Finite Group Databasebeta

← C32×#4808 #13162252

Subgroups — C32×#4808 #13162252

Order 1728.

1080 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#239C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic, complemented
3C2×C32×#204C211yesmaximal, characteristic, complemented
4C32×#897C211yesmaximal, characteristic, complemented
5C33×#103C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7C32×#809C211yesmaximal, characteristic, complemented
8C32×#452—31nomaximal
9C3×#4808C314yesmaximal, direct product factor
10C2×C33×Q8C2211yescharacteristic
11C32×#239—22no—
12C32×#239—22no—
13C2×C32×D12—21no—
14C2×C32×D12C2211yescharacteristic
15C2×C32×C4×D3—21no—
16C2×C32×C4×D3C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C2×C33×C8C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C32×#216—22no—
21C32×(C2×D3)⋊C4—21no—
22C32×C4⋊Dic3—21no—
23C2×C32×C3⋊C8C2211yescharacteristic
24C32×#204—22no—
25C2×C32×#41—31no—
26C2×C3×#239C2×C314yes—
27C2×C32×Q16—31no—
28C2×C32×SD16—31no—
29C32×C22⋊Q8—31no—
30C2×C3×#216C2×C314yescomplemented
31C2×C3×#204C2×C314yescomplemented
32C3×#897C2×C314yescomplemented
33C32×#103—24no—
34C32×#103D311yescharacteristic, complemented
35C32×#103C2×C314yescomplemented
36C32×#102—31no—
37C32×C22⋊C8—31no—
38C3×(C2×D3)⋊C8C2×C314yescomplemented
39C3×#809C2×C314yescomplemented
40C22×C32×D3—21no—
41C22×C32×D3D411yescharacteristic
42C33×Q8—21no—
43C33×Q8D412yes—
44C2×C33×C4—21no—
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C2×C32×Dic3—21no—
48C2×C32×Dic3D411yescharacteristic
49C32×D12—24no—
50C32×D12—21no—
51C32×D12D412yes—
52C32×C4×D3—24no—
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C33×C8—21no—
58C32×C3⋊C8—21no—
59C3×#452—34no—
60#4808C3211yescharacteristic, direct product factor
61C32×#41—62no—
62C32×#41—62no—
63C2×C32×Q8—24no—
64C2×C32×Q8—31no—
65C2×C32×Q8C2×D311yescharacteristic
66C2×C32×Q8C22×C314yes—
67C2×C32×D4—31no—
68C2×C32×D4—61no—
69C22×C32×C4—31no—
70C22×C32×C4—61no—
71C3×#239—28no—
72C3×#239—28no—
73C2×C3×D12—24no—
74C2×C3×D12C22×C314yes—
75C2×C3×C4×D3—24no—
76C2×C3×C4×D3C22×C314yes—
77C2×C3×Dic6C22×C314yes—
78C32×Q16—62no—
79C32×SD16—62no—
80C2×C32×C8—24no—
81C2×C32×C8—31no—
82C2×C32×C8C2×D311yescharacteristic
83C2×C32×C8C22×C314yes—
84C32×C4⋊C4—24no—
85C32×C4⋊C4—61no—
86C32×C4⋊C4C2×D311yescharacteristic
87C32×C4⋊C4C22×C314yes—
88C32×C22⋊C4—61no—
89C3×#216—28no—
90C3×(C2×D3)⋊C4—24no—
91C3×C4⋊Dic3—24no—
92C2×C3×C3⋊C8C22×C314yes—
93C3×#204—28no—
94C22×C33C2×D411yescharacteristic
95C2×C32×D3—21no—
96C2×C32×D3—22no—
97C2×C32×D3—21no—
98C2×C32×D3—21no—
99C2×C32×D3—41no—
100C2×C32×D3—41no—
101C33×C4—22no—
102C33×C4—41no—
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C32×Dic3—21no—
106C32×Dic3—41no—
107C2×C3×#41—34no—
108C2×#239C2×C3211yescharacteristic
109C2×C3×Q16—34no—
110C2×C3×SD16—34no—
111C3×C22⋊Q8—34no—
112C2×#216C2×C3211yescharacteristic, complemented
113C2×#204C2×C3211yescharacteristic, complemented
114#897C2×C3211yescharacteristic, complemented
115C3×#103—24no—
116C3×#103C3×D314yescomplemented
117C3×#103C2×C3211yescharacteristic, complemented
118C3×#102—34no—
119C3×C22⋊C8—34no—
120(C2×D3)⋊C8C2×C3211yescharacteristic, complemented
121#809C2×C3211yescharacteristic, complemented
122C23×C32—31no—
123C23×C32—61no—
124C22×C3×D3—24no—
125C22×C3×D3C3×D414yes—
126C32×Q8—21no—
127C32×Q8—28no—
128C32×Q8—24no—
129C32×Q8—44no—
130C32×Q8—61no—
131C32×Q8—61no—
132C32×Q8D1212yes—
133C32×Q8C3×D418yes—
134C32×D4—32no—
135C32×D4—64no—
136C32×D4—61no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—24no—
142C2×C32×C4—31no—
143C2×C32×C4—44no—
144C2×C32×C4—44no—
145C2×C32×C4—64no—
146C2×C32×C4—61no—
147C2×C32×C4—61no—
148C2×C32×C4—61no—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C23×C314yes—
151C2×C3×Dic3—24no—
152C2×C3×Dic3C3×D414yes—
153C3×D12—216no—
154C3×D12—24no—
155C3×D12C3×D418yes—
156C3×C4×D3—216no—
157C3×C4×D3—24no—
158C3×C4×D3—24no—
159C3×Dic6—24no—
160C3×Dic6—24no—
161C32×C8—21no—
162C32×C8—24no—
163C32×C8—44no—
164C32×C8—61no—
165C3×C3⋊C8—24no—
166#452—31noSylow
167C2×C33#12011yescharacteristic
168C2×C33#13511yescharacteristic
169C2×C33#13711yescharacteristic
170C32×D3—41no—
171C32×D3—42no—
172C3×#41—68no—
173C3×#41—68no—
174C2×C3×Q8—24no—
175C2×C3×Q8—34no—
176C2×C3×Q8C2×C3×D314yes—
177C2×C3×Q8C22×C3211yescharacteristic
178C2×C3×D4—34no—
179C2×C3×D4—64no—
180C22×C3×C4—34no—
181C22×C3×C4—64no—
182#239—22no—
183#239—22no—
184C2×D12—21no—
185C2×D12C22×C3211yescharacteristic
186C2×C4×D3—21no—
187C2×C4×D3C22×C3211yescharacteristic
188C2×Dic6C22×C3211yescharacteristic
189C3×Q16—68no—
190C3×SD16—68no—
191C2×C3×C8—24no—
192C2×C3×C8—34no—
193C2×C3×C8C2×C3×D314yes—
194C2×C3×C8C22×C3211yescharacteristic
195C3×C4⋊C4—24no—
196C3×C4⋊C4—64no—
197C3×C4⋊C4C2×C3×D314yes—
198C3×C4⋊C4C22×C3211yescharacteristic
199C3×C22⋊C4—64no—
200#216—22no—
201(C2×D3)⋊C4—21no—
202C4⋊Dic3—21no—
203C2×C3⋊C8C22×C3211yescharacteristic
204#204—22no—
205C22×C32—24no—
206C22×C32—62no—
207C22×C32—61no—
208C22×C32—61no—
209C22×C32—61no—
210C22×C32—121no—
211C22×C32—121no—
212C22×C32D3×D411yescharacteristic
213C22×C32C2×C3×D414yes—
214C2×C3×D3—24no—
215C2×C3×D3—28no—
216C2×C3×D3—24no—
217C2×C3×D3—24no—
218C2×C3×D3—44no—
219C2×C3×D3—44no—
220C32×C4—22no—
221C32×C4—24no—
222C32×C4—24no—
223C32×C4—28no—
224C32×C4—48no—
225C32×C4—41no—
226C32×C4—44no—
227C32×C4—61no—
228C32×C4—84no—
229C32×C4—121no—
230C32×C4C2×D1211yescharacteristic
231C32×C4D3×D411yescharacteristic
232C32×C4C2×C3×D414yes—
233C32×C4C2×C3×D414yes—
234C3×Dic3—24no—
235C3×Dic3—44no—
236C2×#41—31no—
237C2×Q16—31no—
238C2×SD16—31no—
239C22⋊Q8—31no—
240#103C32×D311yescharacteristic, complemented
241#102—31no—
242C22⋊C8—31no—
243C33#45211yescharacteristic, complemented, Sylow
244C23×C3—34no—
245C23×C3—64no—
246C22×D3—21no—
247C22×D3C32×D411yescharacteristic
248C3×Q8—24no—
249C3×Q8—21no—
250C3×Q8—28no—
251C3×Q8—44no—
252C3×Q8—64no—
253C3×Q8—64no—
254C3×Q8C3×D1218yes—
255C3×Q8C32×D412yes—
256C3×D4—38no—
257C3×D4—616no—
258C3×D4—64no—
259C2×C3×C4—24no—
260C2×C3×C4—21no—
261C2×C3×C4—24no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—34no—
265C2×C3×C4—44no—
266C2×C3×C4—44no—
267C2×C3×C4—616no—
268C2×C3×C4—64no—
269C2×C3×C4—64no—
270C2×C3×C4—64no—
271C2×C3×C4C22×C3×D314yes—
272C2×C3×C4C23×C3211yescharacteristic
273C2×Dic3—21no—
274C2×Dic3C32×D411yescharacteristic
275D12—24no—
276D12—21no—
277D12C32×D412yes—
278C4×D3—24no—
279C4×D3—21no—
280C4×D3—21no—
281Dic6—21no—
282Dic6—21no—
283C3×C8—24no—
284C3×C8—21no—
285C3×C8—44no—
286C3×C8—64no—
287C3⋊C8—21no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—24no—
291C2×C32—121no—
292C2×C32—122no—
293C2×C32#88211yescharacteristic
294C2×C32#91611yescharacteristic
295C2×C32#91811yescharacteristic
296C2×C32C3×#12014yes—
297C2×C32C3×#13514yes—
298C2×C32C3×#13714yes—
299C3×D3—44no—
300C3×D3—48no—
301#41—62no—
302#41—62no—
303C2×Q8—31no—
304C2×Q8C2×C32×D311yescharacteristic
305C2×D4—31no—
306C2×D4—61no—
307C22×C4—31no—
308C22×C4—61no—
309Q16—62no—
310SD16—62no—
311C2×C8—31no—
312C2×C8C2×C32×D311yescharacteristic
313C4⋊C4—61no—
314C4⋊C4C2×C32×D311yescharacteristic
315C22⋊C4—61no—
316C22×C3—24no—
317C22×C3—68no—
318C22×C3—64no—
319C22×C3—64no—
320C22×C3—64no—
321C22×C3—124no—
322C22×C3—124no—
323C22×C3C3×D3×D414yes—
324C22×C3C2×C32×D411yescharacteristic
325C2×D3—21no—
326C2×D3—22no—
327C2×D3—21no—
328C2×D3—21no—
329C2×D3—41no—
330C2×D3—41no—
331C3×C4—28no—
332C3×C4—22no—
333C3×C4—24no—
334C3×C4—24no—
335C3×C4—48no—
336C3×C4—44no—
337C3×C4—41no—
338C3×C4—64no—
339C3×C4—84no—
340C3×C4—124no—
341C3×C4C2×C3×D1214yes—
342C3×C4C3×D3×D414yes—
343C3×C4C2×C32×D411yescharacteristic
344C3×C4C2×C32×D411yescharacteristic
345Dic3—21no—
346Dic3—41no—
347C32—24no—
348C32#480811yescharacteristic, direct product factor
349C32C3×#45214yescomplemented
350C23—31no—
351C23—61no—
352Q8—21no—
353Q8—61no—
354Q8—61no—
355Q8C32×D1212yes—
356D4—32no—
357D4—64no—
358D4—61no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—31no—
362C2×C4—64no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4C22×C32×D311yescharacteristic
367C8—21no—
368C8—61no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3—124no—
373C2×C3—128no—
374C2×C3C3×#88214yes—
375C2×C3C3×#91614yes—
376C2×C3C3×#91814yes—
377C2×C3C32×#12011yescharacteristic
378C2×C3C32×#13511yescharacteristic
379C2×C3C32×#13711yescharacteristic
380D3—41no—
381D3—42no—
382C22—61no—
383C22—61no—
384C22—62no—
385C22—61no—
386C22—121no—
387C22—121no—
388C22C32×D3×D411yescharacteristic
389C4—22no—
390C4—41no—
391C4—61no—
392C4—121no—
393C4C2×C32×D1211yescharacteristic
394C4C32×D3×D411yescharacteristic
395C3—24no—
396C3C3×#480814yesdirect product factor
397C3C32×#45211yescharacteristic, complemented
398C2—121no—
399C2—122no—
400C2C32×#88211yescharacteristic
401C2C32×#91611yescharacteristic
402C2C32×#91811yescharacteristic
subgroupG/H
centerC22×C32D3×D4
commutator, cosocleC2×C3×C4C23×C32
FrattiniC2×C4C22×C32×D3
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC2×C32×#239C2
hypercenterC32×#103D3
hypocenterC3C32×#452
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C2×C32×C4 ◁ C32×#103

Complements

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

NH ≅ G/Ntype
2. C2×C32×#216C2semidirect
3. C2×C32×#204C2semidirect
4. C32×#897C2semidirect
5. C33×#103C2semidirect
6. C32×(C2×D3)⋊C8C2semidirect
7. C32×#809C2semidirect
9. C3×#4808C3direct (N × H)
30. C2×C3×#216C2×C3semidirect
31. C2×C3×#204C2×C3semidirect
32. C3×#897C2×C3semidirect
34. C32×#103D3semidirect
35. C32×#103C2×C3semidirect
38. C3×(C2×D3)⋊C8C2×C3semidirect
39. C3×#809C2×C3semidirect
60. #4808C32direct (N × H)
112. C2×#216C2×C32semidirect
113. C2×#204C2×C32semidirect
114. #897C2×C32semidirect
116. C3×#103C3×D3semidirect
117. C3×#103C2×C32semidirect
120. (C2×D3)⋊C8C2×C32semidirect
121. #809C2×C32semidirect
240. #103C32×D3semidirect
243. C33#452semidirect
348. C32#4808direct (N × H)
349. C32C3×#452semidirect
396. C3C3×#4808direct (N × H)
397. C3C32×#452semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
C2×C32×#2162
C2×C32×#2042
C32×#8972
C33×#1032
C32×(C2×D3)⋊C82
C32×#8092
C32×#4523
C3×#48083