Finite Group Databasebeta

← C2×C3×#5716 #11900749

Subgroups — C2×C3×#5716 #11900749

Order 1152.

3705 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#237C212yesmaximal, complemented
2C2×C32×#127C211yesmaximal, characteristic, complemented
3C2×C3×#938C214yesmaximal, complemented
4C2×C3×#936C212yesmaximal, complemented
5C2×C3×D4×Dic3C214yesmaximal, complemented
6C3×#5716C2116yesmaximal, direct product factor
7C2×C3×C42×D3C211yesmaximal, characteristic, complemented
8C2×C3×#869C211yesmaximal, characteristic, complemented
9C2×C3×#532—31nomaximal
10C2×#5716C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2214yes—
12C22×C32×D4C2212yes—
13C22×C3×#67C2214yes—
14C23×C3×Dic3C2214yes—
15C2×C3×#237—216no—
16C2×C3×#237C22116yescomplemented
17C22×C3×C4×D3C2212yes—
18C22×C3×C4×D3C2211yescharacteristic
19C22×C3×Dic6C2212yes—
20C32×#127C2218yescomplemented
21C2×C32×C42C2211yescharacteristic
22C2×C3×#217C2218yes—
23C3×#938C22132yescomplemented
24C3×#936C22116yescomplemented
25C3×D4×Dic3C22132yescomplemented
26C2×C3×C4⋊Dic3C2214yes—
27C2×C3×C4×Dic3C2212yes—
28C2×C3×C4×Dic3C2211yescharacteristic
29C3×C42×D3C2218yescomplemented
30C3×#869C2218yescomplemented
31C22×C3×#41—32no—
32C22×#237C2×C312yescomplemented
33C3×#532—316no—
34C2×C3×C4⋊Q8—31no—
35C2×C3×#127—21no—
36C2×C3×#127D311yescharacteristic, complemented
37C2×C3×#127C2×C311yescharacteristic, complemented
38C2×C3×#124—32no—
39C2×C3×#121—34no—
40C2×C3×C4×D4—34no—
41C22×C3×C42—31no—
42C2×#938C2×C314yescomplemented
43C2×#936C2×C312yescomplemented
44C2×D4×Dic3C2×C314yescomplemented
45#5716C2×C3116yesdirect product factor
46C2×C42×D3C2×C311yescharacteristic, complemented
47C2×#869C2×C311yescharacteristic, complemented
48C24×C32C2314yes—
49C23×C3×D3C2311yescharacteristic
50C2×C32×D4—28no—
51C2×C32×D4—216no—
52C2×C32×D4C23116yes—
53C2×C32×D4C2318yes—
54C22×C32×C4C2311yescharacteristic
55C22×C32×C4C2312yes—
56C2×C3×#67—216no—
57C2×C3×#67C23116yes—
58C22×C3×Dic3—216no—
59C22×C3×Dic3C2312yes—
60C22×C3×Dic3C2314yes—
61C22×C3×Dic3C23116yes—
62C22×C3×Dic3C2311yescharacteristic
63C3×#237—264no—
64C2×C3×C4×D3—28no—
65C2×C3×C4×D3D418yescomplemented
66C2×C3×C4×D3C2318yes—
67C2×C3×C4×D3C2314yes—
68C2×C3×Dic6—28no—
69C2×C3×Dic6C2318yes—
70C32×C42C2314yes—
71C3×#217C23132yes—
72C3×C4⋊Dic3C23116yes—
73C3×C4×Dic3C2318yes—
74C3×C4×Dic3C2314yes—
75C2×#532—31noSylow
76C2×C3×#41—316no—
77C2×C3×#41—616no—
78C22×C3×Q8—32no—
79C22×C3×D4—24no—
80C22×C3×D4—22no—
81C22×C3×D4—34no—
82C22×C3×D4C2×D312yes—
83C22×C3×D4C2×D314yes—
84C22×C3×D4C22×C312yes—
85C22×C3×D4C22×C314yes—
86C23×C3×C4—31no—
87C23×C3×C4—34no—
88C23×C3×C4—32no—
89C22×#67C22×C314yes—
90C23×Dic3C22×C314yes—
91C2×#237—216no—
92C2×#237C22×C3116yescomplemented
93C22×C4×D3C22×C312yes—
94C22×C4×D3C22×C311yescharacteristic
95C22×Dic6C22×C312yes—
96C3×C4⋊Q8—38no—
97C3×#127—28no—
98C3×#127C2×D318yescomplemented
99C3×#127C22×C318yescomplemented
100C3×#124—316no—
101C3×#121—332no—
102C3×C4×D4—332no—
103C2×C3×C4⋊C4—34no—
104C2×C3×C22⋊C4—38no—
105C2×C3×C42—21no—
106C2×C3×C42—32no—
107C2×C3×C42—31no—
108C2×C3×C42—38no—
109C2×C3×C42C2×D311yescharacteristic
110C2×C3×C42C22×C311yescharacteristic
111C2×#217C22×C318yes—
112#938C22×C3132yescomplemented
113#936C22×C3116yescomplemented
114D4×Dic3C22×C3132yescomplemented
115C2×C4⋊Dic3C22×C314yes—
116C2×C4×Dic3C22×C312yes—
117C2×C4×Dic3C22×C311yescharacteristic
118C42×D3C22×C318yescomplemented
119#869C22×C318yescomplemented
120C23×C32—28no—
121C23×C32—216no—
122C23×C32C2418yes—
123C23×C32C2411yescharacteristic
124C22×C3×D3—24no—
125C22×C3×D3C2×D414yes—
126C22×C3×D3C2412yes—
127C32×D4—216no—
128C32×D4—232no—
129C2×C32×C4C2×D414yes—
130C2×C32×C4#4118yescomplemented
131C2×C32×C4C2412yes—
132C2×C32×C4C2414yes—
133C3×#67—232no—
134C2×C3×Dic3—28no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—232no—
137C2×C3×Dic3C2×D414yes—
138C2×C3×Dic3C2414yes—
139C2×C3×Dic3C2418yes—
140C2×C3×Dic3C2412yes—
141C3×C4×D3—216no—
142C3×C4×D3C2×D4116yescomplemented
143C3×Dic6—216no—
144C22×#41—32no—
145#532—316no—
146C2×C4⋊Q8—31no—
147C2×#127C3×D311yescharacteristic, complemented
148C2×#124—32no—
149C2×#121—34no—
150C2×C4×D4—34no—
151C22×C42—31no—
152C24×C3—24no—
153C24×C3—31no—
154C24×C3C22×D314yes—
155C24×C3C23×C314yes—
156C23×D3C23×C311yescharacteristic
157C3×#41—664no—
158C2×C3×Q8—38no—
159C2×C3×Q8—68no—
160C2×C3×D4—28no—
161C2×C3×D4—216no—
162C2×C3×D4—28no—
163C2×C3×D4—216no—
164C2×C3×D4—216no—
165C2×C3×D4—28no—
166C2×C3×D4—316no—
167C2×C3×D4—48no—
168C2×C3×D4—416no—
169C2×C3×D4—616no—
170C2×C3×D4C22×D3116yes—
171C2×C3×D4C22×D318yes—
172C2×C3×D4C23×C3116yes—
173C2×C3×D4C23×C318yes—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—31no—
177C22×C3×C4—38no—
178C22×C3×C4—34no—
179C22×C3×C4—316no—
180C22×C3×C4—32no—
181C22×C3×C4—34no—
182C22×C3×C4—38no—
183C22×C3×C4—616no—
184C22×C3×C4—68no—
185C22×C3×C4C22×D311yescharacteristic
186C22×C3×C4C22×D312yes—
187C22×C3×C4C23×C311yescharacteristic
188C22×C3×C4C23×C312yes—
189C2×#67—216no—
190C2×#67C23×C3116yes—
191C22×Dic3—216no—
192C22×Dic3C23×C312yes—
193C22×Dic3C23×C314yes—
194C22×Dic3C23×C3116yes—
195C22×Dic3C23×C311yescharacteristic
196#237—264no—
197C2×C4×D3—28no—
198C2×C4×D3C3×D418yescomplemented
199C2×C4×D3C23×C318yes—
200C2×C4×D3C23×C314yes—
201C2×Dic6—28no—
202C2×Dic6C23×C318yes—
203C3×C4⋊C4—316no—
204C3×C22⋊C4—332no—
205C3×C42—24no—
206C3×C42—38no—
207C3×C42—34no—
208C3×C42C22×D314yes—
209C3×C42C23×C314yes—
210#217C23×C3132yes—
211C4⋊Dic3C23×C3116yes—
212C4×Dic3C23×C318yes—
213C4×Dic3C23×C314yes—
214C22×C32—28no—
215C22×C32—216no—
216C22×C32—416no—
217C22×C32C22×D412yes—
218C22×C32C2×#4114yes—
219C22×C32C2511yescharacteristic
220C2×C3×D3—28no—
221C2×C3×D3—24no—
222C2×C3×D3C22×D414yes—
223C32×C4C22×D414yes—
224C32×C4C2×#4118yescomplemented
225C3×Dic3—28no—
226C3×Dic3—24no—
227C3×Dic3C22×D414yes—
228C2×#41—316no—
229C2×#41—616no—
230C22×Q8—32no—
231C22×D4—34no—
232C22×D4C2×C3×D314yes—
233C22×D4C2×C3×D312yes—
234C23×C4—31no—
235C23×C4—34no—
236C23×C4—32no—
237C4⋊Q8—38no—
238#127C2×C3×D318yescomplemented
239#124—316no—
240#121—332no—
241C4×D4—332no—
242C2×C4⋊C4—34no—
243C2×C22⋊C4—38no—
244C2×C42—32no—
245C2×C42—31no—
246C2×C42—38no—
247C2×C42C2×C3×D311yescharacteristic
248C23×C3—28no—
249C23×C3—216no—
250C23×C3—28no—
251C23×C3—216no—
252C23×C3—21no—
253C23×C3—28no—
254C23×C3—34no—
255C23×C3—32no—
256C23×C3—48no—
257C23×C3—416no—
258C23×C3—64no—
259C23×C3C23×D318yes—
260C23×C3C23×D311yescharacteristic
261C23×C3C24×C318yes—
262C23×C3C24×C311yescharacteristic
263C22×D3—24no—
264C22×D3C2×C3×D414yes—
265C22×D3C24×C312yes—
266C3×Q8—616no—
267C3×D4—216no—
268C3×D4—232no—
269C3×D4—216no—
270C3×D4—232no—
271C3×D4—416no—
272C3×D4—432no—
273C3×D4—632no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—28no—
278C2×C3×C4—34no—
279C2×C3×C4—32no—
280C2×C3×C4—316no—
281C2×C3×C4—34no—
282C2×C3×C4—38no—
283C2×C3×C4—632no—
284C2×C3×C4—68no—
285C2×C3×C4—64no—
286C2×C3×C4—616no—
287C2×C3×C4D3×D414yes—
288C2×C3×C4#23718yescomplemented
289C2×C3×C4C2×C3×D414yes—
290C2×C3×C4C3×#4118yescomplemented
291C2×C3×C4C23×D312yes—
292C2×C3×C4C23×D314yes—
293C2×C3×C4C24×C312yes—
294C2×C3×C4C24×C314yes—
295#67—232no—
296C2×Dic3—28no—
297C2×Dic3—24no—
298C2×Dic3—232no—
299C2×Dic3C2×C3×D414yes—
300C2×Dic3C24×C314yes—
301C2×Dic3C24×C318yes—
302C2×Dic3C24×C312yes—
303C4×D3—216no—
304C4×D3C2×C3×D4116yescomplemented
305Dic6—216no—
306C2×C32—48no—
307C2×C32#53214yescomplemented
308C2×C32C23×D411yescharacteristic
309C2×C32C22×#4112yes—
310C3×D3—24no—
311C24—31no—
312C24C22×C3×D314yes—
313#41—664no—
314C2×Q8—38no—
315C2×Q8—68no—
316C2×D4—28no—
317C2×D4—216no—
318C2×D4—316no—
319C2×D4—616no—
320C2×D4C22×C3×D3116yes—
321C2×D4C22×C3×D318yes—
322C22×C4—31no—
323C22×C4—34no—
324C22×C4—38no—
325C22×C4—32no—
326C22×C4—34no—
327C22×C4—316no—
328C22×C4—38no—
329C22×C4—616no—
330C22×C4—68no—
331C22×C4C22×C3×D311yescharacteristic
332C22×C4C22×C3×D312yes—
333C4⋊C4—316no—
334C22⋊C4—332no—
335C42—38no—
336C42—34no—
337C42C22×C3×D314yes—
338C22×C3—28no—
339C22×C3—216no—
340C22×C3—28no—
341C22×C3—216no—
342C22×C3—22no—
343C22×C3—24no—
344C22×C3—21no—
345C22×C3—34no—
346C22×C3—48no—
347C22×C3—416no—
348C22×C3—416no—
349C22×C3—416no—
350C22×C3—64no—
351C22×C3—68no—
352C22×C3—816no—
353C22×C3C2×D3×D412yes—
354C22×C3C2×#23714yes—
355C22×C3C22×C3×D412yes—
356C22×C3C2×C3×#4114yes—
357C22×C3C24×D311yescharacteristic
358C22×C3C25×C311yescharacteristic
359C2×D3—24no—
360C2×D3—28no—
361C2×D3C22×C3×D414yes—
362C3×C4—24no—
363C3×C4—28no—
364C3×C4—34no—
365C3×C4—68no—
366C3×C4—64no—
367C3×C4C2×D3×D414yes—
368C3×C4C2×#23718yescomplemented
369C3×C4C22×C3×D414yes—
370C3×C4C2×C3×#4118yescomplemented
371Dic3—28no—
372Dic3—24no—
373Dic3C22×C3×D414yes—
374C32C2×#53211yescharacteristic, complemented, Sylow
375C23—28no—
376C23—216no—
377C23—34no—
378C23—32no—
379C23—64no—
380C23C23×C3×D318yes—
381C23C23×C3×D311yescharacteristic
382Q8—616no—
383D4—216no—
384D4—232no—
385D4—632no—
386C2×C4—34no—
387C2×C4—32no—
388C2×C4—316no—
389C2×C4—34no—
390C2×C4—38no—
391C2×C4—632no—
392C2×C4—68no—
393C2×C4—64no—
394C2×C4—616no—
395C2×C4C3×D3×D414yes—
396C2×C4C3×#23718yescomplemented
397C2×C4C23×C3×D312yes—
398C2×C4C23×C3×D314yes—
399C2×C3—21no—
400C2×C3—22no—
401C2×C3—24no—
402C2×C3—48no—
403C2×C3—48no—
404C2×C3—64no—
405C2×C3—88no—
406C2×C3#571614yesdirect product factor
407C2×C3C3×#53214yescomplemented
408C2×C3C22×D3×D411yescharacteristic
409C2×C3C22×#23712yes—
410C2×C3C23×C3×D411yescharacteristic
411C2×C3C22×C3×#4112yes—
412D3—24no—
413C22—28no—
414C22—216no—
415C22—34no—
416C22—416no—
417C22—64no—
418C22—68no—
419C22C2×C3×D3×D412yes—
420C22C2×C3×#23714yes—
421C22C24×C3×D311yescharacteristic
422C4—34no—
423C4—68no—
424C4—64no—
425C4C2×C3×D3×D414yes—
426C4C2×C3×#23718yescomplemented
427C3—21no—
428C3C2×#571611yescharacteristic, direct product factor
429C3C2×C3×#53211yescharacteristic, complemented
430C2—48no—
431C2—64no—
432C2C3×#571614yesdirect product factor
433C2C22×C3×D3×D411yescharacteristic
434C2C22×C3×#23712yes—
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×#127C2
socleC23×C32C24
hypercenterC2×C3×#127D3
hypocenterC3C2×C3×#532
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#127

Complements

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

NH ≅ G/Ntype
1. C22×C3×#237C2semidirect
2. C2×C32×#127C2semidirect
3. C2×C3×#938C2semidirect
4. C2×C3×#936C2semidirect
5. C2×C3×D4×Dic3C2semidirect
6. C3×#5716C2direct (N × H)
7. C2×C3×C42×D3C2semidirect
8. C2×C3×#869C2semidirect
10. C2×#5716C3direct (N × H)
16. C2×C3×#237C22semidirect
20. C32×#127C22semidirect
23. C3×#938C22semidirect
24. C3×#936C22semidirect
25. C3×D4×Dic3C22semidirect
29. C3×C42×D3C22semidirect
30. C3×#869C22semidirect
32. C22×#237C2×C3semidirect
36. C2×C3×#127D3semidirect
37. C2×C3×#127C2×C3semidirect
42. C2×#938C2×C3semidirect
43. C2×#936C2×C3semidirect
44. C2×D4×Dic3C2×C3semidirect
45. #5716C2×C3direct (N × H)
46. C2×C42×D3C2×C3semidirect
47. C2×#869C2×C3semidirect
65. C2×C3×C4×D3D4semidirect
92. C2×#237C22×C3semidirect
98. C3×#127C2×D3semidirect
99. C3×#127C22×C3semidirect
112. #938C22×C3semidirect
113. #936C22×C3semidirect
114. D4×Dic3C22×C3semidirect
118. C42×D3C22×C3semidirect
119. #869C22×C3semidirect
130. C2×C32×C4#41semidirect
142. C3×C4×D3C2×D4semidirect
147. C2×#127C3×D3semidirect
198. C2×C4×D3C3×D4semidirect
224. C32×C4C2×#41semidirect
238. #127C2×C3×D3semidirect
288. C2×C3×C4#237semidirect
290. C2×C3×C4C3×#41semidirect
304. C4×D3C2×C3×D4semidirect
307. C2×C32#532semidirect
368. C3×C4C2×#237semidirect
370. C3×C4C2×C3×#41semidirect
374. C32C2×#532semidirect
396. C2×C4C3×#237semidirect
406. C2×C3#5716direct (N × H)
407. C2×C3C3×#532semidirect
426. C4C2×C3×#237semidirect
428. C3C2×#5716direct (N × H)
429. C3C2×C3×#532semidirect
432. C2C3×#5716direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2372
C2×C32×#1272
C2×C3×#9382
C2×C3×#9362
C2×C3×D4×Dic32
C3×#57162
C2×C3×C42×D32
C2×C3×#8692
C2×C3×#5323
C2×#57163