Finite Group Databasebeta

← C2×C3×#5174 #11866886

Subgroups — C2×C3×#5174 #11866886

Order 1152.

1237 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×SD16C211yesmaximal, characteristic, complemented
2C2×C3×#946C211yesmaximal, characteristic, complemented
3C2×C3×#938C211yesmaximal, characteristic
4C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
5C2×C3×#835C211yesmaximal, characteristic, complemented
6C2×C3×#832C211yesmaximal, characteristic, complemented
7C3×#5174C218yesmaximal, direct product factor
8C2×C3×#818C211yesmaximal, characteristic, complemented
9C2×C3×#468—31nomaximal
10C2×#5174C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C22×C3×#67—21no—
14C22×C3×C4×D3C2211yescharacteristic
15C2×C32×SD16—24no—
16C2×C32×SD16C2214yescomplemented
17C22×C32×C8C2211yescharacteristic
18C2×C3×#217—21no—
19C3×#946—22no—
20C3×#946C2214yescomplemented
21C3×#938—22no—
22C3×#938C2214yes—
23C2×C3×(C2×D3)⋊C4—21no—
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C2×C3×Dic3⋊C4—21no—
27C22×C3×C3⋊C8C2211yescharacteristic
28C2×C3×ΓL(1,𝔽25)—24no—
29C2×C3×ΓL(1,𝔽25)C2214yescomplemented
30C3×#835C2214yescomplemented
31C3×#832C2214yescomplemented
32C3×#818C2214yescomplemented
33C22×C3×SD16—21no—
34C22×C3×SD16D311yescharacteristic, complemented
35C22×C3×SD16C2×C311yescharacteristic, complemented
36C22×C3×M16—31no—
37C2×C3×C22⋊Q8—31no—
38C2×C3×#121—31no—
39C2×#946C2×C311yescharacteristic, complemented
40C2×#938C2×C311yescharacteristic
41C22×ΓL(1,𝔽25)C2×C311yescharacteristic, complemented
42C3×#468—38no—
43C2×C3×#107—31no—
44C2×C3×#103—31no—
45C2×C3×#102—31no—
46C2×#835C2×C311yescharacteristic, complemented
47C2×#832C2×C311yescharacteristic, complemented
48#5174C2×C318yesdirect product factor
49C2×#818C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C23×C3×D3D411yescharacteristic
52C2×C32×Q8—21no—
53C2×C32×Q8—22no—
54C2×C32×Q8—22no—
55C2×C32×Q8C2312yes—
56C2×C32×D4—22no—
57C2×C32×D4—21no—
58C2×C32×D4—22no—
59C2×C32×D4C2312yes—
60C22×C32×C4—21no—
61C22×C32×C4C2311yescharacteristic
62C2×C3×#67—24no—
63C2×C3×#67—44no—
64C22×C3×Dic3—21no—
65C22×C3×Dic3—21no—
66C22×C3×Dic3D411yescharacteristic
67C2×C3×C4×D3—22no—
68C2×C3×C4×D3—22no—
69C2×C3×C4×D3D412yes—
70C2×C3×C4×D3C2312yes—
71C32×SD16—28no—
72C2×C32×C8D414yescomplemented
73C2×C32×C8C2312yes—
74C3×#217—24no—
75C3×(C2×D3)⋊C4—24no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—21no—
78C3×C4⋊Dic3C2312yes—
79C3×C4⋊Dic3C2312yes—
80C3×Dic3⋊C4—24no—
81C2×C3×C3⋊C8—22no—
82C2×C3×C3⋊C8C2312yes—
83C3×ΓL(1,𝔽25)—28no—
84C2×#468—31noSylow
85C22×C3×Q8—21no—
86C22×C3×Q8C2×D311yescharacteristic
87C22×C3×Q8C22×C311yescharacteristic
88C22×C3×D4—21no—
89C22×C3×D4—61no—
90C22×C3×D4C2×D311yescharacteristic
91C22×C3×D4C22×C311yescharacteristic
92C23×C3×C4—31no—
93C22×#67—21no—
94C22×C4×D3C22×C311yescharacteristic
95C2×C3×SD16—28no—
96C2×C3×SD16—24no—
97C2×C3×SD16—24no—
98C2×C3×SD16—24no—
99C2×C3×SD16C2×D314yescomplemented
100C2×C3×SD16C22×C314yescomplemented
101C2×C3×M16—34no—
102C2×C3×M16—64no—
103C22×C3×C8—21no—
104C22×C3×C8—31no—
105C22×C3×C8C2×D311yescharacteristic
106C22×C3×C8C22×C311yescharacteristic
107C3×C22⋊Q8—34no—
108C3×C22⋊Q8—62no—
109C3×#121—34no—
110C3×#121—62no—
111C2×C3×C4⋊C4—31no—
112C2×C3×C4⋊C4—31no—
113C2×C3×C4⋊C4—61no—
114C2×C3×C22⋊C4—61no—
115C2×C3×C22⋊C4—61no—
116C2×#217—21no—
117#946—22no—
118#946C22×C314yescomplemented
119#938—22no—
120#938C22×C314yes—
121C2×(C2×D3)⋊C4—21no—
122C2×C4⋊Dic3C22×C311yescharacteristic
123C2×C4⋊Dic3C22×C311yescharacteristic
124C2×Dic3⋊C4—21no—
125C22×C3⋊C8C22×C311yescharacteristic
126C2×ΓL(1,𝔽25)—24no—
127C2×ΓL(1,𝔽25)C22×C314yescomplemented
128C3×#107—34no—
129C3×#103—34no—
130C3×#102—34no—
131#835C22×C314yescomplemented
132#832C22×C314yescomplemented
133#818C22×C314yescomplemented
134C23×C32—22no—
135C23×C32—42no—
136C23×C32—42no—
137C23×C32—42no—
138C23×C32C2×D411yescharacteristic
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3C2×D412yes—
143C32×Q8—22no—
144C32×Q8—22no—
145C32×Q8—42no—
146C32×D4—22no—
147C32×D4—22no—
148C32×D4—42no—
149C2×C32×C4—22no—
150C2×C32×C4—42no—
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2×D412yes—
153C2×C32×C4#4112yes—
154C2×C32×C4C2411yescharacteristic
155C3×#67—48no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—42no—
160C2×C3×Dic3—42no—
161C2×C3×Dic3C2×D412yes—
162C3×C4×D3—24no—
163C3×C4×D3—22no—
164C3×C4×D3—22no—
165C32×C8C2×D414yescomplemented
166C3×C3⋊C8—22no—
167C22×SD16C3×D311yescharacteristic, complemented
168C22×M16—31no—
169C2×C22⋊Q8—31no—
170C2×#121—31no—
171#468—38no—
172C2×#107—31no—
173C2×#103—31no—
174C2×#102—31no—
175C24×C3—21no—
176C24×C3—21no—
177C24×C3—31no—
178C24×C3—41no—
179C23×D3C3×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—24no—
183C2×C3×Q8—22no—
184C2×C3×Q8—22no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—41no—
189C2×C3×Q8—42no—
190C2×C3×Q8C22×D312yes—
191C2×C3×Q8C23×C312yes—
192C2×C3×D4—24no—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—41no—
201C2×C3×D4—42no—
202C2×C3×D4—64no—
203C2×C3×D4—124no—
204C2×C3×D4C22×D312yes—
205C2×C3×D4C23×C312yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—31no—
210C22×C3×C4—32no—
211C22×C3×C4—32no—
212C22×C3×C4—41no—
213C22×C3×C4—62no—
214C22×C3×C4—62no—
215C22×C3×C4—61no—
216C22×C3×C4—61no—
217C22×C3×C4C22×D311yescharacteristic
218C22×C3×C4C23×C311yescharacteristic
219C2×#67—24no—
220C2×#67—44no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3C3×D411yescharacteristic
224C2×C4×D3—22no—
225C2×C4×D3—22no—
226C2×C4×D3C3×D412yes—
227C2×C4×D3C23×C312yes—
228C3×SD16—216no—
229C3×SD16—28no—
230C3×SD16—28no—
231C3×M16—68no—
232C2×C3×C8—24no—
233C2×C3×C8—22no—
234C2×C3×C8—32no—
235C2×C3×C8—62no—
236C2×C3×C8#6714yescomplemented
237C2×C3×C8C3×D414yescomplemented
238C2×C3×C8C22×D312yes—
239C2×C3×C8C23×C312yes—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—32no—
242C3×C4⋊C4—64no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4—61no—
245C3×C22⋊C4—64no—
246C3×C22⋊C4—64no—
247#217—24no—
248(C2×D3)⋊C4—24no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3C23×C312yes—
252C4⋊Dic3C23×C312yes—
253Dic3⋊C4—24no—
254C2×C3⋊C8—22no—
255C2×C3⋊C8C23×C312yes—
256ΓL(1,𝔽25)—28no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—42no—
260C22×C32—84no—
261C22×C32#12112yes—
262C22×C32AGL(1,ℤ/8)12yes—
263C22×C32#13712yes—
264C22×C32C22×D411yescharacteristic
265C2×C3×D3—22no—
266C2×C3×D3—22no—
267C2×C3×D3—22no—
268C2×C3×D3—44no—
269C32×C4—42no—
270C32×C4#12112yes—
271C32×C4C22×D411yescharacteristic
272C32×C4C2×#4111yescharacteristic
273C3×Dic3—22no—
274C3×Dic3—42no—
275C3×Dic3—42no—
276C22×Q8C2×C3×D311yescharacteristic
277C22×D4—61no—
278C22×D4C2×C3×D311yescharacteristic
279C23×C4—31no—
280C2×SD16—24no—
281C2×SD16C2×C3×D314yescomplemented
282C2×M16—34no—
283C2×M16—64no—
284C22×C8—31no—
285C22×C8C2×C3×D311yescharacteristic
286C22⋊Q8—34no—
287C22⋊Q8—62no—
288#121—34no—
289#121—62no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295#107—34no—
296#103—34no—
297#102—34no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—32no—
302C23×C3—42no—
303C23×C3—44no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—62no—
311C23×C3—62no—
312C23×C3—62no—
313C23×C3—82no—
314C23×C3—82no—
315C23×C3D3×D411yescharacteristic
316C23×C3C2×C3×D411yescharacteristic
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—22no—
320C22×D3C2×C3×D412yes—
321C3×Q8—24no—
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—42no—
327C3×Q8—44no—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×D4—24no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—42no—
336C3×D4—44no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—128no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—32no—
347C2×C3×C4—42no—
348C2×C3×C4—44no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—62no—
352C2×C3×C4—64no—
353C2×C3×C4—62no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—122no—
358C2×C3×C4—122no—
359C2×C3×C4D3×D411yescharacteristic
360C2×C3×C4#23712yes—
361C2×C3×C4C2×#6712yes—
362C2×C3×C4C2×C3×D411yescharacteristic
363C2×C3×C4C2×C3×D412yes—
364C2×C3×C4C3×#4112yes—
365C2×C3×C4C23×D311yescharacteristic
366C2×C3×C4C24×C311yescharacteristic
367#67—48no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3C2×C3×D412yes—
374C4×D3—24no—
375C4×D3—22no—
376C4×D3—22no—
377C3×C8—24no—
378C3×C8—62no—
379C3×C8C2×#6714yescomplemented
380C3×C8C2×C3×D414yescomplemented
381C3⋊C8—22no—
382C2×C32—82no—
383C2×C32#46814yescomplemented
384C2×C32C2×#12111yescharacteristic
385C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
386C2×C32C2×#13711yescharacteristic
387C3×D3—42no—
388C24—21no—
389C24—31no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8C22×C3×D312yes—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—64no—
398C2×D4—124no—
399C2×D4C22×C3×D312yes—
400C22×C4—21no—
401C22×C4—31no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4C22×C3×D311yescharacteristic
409SD16—28no—
410M16—68no—
411C2×C8—32no—
412C2×C8—62no—
413C2×C8C3×#6714yescomplemented
414C2×C8C22×C3×D312yes—
415C4⋊C4—32no—
416C4⋊C4—32no—
417C4⋊C4—64no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C22⋊C4—64no—
421C22⋊C4—64no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—21no—
426C22×C3—44no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—62no—
434C22×C3—62no—
435C22×C3—62no—
436C22×C3—82no—
437C22×C3—82no—
438C22×C3—88no—
439C22×C3—84no—
440C22×C3—84no—
441C22×C3—124no—
442C22×C3#91412yes—
443C22×C3#91512yes—
444C22×C3#93812yes—
445C22×C3C3×#12112yes—
446C22×C3C3×AGL(1,ℤ/8)12yes—
447C22×C3C3×#13712yes—
448C22×C3C2×D3×D411yescharacteristic
449C22×C3C22×C3×D411yescharacteristic
450C2×D3—22no—
451C2×D3—22no—
452C2×D3—22no—
453C2×D3—44no—
454C3×C4—22no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—44no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—62no—
461C3×C4—122no—
462C3×C4—122no—
463C3×C4#93812yes—
464C3×C4C3×#12112yes—
465C3×C4C2×#23711yescharacteristic
466C3×C4C22×#6711yescharacteristic
467C3×C4C22×C3×D411yescharacteristic
468C3×C4C2×C3×#4111yescharacteristic
469Dic3—22no—
470Dic3—42no—
471Dic3—42no—
472C32C2×#46811yescharacteristic, complemented, Sylow
473C23—22no—
474C23—32no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—62no—
479C23—62no—
480C23—62no—
481C23C3×D3×D411yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—42no—
485D4—22no—
486D4—22no—
487D4—42no—
488D4—128no—
489C2×C4—22no—
490C2×C4—32no—
491C2×C4—42no—
492C2×C4—62no—
493C2×C4—64no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4C3×D3×D411yescharacteristic
501C2×C4C3×#23712yes—
502C2×C4C2×C3×#6712yes—
503C2×C4C23×C3×D311yescharacteristic
504C8—62no—
505C8C2×C3×#6714yescomplemented
506C2×C3—24no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—84no—
511C2×C3—82no—
512C2×C3—82no—
513C2×C3—122no—
514C2×C3#517414yesdirect product factor
515C2×C3C3×#46814yescomplemented
516C2×C3C2×#91411yescharacteristic
517C2×C3C2×#91511yescharacteristic
518C2×C3C2×#93811yescharacteristic
519C2×C3C2×C3×#12111yescharacteristic
520C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
521C2×C3C2×C3×#13711yescharacteristic
522D3—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—62no—
527C22—62no—
528C22—62no—
529C22—84no—
530C22—124no—
531C22C3×#91412yes—
532C22C3×#91512yes—
533C22C3×#93812yes—
534C22C2×C3×D3×D411yescharacteristic
535C4—42no—
536C4—62no—
537C4—122no—
538C4—122no—
539C4C3×#93812yes—
540C4C2×C3×#23711yescharacteristic
541C4C22×C3×#6711yescharacteristic
542C3—21no—
543C3C2×#517411yescharacteristic, direct product factor
544C3C2×C3×#46811yescharacteristic, complemented
545C2—82no—
546C2—122no—
547C2C3×#517414yesdirect product factor
548C2C2×C3×#91411yescharacteristic
549C2C2×C3×#91511yescharacteristic
550C2C2×C3×#93811yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×SD16C2
socleC23×C32C2×D4
prime radicalC2×C3×#938C2
hypercenterC22×C3×SD16D3
hypocenterC3C2×C3×#468
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C22×C32×SD16C2semidirect
2. C2×C3×#946C2semidirect
4. C22×C3×ΓL(1,𝔽25)C2semidirect
5. C2×C3×#835C2semidirect
6. C2×C3×#832C2semidirect
7. C3×#5174C2direct (N × H)
8. C2×C3×#818C2semidirect
10. C2×#5174C3direct (N × H)
16. C2×C32×SD16C22semidirect
20. C3×#946C22semidirect
29. C2×C3×ΓL(1,𝔽25)C22semidirect
30. C3×#835C22semidirect
31. C3×#832C22semidirect
32. C3×#818C22semidirect
34. C22×C3×SD16D3semidirect
35. C22×C3×SD16C2×C3semidirect
39. C2×#946C2×C3semidirect
41. C22×ΓL(1,𝔽25)C2×C3semidirect
46. C2×#835C2×C3semidirect
47. C2×#832C2×C3semidirect
48. #5174C2×C3direct (N × H)
49. C2×#818C2×C3semidirect
72. C2×C32×C8D4semidirect
99. C2×C3×SD16C2×D3semidirect
100. C2×C3×SD16C22×C3semidirect
118. #946C22×C3semidirect
127. C2×ΓL(1,𝔽25)C22×C3semidirect
131. #835C22×C3semidirect
132. #832C22×C3semidirect
133. #818C22×C3semidirect
165. C32×C8C2×D4semidirect
167. C22×SD16C3×D3semidirect
236. C2×C3×C8#67semidirect
237. C2×C3×C8C3×D4semidirect
281. C2×SD16C2×C3×D3semidirect
379. C3×C8C2×#67semidirect
380. C3×C8C2×C3×D4semidirect
383. C2×C32#468semidirect
413. C2×C8C3×#67semidirect
472. C32C2×#468semidirect
505. C8C2×C3×#67semidirect
514. C2×C3#5174direct (N × H)
515. C2×C3C3×#468semidirect
543. C3C2×#5174direct (N × H)
544. C3C2×C3×#468semidirect
547. C2C3×#5174direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SD162
C2×C3×#9462
C2×C3×#9382
C22×C3×ΓL(1,𝔽25)2
C2×C3×#8352
C2×C3×#8322
C3×#51742
C2×C3×#8182
C2×C3×#4683
C2×#51743