Finite Group Databasebeta

← C3×#70022 #11787925

Subgroups — C3×#70022 #11787925

Order 1152.

1261 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#937C211yesmaximal, characteristic
2C3×#5798C211yesmaximal, characteristic, complemented
3C2×C32×#99C211yesmaximal, characteristic, complemented
4C3×#4955C212yesmaximal, complemented
5C3×#4527C212yesmaximal, complemented
6C3×#1889—31nomaximal
7#70022C311yesmaximal, characteristic, direct product factor
8C24×C3×D3C2211yescharacteristic
9C22×C32×D4C2211yescharacteristic
10C22×C3×#67—21no—
11C2×C32×C22⋊C4C2212yes—
12C2×C3×#217—21no—
13C2×C3×#217C2212yes—
14C2×C3×#217C2211yescharacteristic
15C3×#937—22no—
16C3×#937C414yescomplemented
17C3×D4×Dic3—24no—
18C2×C3×(C2×D3)⋊C4—21no—
19C2×C3×(C2×D3)⋊C4—24no—
20C3×#926—21no—
21C32×#99—22no—
22C2×C3×#120—31no—
23C3×#518—31no—
24C2×#937C2×C311yescharacteristic
25#5798C2×C311yescharacteristic, complemented
26C2×C3×#99—21no—
27C2×C3×#99D311yescharacteristic, complemented
28C2×C3×#99C2×C311yescharacteristic, complemented
29C3×#379—32no—
30#4955C2×C312yescomplemented
31C3×#342—32no—
32#4527C2×C312yescomplemented
33C24×C32D411yescharacteristic
34C24×C32C2311yescharacteristic
35C23×C3×D3—22no—
36C23×C3×D3—21no—
37C23×C3×D3—21no—
38C23×C3×D3—22no—
39C23×C3×D3—44no—
40C23×C3×D3C2×C412yescomplemented
41C2×C32×D4—22no—
42C2×C32×D4—22no—
43C2×C32×D4C2×C412yes—
44C2×C32×D4D412yes—
45C22×C32×C4—22no—
46C22×C32×C4D411yescharacteristic
47C2×C3×#67—24no—
48C2×C3×#67—44no—
49C22×C3×Dic3—21no—
50C22×C3×Dic3—22no—
51C22×C3×Dic3D414yescomplemented
52C32×C22⋊C4—22no—
53C32×C22⋊C4—22no—
54C3×#217—22no—
55C3×#217—22no—
56C3×#217—22no—
57C3×#217—21no—
58C3×#217C2×C412yescomplemented
59C3×(C2×D3)⋊C4—24no—
60C3×(C2×D3)⋊C4—48no—
61C3×C4⋊Dic3—22no—
62C3×C4×Dic3—22no—
63#1889—31noSylow
64C25×C3—31no—
65C24×D3C22×C311yescharacteristic
66C22×C3×D4—21no—
67C22×C3×D4—61no—
68C22×C3×D4C2×D311yescharacteristic
69C22×C3×D4C22×C311yescharacteristic
70C22×#67—21no—
71C3×#120—34no—
72C3×#120—62no—
73C3×C4×D4—64no—
74C3×#117—61no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—31no—
77C2×C3×C22⋊C4—32no—
78C2×C3×C22⋊C4—61no—
79C2×C3×C22⋊C4—61no—
80C2×C3×C22⋊C4—64no—
81C2×C3×C22⋊C4C2×D312yes—
82C2×C3×C22⋊C4C22×C312yes—
83C2×#217—21no—
84C2×#217C22×C312yes—
85C2×#217C22×C311yescharacteristic
86#937—22no—
87#937C3×C414yescomplemented
88D4×Dic3—24no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—24no—
91#926—21no—
92C3×#99—24no—
93C3×#99—22no—
94C3×#99—22no—
95C23×C32—21no—
96C23×C32—21no—
97C23×C32—42no—
98C23×C32—42no—
99C23×C32C22⋊C412yescomplemented
100C23×C32C22×C411yescharacteristic
101C23×C32C2×D411yescharacteristic
102C23×C32C2×D412yes—
103C23×C32C2×D411yescharacteristic
104C23×C32#4112yes—
105C22×C3×D3—28no—
106C22×C3×D3—24no—
107C22×C3×D3—22no—
108C22×C3×D3—24no—
109C22×C3×D3—44no—
110C22×C3×D3—44no—
111C22×C3×D3—48no—
112C22×C3×D3—44no—
113C22×C3×D3—42no—
114C22×C3×D3—41no—
115C22×C3×D3—42no—
116C22×C3×D3—41no—
117C32×D4—24no—
118C32×D4—24no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—42no—
122C2×C32×C4—44no—
123C2×C32×C4C22⋊C412yescomplemented
124C3×#67—48no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—42no—
128C2×C3×Dic3—44no—
129C2×C3×Dic3—82no—
130C2×#120—31no—
131#518—31no—
132C2×#99C3×D311yescharacteristic, complemented
133#379—32no—
134#342—32no—
135C24×C3—21no—
136C24×C3—21no—
137C24×C3—32no—
138C24×C3—61no—
139C24×C3—62no—
140C24×C3—62no—
141C24×C3—61no—
142C24×C3—124no—
143C24×C3D1211yescharacteristic
144C24×C3C3×D411yescharacteristic
145C24×C3C22×D311yescharacteristic
146C24×C3C23×C311yescharacteristic
147C23×D3—22no—
148C23×D3—21no—
149C23×D3—21no—
150C23×D3—22no—
151C23×D3—44no—
152C23×D3C2×C3×C412yescomplemented
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—42no—
160C2×C3×D4—42no—
161C2×C3×D4—64no—
162C2×C3×D4—124no—
163C2×C3×D4C4×D312yes—
164C2×C3×D4#6712yes—
165C2×C3×D4C2×C3×C412yes—
166C2×C3×D4C3×D412yes—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—34no—
171C22×C3×C4—42no—
172C22×C3×C4—61no—
173C22×C3×C4—62no—
174C22×C3×C4D1211yescharacteristic
175C22×C3×C4C3×D411yescharacteristic
176C2×#67—24no—
177C2×#67—44no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3C3×D414yescomplemented
181C3×C4⋊C4—62no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—32no—
188C3×C22⋊C4—42no—
189C3×C22⋊C4—62no—
190C3×C22⋊C4—62no—
191C3×C22⋊C4—62no—
192C3×C22⋊C4—61no—
193C3×C22⋊C4—64no—
194C3×C22⋊C4—128no—
195C3×C42—62no—
196#217—22no—
197#217—22no—
198#217—22no—
199#217—21no—
200#217C2×C3×C412yescomplemented
201(C2×D3)⋊C4—24no—
202(C2×D3)⋊C4—48no—
203C4⋊Dic3—22no—
204C4×Dic3—22no—
205C22×C32—24no—
206C22×C32—22no—
207C22×C32—22no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—42no—
211C22×C32—42no—
212C22×C32—42no—
213C22×C32—81no—
214C22×C32C2×C22⋊C411yescharacteristic
215C22×C32C4×D412yes—
216C22×C32#12011yescharacteristic
217C22×C32#12112yes—
218C22×C32#12311yescharacteristic
219C2×C3×D3—24no—
220C2×C3×D3—28no—
221C2×C3×D3—42no—
222C2×C3×D3—48no—
223C2×C3×D3—44no—
224C2×C3×D3—44no—
225C2×C3×D3—42no—
226C2×C3×D3—44no—
227C32×C4—22no—
228C32×C4—82no—
229C3×Dic3—42no—
230C3×Dic3—44no—
231C25—31no—
232C22×D4—61no—
233C22×D4C2×C3×D311yescharacteristic
234#120—34no—
235#120—62no—
236C4×D4—64no—
237#117—61no—
238C2×C22⋊C4—31no—
239C2×C22⋊C4—32no—
240C2×C22⋊C4—61no—
241C2×C22⋊C4—61no—
242C2×C22⋊C4—64no—
243C2×C22⋊C4C2×C3×D312yes—
244#99—22no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—22no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—21no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—68no—
262C23×C3—64no—
263C23×C3—64no—
264C23×C3—62no—
265C23×C3—82no—
266C23×C3—82no—
267C23×C3—124no—
268C23×C3—122no—
269C23×C3—124no—
270C23×C3—128no—
271C23×C3—124no—
272C23×C3—121no—
273C23×C3—121no—
274C23×C3—122no—
275C23×C3(C2×D3)⋊C412yescomplemented
276C23×C3C3×C22⋊C412yescomplemented
277C23×C3C2×C4×D311yescharacteristic
278C23×C3C2×D1211yescharacteristic
279C23×C3D3×D412yes—
280C23×C3#23712yes—
281C23×C3C2×#6711yescharacteristic
282C23×C3C22×C3×C411yescharacteristic
283C23×C3C2×C3×D411yescharacteristic
284C23×C3C2×C3×D412yes—
285C23×C3C2×C3×D411yescharacteristic
286C23×C3C3×#4112yes—
287C22×D3—28no—
288C22×D3—24no—
289C22×D3—22no—
290C22×D3—24no—
291C22×D3—44no—
292C22×D3—44no—
293C22×D3—48no—
294C22×D3—44no—
295C22×D3—42no—
296C22×D3—41no—
297C22×D3—42no—
298C22×D3—41no—
299C3×D4—24no—
300C3×D4—24no—
301C3×D4—24no—
302C3×D4—24no—
303C3×D4—44no—
304C3×D4—44no—
305C3×D4—128no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—44no—
314C2×C3×C4—44no—
315C2×C3×C4—44no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—64no—
319C2×C3×C4—62no—
320C2×C3×C4—84no—
321C2×C3×C4—122no—
322C2×C3×C4—124no—
323C2×C3×C4—242no—
324C2×C3×C4(C2×D3)⋊C412yescomplemented
325C2×C3×C4C3×C22⋊C412yescomplemented
326#67—48no—
327C2×Dic3—22no—
328C2×Dic3—24no—
329C2×Dic3—42no—
330C2×Dic3—44no—
331C2×Dic3—82no—
332C2×C32—21no—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—21no—
336C2×C32—42no—
337C2×C32#38611yescharacteristic
338C2×C32B(4,𝔽2)12yes—
339C3×D3—44no—
340C24—32no—
341C24—61no—
342C24—62no—
343C24—62no—
344C24—61no—
345C24—124no—
346C24C3×D1211yescharacteristic
347C24C22×C3×D311yescharacteristic
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—64no—
351C2×D4—124no—
352C2×D4C3×C4×D312yes—
353C2×D4C3×#6712yes—
354C22×C4—22no—
355C22×C4—34no—
356C22×C4—61no—
357C22×C4—62no—
358C22×C4C3×D1211yescharacteristic
359C4⋊C4—62no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—32no—
363C22⋊C4—62no—
364C22⋊C4—62no—
365C22⋊C4—62no—
366C22⋊C4—61no—
367C22⋊C4—64no—
368C22⋊C4—128no—
369C42—62no—
370C22×C3—24no—
371C22×C3—22no—
372C22×C3—24no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—22no—
381C22×C3—44no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—42no—
387C22×C3—41no—
388C22×C3—42no—
389C22×C3—41no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—64no—
395C22×C3—68no—
396C22×C3—82no—
397C22×C3—81no—
398C22×C3—82no—
399C22×C3—81no—
400C22×C3—82no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—82no—
404C22×C3—128no—
405C22×C3—122no—
406C22×C3—124no—
407C22×C3—124no—
408C22×C3—122no—
409C22×C3—124no—
410C22×C3#67⋊C412yes—
411C22×C3#88211yescharacteristic
412C22×C3#88611yescharacteristic
413C22×C3C2×(C2×D3)⋊C411yescharacteristic
414C22×C3#93912yes—
415C22×C3C2×C3×C22⋊C411yescharacteristic
416C22×C3C3×C4×D412yes—
417C22×C3C3×#12011yescharacteristic
418C22×C3C3×#12112yes—
419C22×C3C3×#12311yescharacteristic
420C2×D3—24no—
421C2×D3—28no—
422C2×D3—42no—
423C2×D3—48no—
424C2×D3—44no—
425C2×D3—44no—
426C2×D3—42no—
427C2×D3—44no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—42no—
431C3×C4—82no—
432C3×C4—82no—
433C3×C4—84no—
434C3×C4—122no—
435C3×C4—124no—
436Dic3—42no—
437Dic3—44no—
438C32#188911yescharacteristic, complemented, Sylow
439C23—21no—
440C23—21no—
441C23—42no—
442C23—42no—
443C23—68no—
444C23—64no—
445C23—64no—
446C23—62no—
447C23—124no—
448C23—122no—
449C23—124no—
450C23—128no—
451C23—124no—
452C23—121no—
453C23—121no—
454C23—122no—
455C23C3×(C2×D3)⋊C412yescomplemented
456C23C2×C3×C4×D311yescharacteristic
457C23C2×C3×D1211yescharacteristic
458C23C3×D3×D412yes—
459C23C3×#23712yes—
460C23C2×C3×#6711yescharacteristic
461D4—24no—
462D4—24no—
463D4—128no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—64no—
469C2×C4—62no—
470C2×C4—122no—
471C2×C4—124no—
472C2×C4—242no—
473C2×C4C3×(C2×D3)⋊C412yescomplemented
474C2×C3—22no—
475C2×C3—21no—
476C2×C3—22no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—22no—
480C2×C3—22no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—22no—
484C2×C3—42no—
485C2×C3—41no—
486C2×C3—41no—
487C2×C3—42no—
488C2×C3—42no—
489C2×C3—42no—
490C2×C3—82no—
491C2×C3—124no—
492C2×C3#474112yes—
493C2×C3#495811yescharacteristic
494C2×C3C3×#38611yescharacteristic
495C2×C3C3×B(4,𝔽2)12yes—
496D3—44no—
497C22—24no—
498C22—22no—
499C22—22no—
500C22—41no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—42no—
505C22—64no—
506C22—68no—
507C22—81no—
508C22—128no—
509C22—122no—
510C22—124no—
511C22—124no—
512C22—122no—
513C22—124no—
514C22C3×#67⋊C412yes—
515C22C3×#88211yescharacteristic
516C22C3×#88611yescharacteristic
517C22C2×C3×(C2×D3)⋊C411yescharacteristic
518C22C3×#93912yes—
519C4—22no—
520C4—82no—
521C4—122no—
522C4—124no—
523C3—21no—
524C3#7002211yescharacteristic, direct product factor
525C3C3×#188911yescharacteristic, complemented
526C2—22no—
527C2—21no—
528C2—22no—
529C2—21no—
530C2—42no—
531C2—124no—
532C2C3×#474112yes—
533C2C3×#495811yescharacteristic
subgroupG/H
centerC22×C3#882
commutatorC23×C3C22×C3×C4
FrattiniC24C22×C3×D3
FittingC2×C32×#99C2
socleC22×C32#120
prime radicalC2×C3×#937C2
cosocleC24×C3C23×C3
hypercenterC2×C3×#99D3
hypocenterC3C3×#1889
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C2×C3×#99

Complements

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

NH ≅ G/Ntype
2. C3×#5798C2semidirect
3. C2×C32×#99C2semidirect
4. C3×#4955C2semidirect
5. C3×#4527C2semidirect
7. #70022C3direct (N × H)
16. C3×#937C4semidirect
25. #5798C2×C3semidirect
27. C2×C3×#99D3semidirect
28. C2×C3×#99C2×C3semidirect
30. #4955C2×C3semidirect
32. #4527C2×C3semidirect
40. C23×C3×D3C2×C4semidirect
51. C22×C3×Dic3D4semidirect
58. C3×#217C2×C4semidirect
87. #937C3×C4semidirect
99. C23×C32C22⋊C4semidirect
123. C2×C32×C4C22⋊C4semidirect
132. C2×#99C3×D3semidirect
152. C23×D3C2×C3×C4semidirect
180. C22×Dic3C3×D4semidirect
200. #217C2×C3×C4semidirect
275. C23×C3(C2×D3)⋊C4semidirect
276. C23×C3C3×C22⋊C4semidirect
324. C2×C3×C4(C2×D3)⋊C4semidirect
325. C2×C3×C4C3×C22⋊C4semidirect
438. C32#1889semidirect
455. C23C3×(C2×D3)⋊C4semidirect
473. C2×C4C3×(C2×D3)⋊C4semidirect
524. C3#70022direct (N × H)
525. C3C3×#1889semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9372
C3×#57982
C2×C32×#992
C3×#49552
C3×#45272
C3×#18893
#700223