Finite Group Databasebeta

← C2×C3×#5177 #11867844

Subgroups — C2×C3×#5177 #11867844

Order 1152.

1085 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×SD16C211yesmaximal, characteristic, complemented
2C2×C3×#942C211yesmaximal, characteristic, complemented
3C2×C3×#932C211yesmaximal, characteristic
4C2×C3×#901C211yesmaximal, characteristic, complemented
5C2×C3×#836C211yesmaximal, characteristic, complemented
6C2×C3×#833C211yesmaximal, characteristic, complemented
7C3×#5177C218yesmaximal, direct product factor
8C2×C3×#819C211yesmaximal, characteristic, complemented
9C2×C3×#471—31nomaximal
10C2×#5177C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C2×C32×SD16—21no—
15C2×C32×SD16—21no—
16C2×C32×SD16—22no—
17C2×C32×SD16C2214yescomplemented
18C22×C32×C8C2211yescharacteristic
19C2×C3×#216—21no—
20C3×#942—22no—
21C3×#942C2214yescomplemented
22C2×C3×#215—21no—
23C2×C3×#214—21no—
24C3×#932—22no—
25C3×#932C2214yes—
26C2×C3×#213—21no—
27C2×C3×#208C2211yescharacteristic
28C2×C3×#208C2211yescharacteristic
29C2×C3×#208C2211yescharacteristic
30C3×#901—22no—
31C3×#901C2214yescomplemented
32C2×C3×ΓL(1,𝔽25)—21no—
33C2×C3×D3×C8—21no—
34C3×#836C2214yescomplemented
35C3×#833C2214yescomplemented
36C3×#819C2214yescomplemented
37C2×C3×#137—31no—
38C2×C3×AGL(1,ℤ/8)—31no—
39C22×C3×SD16—21no—
40C22×C3×SD16D311yescharacteristic, complemented
41C22×C3×SD16C2×C311yescharacteristic, complemented
42C2×C3×#131—31no—
43C2×#942C2×C311yescharacteristic, complemented
44C2×#932C2×C311yescharacteristic
45C2×#901C2×C311yescharacteristic, complemented
46C3×#471—38no—
47C2×C3×#108—31no—
48C2×C3×#101—31no—
49C2×C3×#100—31no—
50C2×#836C2×C311yescharacteristic, complemented
51C2×#833C2×C311yescharacteristic, complemented
52#5177C2×C318yesdirect product factor
53C2×#819C2×C311yescharacteristic, complemented
54C24×C32—21no—
55C2×C32×Q8—21no—
56C2×C32×Q8—21no—
57C2×C32×Q8—21no—
58C2×C32×Q8—21no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2312yes—
61C2×C32×D4—21no—
62C2×C32×D4—21no—
63C2×C32×D4—21no—
64C2×C32×D4—21no—
65C2×C32×D4—21no—
66C2×C32×D4C2312yes—
67C22×C32×C4—21no—
68C22×C32×C4C2311yescharacteristic
69C2×C3×#67—21no—
70C3×#235—21no—
71C3×#235—21no—
72C3×#235D412yes—
73C3×#235C2312yes—
74C2×C3×D12D411yescharacteristic
75C2×C3×C4×D3—21no—
76C2×C3×Dic6D411yescharacteristic
77C32×SD16—24no—
78C32×SD16—24no—
79C2×C32×C8D411yescharacteristic
80C2×C32×C8D411yescharacteristic
81C2×C32×C8D412yes—
82C2×C32×C8C2312yes—
83C3×#216—24no—
84C3×#215—24no—
85C3×#214—24no—
86C3×#213—24no—
87C3×#208—21no—
88C3×#208—21no—
89C3×#208—21no—
90C3×#208C2312yes—
91C3×#208C2312yes—
92C3×#208C2312yes—
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8—21no—
96C3×ΓL(1,𝔽25)—24no—
97C3×D3×C8—24no—
98C2×#471—31noSylow
99C2×C3×#41—31no—
100C22×C3×Q8—21no—
101C22×C3×Q8C2×D311yescharacteristic
102C22×C3×Q8C22×C311yescharacteristic
103C22×C3×D4—21no—
104C22×C3×D4C2×D311yescharacteristic
105C22×C3×D4C22×C311yescharacteristic
106C2×#235C22×C311yescharacteristic
107C3×#137—34no—
108C3×#137—62no—
109C3×AGL(1,ℤ/8)—34no—
110C3×AGL(1,ℤ/8)—62no—
111C2×C3×Q16—61no—
112C2×C3×SD16—22no—
113C2×C3×SD16—22no—
114C2×C3×SD16—24no—
115C2×C3×SD16—21no—
116C2×C3×SD16—21no—
117C2×C3×SD16—22no—
118C2×C3×SD16—21no—
119C2×C3×SD16—21no—
120C2×C3×SD16—22no—
121C2×C3×SD16—24no—
122C2×C3×SD16—61no—
123C2×C3×SD16—61no—
124C2×C3×SD16C2×D314yescomplemented
125C2×C3×SD16C22×C314yescomplemented
126C2×C3×D8—61no—
127C3×#131—34no—
128C3×#131—62no—
129C2×C3×M16—31no—
130C2×C3×M16—31no—
131C2×C3×M16—31no—
132C2×C3×M16—61no—
133C22×C3×C8—21no—
134C22×C3×C8—61no—
135C22×C3×C8C2×D311yescharacteristic
136C22×C3×C8C22×C311yescharacteristic
137C2×#216—21no—
138#942—22no—
139#942C22×C314yescomplemented
140C2×#215—21no—
141C2×#214—21no—
142#932—22no—
143#932C22×C314yes—
144C2×#213—21no—
145C2×#208C22×C311yescharacteristic
146C2×#208C22×C311yescharacteristic
147C2×#208C22×C311yescharacteristic
148#901—22no—
149#901C22×C314yescomplemented
150C2×ΓL(1,𝔽25)—21no—
151C2×D3×C8—21no—
152C3×#108—34no—
153C3×#101—34no—
154C3×#100—34no—
155#836C22×C314yescomplemented
156#833C22×C314yescomplemented
157#819C22×C314yescomplemented
158C23×C32—22no—
159C23×C32—41no—
160C23×C32—41no—
161C23×C32—82no—
162C23×C32#4111yescharacteristic
163C22×C3×D3—21no—
164C32×Q8—22no—
165C32×Q8—22no—
166C32×Q8—41no—
167C32×Q8—41no—
168C32×D4—22no—
169C32×D4—22no—
170C32×D4—41no—
171C32×D4—41no—
172C2×C32×C4—22no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2×D411yescharacteristic
178C2×C32×C4C2×D411yescharacteristic
179C2×C32×C4#4111yescharacteristic
180C2×C32×C4C2411yescharacteristic
181C3×#67—22no—
182C3×#67—22no—
183C2×C3×Dic3—21no—
184C3×D12—21no—
185C3×D12C2×D412yes—
186C3×C4×D3—22no—
187C3×C4×D3—22no—
188C3×Dic6—21no—
189C3×Dic6C2×D412yes—
190C32×C8C2×D412yes—
191C32×C8C2×D412yes—
192C3×C3⋊C8—22no—
193C3×C3⋊C8—22no—
194C3×C3⋊C8—22no—
195C2×#137—31no—
196C2×AGL(1,ℤ/8)—31no—
197C22×SD16C3×D311yescharacteristic, complemented
198C2×#131—31no—
199#471—38no—
200C2×#108—31no—
201C2×#101—31no—
202C2×#100—31no—
203C24×C3—21no—
204C24×C3—21no—
205C24×C3—41no—
206C3×#41—32no—
207C3×#41—32no—
208C3×#41—61no—
209C3×#41—61no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—22no—
213C2×C3×Q8—22no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—22no—
223C2×C3×Q8—31no—
224C2×C3×Q8—41no—
225C2×C3×Q8—41no—
226C2×C3×Q8—41no—
227C2×C3×Q8C22×D312yes—
228C2×C3×Q8C23×C312yes—
229C2×C3×D4—22no—
230C2×C3×D4—22no—
231C2×C3×D4—21no—
232C2×C3×D4—21no—
233C2×C3×D4—21no—
234C2×C3×D4—21no—
235C2×C3×D4—21no—
236C2×C3×D4—21no—
237C2×C3×D4—21no—
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—22no—
242C2×C3×D4—31no—
243C2×C3×D4—41no—
244C2×C3×D4—41no—
245C2×C3×D4—41no—
246C2×C3×D4—61no—
247C2×C3×D4C22×D312yes—
248C2×C3×D4C23×C312yes—
249C22×C3×C4—21no—
250C22×C3×C4—21no—
251C22×C3×C4—21no—
252C22×C3×C4—41no—
253C22×C3×C4—61no—
254C22×C3×C4C22×D311yescharacteristic
255C22×C3×C4C23×C311yescharacteristic
256C2×#67—21no—
257#235—21no—
258#235—21no—
259#235C3×D412yes—
260#235C23×C312yes—
261C2×D12C3×D411yescharacteristic
262C2×C4×D3—21no—
263C2×Dic6C3×D411yescharacteristic
264C3×Q16—64no—
265C3×SD16—28no—
266C3×SD16—28no—
267C3×SD16—24no—
268C3×SD16—24no—
269C3×SD16—24no—
270C3×SD16—24no—
271C3×SD16—64no—
272C3×SD16—64no—
273C3×D8—64no—
274C3×M16—32no—
275C3×M16—32no—
276C3×M16—32no—
277C3×M16—61no—
278C3×M16—61no—
279C3×M16—61no—
280C3×M16—64no—
281C2×C3×C8—21no—
282C2×C3×C8—21no—
283C2×C3×C8—22no—
284C2×C3×C8—22no—
285C2×C3×C8—61no—
286C2×C3×C8—61no—
287C2×C3×C8—61no—
288C2×C3×C8—64no—
289C2×C3×C8#6711yescharacteristic
290C2×C3×C8#6711yescharacteristic
291C2×C3×C8#6712yes—
292C2×C3×C8C3×D411yescharacteristic
293C2×C3×C8C3×D411yescharacteristic
294C2×C3×C8C3×D412yes—
295C2×C3×C8C22×D312yes—
296C2×C3×C8C23×C312yes—
297#216—24no—
298#215—24no—
299#214—24no—
300#213—24no—
301#208—21no—
302#208—21no—
303#208—21no—
304#208C23×C312yes—
305#208C23×C312yes—
306#208C23×C312yes—
307C2×C3⋊C8—21no—
308C2×C3⋊C8—21no—
309C2×C3⋊C8—21no—
310ΓL(1,𝔽25)—24no—
311D3×C8—24no—
312C22×C32—22no—
313C22×C32—42no—
314C22×C32—82no—
315C22×C32—82no—
316C22×C32—82no—
317C22×C32#12111yescharacteristic
318C22×C32#12111yescharacteristic
319C22×C32C2×#4111yescharacteristic
320C2×C3×D3—22no—
321C2×C3×D3—42no—
322C32×C4—42no—
323C32×C4#12111yescharacteristic
324C32×C4#12111yescharacteristic
325C32×C4C22×D411yescharacteristic
326C32×C4C22×D411yescharacteristic
327C3×Dic3—22no—
328C2×#41—31no—
329C22×Q8C2×C3×D311yescharacteristic
330C22×D4C2×C3×D311yescharacteristic
331#137—34no—
332#137—62no—
333AGL(1,ℤ/8)—34no—
334AGL(1,ℤ/8)—62no—
335C2×Q16—61no—
336C2×SD16—21no—
337C2×SD16—21no—
338C2×SD16—22no—
339C2×SD16—61no—
340C2×SD16—61no—
341C2×SD16C2×C3×D314yescomplemented
342C2×D8—61no—
343#131—34no—
344#131—62no—
345C2×M16—31no—
346C2×M16—31no—
347C2×M16—31no—
348C2×M16—61no—
349C22×C8—61no—
350C22×C8C2×C3×D311yescharacteristic
351#108—34no—
352#101—34no—
353#100—34no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—21no—
357C23×C3—42no—
358C23×C3—42no—
359C23×C3—42no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—61no—
365C23×C3—82no—
366C23×C3—82no—
367C23×C3—84no—
368C23×C3#23711yescharacteristic
369C23×C3C3×#4111yescharacteristic
370C22×D3—21no—
371C3×Q8—24no—
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8—22no—
375C3×Q8—22no—
376C3×Q8—32no—
377C3×Q8—42no—
378C3×Q8—42no—
379C3×Q8—42no—
380C3×Q8—41no—
381C3×Q8—41no—
382C3×Q8—41no—
383C3×Q8—41no—
384C3×Q8—61no—
385C3×D4—24no—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—32no—
391C3×D4—42no—
392C3×D4—42no—
393C3×D4—42no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—41no—
397C3×D4—41no—
398C3×D4—62no—
399C3×D4—62no—
400C3×D4—61no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—42no—
410C2×C3×C4—42no—
411C2×C3×C4—42no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—61no—
417C2×C3×C4—62no—
418C2×C3×C4—62no—
419C2×C3×C4D3×D411yescharacteristic
420C2×C3×C4D3×D411yescharacteristic
421C2×C3×C4#23711yescharacteristic
422C2×C3×C4C2×#6711yescharacteristic
423C2×C3×C4C2×#6711yescharacteristic
424C2×C3×C4C2×C3×D411yescharacteristic
425C2×C3×C4C2×C3×D411yescharacteristic
426C2×C3×C4C2×C3×D411yescharacteristic
427C2×C3×C4C2×C3×D411yescharacteristic
428C2×C3×C4C3×#4111yescharacteristic
429C2×C3×C4C23×D311yescharacteristic
430C2×C3×C4C24×C311yescharacteristic
431#67—22no—
432#67—22no—
433C2×Dic3—21no—
434D12—21no—
435D12C2×C3×D412yes—
436C4×D3—22no—
437C4×D3—22no—
438Dic6—21no—
439Dic6C2×C3×D412yes—
440C3×C8—22no—
441C3×C8—22no—
442C3×C8—62no—
443C3×C8—62no—
444C3×C8—62no—
445C3×C8C2×#6712yes—
446C3×C8C2×#6712yes—
447C3×C8C2×C3×D412yes—
448C3×C8C2×C3×D412yes—
449C3⋊C8—22no—
450C3⋊C8—22no—
451C3⋊C8—22no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—82no—
455C2×C32#47112yescomplemented
456C2×C32C2×#12111yescharacteristic
457C3×D3—42no—
458C24—21no—
459#41—32no—
460#41—32no—
461#41—61no—
462#41—61no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—31no—
469C2×Q8C22×C3×D312yes—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—31no—
476C2×D4—61no—
477C2×D4C22×C3×D312yes—
478C22×C4—21no—
479C22×C4—61no—
480C22×C4C22×C3×D311yescharacteristic
481Q16—64no—
482SD16—24no—
483SD16—24no—
484SD16—64no—
485SD16—64no—
486D8—64no—
487M16—32no—
488M16—32no—
489M16—32no—
490M16—61no—
491M16—61no—
492M16—61no—
493M16—64no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—64no—
498C2×C8C3×#6711yescharacteristic
499C2×C8C3×#6711yescharacteristic
500C2×C8C3×#6712yes—
501C2×C8C22×C3×D312yes—
502C22×C3—24no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—44no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—62no—
512C22×C3—82no—
513C22×C3—82no—
514C22×C3—84no—
515C22×C3—84no—
516C22×C3—82no—
517C22×C3—82no—
518C22×C3—84no—
519C22×C3—82no—
520C22×C3—82no—
521C22×C3—122no—
522C22×C3#93811yescharacteristic
523C22×C3#93811yescharacteristic
524C22×C3C3×#12111yescharacteristic
525C22×C3C3×#12111yescharacteristic
526C22×C3C2×#23711yescharacteristic
527C22×C3C2×C3×#4111yescharacteristic
528C2×D3—22no—
529C2×D3—42no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—44no—
535C3×C4—42no—
536C3×C4—42no—
537C3×C4—62no—
538C3×C4#93811yescharacteristic
539C3×C4#93811yescharacteristic
540C3×C4C3×#12111yescharacteristic
541C3×C4C3×#12111yescharacteristic
542C3×C4C2×D3×D411yescharacteristic
543C3×C4C22×#6711yescharacteristic
544C3×C4C22×C3×D411yescharacteristic
545C3×C4C22×C3×D411yescharacteristic
546Dic3—22no—
547C32C2×#47111yescharacteristic, complemented, Sylow
548C23—22no—
549C23—41no—
550C23—41no—
551C23—61no—
552C23—82no—
553C23C3×#23711yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—32no—
557Q8—41no—
558Q8—41no—
559Q8—61no—
560D4—22no—
561D4—22no—
562D4—32no—
563D4—41no—
564D4—41no—
565D4—62no—
566D4—62no—
567D4—61no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—61no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4C3×D3×D411yescharacteristic
575C2×C4C3×D3×D411yescharacteristic
576C2×C4C3×#23711yescharacteristic
577C2×C4C2×C3×#6711yescharacteristic
578C2×C4C2×C3×#6711yescharacteristic
579C2×C4C23×C3×D311yescharacteristic
580C8—62no—
581C8—62no—
582C8—62no—
583C8C2×C3×#6712yes—
584C8C2×C3×#6712yes—
585C2×C3—22no—
586C2×C3—22no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—22no—
592C2×C3—21no—
593C2×C3—84no—
594C2×C3—82no—
595C2×C3—82no—
596C2×C3—122no—
597C2×C3#517712yesdirect product factor
598C2×C3C3×#47112yescomplemented
599C2×C3C2×#93811yescharacteristic
600C2×C3C2×C3×#12111yescharacteristic
601D3—42no—
602C22—22no—
603C22—42no—
604C22—62no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—122no—
609C22C3×#93811yescharacteristic
610C22C3×#93811yescharacteristic
611C22C2×C3×#23711yescharacteristic
612C4—42no—
613C4—62no—
614C4C3×#93811yescharacteristic
615C4C3×#93811yescharacteristic
616C4C2×C3×D3×D411yescharacteristic
617C4C22×C3×#6711yescharacteristic
618C3—21no—
619C3C2×#517711yescharacteristic, direct product factor
620C3C2×C3×#47111yescharacteristic, complemented
621C2—21no—
622C2—21no—
623C2—82no—
624C2—122no—
625C2C3×#517712yesdirect product factor
626C2C2×C3×#93811yescharacteristic
subgroupG/H
centerC22×C3#938
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×SD16C2
socleC22×C32#121
prime radicalC2×C3×#932C2
hypercenterC22×C3×SD16D3
hypocenterC3C2×C3×#471
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×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×#942C2semidirect
4. C2×C3×#901C2semidirect
5. C2×C3×#836C2semidirect
6. C2×C3×#833C2semidirect
7. C3×#5177C2direct (N × H)
8. C2×C3×#819C2semidirect
10. C2×#5177C3direct (N × H)
17. C2×C32×SD16C22semidirect
21. C3×#942C22semidirect
31. C3×#901C22semidirect
34. C3×#836C22semidirect
35. C3×#833C22semidirect
36. C3×#819C22semidirect
40. C22×C3×SD16D3semidirect
41. C22×C3×SD16C2×C3semidirect
43. C2×#942C2×C3semidirect
45. C2×#901C2×C3semidirect
50. C2×#836C2×C3semidirect
51. C2×#833C2×C3semidirect
52. #5177C2×C3direct (N × H)
53. C2×#819C2×C3semidirect
124. C2×C3×SD16C2×D3semidirect
125. C2×C3×SD16C22×C3semidirect
139. #942C22×C3semidirect
149. #901C22×C3semidirect
155. #836C22×C3semidirect
156. #833C22×C3semidirect
157. #819C22×C3semidirect
197. C22×SD16C3×D3semidirect
341. C2×SD16C2×C3×D3semidirect
455. C2×C32#471semidirect
547. C32C2×#471semidirect
597. C2×C3#5177direct (N × H)
598. C2×C3C3×#471semidirect
619. C3C2×#5177direct (N × H)
620. C3C2×C3×#471semidirect
625. C2C3×#5177direct (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×#9422
C2×C3×#9322
C2×C3×#9012
C2×C3×#8362
C2×C3×#8332
C3×#51772
C2×C3×#8192
C2×C3×#4713
C2×#51773