Finite Group Databasebeta

← C3×#80178 #11878882

Subgroups — C3×#80178 #11878882

Order 1152.

842 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#944C211yesmaximal, characteristic
2C3×#5622C211yesmaximal, characteristic
3C32×#451C211yesmaximal, characteristic
4C3×#5228C211yesmaximal, characteristic
5C3×#5183C212yesmaximal, complemented
6C3×#5182C212yesmaximal, complemented
7C3×#5050C211yesmaximal, characteristic
8C3×#4795C212yesmaximal, complemented
9C3×#4792C212yesmaximal, complemented
10C3×#4739C211yesmaximal, characteristic
11C3×#4718C211yesmaximal, characteristic
12C3×#3209—31nomaximal
13#80178C311yesmaximal, characteristic, direct product factor
14C22×C32×Q8C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C2×C32×Q16C2212yes—
17C32×C22⋊Q8C2211yescharacteristic
18C3×Q8×Dic3C2211yescharacteristic
19C3×#944—21no—
20C3×#944—21no—
21C3×#944C2212yes—
22C3×#944C2211yescharacteristic
23C3×#944C2212yes—
24C2×C3×#216C2212yes—
25C2×C3×#216C2212yes—
26C3×#924C2211yescharacteristic
27C2×C3×Dic3⋊C4—21no—
28C2×C3×Dic3⋊C4—21no—
29C2×C3×C4×Dic3C2211yescharacteristic
30C2×C3×Dic12C2212yes—
31C3×#889—21no—
32C3×#888C2211yescharacteristic
33C3×#887—21no—
34C3×#887C2211yescharacteristic
35C3×#878—21no—
36C3×#877—21no—
37C32×#103C2212yes—
38C32×C22⋊C8C2211yescharacteristic
39C3×#835C2212yes—
40C3×#830C2211yescharacteristic
41C3×#816C2212yes—
42C3×Dic3⋊C8C2212yes—
43C3×C8×Dic3C2212yes—
44C3×#810C2212yes—
45C3×#533—31no—
46C2×C3×C4⋊Q8—31no—
47C2×#944C2×C311yescharacteristic
48#5622C2×C311yescharacteristic
49C3×#494—32no—
50C3×#487—32no—
51C3×#462—32no—
52C3×#462—32no—
53C3×#451—21no—
54C3×#451—31no—
55C3×#451—31no—
56C3×#451—31no—
57C3×#451D311yescharacteristic
58C3×#451C2×C311yescharacteristic
59C3×#431—31no—
60#5228C2×C311yescharacteristic
61#5183C2×C312yescomplemented
62#5182C2×C312yescomplemented
63#5050C2×C311yescharacteristic
64#4795C2×C312yescomplemented
65#4792C2×C312yescomplemented
66#4739C2×C311yescharacteristic
67#4718C2×C311yescharacteristic
68C2×C32×Q8—21no—
69C2×C32×Q8—42no—
70C2×C32×Q8C2311yescharacteristic
71C2×C32×Q8C2312yes—
72C22×C32×C4—21no—
73C22×C32×C4C2311yescharacteristic
74C22×C3×Dic3—21no—
75C22×C3×Dic3D411yescharacteristic
76C22×C3×Dic3D411yescharacteristic
77C2×C3×Dic6—21no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6—42no—
80C2×C3×Dic6C2311yescharacteristic
81C2×C3×Dic6C2312yes—
82C32×Q16—42no—
83C2×C32×C8C2312yes—
84C32×C4⋊C4—21no—
85C32×C4⋊C4C2311yescharacteristic
86C32×C22⋊C4—21no—
87C3×#217—21no—
88C3×#216—42no—
89C3×#216—42no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3C2311yescharacteristic
92C3×Dic3⋊C4—21no—
93C3×Dic3⋊C4—21no—
94C3×Dic3⋊C4—21no—
95C3×Dic3⋊C4—21no—
96C3×Dic3⋊C4—21no—
97C3×Dic3⋊C4—22no—
98C3×Dic3⋊C4—22no—
99C3×Dic3⋊C4—22no—
100C3×Dic3⋊C4—22no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3—21no—
103C3×C4×Dic3D411yescharacteristic
104C3×C4×Dic3D411yescharacteristic
105C3×C4×Dic3C2311yescharacteristic
106C3×C4×Dic3C2311yescharacteristic
107C2×C3×C3⋊C8C2312yes—
108C3×Dic12—42no—
109#3209—31noSylow
110C22×C3×Q8—21no—
111C22×C3×Q8—31no—
112C22×C3×Q8C2×D311yescharacteristic
113C22×C3×Q8C22×C311yescharacteristic
114C22×Dic6C22×C311yescharacteristic
115C2×C3×Q16—22no—
116C2×C3×Q16—32no—
117C2×C3×Q16—32no—
118C2×C3×Q16—32no—
119C2×C3×Q16C2×D312yes—
120C2×C3×Q16C22×C312yes—
121C3×C4⋊Q8—31no—
122C3×C4⋊Q8—32no—
123C3×C4⋊Q8—32no—
124C3×C4⋊Q8—31no—
125C3×C4⋊Q8—61no—
126C3×C4⋊Q8—61no—
127C3×#125—61no—
128C3×C22⋊Q8—21no—
129C3×C22⋊Q8—31no—
130C3×C22⋊Q8—61no—
131C3×C22⋊Q8C2×D311yescharacteristic
132C3×C22⋊Q8C22×C311yescharacteristic
133C3×C4×Q8—31no—
134C3×C4×Q8—31no—
135C3×C4×Q8—61no—
136C3×#117—61no—
137C2×C3×C4⋊C4—61no—
138C2×C3×C4⋊C4—61no—
139C2×C3×C42—31no—
140Q8×Dic3C22×C311yescharacteristic
141#944—21no—
142#944—21no—
143#944C22×C312yes—
144#944C22×C311yescharacteristic
145#944C22×C312yes—
146C2×#216C22×C312yes—
147C2×#216C22×C312yes—
148#924C22×C311yescharacteristic
149C2×Dic3⋊C4—21no—
150C2×Dic3⋊C4—21no—
151C2×C4×Dic3C22×C311yescharacteristic
152C2×Dic12C22×C312yes—
153#889—21no—
154#888C22×C311yescharacteristic
155#887—21no—
156#887C22×C311yescharacteristic
157#878—21no—
158#877—21no—
159C3×C4⋊C8—32no—
160C3×#103—22no—
161C3×#103—32no—
162C3×#103—32no—
163C3×#103—32no—
164C3×#103C2×D312yes—
165C3×#103C22×C312yes—
166C3×C22⋊C8—21no—
167C3×C22⋊C8—31no—
168C3×C22⋊C8C2×D311yescharacteristic
169C3×C22⋊C8C22×C311yescharacteristic
170C3×C4×C8—32no—
171#835C22×C312yes—
172#830C22×C311yescharacteristic
173#816C22×C312yes—
174Dic3⋊C8C22×C312yes—
175C8×Dic3C22×C312yes—
176#810C22×C312yes—
177C23×C32C2×D411yescharacteristic
178C32×Q8—22no—
179C32×Q8—22no—
180C32×Q8—22no—
181C32×Q8—42no—
182C2×C32×C4—22no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—22no—
186C2×C32×C4—42no—
187C2×C32×C4C2×D411yescharacteristic
188C2×C32×C4C2411yescharacteristic
189C2×C3×Dic3—22no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—42no—
194C2×C3×Dic3Q1614yescomplemented
195C2×C3×Dic3C2×D411yescharacteristic
196C2×C3×Dic3C2×D411yescharacteristic
197C2×C3×Dic3C2×D411yescharacteristic
198C2×C3×Dic3C2×D411yescharacteristic
199C3×Dic6—22no—
200C3×Dic6—22no—
201C3×Dic6—22no—
202C3×Dic6—42no—
203C3×Dic6—42no—
204C32×C8—22no—
205C3×C3⋊C8—22no—
206#533—31no—
207C2×C4⋊Q8—31no—
208#494—32no—
209#487—32no—
210#462—32no—
211#462—32no—
212#451—31no—
213#451—31no—
214#451—31no—
215#451C3×D311yescharacteristic
216#431—31no—
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—22no—
220C2×C3×Q8—21no—
221C2×C3×Q8—32no—
222C2×C3×Q8—31no—
223C2×C3×Q8—41no—
224C2×C3×Q8—42no—
225C2×C3×Q8—42no—
226C2×C3×Q8—44no—
227C2×C3×Q8—61no—
228C2×C3×Q8—61no—
229C2×C3×Q8—122no—
230C2×C3×Q8C22×D312yes—
231C2×C3×Q8C22×D311yescharacteristic
232C2×C3×Q8C23×C312yes—
233C2×C3×Q8C23×C311yescharacteristic
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4—31no—
238C22×C3×C4—31no—
239C22×C3×C4—41no—
240C22×C3×C4—61no—
241C22×C3×C4C22×D311yescharacteristic
242C22×C3×C4C23×C311yescharacteristic
243C22×Dic3—21no—
244C22×Dic3C3×D411yescharacteristic
245C22×Dic3C3×D411yescharacteristic
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—42no—
249C2×Dic6C23×C312yes—
250C2×Dic6C23×C311yescharacteristic
251C3×Q16—42no—
252C3×Q16—42no—
253C3×Q16—44no—
254C3×Q16—122no—
255C3×Q16—122no—
256C3×Q16—122no—
257C2×C3×C8—22no—
258C2×C3×C8—32no—
259C2×C3×C8C22×D312yes—
260C2×C3×C8C23×C312yes—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—31no—
265C3×C4⋊C4—41no—
266C3×C4⋊C4—62no—
267C3×C4⋊C4—62no—
268C3×C4⋊C4—62no—
269C3×C4⋊C4—62no—
270C3×C4⋊C4—61no—
271C3×C4⋊C4—61no—
272C3×C4⋊C4—61no—
273C3×C4⋊C4—61no—
274C3×C4⋊C4—61no—
275C3×C4⋊C4—61no—
276C3×C4⋊C4C22×D311yescharacteristic
277C3×C4⋊C4C23×C311yescharacteristic
278C3×C22⋊C4—21no—
279C3×C22⋊C4—21no—
280C3×C22⋊C4—41no—
281C3×C22⋊C4—61no—
282C3×C42—31no—
283C3×C42—31no—
284C3×C42—31no—
285C3×C42—31no—
286C3×C42—61no—
287C3×C42—61no—
288#217—21no—
289#216—42no—
290#216—42no—
291C4⋊Dic3—21no—
292C4⋊Dic3C23×C311yescharacteristic
293Dic3⋊C4—21no—
294Dic3⋊C4—21no—
295Dic3⋊C4—21no—
296Dic3⋊C4—21no—
297Dic3⋊C4—21no—
298Dic3⋊C4—22no—
299Dic3⋊C4—22no—
300Dic3⋊C4—22no—
301Dic3⋊C4—22no—
302C4×Dic3—21no—
303C4×Dic3—21no—
304C4×Dic3C3×D411yescharacteristic
305C4×Dic3C3×D411yescharacteristic
306C4×Dic3C23×C311yescharacteristic
307C4×Dic3C23×C311yescharacteristic
308C2×C3⋊C8C23×C312yes—
309Dic12—42no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32C2×Q1612yes—
313C22×C32C22×D411yescharacteristic
314C32×C4—21no—
315C32×C4—42no—
316C32×C4—41no—
317C32×C4—41no—
318C32×C4H(5,𝔽2)12yes—
319C3×Dic3—21no—
320C3×Dic3—42no—
321C3×Dic3—41no—
322C3×Dic3—41no—
323C3×Dic3C2×Q1612yes—
324C3×Dic3C2×Q1612yes—
325C3×Dic3#13712yes—
326C22×Q8—31no—
327C22×Q8C2×C3×D311yescharacteristic
328C2×Q16—32no—
329C2×Q16—32no—
330C2×Q16—32no—
331C2×Q16C2×C3×D312yes—
332C4⋊Q8—31no—
333C4⋊Q8—32no—
334C4⋊Q8—32no—
335C4⋊Q8—31no—
336C4⋊Q8—61no—
337C4⋊Q8—61no—
338#125—61no—
339C22⋊Q8—31no—
340C22⋊Q8—61no—
341C22⋊Q8C2×C3×D311yescharacteristic
342C4×Q8—31no—
343C4×Q8—31no—
344C4×Q8—61no—
345#117—61no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4—61no—
348C2×C42—31no—
349C4⋊C8—32no—
350#103—32no—
351#103—32no—
352#103—32no—
353#103C2×C3×D312yes—
354C22⋊C8—31no—
355C22⋊C8C2×C3×D311yescharacteristic
356C4×C8—32no—
357C23×C3—21no—
358C23×C3D3×D411yescharacteristic
359C23×C3C2×C3×D411yescharacteristic
360C3×Q8—24no—
361C3×Q8—22no—
362C3×Q8—22no—
363C3×Q8—22no—
364C3×Q8—22no—
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—42no—
368C3×Q8—42no—
369C3×Q8—42no—
370C3×Q8—42no—
371C3×Q8—44no—
372C3×Q8—62no—
373C3×Q8—62no—
374C3×Q8—62no—
375C3×Q8—122no—
376C3×Q8—122no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—34no—
388C2×C3×C4—31no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—31no—
392C2×C3×C4—42no—
393C2×C3×C4—42no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—42no—
397C2×C3×C4—42no—
398C2×C3×C4—44no—
399C2×C3×C4—62no—
400C2×C3×C4—61no—
401C2×C3×C4—61no—
402C2×C3×C4—62no—
403C2×C3×C4—122no—
404C2×C3×C4D3×D411yescharacteristic
405C2×C3×C4C2×C3×D411yescharacteristic
406C2×C3×C4C23×D311yescharacteristic
407C2×C3×C4C24×C311yescharacteristic
408C2×Dic3—22no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—42no—
413C2×Dic3C3×Q1614yescomplemented
414C2×Dic3C2×C3×D411yescharacteristic
415C2×Dic3C2×C3×D411yescharacteristic
416C2×Dic3C2×C3×D411yescharacteristic
417C2×Dic3C2×C3×D411yescharacteristic
418Dic6—22no—
419Dic6—22no—
420Dic6—22no—
421Dic6—42no—
422Dic6—42no—
423C3×C8—22no—
424C3×C8—22no—
425C3×C8—42no—
426C3×C8—62no—
427C3⋊C8—22no—
428C2×C32—22no—
429C2×C32#53511yescharacteristic
430C2×C32C22×Q1611yescharacteristic
431C2×C32C2×#13711yescharacteristic
432C2×Q8—21no—
433C2×Q8—32no—
434C2×Q8—31no—
435C2×Q8—42no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—122no—
439C2×Q8C22×C3×D312yes—
440C2×Q8C22×C3×D311yescharacteristic
441C22×C4—21no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—61no—
445C22×C4C22×C3×D311yescharacteristic
446Q16—42no—
447Q16—122no—
448Q16—122no—
449Q16—122no—
450C2×C8—32no—
451C2×C8C22×C3×D312yes—
452C4⋊C4—21no—
453C4⋊C4—31no—
454C4⋊C4—62no—
455C4⋊C4—62no—
456C4⋊C4—62no—
457C4⋊C4—62no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4C22×C3×D311yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—61no—
467C42—31no—
468C42—31no—
469C42—31no—
470C42—31no—
471C42—61no—
472C42—61no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—22no—
478C22×C3—21no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3D3×Q1612yes—
482C22×C3C2×C3×Q1612yes—
483C22×C3C2×D3×D411yescharacteristic
484C22×C3C22×C3×D411yescharacteristic
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—22no—
488C3×C4—32no—
489C3×C4—32no—
490C3×C4—32no—
491C3×C4—41no—
492C3×C4—44no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—42no—
496C3×C4—41no—
497C3×C4—42no—
498C3×C4—41no—
499C3×C4—61no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—121no—
503C3×C4—122no—
504C3×C4—121no—
505C3×C4#100412yes—
506C3×C4C3×H(5,𝔽2)12yes—
507Dic3—21no—
508Dic3—41no—
509Dic3—42no—
510Dic3—41no—
511Dic3C2×C3×Q1612yes—
512Dic3C2×C3×Q1612yes—
513Dic3C3×#13712yes—
514C32#320911yescharacteristic, complemented, Sylow
515C23C3×D3×D411yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—42no—
520Q8—62no—
521Q8—62no—
522Q8—62no—
523Q8—122no—
524Q8—122no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—34no—
530C2×C4—31no—
531C2×C4—31no—
532C2×C4—31no—
533C2×C4—31no—
534C2×C4—42no—
535C2×C4—62no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—62no—
539C2×C4—122no—
540C2×C4C3×D3×D411yescharacteristic
541C2×C4C23×C3×D311yescharacteristic
542C8—22no—
543C8—62no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—42no—
550C2×C3#559211yescharacteristic
551C2×C3C2×D3×Q1611yescharacteristic
552C2×C3D3×#13711yescharacteristic
553C2×C3C3×#53511yescharacteristic
554C2×C3C22×C3×Q1611yescharacteristic
555C2×C3C2×C3×#13711yescharacteristic
556C22—21no—
557C22—21no—
558C22C3×D3×Q1612yes—
559C22C2×C3×D3×D411yescharacteristic
560C4—21no—
561C4—32no—
562C4—32no—
563C4—32no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—61no—
568C4—121no—
569C4—122no—
570C4—121no—
571C4C3×#100412yes—
572C3—21no—
573C3#8017811yescharacteristic, direct product factor
574C3C3×#320911yescharacteristic, complemented
575C2—22no—
576C2C3×#559211yescharacteristic
577C2C2×C3×D3×Q1611yescharacteristic
578C2C3×D3×#13711yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#451C2
socleC22×C32C22×D4
prime radicalC23×C32C2×D4
hypercenterC3×#451D3
hypocenterC3C3×#3209
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#451

Complements

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

NH ≅ G/Ntype
5. C3×#5183C2semidirect
6. C3×#5182C2semidirect
8. C3×#4795C2semidirect
9. C3×#4792C2semidirect
13. #80178C3direct (N × H)
61. #5183C2×C3semidirect
62. #5182C2×C3semidirect
64. #4795C2×C3semidirect
65. #4792C2×C3semidirect
194. C2×C3×Dic3Q16semidirect
413. C2×Dic3C3×Q16semidirect
514. C32#3209semidirect
573. C3#80178direct (N × H)
574. C3C3×#3209semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9442
C3×#56222
C32×#4512
C3×#52282
C3×#51832
C3×#51822
C3×#50502
C3×#47952
C3×#47922
C3×#47392
C3×#47182
C3×#32093
#801783