Finite Group Databasebeta

← C2×C3×#5182 #11865625

Subgroups — C2×C3×#5182 #11865625

Order 1152.

1089 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×Q16C211yesmaximal, characteristic
2C2×C3×Q8×Dic3C211yesmaximal, characteristic
3C2×C3×#944C211yesmaximal, characteristic
4C22×C3×#216C211yesmaximal, characteristic
5C2×C3×#835C211yesmaximal, characteristic
6C3×#5182C218yesmaximal, direct product factor
7C2×C3×#816C211yesmaximal, characteristic
8C2×C3×Dic3⋊C8C211yesmaximal, characteristic
9C2×C3×#462—31nomaximal
10C2×#5182C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×Q8C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C2×C32×Q16—24no—
15C2×C32×Q16C2214yes—
16C22×C32×C8C2211yescharacteristic
17C3×Q8×Dic3—22no—
18C3×Q8×Dic3C2214yes—
19C3×#944—22no—
20C3×#944C2214yes—
21C2×C3×#216—24no—
22C2×C3×#216C2214yes—
23C2×C3×C4⋊Dic3—21no—
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×Dic3⋊C4—21no—
26C2×C3×Dic3⋊C4—21no—
27C2×C3×C4×Dic3—21no—
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C3×#835C2214yes—
31C3×#816C2214yes—
32C3×Dic3⋊C8C2214yes—
33C22×C3×Q16—21no—
34C22×C3×Q16—31no—
35C22×C3×Q16D311yescharacteristic
36C22×C3×Q16C2×C311yescharacteristic
37C2×C3×C4⋊Q8—31no—
38C2×C3×C4×Q8—31no—
39C2×Q8×Dic3C2×C311yescharacteristic
40C2×#944C2×C311yescharacteristic
41C22×#216C2×C311yescharacteristic
42C3×#462—38no—
43C2×C3×C4⋊C8—31no—
44C2×C3×#103—31no—
45C2×C3×#103—31no—
46C2×#835C2×C311yescharacteristic
47#5182C2×C318yesdirect product factor
48C2×#816C2×C311yescharacteristic
49C2×Dic3⋊C8C2×C311yescharacteristic
50C2×C32×Q8—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—22no—
53C2×C32×Q8—22no—
54C2×C32×Q8—22no—
55C2×C32×Q8D414yes—
56C2×C32×Q8C2312yes—
57C2×C32×Q8C2312yes—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4C2311yescharacteristic
61C22×C3×Dic3—21no—
62C22×C3×Dic3—21no—
63C22×C3×Dic3D411yescharacteristic
64C22×C3×Dic3D411yescharacteristic
65C2×C3×Dic6—22no—
66C2×C3×Dic6—22no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2312yes—
69C32×Q16—28no—
70C2×C32×C8—22no—
71C2×C32×C8C2312yes—
72C3×#216—28no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—24no—
75C3×C4⋊Dic3C2312yes—
76C3×Dic3⋊C4—24no—
77C3×Dic3⋊C4—24no—
78C3×C4×Dic3—24no—
79C3×C4×Dic3D412yes—
80C3×C4×Dic3C2312yes—
81C2×C3×C3⋊C8—22no—
82C2×C3×C3⋊C8C2312yes—
83C2×#462—31noSylow
84C22×C3×Q8—21no—
85C22×C3×Q8—21no—
86C22×C3×Q8—31no—
87C22×C3×Q8C2×D311yescharacteristic
88C22×C3×Q8C2×D311yescharacteristic
89C22×C3×Q8C22×C311yescharacteristic
90C22×C3×Q8C22×C311yescharacteristic
91C22×Dic6C22×C311yescharacteristic
92C2×C3×Q16—28no—
93C2×C3×Q16—24no—
94C2×C3×Q16—24no—
95C2×C3×Q16—24no—
96C2×C3×Q16—34no—
97C2×C3×Q16—64no—
98C2×C3×Q16C2×D314yes—
99C2×C3×Q16C22×C314yes—
100C22×C3×C8—21no—
101C22×C3×C8—31no—
102C22×C3×C8C2×D311yescharacteristic
103C22×C3×C8C22×C311yescharacteristic
104C3×C4⋊Q8—34no—
105C3×C4⋊Q8—62no—
106C3×C4×Q8—34no—
107C3×C4×Q8—62no—
108C2×C3×C4⋊C4—31no—
109C2×C3×C4⋊C4—61no—
110C2×C3×C4⋊C4—61no—
111C2×C3×C4⋊C4—61no—
112C2×C3×C42—31no—
113C2×C3×C42—61no—
114Q8×Dic3—22no—
115Q8×Dic3C22×C314yes—
116#944—22no—
117#944C22×C314yes—
118C2×#216—24no—
119C2×#216C22×C314yes—
120C2×C4⋊Dic3—21no—
121C2×C4⋊Dic3C22×C311yescharacteristic
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×C4×Dic3—21no—
125C2×C4×Dic3C22×C311yescharacteristic
126C22×C3⋊C8C22×C311yescharacteristic
127C3×C4⋊C8—34no—
128C3×#103—34no—
129C3×#103—34no—
130#835C22×C314yes—
131#816C22×C314yes—
132Dic3⋊C8C22×C314yes—
133C23×C32C2×D411yescharacteristic
134C32×Q8—22no—
135C32×Q8—24no—
136C32×Q8—22no—
137C32×Q8—22no—
138C32×Q8—42no—
139C32×Q8C2×D414yes—
140C2×C32×C4—24no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—42no—
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D412yes—
146C2×C32×C4#4112yes—
147C2×C32×C4C2411yescharacteristic
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—42no—
152C2×C3×Dic3—42no—
153C2×C3×Dic3Q1614yescomplemented
154C2×C3×Dic3C2×D412yes—
155C2×C3×Dic3C2×D412yes—
156C3×Dic6—22no—
157C3×Dic6—22no—
158C3×Dic6—42no—
159C32×C8—22no—
160C3×C3⋊C8—22no—
161C22×Q16—31no—
162C22×Q16C3×D311yescharacteristic
163C2×C4⋊Q8—31no—
164C2×C4×Q8—31no—
165#462—38no—
166C2×C4⋊C8—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—24no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—22no—
180C2×C3×Q8—24no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—32no—
184C2×C3×Q8—41no—
185C2×C3×Q8—41no—
186C2×C3×Q8—42no—
187C2×C3×Q8—42no—
188C2×C3×Q8—61no—
189C2×C3×Q8—62no—
190C2×C3×Q8—62no—
191C2×C3×Q8#6714yes—
192C2×C3×Q8C3×D414yes—
193C2×C3×Q8C22×D312yes—
194C2×C3×Q8C22×D312yes—
195C2×C3×Q8C23×C312yes—
196C2×C3×Q8C23×C312yes—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—41no—
205C22×C3×C4—41no—
206C22×C3×C4—61no—
207C22×C3×C4—61no—
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C23×C311yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3C3×D411yescharacteristic
213C22×Dic3C3×D411yescharacteristic
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—21no—
217C2×Dic6C23×C312yes—
218C3×Q16—216no—
219C3×Q16—28no—
220C3×Q16—28no—
221C3×Q16—68no—
222C2×C3×C8—24no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—32no—
227C2×C3×C8—62no—
228C2×C3×C8C22×D312yes—
229C2×C3×C8C23×C312yes—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—64no—
232C3×C4⋊C4—64no—
233C3×C4⋊C4—64no—
234C3×C4⋊C4—61no—
235C3×C42—32no—
236C3×C42—32no—
237C3×C42—64no—
238#216—28no—
239C4⋊Dic3—21no—
240C4⋊Dic3—24no—
241C4⋊Dic3C23×C312yes—
242Dic3⋊C4—24no—
243Dic3⋊C4—24no—
244C4×Dic3—24no—
245C4×Dic3C3×D412yes—
246C4×Dic3C23×C312yes—
247C2×C3⋊C8—22no—
248C2×C3⋊C8C23×C312yes—
249C22×C32#12112yes—
250C22×C32C2×Q1612yes—
251C22×C32#13712yes—
252C22×C32C22×D411yescharacteristic
253C32×C4—24no—
254C32×C4—42no—
255C32×C4#12112yes—
256C32×C4C22×D411yescharacteristic
257C32×C4C2×#4111yescharacteristic
258C3×Dic3—22no—
259C3×Dic3—42no—
260C3×Dic3—42no—
261C3×Dic3C2×Q1614yescomplemented
262C22×Q8—31no—
263C22×Q8C2×C3×D311yescharacteristic
264C22×Q8C2×C3×D311yescharacteristic
265C2×Q16—24no—
266C2×Q16—34no—
267C2×Q16—64no—
268C2×Q16C2×C3×D314yes—
269C22×C8—31no—
270C22×C8C2×C3×D311yescharacteristic
271C4⋊Q8—34no—
272C4⋊Q8—62no—
273C4×Q8—34no—
274C4×Q8—62no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C42—31no—
280C2×C42—61no—
281C4⋊C8—34no—
282#103—34no—
283#103—34no—
284C23×C3—21no—
285C23×C3D3×D411yescharacteristic
286C23×C3C2×C3×D411yescharacteristic
287C3×Q8—24no—
288C3×Q8—24no—
289C3×Q8—22no—
290C3×Q8—22no—
291C3×Q8—24no—
292C3×Q8—22no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×Q8—22no—
296C3×Q8—24no—
297C3×Q8—44no—
298C3×Q8—42no—
299C3×Q8—42no—
300C3×Q8—44no—
301C3×Q8—42no—
302C3×Q8—42no—
303C3×Q8—62no—
304C3×Q8—62no—
305C3×Q8—122no—
306C3×Q8C2×#6714yes—
307C3×Q8C2×C3×D414yes—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—34no—
319C2×C3×C4—32no—
320C2×C3×C4—32no—
321C2×C3×C4—44no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—44no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—122no—
331C2×C3×C4—122no—
332C2×C3×C4D3×D411yescharacteristic
333C2×C3×C4#23712yes—
334C2×C3×C4C2×#6712yes—
335C2×C3×C4C2×C3×D411yescharacteristic
336C2×C3×C4C2×C3×D412yes—
337C2×C3×C4C3×#4112yes—
338C2×C3×C4C23×D311yescharacteristic
339C2×C3×C4C24×C311yescharacteristic
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3C3×Q1614yescomplemented
346C2×Dic3C2×C3×D412yes—
347C2×Dic3C2×C3×D412yes—
348Dic6—22no—
349Dic6—22no—
350Dic6—42no—
351C3×C8—24no—
352C3×C8—22no—
353C3×C8—22no—
354C3×C8—62no—
355C3⋊C8—22no—
356C2×C32#46214yescomplemented
357C2×C32C2×#12111yescharacteristic
358C2×C32C22×Q1611yescharacteristic
359C2×C32C2×#13711yescharacteristic
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—32no—
366C2×Q8—61no—
367C2×Q8—62no—
368C2×Q8—62no—
369C2×Q8C3×#6714yes—
370C2×Q8C22×C3×D312yes—
371C2×Q8C22×C3×D312yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—31no—
375C22×C4—31no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4C22×C3×D311yescharacteristic
379Q16—28no—
380Q16—68no—
381C2×C8—22no—
382C2×C8—32no—
383C2×C8—62no—
384C2×C8C22×C3×D312yes—
385C4⋊C4—32no—
386C4⋊C4—64no—
387C4⋊C4—64no—
388C4⋊C4—64no—
389C4⋊C4—61no—
390C42—32no—
391C42—32no—
392C42—64no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—21no—
397C22×C3D3×Q1612yes—
398C22×C3#91812yes—
399C22×C3#93912yes—
400C22×C3C3×#12112yes—
401C22×C3C2×C3×Q1612yes—
402C22×C3C3×#13712yes—
403C22×C3C2×D3×D411yescharacteristic
404C22×C3C22×C3×D411yescharacteristic
405C3×C4—24no—
406C3×C4—24no—
407C3×C4—22no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—34no—
411C3×C4—44no—
412C3×C4—44no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—62no—
416C3×C4—122no—
417C3×C4—122no—
418C3×C4#93912yes—
419C3×C4C3×#12112yes—
420C3×C4C2×#23711yescharacteristic
421C3×C4C22×#6711yescharacteristic
422C3×C4C22×C3×D411yescharacteristic
423C3×C4C2×C3×#4111yescharacteristic
424Dic3—22no—
425Dic3—42no—
426Dic3—42no—
427Dic3C2×C3×Q1614yescomplemented
428C32C2×#46211yescharacteristic, complemented, Sylow
429C23C3×D3×D411yescharacteristic
430Q8—24no—
431Q8—22no—
432Q8—22no—
433Q8—22no—
434Q8—42no—
435Q8—62no—
436Q8—62no—
437Q8—122no—
438Q8C2×C3×#6714yes—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—34no—
443C2×C4—32no—
444C2×C4—32no—
445C2×C4—42no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4C3×D3×D411yescharacteristic
452C2×C4C3×#23712yes—
453C2×C4C2×C3×#6712yes—
454C2×C4C23×C3×D311yescharacteristic
455C8—22no—
456C8—62no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—24no—
460C2×C3—21no—
461C2×C3#518214yesdirect product factor
462C2×C3C3×#46214yescomplemented
463C2×C3C2×D3×Q1611yescharacteristic
464C2×C3C2×#91811yescharacteristic
465C2×C3C2×#93911yescharacteristic
466C2×C3C2×C3×#12111yescharacteristic
467C2×C3C22×C3×Q1611yescharacteristic
468C2×C3C2×C3×#13711yescharacteristic
469C22C3×D3×Q1612yes—
470C22C3×#91812yes—
471C22C3×#93912yes—
472C22C2×C3×D3×D411yescharacteristic
473C4—24no—
474C4—34no—
475C4—42no—
476C4—62no—
477C4—122no—
478C4—122no—
479C4C3×#93912yes—
480C4C2×C3×#23711yescharacteristic
481C4C22×C3×#6711yescharacteristic
482C3—21no—
483C3C2×#518211yescharacteristic, direct product factor
484C3C2×C3×#46211yescharacteristic, complemented
485C2C3×#518214yesdirect product factor
486C2C2×C3×D3×Q1611yescharacteristic
487C2C2×C3×#91811yescharacteristic
488C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×Q16C2
socle, prime radicalC23×C32C2×D4
hypercenterC22×C3×Q16D3
hypocenterC3C2×C3×#462
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×Q16

Complements

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

NH ≅ G/Ntype
6. C3×#5182C2direct (N × H)
10. C2×#5182C3direct (N × H)
47. #5182C2×C3direct (N × H)
153. C2×C3×Dic3Q16semidirect
261. C3×Dic3C2×Q16semidirect
345. C2×Dic3C3×Q16semidirect
356. C2×C32#462semidirect
427. Dic3C2×C3×Q16semidirect
428. C32C2×#462semidirect
461. C2×C3#5182direct (N × H)
462. C2×C3C3×#462semidirect
483. C3C2×#5182direct (N × H)
484. C3C2×C3×#462semidirect
485. C2C3×#5182direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×Q162
C2×C3×Q8×Dic32
C2×C3×#9442
C22×C3×#2162
C2×C3×#8352
C3×#51822
C2×C3×#8162
C2×C3×Dic3⋊C82
C2×C3×#4623
C2×#51823