Finite Group Databasebeta

← C2×C3×#4993 #11808472

Subgroups — C2×C3×#4993 #11808472

Order 1152.

2025 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4⋊C4C211yesmaximal, characteristic, complemented
2C22×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
3C22×C3×(C2×D3)⋊C4C212yesmaximal
4C22×C3×Dic3⋊C4C212yesmaximal, complemented
5C2×C3×C3⋊#95C211yesmaximal, characteristic, complemented
6C3×#4993C218yesmaximal, direct product factor
7C2×C3×#399—31nomaximal
8C2×#4993C311yesmaximal, characteristic, direct product factor
9C24×C3×D3C2211yescharacteristic
10C23×C32×C4C2212yes—
11C23×C32×C4C2211yescharacteristic
12C23×C3×Dic3C2211yescharacteristic
13C23×C3×Dic3C2212yes—
14C2×C32×C4⋊C4—24no—
15C2×C32×C4⋊C4C2214yescomplemented
16C2×C3×(C2×D3)⋊C4—24no—
17C2×C3×(C2×D3)⋊C4—28no—
18C2×C3×(C2×D3)⋊C4C2214yes—
19C2×C3×(C2×D3)⋊C4C2218yes—
20C2×C3×Dic3⋊C4—28no—
21C2×C3×Dic3⋊C4C2218yescomplemented
22C3×C3⋊#95C2214yescomplemented
23C22×C3×C4⋊C4—21no—
24C22×C3×C4⋊C4—32no—
25C22×C3×C4⋊C4D311yescharacteristic, complemented
26C22×C3×C4⋊C4C2×C311yescharacteristic, complemented
27C22×C3×C22⋊C4—32no—
28C22×C3×C22⋊C4—31no—
29C22×(C2×D3)⋊C4C2×C311yescharacteristic
30C22×(C2×D3)⋊C4C2×C312yes—
31C22×Dic3⋊C4C2×C312yescomplemented
32C3×#399—38no—
33C2×C3×#95—31no—
34C2×C3⋊#95C2×C311yescharacteristic, complemented
35#4993C2×C318yesdirect product factor
36C24×C32C2311yescharacteristic
37C23×C3×D3—24no—
38C23×C3×D3—22no—
39C23×C3×D3—24no—
40C23×C3×D3—22no—
41C23×C3×D3Q814yes—
42C23×C3×D3C2312yes—
43C22×C32×C4—24no—
44C22×C32×C4—24no—
45C22×C32×C4—24no—
46C22×C32×C4—24no—
47C22×C32×C4D414yes—
48C22×C32×C4C2314yes—
49C22×C32×C4C2312yes—
50C22×C3×Dic3—24no—
51C22×C3×Dic3—24no—
52C22×C3×Dic3—24no—
53C22×C3×Dic3—22no—
54C22×C3×Dic3D418yes—
55C22×C3×Dic3C2312yes—
56C22×C3×Dic3C2314yes—
57C32×C4⋊C4—28no—
58C3×(C2×D3)⋊C4—28no—
59C3×(C2×D3)⋊C4—216no—
60C3×Dic3⋊C4—216no—
61C2×#399—31noSylow
62C25×C3—31no—
63C24×D3C22×C311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4—22no—
66C23×C3×C4—32no—
67C23×C3×C4—31no—
68C23×C3×C4C2×D312yes—
69C23×C3×C4C2×D311yescharacteristic
70C23×C3×C4C22×C312yes—
71C23×C3×C4C22×C311yescharacteristic
72C23×Dic3C22×C311yescharacteristic
73C23×Dic3C22×C312yes—
74C2×C3×C4⋊C4—28no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—38no—
79C2×C3×C4⋊C4—68no—
80C2×C3×C4⋊C4C2×D314yescomplemented
81C2×C3×C4⋊C4C22×C314yescomplemented
82C2×C3×C22⋊C4—38no—
83C2×C3×C22⋊C4—34no—
84C2×C3×C22⋊C4—68no—
85C2×C3×C22⋊C4—64no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—28no—
88C2×(C2×D3)⋊C4C22×C314yes—
89C2×(C2×D3)⋊C4C22×C318yes—
90C2×Dic3⋊C4—28no—
91C2×Dic3⋊C4C22×C318yescomplemented
92C3×#95—34no—
93C3⋊#95C22×C314yescomplemented
94C23×C32C2×D414yes—
95C23×C32C2×D412yes—
96C23×C32C2×Q812yes—
97C23×C32#4114yes—
98C23×C32#4112yes—
99C23×C32C2411yescharacteristic
100C22×C3×D3—216no—
101C22×C3×D3—28no—
102C22×C3×D3—24no—
103C22×C3×D3—22no—
104C22×C3×D3—24no—
105C22×C3×D3—22no—
106C22×C3×D3—48no—
107C22×C3×D3—44no—
108C22×C3×D3—44no—
109C22×C3×D3C2×Q814yes—
110C2×C32×C4—28no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—48no—
116C2×C32×C4C2×D414yes—
117C2×C3×Dic3—216no—
118C2×C3×Dic3—24no—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—44no—
123C2×C3×Dic3C2×D418yes—
124C22×C4⋊C4—32no—
125C22×C4⋊C4C3×D311yescharacteristic, complemented
126C22×C22⋊C4—32no—
127C22×C22⋊C4—31no—
128#399—38no—
129C2×#95—31no—
130C24×C3—21no—
131C24×C3—34no—
132C24×C3—32no—
133C24×C3—64no—
134C24×C3—64no—
135C24×C3—62no—
136C24×C3—62no—
137C24×C3C22×D311yescharacteristic
138C24×C3C23×C311yescharacteristic
139C23×D3—24no—
140C23×D3—22no—
141C23×D3—24no—
142C23×D3—22no—
143C23×D3C3×Q814yes—
144C23×D3C23×C312yes—
145C22×C3×C4—28no—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4—28no—
149C22×C3×C4—24no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—24no—
158C22×C3×C4—38no—
159C22×C3×C4—34no—
160C22×C3×C4—32no—
161C22×C3×C4—44no—
162C22×C3×C4—44no—
163C22×C3×C4—64no—
164C22×C3×C4—64no—
165C22×C3×C4—64no—
166C22×C3×C4—62no—
167C22×C3×C4#6714yes—
168C22×C3×C4C3×D414yes—
169C22×C3×C4C22×D314yes—
170C22×C3×C4C22×D312yes—
171C22×C3×C4C23×C314yes—
172C22×C3×C4C23×C312yes—
173C22×Dic3—24no—
174C22×Dic3—24no—
175C22×Dic3—24no—
176C22×Dic3—22no—
177C22×Dic3C3×D418yes—
178C22×Dic3C23×C312yes—
179C22×Dic3C23×C314yes—
180C3×C4⋊C4—216no—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—616no—
184C3×C22⋊C4—616no—
185C3×C22⋊C4—68no—
186(C2×D3)⋊C4—28no—
187(C2×D3)⋊C4—216no—
188Dic3⋊C4—216no—
189C22×C32C22⋊Q818yes—
190C22×C32C22⋊Q814yes—
191C22×C32#12318yes—
192C22×C32#12314yes—
193C22×C32#12714yes—
194C22×C32C22×D412yes—
195C22×C32C22×D411yescharacteristic
196C22×C32C22×Q811yescharacteristic
197C22×C32C2×#4112yes—
198C22×C32C2×#4111yescharacteristic
199C2×C3×D3—28no—
200C2×C3×D3—24no—
201C2×C3×D3—44no—
202C2×C3×D3—42no—
203C2×C3×D3—42no—
204C2×C3×D3—416no—
205C32×C4—24no—
206C32×C4—44no—
207C3×Dic3—28no—
208C3×Dic3—42no—
209C25—31no—
210C23×C4—32no—
211C23×C4—31no—
212C23×C4C2×C3×D312yes—
213C23×C4C2×C3×D311yescharacteristic
214C2×C4⋊C4—24no—
215C2×C4⋊C4—38no—
216C2×C4⋊C4—68no—
217C2×C4⋊C4C2×C3×D314yescomplemented
218C2×C22⋊C4—38no—
219C2×C22⋊C4—34no—
220C2×C22⋊C4—68no—
221C2×C22⋊C4—64no—
222#95—34no—
223C23×C3—21no—
224C23×C3—24no—
225C23×C3—24no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—34no—
230C23×C3—616no—
231C23×C3—68no—
232C23×C3—64no—
233C23×C3—64no—
234C23×C3—62no—
235C23×C3—62no—
236C23×C3—128no—
237C23×C3—124no—
238C23×C3—124no—
239C23×C3#23514yes—
240C23×C3D3×D414yes—
241C23×C3D3×Q812yes—
242C23×C3#23912yes—
243C23×C3C2×#6712yes—
244C23×C3C2×C3×D414yes—
245C23×C3C2×C3×D412yes—
246C23×C3C2×C3×Q812yes—
247C23×C3C3×#4114yes—
248C23×C3C3×#4112yes—
249C23×C3C23×D311yescharacteristic
250C23×C3C24×C311yescharacteristic
251C22×D3—216no—
252C22×D3—28no—
253C22×D3—24no—
254C22×D3—22no—
255C22×D3—24no—
256C22×D3—22no—
257C22×D3—48no—
258C22×D3—44no—
259C22×D3—44no—
260C22×D3C2×C3×Q814yes—
261C2×C3×C4—216no—
262C2×C3×C4—28no—
263C2×C3×C4—28no—
264C2×C3×C4—28no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—28no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—38no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—416no—
279C2×C3×C4—48no—
280C2×C3×C4—48no—
281C2×C3×C4—616no—
282C2×C3×C4—64no—
283C2×C3×C4—64no—
284C2×C3×C4—64no—
285C2×C3×C4—62no—
286C2×C3×C4—124no—
287C2×C3×C4C2×#6714yes—
288C2×C3×C4C2×C3×D414yes—
289C2×Dic3—216no—
290C2×Dic3—24no—
291C2×Dic3—24no—
292C2×Dic3—24no—
293C2×Dic3—22no—
294C2×Dic3—44no—
295C2×Dic3C2×C3×D418yes—
296C2×C32#39918yescomplemented
297C2×C32C2×C22⋊Q812yes—
298C2×C32C2×C22⋊Q811yescharacteristic
299C2×C32C2×#12312yes—
300C2×C32C2×#12311yescharacteristic
301C2×C32C2×#12711yescharacteristic
302C3×D3—44no—
303C24—32no—
304C24—34no—
305C24—64no—
306C24—64no—
307C24—62no—
308C24—62no—
309C24C22×C3×D311yescharacteristic
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—34no—
315C22×C4—38no—
316C22×C4—32no—
317C22×C4—64no—
318C22×C4—64no—
319C22×C4—64no—
320C22×C4—62no—
321C22×C4C3×#6714yes—
322C22×C4C22×C3×D314yes—
323C22×C4C22×C3×D312yes—
324C4⋊C4—28no—
325C4⋊C4—616no—
326C22⋊C4—616no—
327C22⋊C4—68no—
328C22×C3—28no—
329C22×C3—28no—
330C22×C3—24no—
331C22×C3—24no—
332C22×C3—24no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—68no—
339C22×C3—64no—
340C22×C3—1216no—
341C22×C3—124no—
342C22×C3—122no—
343C22×C3—122no—
344C22×C3#89418yes—
345C22×C3#89618yes—
346C22×C3#92914yes—
347C22×C3#94014yes—
348C22×C3#94614yes—
349C22×C3C3×C22⋊Q818yes—
350C22×C3C3×C22⋊Q814yes—
351C22×C3C3×#12318yes—
352C22×C3C3×#12314yes—
353C22×C3C3×#12714yes—
354C22×C3C2×#23512yes—
355C22×C3C2×D3×D412yes—
356C22×C3C2×D3×Q811yescharacteristic
357C22×C3C2×#23911yescharacteristic
358C22×C3C22×#6711yescharacteristic
359C22×C3C22×C3×D412yes—
360C22×C3C22×C3×D411yescharacteristic
361C22×C3C22×C3×Q811yescharacteristic
362C22×C3C2×C3×#4112yes—
363C22×C3C2×C3×#4111yescharacteristic
364C2×D3—28no—
365C2×D3—24no—
366C2×D3—416no—
367C2×D3—44no—
368C2×D3—42no—
369C2×D3—42no—
370C3×C4—28no—
371C3×C4—24no—
372C3×C4—24no—
373C3×C4—48no—
374C3×C4—44no—
375C3×C4—44no—
376C3×C4—68no—
377C3×C4—122no—
378Dic3—28no—
379Dic3—42no—
380C32C2×#39911yescharacteristic, complemented, Sylow
381C23—34no—
382C23—616no—
383C23—68no—
384C23—64no—
385C23—64no—
386C23—62no—
387C23—62no—
388C23—128no—
389C23—124no—
390C23—124no—
391C23C3×#23514yes—
392C23C3×D3×D414yes—
393C23C3×D3×Q812yes—
394C23C3×#23912yes—
395C23C2×C3×#6712yes—
396C23C23×C3×D311yescharacteristic
397C2×C4—28no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—38no—
403C2×C4—48no—
404C2×C4—616no—
405C2×C4—64no—
406C2×C4—64no—
407C2×C4—64no—
408C2×C4—62no—
409C2×C4—124no—
410C2×C4C2×C3×#6714yes—
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—28no—
417C2×C3—124no—
418C2×C3#499318yesdirect product factor
419C2×C3C3×#39918yescomplemented
420C2×C3C2×#89412yes—
421C2×C3C2×#89612yes—
422C2×C3C2×#92911yescharacteristic
423C2×C3C2×#94011yescharacteristic
424C2×C3C2×#94611yescharacteristic
425C2×C3C2×C3×C22⋊Q812yes—
426C2×C3C2×C3×C22⋊Q811yescharacteristic
427C2×C3C2×C3×#12312yes—
428C2×C3C2×C3×#12311yescharacteristic
429C2×C3C2×C3×#12711yescharacteristic
430D3—44no—
431C22—68no—
432C22—64no—
433C22—124no—
434C22—122no—
435C22—122no—
436C22—1216no—
437C22C3×#89418yes—
438C22C3×#89618yes—
439C22C3×#92914yes—
440C22C3×#94014yes—
441C22C3×#94614yes—
442C22C2×C3×#23512yes—
443C22C2×C3×D3×D412yes—
444C22C2×C3×D3×Q811yescharacteristic
445C22C2×C3×#23911yescharacteristic
446C22C22×C3×#6711yescharacteristic
447C4—24no—
448C4—44no—
449C4—68no—
450C4—122no—
451C3—21no—
452C3C2×#499311yescharacteristic, direct product factor
453C3C2×C3×#39911yescharacteristic, complemented
454C2—124no—
455C2C3×#499318yesdirect product factor
456C2C2×C3×#89412yes—
457C2C2×C3×#89612yes—
458C2C2×C3×#92911yescharacteristic
459C2C2×C3×#94011yescharacteristic
460C2C2×C3×#94611yescharacteristic
subgroupG/H
centerC24×C3C22×D3
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC22×C32×C4⋊C4C2
socleC24×C32C23
prime radicalC24×C3×D3C22
hypercenterC22×C3×C4⋊C4D3
hypocenterC3C2×C3×#399
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24×C3 ◁ C22×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C32×C4⋊C4C2semidirect
4. C22×C3×Dic3⋊C4C2semidirect
5. C2×C3×C3⋊#95C2semidirect
6. C3×#4993C2direct (N × H)
8. C2×#4993C3direct (N × H)
15. C2×C32×C4⋊C4C22semidirect
21. C2×C3×Dic3⋊C4C22semidirect
22. C3×C3⋊#95C22semidirect
25. C22×C3×C4⋊C4D3semidirect
26. C22×C3×C4⋊C4C2×C3semidirect
31. C22×Dic3⋊C4C2×C3semidirect
34. C2×C3⋊#95C2×C3semidirect
35. #4993C2×C3direct (N × H)
80. C2×C3×C4⋊C4C2×D3semidirect
81. C2×C3×C4⋊C4C22×C3semidirect
91. C2×Dic3⋊C4C22×C3semidirect
93. C3⋊#95C22×C3semidirect
125. C22×C4⋊C4C3×D3semidirect
217. C2×C4⋊C4C2×C3×D3semidirect
296. C2×C32#399semidirect
380. C32C2×#399semidirect
418. C2×C3#4993direct (N × H)
419. C2×C3C3×#399semidirect
452. C3C2×#4993direct (N × H)
453. C3C2×C3×#399semidirect
455. C2C3×#4993direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C4⋊C42
C22×C3×(C2×D3)⋊C42
C22×C3×Dic3⋊C42
C2×C3×C3⋊#952
C3×#49932
C2×C3×#3993
C2×#49933