Finite Group Databasebeta

← C2×#63440⋊C3 #13152372

Subgroups — C2×#63440⋊C3 #13152372

Order 1728.

1032 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×3−1+2⋊C4C211yesmaximal, characteristic
2C22×3−1+2⋊Q8C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×3−1+2C211yesmaximal, characteristic
4C2×#3676⋊C3C211yesmaximal, characteristic
5C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
6C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
7C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
8#63440⋊C3C218yesmaximal, direct product factor
9C2×C3×#878—31nomaximal
10C2×#63440C311yesmaximal, characteristic, complemented
11C24×3−1+2C2211yescharacteristic
12C22×C4×3−1+2C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C22×3−1+2⋊C4—24no—
15C22×3−1+2⋊C4C2211yescharacteristic
16C22×3−1+2⋊C4C2211yescharacteristic
17C22×3−1+2⋊C4C2211yescharacteristic
18C22×3−1+2⋊C4C2214yes—
19C22×3−1+2⋊C4C2211yescharacteristic
20C2×3−1+2⋊Q8—24no—
21C2×3−1+2⋊Q8C2214yescomplemented
22C22⋊C4×3−1+2C2214yes—
23#3676⋊C3C2214yes—
24C4⋊(3−1+2⋊C4)C2214yescomplemented
25(3−1+2⋊C4)⋊C4C2214yescomplemented
26(3−1+2⋊C4)⋊C4C2214yescomplemented
27C23×C3×Dic3—31no—
28C22×C3×Dic6—31no—
29C2×C32×C22⋊C4D311yescharacteristic
30C2×C3×#217—31no—
31C2×C3×C4⋊Dic3—31no—
32C2×C3×Dic3⋊C4—31no—
33C2×C3×Dic3⋊C4—31no—
34C3×#878—38no—
35C23×Dic9C2×C311yescharacteristic
36C22×Dic18C2×C311yescharacteristic, complemented
37C2×C9×C22⋊C4—21no—
38C2×C9×C22⋊C4C2×C311yescharacteristic
39C2×#3676C2×C311yescharacteristic
40C2×C4⋊Dic9C2×C311yescharacteristic, complemented
41C2×Dic9⋊C4C2×C311yescharacteristic, complemented
42C2×Dic9⋊C4C2×C311yescharacteristic, complemented
43#63440C2×C318yescomplemented
44C23×3−1+2—22no—
45C23×3−1+2—22no—
46C23×3−1+2Q814yescomplemented
47C23×3−1+2C2312yes—
48C23×3−1+2C2311yescharacteristic
49C2×C4×3−1+2—22no—
50C2×C4×3−1+2—22no—
51C2×C4×3−1+2C2312yes—
52C2×C4×3−1+2C2312yes—
53C2×3−1+2⋊C4—22no—
54C2×3−1+2⋊C4—22no—
55C2×3−1+2⋊C4—22no—
56C2×3−1+2⋊C4—28no—
57C2×3−1+2⋊C4D414yes—
58C2×3−1+2⋊C4C2312yes—
59C2×3−1+2⋊C4C2312yes—
60C2×3−1+2⋊C4C2312yes—
61C2×3−1+2⋊C4C2312yes—
623−1+2⋊Q8—28no—
63C2×C3×C22⋊Q8—91no—
64C2×#878—31no—
65C24×C32C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C3×Dic3—31no—
69C22×C3×Dic3—34no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3—64no—
74C2×C3×Dic6—34no—
75C2×C3×Dic6—64no—
76C24×C9—21no—
77C24×C9C22×C311yescharacteristic
78C32×C22⋊C4C2×D314yes—
79C3×#217—34no—
80C3×C4⋊Dic3—34no—
81C3×Dic3⋊C4—34no—
82C3×Dic3⋊C4—34no—
83C22×C4×C9—21no—
84C22×C4×C9—21no—
85C22×C4×C9C22×C311yescharacteristic
86C22×C4×C9C22×C311yescharacteristic
87C22×Dic9—24no—
88C22×Dic9C22×C311yescharacteristic
89C22×Dic9C22×C311yescharacteristic
90C22×Dic9C22×C314yes—
91C22×Dic9C22×C311yescharacteristic
92C22×Dic9C22×C311yescharacteristic
93C2×Dic18—24no—
94C2×Dic18C22×C314yescomplemented
95C9×C22⋊C4—24no—
96C9×C22⋊C4C22×C314yes—
97#3676C22×C314yes—
98C4⋊Dic9C22×C314yescomplemented
99Dic9⋊C4C22×C314yescomplemented
100Dic9⋊C4C22×C314yescomplemented
101C22×3−1+2—28no—
102C22×3−1+2—22no—
103C22×3−1+2—22no—
104C22×3−1+2C2×D412yes—
105C22×3−1+2C2×Q812yes—
106C22×3−1+2C2×Q814yescomplemented
107C22×3−1+2#4112yes—
108C22×3−1+2C2411yescharacteristic
109C4×3−1+2—22no—
110C4×3−1+2—22no—
1113−1+2⋊C4—22no—
1123−1+2⋊C4—22no—
1133−1+2⋊C4—22no—
1143−1+2⋊C4C2×D414yes—
115C22×C3×Q8—91no—
116C23×C3×C4—91no—
117C23×Dic3—31no—
118C22×Dic6—31no—
119C3×C22⋊Q8—98no—
120C2×C3×C4⋊C4—91no—
121C2×C3×C4⋊C4—91no—
122C2×C3×C4⋊C4—91no—
123C2×C3×C22⋊C4—31no—
124C2×C3×C22⋊C4—91no—
125C2×C3×C22⋊C4C3×D311yescharacteristic
126C2×#217—31no—
127C2×C4⋊Dic3—31no—
128C2×Dic3⋊C4—31no—
129C2×Dic3⋊C4—31no—
130#878—38no—
131C23×C32—22no—
132C23×C32—22no—
133C23×C32Dic614yes—
134C23×C32C22×D312yes—
135C23×C32C22×D311yescharacteristic
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4C22×D312yes—
139C2×C32×C4C22×D312yes—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—32no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—68no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C3×Dic6—68no—
150C23×C9—22no—
151C23×C9—22no—
152C23×C9—24no—
153C23×C9—21no—
154C23×C9—22no—
155C23×C9—42no—
156C23×C9—42no—
157C23×C9C3×Q814yescomplemented
158C23×C9C23×C312yes—
159C23×C9C23×C311yescharacteristic
160C2×C4×C9—22no—
161C2×C4×C9—22no—
162C2×C4×C9—22no—
163C2×C4×C9—22no—
164C2×C4×C9—42no—
165C2×C4×C9—42no—
166C2×C4×C9C23×C312yes—
167C2×C4×C9C23×C312yes—
168C2×Dic9—22no—
169C2×Dic9—22no—
170C2×Dic9—22no—
171C2×Dic9—28no—
172C2×Dic9C3×D414yes—
173C2×Dic9C23×C312yes—
174C2×Dic9C23×C312yes—
175C2×Dic9C23×C312yes—
176C2×Dic9C23×C312yes—
177Dic18—28no—
178C2×C22⋊Q8—91noSylow
179C2×3−1+2—24no—
180C2×3−1+2C22⋊Q814yescomplemented
181C2×3−1+2C22×D411yescharacteristic
182C2×3−1+2C22×Q811yescharacteristic
183C2×3−1+2C2×#4111yescharacteristic
184C24×C3—31no—
185C24×C3C2×C3×D311yescharacteristic
186C2×C3×Q8—94no—
187C2×C3×Q8—184no—
188C22×C3×C4—31no—
189C22×C3×C4—31no—
190C22×C3×C4—91no—
191C22×C3×C4—91no—
192C22×C3×C4—91no—
193C22×C3×C4—91no—
194C22×C3×C4—94no—
195C22×C3×C4—184no—
196C22×C3×C4C2×C3×D311yescharacteristic
197C22×C3×C4C2×C3×D311yescharacteristic
198C22×Dic3—31no—
199C22×Dic3—34no—
200C22×Dic3—31no—
201C22×Dic3—31no—
202C22×Dic3—31no—
203C22×Dic3—64no—
204C2×Dic6—34no—
205C2×Dic6—64no—
206C3×C4⋊C4—94no—
207C3×C4⋊C4—94no—
208C3×C4⋊C4—94no—
209C3×C22⋊C4—34no—
210C3×C22⋊C4—94no—
211C3×C22⋊C4C2×C3×D314yes—
212#217—34no—
213C4⋊Dic3—34no—
214Dic3⋊C4—34no—
215Dic3⋊C4—34no—
216C22×C32—28no—
217C22×C32—22no—
218C22×C32—22no—
219C22×C32C2×Dic614yes—
220C22×C32C2×Dic612yes—
221C22×C32D3×D412yes—
222C22×C32#23712yes—
223C22×C32C23×D311yescharacteristic
224C32×C4—22no—
225C32×C4—22no—
226C3×Dic3—34no—
227C3×Dic3—62no—
228C3×Dic3—62no—
229C3×Dic3—62no—
230C22×C9—28no—
231C22×C9—22no—
232C22×C9—22no—
233C22×C9—21no—
234C22×C9—22no—
235C22×C9—24no—
236C22×C9—22no—
237C22×C9—22no—
238C22×C9—48no—
239C22×C9—42no—
240C22×C9—42no—
241C22×C9C2×C3×D412yes—
242C22×C9C2×C3×Q812yes—
243C22×C9C2×C3×Q814yescomplemented
244C22×C9C3×#4112yes—
245C22×C9C24×C311yescharacteristic
246C4×C9—22no—
247C4×C9—22no—
248C4×C9—42no—
249C4×C9—42no—
250Dic9—22no—
251Dic9—22no—
252Dic9—22no—
253Dic9C2×C3×D414yes—
254C22×Q8—91no—
255C23×C4—91no—
256C22⋊Q8—98no—
257C2×C4⋊C4—91no—
258C2×C4⋊C4—91no—
259C2×C4⋊C4—91no—
260C2×C22⋊C4—91no—
261C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic
2623−1+2C2×C22⋊Q811yescharacteristic, complemented, Sylow
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—34no—
266C23×C3—32no—
267C23×C3—31no—
268C23×C3—62no—
269C23×C3—62no—
270C23×C3C3×Dic614yes—
271C23×C3C22×C3×D312yes—
272C23×C3C22×C3×D311yescharacteristic
273C3×Q8—188no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—32no—
277C2×C3×C4—32no—
278C2×C3×C4—62no—
279C2×C3×C4—62no—
280C2×C3×C4—92no—
281C2×C3×C4—94no—
282C2×C3×C4—92no—
283C2×C3×C4—92no—
284C2×C3×C4—92no—
285C2×C3×C4—188no—
286C2×C3×C4—182no—
287C2×C3×C4—182no—
288C2×C3×C4—182no—
289C2×C3×C4C22×C3×D312yes—
290C2×C3×C4C22×C3×D312yes—
291C2×Dic3—32no—
292C2×Dic3—34no—
293C2×Dic3—32no—
294C2×Dic3—32no—
295C2×Dic3—32no—
296C2×Dic3—68no—
297C2×Dic3—62no—
298C2×Dic3—62no—
299C2×Dic3—62no—
300Dic6—68no—
301C2×C32—24no—
302C2×C32#87814yes—
303C2×C32C22×Dic611yescharacteristic
304C2×C32C2×D3×D411yescharacteristic
305C2×C32C2×#23711yescharacteristic
306C2×C9—24no—
307C2×C9—24no—
308C2×C9—21no—
309C2×C9—21no—
310C2×C9—21no—
311C2×C9—44no—
312C2×C9C3×C22⋊Q814yescomplemented
313C2×C9C22×C3×D411yescharacteristic
314C2×C9C22×C3×Q811yescharacteristic
315C2×C9C2×C3×#4111yescharacteristic
316C24C2×AGL(1,ℤ/9)11yescharacteristic
317C2×Q8—94no—
318C2×Q8—184no—
319C22×C4—91no—
320C22×C4—94no—
321C22×C4—91no—
322C22×C4—91no—
323C22×C4—91no—
324C22×C4—184no—
325C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
326C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
327C4⋊C4—94no—
328C4⋊C4—94no—
329C4⋊C4—94no—
330C22⋊C4—94no—
331C22⋊C4C2×AGL(1,ℤ/9)14yes—
332C22×C3—28no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—31no—
336C22×C3—32no—
337C22×C3—34no—
338C22×C3—32no—
339C22×C3—32no—
340C22×C3—62no—
341C22×C3—62no—
342C22×C3—68no—
343C22×C3C2×C3×Dic612yes—
344C22×C3C2×C3×Dic614yes—
345C22×C3C3×D3×D412yes—
346C22×C3C3×#23712yes—
347C22×C3C23×C3×D311yescharacteristic
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—62no—
351C3×C4—62no—
352C3×C4—94no—
353C3×C4—182no—
354C3×C4—182no—
355C3×C4—182no—
356Dic3—34no—
357Dic3—62no—
358Dic3—62no—
359Dic3—62no—
360C32C2×#87811yescharacteristic
361C9—21no—
362C9C2×C3×C22⋊Q811yescharacteristic, complemented
363C23—22no—
364C23—22no—
365C233−1+2⋊Q814yescomplemented
366C23C22×AGL(1,ℤ/9)12yes—
367C23C22×AGL(1,ℤ/9)11yescharacteristic
368Q8—188no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—92no—
372C2×C4—94no—
373C2×C4—92no—
374C2×C4—92no—
375C2×C4—92no—
376C2×C4—188no—
377C2×C4—182no—
378C2×C4—182no—
379C2×C4—182no—
380C2×C4C22×AGL(1,ℤ/9)12yes—
381C2×C4C22×AGL(1,ℤ/9)12yes—
382C2×C3—24no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—34no—
387C2×C3—64no—
388C2×C3C3×#87814yes—
389C2×C3C22×C3×Dic611yescharacteristic
390C2×C3C2×C3×D3×D411yescharacteristic
391C2×C3C2×C3×#23711yescharacteristic
392C22—22no—
393C22—22no—
394C22—28no—
395C22C2×3−1+2⋊Q812yes—
396C22C2×3−1+2⋊Q814yescomplemented
397C22D4×AGL(1,ℤ/9)12yes—
398C22#3699⋊C312yes—
399C22C23×AGL(1,ℤ/9)11yescharacteristic
400C4—22no—
401C4—22no—
402C4—94no—
403C4—182no—
404C4—182no—
405C4—182no—
406C3—31no—
407C3C2×C3×#87811yescharacteristic
408C2—24no—
409C2#63440⋊C314yesdirect product factor
410C2C22×3−1+2⋊Q811yescharacteristic
411C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
412C2C2×#3699⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C22⋊C4×3−1+2C2
socleC23×C3C22×C3×D3
prime radicalC24×C32C2×D3
hypercenterC2×C22⋊C4AGL(1,ℤ/9)
hypocenterC9C2×C3×C22⋊Q8
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×3−1+2⋊Q8C2semidirect
5. C2×C4⋊(3−1+2⋊C4)C2semidirect
6. C2×(3−1+2⋊C4)⋊C4C2semidirect
7. C2×(3−1+2⋊C4)⋊C4C2semidirect
8. #63440⋊C3C2direct (N × H)
10. C2×#63440C3semidirect
21. C2×3−1+2⋊Q8C22semidirect
24. C4⋊(3−1+2⋊C4)C22semidirect
25. (3−1+2⋊C4)⋊C4C22semidirect
26. (3−1+2⋊C4)⋊C4C22semidirect
36. C22×Dic18C2×C3semidirect
40. C2×C4⋊Dic9C2×C3semidirect
41. C2×Dic9⋊C4C2×C3semidirect
42. C2×Dic9⋊C4C2×C3semidirect
43. #63440C2×C3semidirect
46. C23×3−1+2Q8semidirect
94. C2×Dic18C22×C3semidirect
98. C4⋊Dic9C22×C3semidirect
99. Dic9⋊C4C22×C3semidirect
100. Dic9⋊C4C22×C3semidirect
106. C22×3−1+2C2×Q8semidirect
157. C23×C9C3×Q8semidirect
180. C2×3−1+2C22⋊Q8semidirect
243. C22×C9C2×C3×Q8semidirect
262. 3−1+2C2×C22⋊Q8semidirect
312. C2×C9C3×C22⋊Q8semidirect
362. C9C2×C3×C22⋊Q8semidirect
365. C233−1+2⋊Q8semidirect
396. C22C2×3−1+2⋊Q8semidirect
409. C2#63440⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×3−1+2⋊C42
C22×3−1+2⋊Q82
C2×C22⋊C4×3−1+22
C2×#3676⋊C32
C2×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
#63440⋊C32
C2×C3×#8783
C2×#634403