no—
186C22×C3×C4C2×C3×D311yescharacteristic
187C22×C3×C4C2×C3×D311yescharacteristic
188C22×C3×C4C2×C3×D311yescharacteristic
189C22×Dic3—31no—
190C22×Dic3—31no—
191C22×Dic3—31no—
192C22×Dic3—31no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—62no—
195C3×C4⋊C4—94no—
196C3×C4⋊C4—182no—
197C3×C4⋊C4—182no—
198C3×C42—182no—
199Dic3⋊C4—34no—
200Dic3⋊C4—62no—
201Dic3⋊C4—62no—
202C4×Dic3—62no—
203C22×C32C2×C4×D311yescharacteristic
204C22×C32#23511yescharacteristic
205C22×C32D3×D411yescharacteristic
206C22×C32#23711yescharacteristic
207C22×C32D3×Q811yescharacteristic
208C22×C32D3×Q811yescharacteristic
209C22×C32C2×#6711yescharacteristic
210C32×C4—22no—
211C32×C4—22no—
212C32×C4—41no—
213C3×Dic3—34no—
214C3×Dic3—62no—
215C3×Dic3—62no—
216C3×Dic3—121no—
217C22×C9—21no—
218C22×C9—21no—
219C22×C9—21no—
220C22×C9—21no—
221C22×C9—21no—
222C22×C9—21no—
223C22×C9—21no—
224C22×C9C22×C3×C411yescharacteristic
225C22×C9C2×C3×D411yescharacteristic
226C22×C9C2×C3×D411yescharacteristic
227C22×C9C2×C3×Q811yescharacteristic
228C22×C9C2×C3×Q811yescharacteristic
229C22×C9C3×#4111yescharacteristic
230C22×C9C3×#4111yescharacteristic
231C4×C9—24no—
232C4×C9—22no—
233C4×C9—22no—
234C4×C9—42no—
235C4×C9—42no—
236C4×C9—41no—
237Dic9—22no—
238Dic9—22no—
239Dic9—41no—
240Dic9C3×C4⋊C414yescomplemented
241C2×C4⋊C4—91no—
242C2×C4⋊C4—91no—
243C2×C4⋊C4—91no—
244C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic
245C2×C42—91no—
246#95—91no—
247#95—91no—
2483−1+2#38911yescharacteristic, complemented, Sylow
249C23×C3—31no—
250C23×C3C22×C3×D311yescharacteristic
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—32no—
259C2×C3×C4—32no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—92no—
268C2×C3×C4—92no—
269C2×C3×C4—92no—
270C2×C3×C4—92no—
271C2×C3×C4—92no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4C3×C4×D312yes—
280C2×C3×C4C3×#6712yes—
281C2×Dic3—32no—
282C2×Dic3—32no—
283C2×Dic3—32no—
284C2×Dic3—32no—
285C2×Dic3—32no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×C32#88711yescharacteristic
294C2×C32#88911yescharacteristic
295C2×C32D3×C4⋊C411yescharacteristic
296C2×C32#89611yescharacteristic
297C2×C32C4×#6711yescharacteristic
298C2×C32#93911yescharacteristic
299C2×C32#94411yescharacteristic
300C2×C9—21no—
301C2×C9—21no—
302C2×C9—21no—
303C2×C9—21no—
304C2×C9—21no—
305C2×C9—21no—
306C2×C9—21no—
307C2×C9C2×C3×C4⋊C411yescharacteristic
308C2×C9C3×C4×D411yescharacteristic
309C2×C9C3×C4×Q811yescharacteristic
310C2×C9C3×#12111yescharacteristic
311C2×C9C3×C22⋊Q811yescharacteristic
312C2×C9C3×#12511yescharacteristic
313C2×C9C3×C4⋊Q811yescharacteristic
314C22×C4—91no—
315C22×C4—91no—
316C22×C4—91no—
317C22×C4—91no—
318C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
319C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
320C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—94no—
323C4⋊C4—182no—
324C4⋊C4—182no—
325C42—182no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3C2×C3×C4×D311yescharacteristic
334C22×C3C3×#23511yescharacteristic
335C22×C3C3×D3×D411yescharacteristic
336C22×C3C3×#23711yescharacteristic
337C22×C3C3×D3×Q811yescharacteristic
338C22×C3C3×D3×Q811yescharacteristic
339C22×C3C2×C3×#6711yescharacteristic
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—41no—
343C3×C4—62no—
344C3×C4—62no—
345C3×C4—94no—
346C3×C4—121no—
347C3×C4—182no—
348C3×C4—182no—
349C3×C4—361no—
350Dic3—34no—
351Dic3—62no—
352Dic3—62no—
353Dic3—121no—
354C32#497611yescharacteristic
355C9—21no—
356C9C3×#38911yescharacteristic, complemented
357C23C22×AGL(1,ℤ/9)11yescharacteristic
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—92no—
366C2×C4—92no—
367C2×C4—92no—
368C2×C4—92no—
369C2×C4—92no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4—181no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—181no—
376C2×C4—181no—
377C2×C4C4×AGL(1,ℤ/9)12yes—
378C2×C4#611⋊C312yes—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3C3×#88711yescharacteristic
387C2×C3C3×#88911yescharacteristic
388C2×C3C3×D3×C4⋊C411yescharacteristic
389C2×C3C3×#89611yescharacteristic
390C2×C3C3×C4×#6711yescharacteristic
391C2×C3C3×#93911yescharacteristic
392C2×C3C3×#94411yescharacteristic
393C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
394C22#3697⋊C311yescharacteristic
395C22D4×AGL(1,ℤ/9)11yescharacteristic
396C22#3699⋊C311yescharacteristic
397C22Q8×AGL(1,ℤ/9)11yescharacteristic
398C22Q8×AGL(1,ℤ/9)11yescharacteristic
399C22C2×#611⋊C311yescharacteristic
400C4—22no—
401C4—22no—
402C4—41no—
403C4—94no—
404C4—182no—
405C4—182no—
406C4—361no—
407C3—31no—
408C3C3×#497611yescharacteristic
409C2#63449⋊C311yescharacteristic
410C2#63451⋊C311yescharacteristic
411C2C4⋊C4×AGL(1,ℤ/9)11yescharacteristic
412C2#63458⋊C311yescharacteristic
413C2C4×#611⋊C311yescharacteristic
414C2#63501⋊C311yescharacteristic
415C2#63506⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×3−1+2C2
prime radicalC23×C32C22×D3
cosocleC23×C9C23×C3
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#389
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
9. #10588656C3semidirect
20. (3−1+2⋊C4)⋊C4C4semidirect
87. Dic9⋊C4C3×C4semidirect
102. 3−1+2⋊C4C4⋊C4semidirect
240. Dic9C3×C4⋊C4semidirect
248. 3−1+2#389semidirect
356. C9C3×#389semidirect

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×(3−1+2⋊C4)⋊C42
C2×C4×3−1+2⋊C42
3−1+2⋊#952
C3×#49763
#105886563