Finite Group Databasebeta

← C3×#84123 #11865198

Subgroups — C3×#84123 #11865198

Order 1152.

1813 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#585C211yesmaximal, characteristic
2C3×#5825C211yesmaximal, characteristic, complemented
3C2×C3×#951C212yesmaximal, complemented
4C3×#5246C214yesmaximal, complemented
5C3×#5242C214yesmaximal, complemented
6C3×#5237C211yesmaximal, characteristic, complemented
7C3×#5236C212yesmaximal, complemented
8C3×#3022—31nomaximal
9#84123C311yesmaximal, characteristic, direct product factor
10C32×2−1+4—22no—
11C32×2−1+4C2212yes—
12C32×H(5,𝔽2)—22no—
13C32×H(5,𝔽2)—22no—
14C32×H(5,𝔽2)C2212yes—
15C2×C32×#41—24no—
16C2×C32×#41—22no—
17C2×C32×#41C2212yes—
18C2×C32×#41C2211yescharacteristic
19C22×C3×Dic6C2211yescharacteristic, complemented
20C3×#951—24no—
21C3×#951C2218yescomplemented
22C3×#944C2212yescomplemented
23C2×C3×#216—22no—
24C3×#936C2212yescomplemented
25C3×#935—22no—
26C2×C3×#214—22no—
27C3×#926C2211yescharacteristic, complemented
28C3×#924—22no—
29C2×C3×#208C2212yescomplemented
30C3×#837C2218yescomplemented
31C3×#836C2212yescomplemented
32C3×#833C2212yescomplemented
33C3×#585—21no—
34C3×#585D311yescharacteristic
35C3×#585C2×C311yescharacteristic
36C2×C3×#137—32no—
37C3×#536—31no—
38#5825C2×C311yescharacteristic, complemented
39C2×#951C2×C312yescomplemented
40C3×#456—34no—
41C3×#455—34no—
42C3×#421—32no—
43C3×#413—31no—
44#5246C2×C314yescomplemented
45#5242C2×C314yescomplemented
46#5237C2×C311yescharacteristic, complemented
47#5236C2×C312yescomplemented
48C32×#41—24no—
49C32×#41—28no—
50C32×#41—24no—
51C32×#41—24no—
52C32×#41—24no—
53C32×#41—22no—
54C32×#41—22no—
55C32×#41—24no—
56C32×#41—22no—
57C32×#41—22no—
58C32×#41D414yes—
59C32×#41C2314yes—
60C2×C32×Q8—22no—
61C2×C32×Q8—24no—
62C2×C32×Q8D412yes—
63C2×C32×Q8C2311yescharacteristic
64C2×C32×D4—28no—
65C2×C32×D4—22no—
66C2×C32×D4—24no—
67C2×C32×D4—22no—
68C2×C32×D4—24no—
69C2×C32×D4D412yes—
70C2×C32×D4D412yes—
71C2×C32×D4C2311yescharacteristic
72C22×C32×C4—24no—
73C22×C32×C4—22no—
74C22×C32×C4D412yes—
75C22×C32×C4C2311yescharacteristic
76C22×C3×Dic3—21no—
77C2×C3×Dic6—22no—
78C2×C3×Dic6—22no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6C2312yes—
81C3×#217—24no—
82C3×#217—21no—
83C3×#216—28no—
84C3×#214—28no—
85C3×C4⋊Dic3—21no—
86C3×Dic3⋊C4—24no—
87C3×C4×Dic3C2312yes—
88C3×#208—22no—
89C3×#208C2314yes—
90C2×C3×C3⋊C8—22no—
91#3022—31noSylow
92C3×2−1+4—24no—
93C3×2−1+4—22no—
94C3×2−1+4—22no—
95C3×2−1+4—22no—
96C3×2−1+4C2×D312yes—
97C3×2−1+4C22×C312yes—
98C3×H(5,𝔽2)—24no—
99C3×H(5,𝔽2)—24no—
100C3×H(5,𝔽2)—22no—
101C3×H(5,𝔽2)—22no—
102C3×H(5,𝔽2)—22no—
103C3×H(5,𝔽2)—22no—
104C3×H(5,𝔽2)—22no—
105C3×H(5,𝔽2)C2×D312yes—
106C3×H(5,𝔽2)C22×C312yes—
107C2×C3×#41—28no—
108C2×C3×#41—24no—
109C2×C3×#41—24no—
110C2×C3×#41—24no—
111C2×C3×#41—22no—
112C2×C3×#41—22no—
113C2×C3×#41—22no—
114C2×C3×#41—21no—
115C2×C3×#41C2×D312yes—
116C2×C3×#41C2×D311yescharacteristic
117C2×C3×#41C22×C312yes—
118C2×C3×#41C22×C311yescharacteristic
119C22×C3×Q8—31no—
120C22×Dic6C22×C311yescharacteristic, complemented
121C3×#137—38no—
122C3×#137—64no—
123C2×C3×Q16—62no—
124C2×C3×SD16—62no—
125C2×C3×M16—32no—
126C3×C4⋊Q8—32no—
127C3×#124—32no—
128C3×#123—62no—
129C3×C22⋊Q8—62no—
130C3×#117—31no—
131#951—24no—
132#951C22×C318yescomplemented
133#944C22×C312yescomplemented
134C2×#216—22no—
135#936C22×C312yescomplemented
136#935—22no—
137C2×#214—22no—
138#926C22×C311yescharacteristic, complemented
139#924—22no—
140C2×#208C22×C312yescomplemented
141C3×#104—38no—
142C3×#101—32no—
143C3×#100—32no—
144#837C22×C318yescomplemented
145#836C22×C312yescomplemented
146#833C22×C312yescomplemented
147C23×C32—24no—
148C23×C32—22no—
149C23×C32C2×D412yes—
150C23×C32C2×D411yescharacteristic
151C32×Q8—24no—
152C32×Q8—22no—
153C32×Q8—22no—
154C32×Q8C2×D414yes—
155C32×D4—28no—
156C32×D4—24no—
157C32×D4—24no—
158C32×D4—24no—
159C32×D4—24no—
160C32×D4—22no—
161C32×D4—22no—
162C32×D4C2×D414yes—
163C2×C32×C4—24no—
164C2×C32×C4—28no—
165C2×C32×C4—22no—
166C2×C32×C4—24no—
167C2×C32×C4—22no—
168C2×C32×C4—22no—
169C2×C32×C4—24no—
170C2×C32×C4C2×D412yes—
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2×D412yes—
173C2×C32×C4C2×D412yes—
174C2×C32×C4C2411yescharacteristic
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—42no—
178C3×Dic6—24no—
179C3×Dic6—41no—
180C3×Dic6—41no—
181C3×C3⋊C8—24no—
182#585C3×D311yescharacteristic
183C2×#137—32no—
184#536—31no—
185#456—34no—
186#455—34no—
187#421—32no—
188#413—31no—
189C3×#41—28no—
190C3×#41—24no—
191C3×#41—24no—
192C3×#41—24no—
193C3×#41—24no—
194C3×#41—216no—
195C3×#41—28no—
196C3×#41—28no—
197C3×#41—28no—
198C3×#41—24no—
199C3×#41—24no—
200C3×#41—28no—
201C3×#41—28no—
202C3×#41—28no—
203C3×#41—24no—
204C3×#41—24no—
205C3×#41—24no—
206C3×#41—24no—
207C3×#41—24no—
208C3×#41—24no—
209C3×#41—22no—
210C3×#41—22no—
211C3×#41—24no—
212C3×#41—22no—
213C3×#41—22no—
214C3×#41—22no—
215C3×#41—22no—
216C3×#41—24no—
217C3×#41—22no—
218C3×#41—22no—
219C3×#41—24no—
220C3×#41—24no—
221C3×#41#6714yes—
222C3×#41C3×D414yes—
223C3×#41C22×D314yes—
224C3×#41C23×C314yes—
225C2×C3×Q8—24no—
226C2×C3×Q8—22no—
227C2×C3×Q8—22no—
228C2×C3×Q8—28no—
229C2×C3×Q8—24no—
230C2×C3×Q8—24no—
231C2×C3×Q8—22no—
232C2×C3×Q8—21no—
233C2×C3×Q8—32no—
234C2×C3×Q8—61no—
235C2×C3×Q8—62no—
236C2×C3×Q8—62no—
237C2×C3×Q8#6712yes—
238C2×C3×Q8C3×D412yes—
239C2×C3×Q8C22×D311yescharacteristic
240C2×C3×Q8C23×C311yescharacteristic
241C2×C3×D4—216no—
242C2×C3×D4—24no—
243C2×C3×D4—28no—
244C2×C3×D4—28no—
245C2×C3×D4—22no—
246C2×C3×D4—22no—
247C2×C3×D4—28no—
248C2×C3×D4—24no—
249C2×C3×D4—28no—
250C2×C3×D4—24no—
251C2×C3×D4—22no—
252C2×C3×D4—24no—
253C2×C3×D4—22no—
254C2×C3×D4—24no—
255C2×C3×D4—24no—
256C2×C3×D4—22no—
257C2×C3×D4—22no—
258C2×C3×D4—21no—
259C2×C3×D4#6712yes—
260C2×C3×D4#6712yes—
261C2×C3×D4C3×D412yes—
262C2×C3×D4C3×D412yes—
263C2×C3×D4C22×D311yescharacteristic
264C2×C3×D4C23×C311yescharacteristic
265C22×C3×C4—28no—
266C22×C3×C4—24no—
267C22×C3×C4—24no—
268C22×C3×C4—24no—
269C22×C3×C4—22no—
270C22×C3×C4—22no—
271C22×C3×C4—22no—
272C22×C3×C4—21no—
273C22×C3×C4—61no—
274C22×C3×C4#6712yes—
275C22×C3×C4C3×D412yes—
276C22×C3×C4C22×D311yescharacteristic
277C22×C3×C4C23×C311yescharacteristic
278C22×Dic3—21no—
279C2×Dic6—22no—
280C2×Dic6—22no—
281C2×Dic6—21no—
282C2×Dic6C23×C312yes—
283C3×Q16—68no—
284C3×SD16—68no—
285C3×M16—34no—
286C3×M16—62no—
287C2×C3×C8—62no—
288C3×C4⋊C4—64no—
289C3×C4⋊C4—61no—
290C3×C22⋊C4—61no—
291C3×C22⋊C4—64no—
292C3×C42—32no—
293#217—24no—
294#217—21no—
295#216—28no—
296#214—28no—
297C4⋊Dic3—21no—
298Dic3⋊C4—24no—
299C4×Dic3C23×C312yes—
300#208—22no—
301#208C23×C314yes—
302C2×C3⋊C8—22no—
303C22×C32—22no—
304C22×C32—24no—
305C22×C32—41no—
306C22×C32—42no—
307C22×C32—82no—
308C22×C32—84no—
309C22×C32#12012yes—
310C22×C32C22×D411yescharacteristic
311C32×C4—24no—
312C32×C4—22no—
313C32×C4#12011yescharacteristic
314C32×C4#12011yescharacteristic
315C32×C4C22×D412yes—
316C3×Dic3—42no—
317C3×Dic3—42no—
3182−1+4—22no—
3192−1+4C2×C3×D312yes—
320H(5,𝔽2)—22no—
321H(5,𝔽2)—22no—
322H(5,𝔽2)C2×C3×D312yes—
323C2×#41—24no—
324C2×#41—22no—
325C2×#41C2×C3×D312yes—
326C2×#41C2×C3×D311yescharacteristic
327C22×Q8—31no—
328#137—38no—
329#137—64no—
330C2×Q16—62no—
331C2×SD16—62no—
332C2×M16—32no—
333C4⋊Q8—32no—
334#124—32no—
335#123—62no—
336C22⋊Q8—62no—
337#117—31no—
338#104—38no—
339#101—32no—
340#100—32no—
341C23×C3—28no—
342C23×C3—24no—
343C23×C3—24no—
344C23×C3—24no—
345C23×C3—22no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—21no—
349C23×C3C2×#6712yes—
350C23×C3C2×#6711yescharacteristic
351C23×C3C2×C3×D412yes—
352C23×C3C2×C3×D411yescharacteristic
353C3×Q8—28no—
354C3×Q8—24no—
355C3×Q8—24no—
356C3×Q8—24no—
357C3×Q8—24no—
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—24no—
363C3×Q8—64no—
364C3×Q8—121no—
365C3×Q8—121no—
366C3×Q8C2×#6714yes—
367C3×Q8C2×C3×D414yes—
368C3×D4—216no—
369C3×D4—28no—
370C3×D4—28no—
371C3×D4—24no—
372C3×D4—28no—
373C3×D4—24no—
374C3×D4—28no—
375C3×D4—28no—
376C3×D4—28no—
377C3×D4—24no—
378C3×D4—24no—
379C3×D4—24no—
380C3×D4—24no—
381C3×D4—24no—
382C3×D4—24no—
383C3×D4—24no—
384C3×D4—24no—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—24no—
390C3×D4C2×#6714yes—
391C3×D4C2×C3×D414yes—
392C2×C3×C4—28no—
393C2×C3×C4—24no—
394C2×C3×C4—28no—
395C2×C3×C4—216no—
396C2×C3×C4—24no—
397C2×C3×C4—28no—
398C2×C3×C4—24no—
399C2×C3×C4—28no—
400C2×C3×C4—24no—
401C2×C3×C4—24no—
402C2×C3×C4—28no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—24no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—24no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—24no—
412C2×C3×C4—24no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—62no—
419C2×C3×C4—62no—
420C2×C3×C4—122no—
421C2×C3×C4C2×#6712yes—
422C2×C3×C4C2×#6711yescharacteristic
423C2×C3×C4C2×#6712yes—
424C2×C3×C4C2×#6712yes—
425C2×C3×C4C2×C3×D412yes—
426C2×C3×C4C2×C3×D411yescharacteristic
427C2×C3×C4C2×C3×D412yes—
428C2×C3×C4C2×C3×D412yes—
429C2×C3×C4C23×D311yescharacteristic
430C2×C3×C4C24×C311yescharacteristic
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—42no—
434Dic6—24no—
435Dic6—41no—
436Dic6—41no—
437C3×C8—64no—
438C3⋊C8—24no—
439C2×C32—21no—
440C2×C32—22no—
441C2×C32—42no—
442C2×C32—44no—
443C2×C32C2×#12011yescharacteristic
444#41—24no—
445#41—28no—
446#41—24no—
447#41—24no—
448#41—24no—
449#41—22no—
450#41—22no—
451#41—24no—
452#41—22no—
453#41—22no—
454#41C3×#6714yes—
455#41C22×C3×D314yes—
456C2×Q8—22no—
457C2×Q8—24no—
458C2×Q8—32no—
459C2×Q8—61no—
460C2×Q8—62no—
461C2×Q8—62no—
462C2×Q8C3×#6712yes—
463C2×Q8C22×C3×D311yescharacteristic
464C2×D4—28no—
465C2×D4—22no—
466C2×D4—24no—
467C2×D4—22no—
468C2×D4—24no—
469C2×D4C3×#6712yes—
470C2×D4C3×#6712yes—
471C2×D4C22×C3×D311yescharacteristic
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—61no—
475C22×C4C3×#6712yes—
476C22×C4C22×C3×D311yescharacteristic
477Q16—68no—
478SD16—68no—
479M16—34no—
480M16—62no—
481C2×C8—62no—
482C4⋊C4—61no—
483C4⋊C4—64no—
484C22⋊C4—61no—
485C22⋊C4—64no—
486C42—32no—
487C22×C3—24no—
488C22×C3—28no—
489C22×C3—22no—
490C22×C3—24no—
491C22×C3—22no—
492C22×C3—24no—
493C22×C3—22no—
494C22×C3—21no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—81no—
500C22×C3—82no—
501C22×C3—82no—
502C22×C3—82no—
503C22×C3—84no—
504C22×C3—88no—
505C22×C3—84no—
506C22×C3—84no—
507C22×C3#95312yes—
508C22×C3C3×#12012yes—
509C22×C3C22×#6711yescharacteristic
510C22×C3C22×C3×D411yescharacteristic
511C3×C4—28no—
512C3×C4—24no—
513C3×C4—24no—
514C3×C4—22no—
515C3×C4—24no—
516C3×C4—22no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—22no—
520C3×C4—122no—
521C3×C4—122no—
522C3×C4#95311yescharacteristic
523C3×C4#95311yescharacteristic
524C3×C4C3×#12011yescharacteristic
525C3×C4C3×#12011yescharacteristic
526C3×C4C22×#6712yes—
527C3×C4C22×C3×D412yes—
528Dic3—42no—
529Dic3—42no—
530C32#302211yescharacteristic, complemented, Sylow
531C23—24no—
532C23—22no—
533C23C2×C3×#6712yes—
534C23C2×C3×#6711yescharacteristic
535Q8—24no—
536Q8—22no—
537Q8—22no—
538Q8—64no—
539Q8—121no—
540Q8—121no—
541Q8C2×C3×#6714yes—
542D4—28no—
543D4—24no—
544D4—24no—
545D4—24no—
546D4—24no—
547D4—22no—
548D4—22no—
549D4C2×C3×#6714yes—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—122no—
560C2×C4C2×C3×#6712yes—
561C2×C4C2×C3×#6711yescharacteristic
562C2×C4C2×C3×#6712yes—
563C2×C4C2×C3×#6712yes—
564C2×C4C23×C3×D311yescharacteristic
565C8—64no—
566C2×C3—21no—
567C2×C3—22no—
568C2×C3—21no—
569C2×C3—22no—
570C2×C3—21no—
571C2×C3—41no—
572C2×C3—42no—
573C2×C3—44no—
574C2×C3—48no—
575C2×C3—42no—
576C2×C3—44no—
577C2×C3—42no—
578C2×C3—44no—
579C2×C3C2×#95311yescharacteristic
580C2×C3C2×C3×#12011yescharacteristic
581C22—22no—
582C22—24no—
583C22—41no—
584C22—42no—
585C22—82no—
586C22—84no—
587C22C3×#95312yes—
588C22C22×C3×#6711yescharacteristic
589C4—24no—
590C4—22no—
591C4—122no—
592C4—122no—
593C4C3×#95311yescharacteristic
594C4C3×#95311yescharacteristic
595C4C22×C3×#6712yes—
596C3—21no—
597C3#8412311yescharacteristic, direct product factor
598C3C3×#302211yescharacteristic, complemented
599C2—21no—
600C2—22no—
601C2—42no—
602C2—44no—
603C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC2×C3C2×#953
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC32×#585C2
socleC2×C32C2×#120
hypercenterC3×#585D3
hypocenterC3C3×#3022
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#585

Complements

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

NH ≅ G/Ntype
2. C3×#5825C2semidirect
3. C2×C3×#951C2semidirect
4. C3×#5246C2semidirect
5. C3×#5242C2semidirect
6. C3×#5237C2semidirect
7. C3×#5236C2semidirect
9. #84123C3direct (N × H)
19. C22×C3×Dic6C22semidirect
21. C3×#951C22semidirect
22. C3×#944C22semidirect
24. C3×#936C22semidirect
27. C3×#926C22semidirect
29. C2×C3×#208C22semidirect
30. C3×#837C22semidirect
31. C3×#836C22semidirect
32. C3×#833C22semidirect
38. #5825C2×C3semidirect
39. C2×#951C2×C3semidirect
44. #5246C2×C3semidirect
45. #5242C2×C3semidirect
46. #5237C2×C3semidirect
47. #5236C2×C3semidirect
120. C22×Dic6C22×C3semidirect
132. #951C22×C3semidirect
133. #944C22×C3semidirect
135. #936C22×C3semidirect
138. #926C22×C3semidirect
140. C2×#208C22×C3semidirect
144. #837C22×C3semidirect
145. #836C22×C3semidirect
146. #833C22×C3semidirect
530. C32#3022semidirect
597. C3#84123direct (N × H)
598. C3C3×#3022semidirect

Maximal subgroups

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

subgroupindex
C32×#5852
C3×#58252
C2×C3×#9512
C3×#52462
C3×#52422
C3×#52372
C3×#52362
C3×#30223
#841233