Finite Group Databasebeta

← C32×#5914 #13181356

Subgroups — C32×#5914 #13181356

Order 1728.

1166 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D4×A4C211yesmaximal, characteristic, complemented
2C32×GL(2,ℤ/4)C212yesmaximal, complemented
3C2×C32×SL(2,ℤ/4)C212yesmaximal, complemented
4C32×C4×S4C211yesmaximal, characteristic, complemented
5C32×A4⋊Q8C211yesmaximal, characteristic, complemented
6C3×#5914C314yesmaximal, direct product factor
7C32×#546—31nomaximal
8C22×C32×A4C2212yes—
9C2×C32×S4C2211yescharacteristic
10C32×#237—41nomaximal
11C32×C4×A4C2211yescharacteristic
12C32×SL(2,ℤ/4)C2212yes—
13C32×SL(2,ℤ/4)C2211yescharacteristic
14C2×C32×#41—31no—
15C22×C32×D4D311yescharacteristic, complemented
16C3×D4×A4—24no—
17C3×D4×A4C2×C314yescomplemented
18C3×GL(2,ℤ/4)C2×C318yescomplemented
19C2×C3×SL(2,ℤ/4)C2×C318yescomplemented
20C3×C4×S4C2×C314yescomplemented
21C3×A4⋊Q8C2×C314yescomplemented
22C32×#124—31no—
23C32×#123—32no—
24C32×C22⋊Q8—31no—
25C32×#121—31no—
26C32×#121—32no—
27C32×#120—32no—
28C32×C4×D4—31no—
29C32×C4×D4—31no—
30C2×C32×C22⋊C4—32no—
31C2×C32×A4—22no—
32C2×C32×A4C2311yescharacteristic
33C32×S4—21no—
34C33×D4—41no—
35C32×#67—42no—
36C2×C32×Dic3—42no—
37C32×C4×D3—41no—
38C32×Dic6—41no—
39#5914C3211yescharacteristic, direct product factor
40C3×#546—34no—
41C24×C32C2×D312yes—
42C22×C3×A4—28no—
43C22×C3×A4C22×C318yes—
44C2×C3×S4C22×C314yes—
45C32×#41—62no—
46C32×#41—61no—
47C32×#41—61no—
48C2×C32×Q8—31no—
49C2×C32×D4—31no—
50C2×C32×D4—32no—
51C2×C32×D4—31no—
52C2×C32×D4—32no—
53C2×C32×D4—31no—
54C2×C32×D4—32no—
55C22×C32×C4—32no—
56C22×C32×C4—31no—
57C22×C32×C4—32no—
58C22×C32×C4C2×D311yescharacteristic
59C3×#237—44no—
60C3×C4×A4—24no—
61C3×C4×A4C22×C314yes—
62C3×SL(2,ℤ/4)C22×C314yes—
63C3×SL(2,ℤ/4)C22×C318yes—
64C32×C4⋊C4—31no—
65C32×C4⋊C4—31no—
66C32×C4⋊C4—32no—
67C32×C22⋊C4—32no—
68C32×C22⋊C4—32no—
69C32×C22⋊C4—32no—
70C32×C22⋊C4—31no—
71C32×C22⋊C4—32no—
72C32×C22⋊C4—31no—
73C32×C22⋊C4—32no—
74C32×C42—31no—
75C22×C33—42no—
76C2×C32×D3—41no—
77C32×A4#4111yescharacteristic, complemented
78C33×C4—41no—
79C32×Dic3—42no—
80C32×Dic3—41no—
81C2×C3×#41—34no—
82C22×C3×D4C3×D314yescomplemented
83D4×A4—24no—
84D4×A4C2×C3211yescharacteristic, complemented
85GL(2,ℤ/4)C2×C3212yescomplemented
86C2×SL(2,ℤ/4)C2×C3212yescomplemented
87C4×S4C2×C3211yescharacteristic, complemented
88A4⋊Q8C2×C3211yescharacteristic, complemented
89C3×#124—34no—
90C3×#123—38no—
91C3×C22⋊Q8—34no—
92C3×#121—34no—
93C3×#121—38no—
94C3×#120—38no—
95C3×C4×D4—34no—
96C3×C4×D4—34no—
97C2×C3×C22⋊C4—38no—
98C23×C32—22no—
99C23×C32—32no—
100C23×C32—32no—
101C23×C32—31no—
102C23×C32—62no—
103C23×C32C22×D311yescharacteristic
104C2×C3×A4—24no—
105C2×C3×A4—28no—
106C2×C3×A4—48no—
107C2×C3×A4C23×C314yes—
108C3×S4—24no—
109C32×Q8—61no—
110C32×Q8—61no—
111C32×D4—32no—
112C32×D4—31no—
113C32×D4—44no—
114C32×D4—61no—
115C32×D4—61no—
116C32×D4—62no—
117C32×D4—61no—
118C32×D4—62no—
119C32×D4—84no—
120C32×D4—122no—
121C32×D4S411yescharacteristic, complemented
122C2×C32×C4—31no—
123C2×C32×C4—31no—
124C2×C32×C4—31no—
125C2×C32×C4—31no—
126C2×C32×C4—32no—
127C2×C32×C4—31no—
128C2×C32×C4—32no—
129C2×C32×C4—61no—
130C2×C32×C4—62no—
131C2×C32×C4—62no—
132C2×C32×C4—61no—
133C2×C32×C4—122no—
134C3×#67—48no—
135C2×C3×Dic3—48no—
136C3×C4×D3—44no—
137C3×Dic6—44no—
138#546—31noSylow
139C2×C33—41no—
140C2×C33—82no—
141C32×D3—81no—
142C24×C3C2×C3×D318yes—
143C22×A4—28no—
144C22×A4C22×C3212yes—
145C2×S4C22×C3211yescharacteristic
146C3×#41—68no—
147C3×#41—64no—
148C3×#41—64no—
149C2×C3×Q8—34no—
150C2×C3×D4—34no—
151C2×C3×D4—38no—
152C2×C3×D4—34no—
153C2×C3×D4—38no—
154C2×C3×D4—34no—
155C2×C3×D4—38no—
156C22×C3×C4—38no—
157C22×C3×C4—34no—
158C22×C3×C4—38no—
159C22×C3×C4C2×C3×D314yes—
160#237—41no—
161C4×A4—24no—
162C4×A4C22×C3211yescharacteristic
163SL(2,ℤ/4)C22×C3212yes—
164SL(2,ℤ/4)C22×C3211yescharacteristic
165C3×C4⋊C4—34no—
166C3×C4⋊C4—34no—
167C3×C4⋊C4—38no—
168C3×C22⋊C4—38no—
169C3×C22⋊C4—38no—
170C3×C22⋊C4—38no—
171C3×C22⋊C4—34no—
172C3×C22⋊C4—38no—
173C3×C22⋊C4—34no—
174C3×C22⋊C4—38no—
175C3×C42—34no—
176C22×C32—31no—
177C22×C32—31no—
178C22×C32—32no—
179C22×C32—48no—
180C22×C32—62no—
181C22×C32—62no—
182C22×C32—62no—
183C22×C32—62no—
184C22×C32—61no—
185C22×C32—61no—
186C22×C32—61no—
187C22×C32—88no—
188C22×C32#23711yescharacteristic, complemented
189C22×C32C2×S412yes—
190C2×C3×D3—44no—
191C3×A4—24no—
192C3×A4C3×#4114yescomplemented
193C32×C4—31no—
194C32×C4—44no—
195C32×C4—61no—
196C32×C4—62no—
197C32×C4—61no—
198C32×C4—61no—
199C32×C4—62no—
200C32×C4—84no—
201C32×C4C2×S411yescharacteristic
202C3×Dic3—48no—
203C3×Dic3—44no—
204C2×#41—31no—
205C22×D4C32×D311yescharacteristic, complemented
206#124—31no—
207#123—32no—
208C22⋊Q8—31no—
209#121—31no—
210#121—32no—
211#120—32no—
212C4×D4—31no—
213C4×D4—31no—
214C2×C22⋊C4—32no—
215C33—41noSylow
216C23×C3—28no—
217C23×C3—38no—
218C23×C3—38no—
219C23×C3—34no—
220C23×C3—68no—
221C23×C3C22×C3×D314yes—
222C2×A4—22no—
223C2×A4—24no—
224C2×A4—48no—
225C2×A4C23×C3211yescharacteristic
226S4—21no—
227C3×Q8—64no—
228C3×Q8—64no—
229C3×D4—38no—
230C3×D4—34no—
231C3×D4—41no—
232C3×D4—64no—
233C3×D4—64no—
234C3×D4—68no—
235C3×D4—68no—
236C3×D4—64no—
237C3×D4—84no—
238C3×D4—128no—
239C3×D4C3×S414yescomplemented
240C2×C3×C4—34no—
241C2×C3×C4—34no—
242C2×C3×C4—34no—
243C2×C3×C4—34no—
244C2×C3×C4—38no—
245C2×C3×C4—34no—
246C2×C3×C4—38no—
247C2×C3×C4—64no—
248C2×C3×C4—68no—
249C2×C3×C4—68no—
250C2×C3×C4—64no—
251C2×C3×C4—128no—
252#67—42no—
253C2×Dic3—42no—
254C4×D3—41no—
255Dic6—41no—
256C2×C32—22no—
257C2×C32—31no—
258C2×C32—31no—
259C2×C32—44no—
260C2×C32—62no—
261C2×C32—88no—
262C2×C32—84no—
263C2×C32—121no—
264C2×C32—168no—
265C2×C32C22×S411yescharacteristic
266C3×D3—84no—
267C24C2×C32×D312yes—
268#41—62no—
269#41—61no—
270#41—61no—
271C2×Q8—31no—
272C2×D4—31no—
273C2×D4—32no—
274C2×D4—31no—
275C2×D4—32no—
276C2×D4—31no—
277C2×D4—32no—
278C22×C4—32no—
279C22×C4—31no—
280C22×C4—32no—
281C22×C4C2×C32×D311yescharacteristic
282C4⋊C4—31no—
283C4⋊C4—31no—
284C4⋊C4—32no—
285C22⋊C4—32no—
286C22⋊C4—32no—
287C22⋊C4—32no—
288C22⋊C4—31no—
289C22⋊C4—32no—
290C22⋊C4—31no—
291C22⋊C4—32no—
292C42—31no—
293C22×C3—34no—
294C22×C3—34no—
295C22×C3—38no—
296C22×C3—42no—
297C22×C3—68no—
298C22×C3—68no—
299C22×C3—68no—
300C22×C3—68no—
301C22×C3—64no—
302C22×C3—64no—
303C22×C3—64no—
304C22×C3—88no—
305C22×C3C3×#23714yescomplemented
306C22×C3C2×C3×S418yes—
307C2×D3—41no—
308A4—24no—
309A4C32×#4111yescharacteristic, complemented
310C3×C4—34no—
311C3×C4—41no—
312C3×C4—64no—
313C3×C4—68no—
314C3×C4—64no—
315C3×C4—64no—
316C3×C4—68no—
317C3×C4—84no—
318C3×C4C2×C3×S414yes—
319Dic3—42no—
320Dic3—41no—
321C32—44no—
322C32—84no—
323C32#591411yescharacteristic, direct product factor
324C23—22no—
325C23—32no—
326C23—32no—
327C23—31no—
328C23—62no—
329C23C22×C32×D311yescharacteristic
330Q8—61no—
331Q8—61no—
332D4—32no—
333D4—31no—
334D4—61no—
335D4—62no—
336D4—61no—
337D4—62no—
338D4—61no—
339D4—122no—
340D4C32×S411yescharacteristic, complemented
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—32no—
346C2×C4—31no—
347C2×C4—32no—
348C2×C4—61no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—61no—
352C2×C4—122no—
353C2×C3—28no—
354C2×C3—34no—
355C2×C3—34no—
356C2×C3—41no—
357C2×C3—68no—
358C2×C3—82no—
359C2×C3—84no—
360C2×C3—124no—
361C2×C3—168no—
362C2×C3C22×C3×S414yes—
363D3—81no—
364C22—31no—
365C22—31no—
366C22—32no—
367C22—62no—
368C22—62no—
369C22—62no—
370C22—62no—
371C22—61no—
372C22—61no—
373C22—61no—
374C22C32×#23711yescharacteristic, complemented
375C22C2×C32×S412yes—
376C4—31no—
377C4—61no—
378C4—62no—
379C4—61no—
380C4—61no—
381C4—62no—
382C4C2×C32×S411yescharacteristic
383C3—41no—
384C3—84no—
385C3C3×#591414yesdirect product factor
386C2—22no—
387C2—31no—
388C2—31no—
389C2—62no—
390C2—121no—
391C2C22×C32×S411yescharacteristic
subgroupG/H
centerC2×C32C22×S4
commutator, cosocleC2×A4C23×C32
FrattiniC2C22×C32×S4
FittingC22×C32×D4D3
socleC23×C32C22×D3
hypercenterC32×D4S4
hypocenterA4C32×#41
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C32 ◁ C32×D4

Complements

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

NH ≅ G/Ntype
1. C32×D4×A4C2semidirect
2. C32×GL(2,ℤ/4)C2semidirect
3. C2×C32×SL(2,ℤ/4)C2semidirect
4. C32×C4×S4C2semidirect
5. C32×A4⋊Q8C2semidirect
6. C3×#5914C3direct (N × H)
15. C22×C32×D4D3semidirect
17. C3×D4×A4C2×C3semidirect
18. C3×GL(2,ℤ/4)C2×C3semidirect
19. C2×C3×SL(2,ℤ/4)C2×C3semidirect
20. C3×C4×S4C2×C3semidirect
21. C3×A4⋊Q8C2×C3semidirect
39. #5914C32direct (N × H)
77. C32×A4#41semidirect
82. C22×C3×D4C3×D3semidirect
84. D4×A4C2×C32semidirect
85. GL(2,ℤ/4)C2×C32semidirect
86. C2×SL(2,ℤ/4)C2×C32semidirect
87. C4×S4C2×C32semidirect
88. A4⋊Q8C2×C32semidirect
121. C32×D4S4semidirect
188. C22×C32#237semidirect
192. C3×A4C3×#41semidirect
205. C22×D4C32×D3semidirect
239. C3×D4C3×S4semidirect
305. C22×C3C3×#237semidirect
309. A4C32×#41semidirect
323. C32#5914direct (N × H)
340. D4C32×S4semidirect
374. C22C32×#237semidirect
385. C3C3×#5914direct (N × H)

Maximal subgroups

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

subgroupindex
C32×D4×A42
C32×GL(2,ℤ/4)2
C2×C32×SL(2,ℤ/4)2
C32×C4×S42
C32×A4⋊Q82
C3×#59143
C32×#5463
C32×#2374