Finite Group Databasebeta

← C3×#70417 #11787759

Subgroups — C3×#70417 #11787759

Order 1152.

741 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5832C211yesmaximal, characteristic
2C3×#5745C211yesmaximal, characteristic, complemented
3C32×#421C211yesmaximal, characteristic, complemented
4C3×#5236C211yesmaximal, characteristic, complemented
5C3×#5006C211yesmaximal, characteristic, complemented
6C3×#4555C211yesmaximal, characteristic, complemented
7C3×#4532C211yesmaximal, characteristic, complemented
8C3×#1884—31nomaximal
9#70417C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C3×#947C411yescharacteristic, complemented
15C3×#944C411yescharacteristic, complemented
16C3×#940C411yescharacteristic, complemented
17C3×#939—22no—
18C3×#936C411yescharacteristic, complemented
19C3×C4×#67—22no—
20C3×#926C2211yescharacteristic
21C2×C3×C4×Dic3C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C3×#907—22no—
24C3×D3×M16—22no—
25C3×#901—22no—
26C32×#104—22no—
27C3×#837—22no—
28C3×#806—22no—
29C3×#568—31no—
30C3×#532—31no—
31#5832C2×C311yescharacteristic
32#5745C2×C311yescharacteristic, complemented
33C3×#421—21no—
34C3×#421—31no—
35C3×#421D311yescharacteristic, complemented
36C3×#421C2×C311yescharacteristic, complemented
37C3×#410—31no—
38#5236C2×C311yescharacteristic, complemented
39#5006C2×C311yescharacteristic, complemented
40C3×#339—31no—
41C3×#339—31no—
42#4555C2×C311yescharacteristic, complemented
43#4532C2×C311yescharacteristic, complemented
44C32×#41—21no—
45C32×#41—21no—
46C32×#41—21no—
47C32×#41—21no—
48C2×C32×Q8C2×C411yescharacteristic
49C2×C32×D4—21no—
50C2×C32×D4C2×C411yescharacteristic
51C22×C32×C4—21no—
52C22×C32×C4C2311yescharacteristic
53C2×C3×#67—22no—
54C2×C3×#67D412yes—
55C22×C3×Dic3—21no—
56C3×#235—21no—
57C3×#235—21no—
58C3×#235—21no—
59C3×#235—21no—
60C2×C3×D12C2×C411yescharacteristic
61C2×C3×C4×D3D412yes—
62C2×C3×Dic6C2×C411yescharacteristic
63C32×M16—21no—
64C32×M16D412yescomplemented
65C2×C32×C8—21no—
66C32×C4⋊C4—21no—
67C32×C22⋊C4—21no—
68C32×C42—21no—
69C3×#217—22no—
70C3×#217—21no—
71C3×(C2×D3)⋊C4—22no—
72C3×C4⋊Dic3—21no—
73C3×Dic3⋊C4—22no—
74C3×C4×Dic3—21no—
75C3×C4×Dic3—21no—
76C3×C4×Dic3C2×C411yescharacteristic
77C3×C4×Dic3C2×C411yescharacteristic
78C3×#208—21no—
79C3×#208D412yescomplemented
80C2×C3×C3⋊C8—21no—
81C3×ΓL(1,𝔽25)—24no—
82C3×D3×C8—24no—
83#1884—31noSylow
84C2×C3×#41—21no—
85C2×C3×#41—31no—
86C2×C3×#41C2×D311yescharacteristic
87C2×C3×#41C22×C311yescharacteristic
88C2×#235C22×C311yescharacteristic
89C3×#131—62no—
90C3×#131—62no—
91C2×C3×M16—21no—
92C2×C3×M16—31no—
93C2×C3×M16—62no—
94C2×C3×M16C2×D311yescharacteristic
95C2×C3×M16C22×C311yescharacteristic
96C3×C4⋊Q8—31no—
97C3×#127—31no—
98C3×#124—31no—
99C3×#124—31no—
100C3×#121—62no—
101C3×C4×D4—62no—
102C3×#117—21no—
103C3×#117—31no—
104C3×#117C2×D311yescharacteristic
105C3×#117C22×C311yescharacteristic
106C2×C3×C42—31no—
107#947C3×C411yescharacteristic, complemented
108#944C3×C411yescharacteristic, complemented
109#940C3×C411yescharacteristic, complemented
110#939—22no—
111#936C3×C411yescharacteristic, complemented
112C4×#67—22no—
113#926C22×C311yescharacteristic
114C2×C4×Dic3C22×C311yescharacteristic
115C2×#208C22×C311yescharacteristic
116#907—22no—
117D3×M16—22no—
118#901—22no—
119C3×#104—24no—
120C3×#104—22no—
121C3×#104—22no—
122C3×#104—62no—
123C3×#99—62no—
124#837—22no—
125#806—22no—
126C23×C32—21no—
127C23×C32C2×D411yescharacteristic
128C22×C3×D3C22⋊C412yes—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—41no—
134C32×D4—41no—
135C2×C32×C4—21no—
136C2×C32×C4—41no—
137C2×C32×C4—41no—
138C2×C32×C4—41no—
139C2×C32×C4C22×C411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4#4111yescharacteristic
144C2×C32×C4#4111yescharacteristic
145C3×#67—22no—
146C3×#67—22no—
147C3×#67—44no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—21no—
150C2×C3×Dic3—42no—
151C2×C3×Dic3—41no—
152C2×C3×Dic3C22⋊C412yes—
153C3×D12—21no—
154C3×D12—21no—
155C3×C4×D3—22no—
156C3×C4×D3—22no—
157C3×Dic6—21no—
158C3×Dic6—21no—
159C32×C8—22no—
160C3×C3⋊C8—22no—
161#568—31no—
162#532—31no—
163#421—31no—
164#421C3×D311yescharacteristic, complemented
165#410—31no—
166#339—31no—
167#339—31no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—21no—
171C3×#41—21no—
172C3×#41—21no—
173C3×#41—21no—
174C3×#41—21no—
175C3×#41—21no—
176C3×#41—21no—
177C3×#41—21no—
178C3×#41—41no—
179C3×#41—41no—
180C3×#41—61no—
181C3×#41—61no—
182C3×#41—61no—
183C3×#41—61no—
184C2×C3×Q8—21no—
185C2×C3×Q8—31no—
186C2×C3×Q8C4×D311yescharacteristic
187C2×C3×Q8C2×C3×C411yescharacteristic
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—32no—
192C2×C3×D4—31no—
193C2×C3×D4—41no—
194C2×C3×D4—62no—
195C2×C3×D4C4×D311yescharacteristic
196C2×C3×D4C2×C3×C411yescharacteristic
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—32no—
201C22×C3×C4—41no—
202C22×C3×C4—61no—
203C22×C3×C4C22×D311yescharacteristic
204C22×C3×C4C23×C311yescharacteristic
205C2×#67—22no—
206C2×#67C3×D412yes—
207C22×Dic3—21no—
208#235—21no—
209#235—21no—
210#235—21no—
211#235—21no—
212C2×D12C2×C3×C411yescharacteristic
213C2×C4×D3C3×D412yes—
214C2×Dic6C2×C3×C411yescharacteristic
215C3×M16—21no—
216C3×M16—21no—
217C3×M16—22no—
218C3×M16—32no—
219C3×M16—41no—
220C3×M16—64no—
221C3×M16—61no—
222C3×M16#6712yescomplemented
223C3×M16C3×D412yescomplemented
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—41no—
227C2×C3×C8—64no—
228C2×C3×C8—61no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—41no—
232C3×C4⋊C4—62no—
233C3×C4⋊C4—61no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—41no—
237C3×C22⋊C4—62no—
238C3×C22⋊C4—62no—
239C3×C22⋊C4—61no—
240C3×C42—22no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—31no—
244C3×C42—31no—
245C3×C42—61no—
246C3×C42—61no—
247#217—22no—
248#217—21no—
249(C2×D3)⋊C4—22no—
250C4⋊Dic3—21no—
251Dic3⋊C4—22no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3C2×C3×C411yescharacteristic
255C4×Dic3C2×C3×C411yescharacteristic
256#208—21no—
257#208C3×D412yescomplemented
258C2×C3⋊C8—21no—
259ΓL(1,𝔽25)—24no—
260D3×C8—24no—
261C22×C32—41no—
262C22×C32—41no—
263C22×C32—42no—
264C22×C32C2×C22⋊C411yescharacteristic
265C22×C32#12111yescharacteristic
266C22×C32#12111yescharacteristic
267C2×C3×D3—22no—
268C2×C3×D3—42no—
269C32×C4—41no—
270C32×C4—81no—
271C32×C4C4×D411yescharacteristic
272C32×C4C4×D411yescharacteristic
273C32×C4#12011yescharacteristic
274C32×C4#12311yescharacteristic
275C3×Dic3—22no—
276C3×Dic3—22no—
277C3×Dic3—41no—
278C3×Dic3—81no—
279C2×#41—31no—
280C2×#41C2×C3×D311yescharacteristic
281#131—62no—
282#131—62no—
283C2×M16—31no—
284C2×M16—62no—
285C2×M16C2×C3×D311yescharacteristic
286C4⋊Q8—31no—
287#127—31no—
288#124—31no—
289#124—31no—
290#121—62no—
291C4×D4—62no—
292#117—31no—
293#117C2×C3×D311yescharacteristic
294C2×C42—31no—
295#104—22no—
296#104—62no—
297#99—62no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—32no—
302C23×C3—41no—
303C23×C3D3×D411yescharacteristic
304C23×C3C2×C3×D411yescharacteristic
305C22×D3C3×C22⋊C412yes—
306C3×Q8—22no—
307C3×Q8—21no—
308C3×Q8—21no—
309C3×Q8—21no—
310C3×Q8—21no—
311C3×Q8—41no—
312C3×Q8—61no—
313C3×Q8—61no—
314C3×D4—22no—
315C3×D4—21no—
316C3×D4—21no—
317C3×D4—21no—
318C3×D4—21no—
319C3×D4—41no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—41no—
323C3×D4—41no—
324C3×D4—41no—
325C3×D4—41no—
326C3×D4—62no—
327C3×D4—62no—
328C3×D4—61no—
329C3×D4—61no—
330C3×D4—124no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—32no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—61no—
351C2×C3×C4—62no—
352C2×C3×C4—62no—
353C2×C3×C4—61no—
354C2×C3×C4—122no—
355C2×C3×C4—121no—
356C2×C3×C4C2×C4×D311yescharacteristic
357C2×C3×C4#23511yescharacteristic
358C2×C3×C4D3×D411yescharacteristic
359C2×C3×C4D3×D411yescharacteristic
360C2×C3×C4#23711yescharacteristic
361C2×C3×C4C2×#6711yescharacteristic
362C2×C3×C4C22×C3×C411yescharacteristic
363C2×C3×C4C2×C3×D411yescharacteristic
364C2×C3×C4C2×C3×D411yescharacteristic
365C2×C3×C4C2×C3×D411yescharacteristic
366C2×C3×C4C3×#4111yescharacteristic
367C2×C3×C4C3×#4111yescharacteristic
368#67—22no—
369#67—22no—
370#67—44no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—42no—
374C2×Dic3—41no—
375C2×Dic3C3×C22⋊C412yes—
376D12—21no—
377D12—21no—
378C4×D3—22no—
379C4×D3—22no—
380Dic6—21no—
381Dic6—21no—
382C3×C8—22no—
383C3×C8—22no—
384C3×C8—42no—
385C3×C8—62no—
386C3⋊C8—22no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—81no—
391C2×C32#38611yescharacteristic
392C3×D3—42no—
393#41—21no—
394#41—21no—
395#41—21no—
396#41—21no—
397#41—61no—
398#41—61no—
399#41—61no—
400#41—61no—
401C2×Q8—31no—
402C2×Q8C3×C4×D311yescharacteristic
403C2×D4—21no—
404C2×D4—32no—
405C2×D4—31no—
406C2×D4—62no—
407C2×D4C3×C4×D311yescharacteristic
408C22×C4—21no—
409C22×C4—32no—
410C22×C4—61no—
411C22×C4C22×C3×D311yescharacteristic
412M16—21no—
413M16—32no—
414M16—64no—
415M16—61no—
416M16C3×#6712yescomplemented
417C2×C8—21no—
418C2×C8—64no—
419C2×C8—61no—
420C4⋊C4—21no—
421C4⋊C4—62no—
422C4⋊C4—61no—
423C22⋊C4—21no—
424C22⋊C4—62no—
425C22⋊C4—62no—
426C22⋊C4—61no—
427C42—21no—
428C42—31no—
429C42—31no—
430C42—61no—
431C42—61no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—62no—
443C22×C3—81no—
444C22×C3—82no—
445C22×C3—122no—
446C22×C3D3×C22⋊C411yescharacteristic
447C22×C3#88411yescharacteristic
448C22×C3#93811yescharacteristic
449C22×C3C2×C3×C22⋊C411yescharacteristic
450C22×C3C3×#12111yescharacteristic
451C22×C3C3×#12111yescharacteristic
452C2×D3—22no—
453C2×D3—42no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—42no—
459C3×C4—41no—
460C3×C4—41no—
461C3×C4—62no—
462C3×C4—62no—
463C3×C4—81no—
464C3×C4—81no—
465C3×C4—82no—
466C3×C4—121no—
467C3×C4—241no—
468C3×C4#67⋊C411yescharacteristic
469C3×C4#88311yescharacteristic
470C3×C4C4×#6711yescharacteristic
471C3×C4#93711yescharacteristic
472C3×C4C3×C4×D411yescharacteristic
473C3×C4C3×C4×D411yescharacteristic
474C3×C4C3×#12011yescharacteristic
475C3×C4C3×#12311yescharacteristic
476Dic3—22no—
477Dic3—22no—
478Dic3—41no—
479Dic3—81no—
480C32#188411yescharacteristic, complemented, Sylow
481C23—21no—
482C23—32no—
483C23C3×D3×D411yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—61no—
487Q8—61no—
488D4—21no—
489D4—21no—
490D4—41no—
491D4—41no—
492D4—62no—
493D4—62no—
494D4—61no—
495D4—61no—
496D4—124no—
497C2×C4—21no—
498C2×C4—32no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—61no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—61no—
506C2×C4—122no—
507C2×C4—121no—
508C2×C4C2×C3×C4×D311yescharacteristic
509C2×C4C3×#23511yescharacteristic
510C2×C4C3×D3×D411yescharacteristic
511C2×C4C3×D3×D411yescharacteristic
512C2×C4C3×#23711yescharacteristic
513C2×C4C2×C3×#6711yescharacteristic
514C8—22no—
515C8—62no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—82no—
527C2×C3—81no—
528C2×C3—81no—
529C2×C3—122no—
530C2×C3#495611yescharacteristic
531C2×C3C3×#38611yescharacteristic
532D3—42no—
533C22—41no—
534C22—42no—
535C22—41no—
536C22—62no—
537C22—122no—
538C22C3×D3×C22⋊C411yescharacteristic
539C22C3×#88411yescharacteristic
540C22C3×#93811yescharacteristic
541C4—41no—
542C4—62no—
543C4—62no—
544C4—81no—
545C4—121no—
546C4—241no—
547C4C3×#67⋊C411yescharacteristic
548C4C3×#88311yescharacteristic
549C4C3×C4×#6711yescharacteristic
550C4C3×#93711yescharacteristic
551C3—21no—
552C3#7041711yescharacteristic, direct product factor
553C3C3×#188411yescharacteristic, complemented
554C2—21no—
555C2—21no—
556C2—21no—
557C2—81no—
558C2—122no—
559C2C3×#495611yescharacteristic
subgroupG/H
centerC3×C4#937
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#421C2
socleC2×C32#386
prime radicalC3×#5832C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#421D3
hypocenterC3C3×#1884
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#421

Complements

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

NH ≅ G/Ntype
2. C3×#5745C2semidirect
3. C32×#421C2semidirect
4. C3×#5236C2semidirect
5. C3×#5006C2semidirect
6. C3×#4555C2semidirect
7. C3×#4532C2semidirect
9. #70417C3direct (N × H)
14. C3×#947C4semidirect
15. C3×#944C4semidirect
16. C3×#940C4semidirect
18. C3×#936C4semidirect
32. #5745C2×C3semidirect
35. C3×#421D3semidirect
36. C3×#421C2×C3semidirect
38. #5236C2×C3semidirect
39. #5006C2×C3semidirect
42. #4555C2×C3semidirect
43. #4532C2×C3semidirect
64. C32×M16D4semidirect
79. C3×#208D4semidirect
107. #947C3×C4semidirect
108. #944C3×C4semidirect
109. #940C3×C4semidirect
111. #936C3×C4semidirect
164. #421C3×D3semidirect
222. C3×M16#67semidirect
223. C3×M16C3×D4semidirect
257. #208C3×D4semidirect
416. M16C3×#67semidirect
480. C32#1884semidirect
552. C3#70417direct (N × H)
553. C3C3×#1884semidirect

Maximal subgroups

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

subgroupindex
C3×#58322
C3×#57452
C32×#4212
C3×#52362
C3×#50062
C3×#45552
C3×#45322
C3×#18843
#704173