Finite Group Databasebeta

← C3×#70115 #11788327

Subgroups — C3×#70115 #11788327

Order 1152.

749 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#946C211yesmaximal, characteristic
2C2×C3×D3×M16C211yesmaximal, characteristic
3C2×C32×#101C211yesmaximal, characteristic, complemented
4C2×C3×#836C211yesmaximal, characteristic, complemented
5C3×#5126C211yesmaximal, characteristic
6C3×#4554C211yesmaximal, characteristic, complemented
7C3×#4554C211yesmaximal, characteristic, complemented
8C3×#1901—31nomaximal
9#70115C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C22×C3×C4×D3C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C3×#946—21no—
15C3×#946—21no—
16C3×#946—21no—
17C3×#946—21no—
18C2×C3×(C2×D3)⋊C4C412yes—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×Dic3⋊C4C412yes—
21C2×C3×#208C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C3×D3×M16—22no—
24C3×D3×M16—22no—
25C2×C3×ΓL(1,𝔽25)—21no—
26C2×C3×D3×C8—21no—
27C32×#101—22no—
28C3×#836—22no—
29C3×(C2×D3)⋊C8—21no—
30C3×(C2×D3)⋊C8—21no—
31C22×C3×M16—31no—
32C2×C3×C22⋊Q8—31no—
33C2×#946C2×C311yescharacteristic
34C2×D3×M16C2×C311yescharacteristic
35C2×C3×#101—21no—
36C2×C3×#101—31no—
37C2×C3×#101D311yescharacteristic, complemented
38C2×C3×#101C2×C311yescharacteristic, complemented
39C3×#407—31no—
40C2×#836C2×C311yescharacteristic, complemented
41#5126C2×C311yescharacteristic
42C3×#343—31no—
43C3×#343—31no—
44#4554C2×C311yescharacteristic, complemented
45#4554C2×C311yescharacteristic, complemented
46C23×C3×D3C2×C411yescharacteristic
47C2×C32×Q8—21no—
48C2×C32×Q8—21no—
49C2×C32×Q8—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—21no—
53C22×C32×C4C2×C412yes—
54C22×C32×C4C2311yescharacteristic
55C22×C3×Dic3C2×C411yescharacteristic
56C22×C3×Dic3C2×C412yes—
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3—21no—
60C2×C3×C4×D3—21no—
61C2×C3×C4×D3D411yescharacteristic
62C2×C3×C4×D3D411yescharacteristic
63C2×C3×C4×D3D411yescharacteristic
64C2×C3×C4×D3D411yescharacteristic
65C32×M16—21no—
66C32×M16—21no—
67C32×M16—21no—
68C32×M16D412yes—
69C2×C32×C8—21no—
70C2×C32×C8—21no—
71C3×(C2×D3)⋊C4—22no—
72C3×(C2×D3)⋊C4—22no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—21no—
75C3×Dic3⋊C4—22no—
76C3×Dic3⋊C4—22no—
77C3×#208—21no—
78C3×#208—21no—
79C3×#208—21no—
80C3×#208D412yes—
81C2×C3×C3⋊C8—21no—
82C2×C3×C3⋊C8—21no—
83C3×ΓL(1,𝔽25)—24no—
84C3×D3×C8—24no—
85#1901—31noSylow
86C22×C3×Q8—21no—
87C22×C3×Q8C2×D311yescharacteristic
88C22×C3×Q8C22×C311yescharacteristic
89C23×C3×C4—31no—
90C22×C4×D3C22×C311yescharacteristic
91C2×C3×M16—21no—
92C2×C3×M16—21no—
93C2×C3×M16—31no—
94C2×C3×M16—31no—
95C2×C3×M16—61no—
96C2×C3×M16—62no—
97C2×C3×M16—62no—
98C2×C3×M16C2×D311yescharacteristic
99C2×C3×M16C2×D311yescharacteristic
100C2×C3×M16C22×C311yescharacteristic
101C2×C3×M16C22×C311yescharacteristic
102C22×C3×C8—61no—
103C3×C22⋊Q8—61no—
104C3×C22⋊Q8—61no—
105C3×C22⋊Q8—61no—
106C3×C22⋊Q8—61no—
107C2×C3×C4⋊C4—32no—
108C2×C3×C4⋊C4—31no—
109C2×C3×C22⋊C4—32no—
110#946—21no—
111#946—21no—
112#946—21no—
113#946—21no—
114C2×(C2×D3)⋊C4C3×C412yes—
115C2×C4⋊Dic3C22×C311yescharacteristic
116C2×Dic3⋊C4C3×C412yes—
117C2×#208C22×C311yescharacteristic
118C2×#208C22×C311yescharacteristic
119D3×M16—22no—
120D3×M16—22no—
121C2×ΓL(1,𝔽25)—21no—
122C2×D3×C8—21no—
123C3×#101—24no—
124C3×#101—22no—
125C3×#101—22no—
126C3×#101—62no—
127C3×C22⋊C8—61no—
128C3×C22⋊C8—61no—
129#836—22no—
130(C2×D3)⋊C8—21no—
131(C2×D3)⋊C8—21no—
132C23×C32C22×C411yescharacteristic
133C22×C3×D3—21no—
134C22×C3×D3—21no—
135C22×C3×D3—22no—
136C22×C3×D3—41no—
137C22×C3×D3C22⋊C411yescharacteristic
138C22×C3×D3C22⋊C411yescharacteristic
139C32×Q8—41no—
140C32×Q8—41no—
141C32×Q8—41no—
142C32×Q8—41no—
143C2×C32×C4—22no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4#4111yescharacteristic
151C2×C32×C4#4111yescharacteristic
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—42no—
156C2×C3×Dic3C22⋊C411yescharacteristic
157C2×C3×Dic3C22⋊C411yescharacteristic
158C3×C4×D3—21no—
159C3×C4×D3—21no—
160C3×C4×D3—21no—
161C3×C4×D3—21no—
162C3×C4×D3—21no—
163C3×C4×D3—21no—
164C3×C4×D3—21no—
165C3×C4×D3—21no—
166C32×C8—22no—
167C32×C8—41no—
168C3×C3⋊C8—22no—
169C3×C3⋊C8—41no—
170C22×M16—31no—
171C2×C22⋊Q8—31no—
172C2×#101—31no—
173C2×#101C3×D311yescharacteristic, complemented
174#407—31no—
175#343—31no—
176#343—31no—
177C24×C3—31no—
178C23×D3C2×C3×C411yescharacteristic
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—41no—
194C2×C3×Q8—41no—
195C2×C3×Q8—41no—
196C2×C3×Q8—41no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—31no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—32no—
205C22×C3×C4—61no—
206C22×C3×C4—61no—
207C22×C3×C4—61no—
208C22×C3×C4—61no—
209C22×C3×C4C4×D312yes—
210C22×C3×C4C2×C3×C412yes—
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C23×C311yescharacteristic
213C22×Dic3C2×C3×C411yescharacteristic
214C22×Dic3C2×C3×C412yes—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3C3×D411yescharacteristic
220C2×C4×D3C3×D411yescharacteristic
221C2×C4×D3C3×D411yescharacteristic
222C2×C4×D3C3×D411yescharacteristic
223C3×M16—21no—
224C3×M16—21no—
225C3×M16—22no—
226C3×M16—21no—
227C3×M16—21no—
228C3×M16—21no—
229C3×M16—21no—
230C3×M16—22no—
231C3×M16—32no—
232C3×M16—41no—
233C3×M16—41no—
234C3×M16—64no—
235C3×M16—61no—
236C3×M16—61no—
237C3×M16—61no—
238C3×M16#6712yes—
239C3×M16C3×D412yes—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—41no—
245C2×C3×C8—41no—
246C2×C3×C8—64no—
247C2×C3×C8—61no—
248C2×C3×C8—61no—
249C3×C4⋊C4—62no—
250C3×C4⋊C4—62no—
251C3×C4⋊C4—61no—
252C3×C4⋊C4—61no—
253C3×C22⋊C4—62no—
254C3×C22⋊C4—62no—
255(C2×D3)⋊C4—22no—
256(C2×D3)⋊C4—22no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259Dic3⋊C4—22no—
260Dic3⋊C4—22no—
261#208—21no—
262#208—21no—
263#208—21no—
264#208C3×D412yes—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267ΓL(1,𝔽25)—24no—
268D3×C8—24no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32C2×C22⋊C411yescharacteristic
272C22×C32C4×D411yescharacteristic
273C22×C32C4×D411yescharacteristic
274C2×C3×D3—21no—
275C2×C3×D3—21no—
276C2×C3×D3—21no—
277C2×C3×D3—21no—
278C2×C3×D3—21no—
279C2×C3×D3—42no—
280C2×C3×D3—42no—
281C2×C3×D3#10112yescomplemented
282C32×C4—42no—
283C32×C4#12011yescharacteristic
284C32×C4#12111yescharacteristic
285C32×C4#12111yescharacteristic
286C32×C4#12311yescharacteristic
287C3×Dic3—21no—
288C3×Dic3—21no—
289C3×Dic3—42no—
290C22×Q8C2×C3×D311yescharacteristic
291C23×C4—31no—
292C2×M16—31no—
293C2×M16—31no—
294C2×M16—61no—
295C2×M16—62no—
296C2×M16—62no—
297C2×M16C2×C3×D311yescharacteristic
298C2×M16C2×C3×D311yescharacteristic
299C22×C8—61no—
300C22⋊Q8—61no—
301C22⋊Q8—61no—
302C22⋊Q8—61no—
303C22⋊Q8—61no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—31no—
306C2×C22⋊C4—32no—
307#101—22no—
308#101—62no—
309C22⋊C8—61no—
310C22⋊C8—61no—
311C23×C3—21no—
312C23×C3—31no—
313C23×C3—31no—
314C23×C3—62no—
315C23×C3—61no—
316C23×C3—61no—
317C23×C3—121no—
318C23×C3C2×C4×D311yescharacteristic
319C23×C3C22×C3×C411yescharacteristic
320C22×D3—21no—
321C22×D3—21no—
322C22×D3—22no—
323C22×D3—41no—
324C22×D3C3×C22⋊C411yescharacteristic
325C22×D3C3×C22⋊C411yescharacteristic
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8—41no—
331C3×Q8—41no—
332C3×Q8—41no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×Q8—41no—
337C3×Q8—41no—
338C2×C3×C4—24no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—62no—
366C2×C3×C4—122no—
367C2×C3×C4#23511yescharacteristic
368C2×C3×C4D3×D411yescharacteristic
369C2×C3×C4D3×D411yescharacteristic
370C2×C3×C4D3×D411yescharacteristic
371C2×C3×C4#23711yescharacteristic
372C2×C3×C4C2×#6711yescharacteristic
373C2×C3×C4C2×C3×D411yescharacteristic
374C2×C3×C4C2×C3×D411yescharacteristic
375C2×C3×C4C2×C3×D411yescharacteristic
376C2×C3×C4C2×C3×D411yescharacteristic
377C2×C3×C4C3×#4111yescharacteristic
378C2×C3×C4C3×#4111yescharacteristic
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—22no—
382C2×Dic3—42no—
383C2×Dic3C3×C22⋊C411yescharacteristic
384C2×Dic3C3×C22⋊C411yescharacteristic
385C4×D3—21no—
386C4×D3—21no—
387C4×D3—21no—
388C4×D3—21no—
389C4×D3—21no—
390C4×D3—21no—
391C4×D3—21no—
392C4×D3—21no—
393C3×C8—22no—
394C3×C8—22no—
395C3×C8—42no—
396C3×C8—42no—
397C3×C8—41no—
398C3×C8—41no—
399C3×C8—62no—
400C3×C8—121no—
401C3⋊C8—22no—
402C3⋊C8—41no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32#38611yescharacteristic
406C2×C32C2×#10111yescharacteristic
407C2×C32#41311yescharacteristic
408C3×D3—22no—
409C3×D3—41no—
410C24—31no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—32no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4C3×C4×D312yes—
428C22×C4C22×C3×D311yescharacteristic
429M16—21no—
430M16—21no—
431M16—21no—
432M16—32no—
433M16—64no—
434M16—61no—
435M16—61no—
436M16—61no—
437M16C3×#6712yes—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—64no—
441C2×C8—61no—
442C2×C8—61no—
443C4⋊C4—62no—
444C4⋊C4—62no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C22⋊C4—62no—
448C22⋊C4—62no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—32no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—61no—
460C22×C3—61no—
461C22×C3—61no—
462C22×C3—61no—
463C22×C3—61no—
464C22×C3—122no—
465C22×C3—122no—
466C22×C3D3×C22⋊C411yescharacteristic
467C22×C3#67⋊C411yescharacteristic
468C22×C3C4×#6711yescharacteristic
469C22×C3C2×C3×C22⋊C411yescharacteristic
470C22×C3C3×C4×D411yescharacteristic
471C22×C3C3×C4×D411yescharacteristic
472C2×D3—21no—
473C2×D3—21no—
474C2×D3—21no—
475C2×D3—21no—
476C2×D3—21no—
477C2×D3—42no—
478C2×D3—42no—
479C2×D3C3×#10112yescomplemented
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—44no—
485C3×C4—42no—
486C3×C4—42no—
487C3×C4—61no—
488C3×C4—61no—
489C3×C4—122no—
490C3×C4#88311yescharacteristic
491C3×C4#88411yescharacteristic
492C3×C4#93711yescharacteristic
493C3×C4#93811yescharacteristic
494C3×C4C3×#12011yescharacteristic
495C3×C4C3×#12111yescharacteristic
496C3×C4C3×#12111yescharacteristic
497C3×C4C3×#12311yescharacteristic
498Dic3—21no—
499Dic3—21no—
500Dic3—42no—
501C32#190111yescharacteristic, complemented, Sylow
502C23—31no—
503C23—31no—
504C23—62no—
505C23—61no—
506C23—61no—
507C23—121no—
508C23C2×C3×C4×D311yescharacteristic
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—31no—
517C2×C4—31no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—62no—
529C2×C4—122no—
530C2×C4C3×#23511yescharacteristic
531C2×C4C3×D3×D411yescharacteristic
532C2×C4C3×D3×D411yescharacteristic
533C2×C4C3×D3×D411yescharacteristic
534C2×C4C3×#23711yescharacteristic
535C2×C4C2×C3×#6711yescharacteristic
536C8—22no—
537C8—41no—
538C8—62no—
539C8—121no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—62no—
550C2×C3—121no—
551C2×C3D3×#10111yescharacteristic
552C2×C3#475111yescharacteristic
553C2×C3#495611yescharacteristic
554C2×C3C3×#38611yescharacteristic
555C2×C3C2×C3×#10111yescharacteristic
556C2×C3C3×#41311yescharacteristic
557D3—22no—
558D3—41no—
559C22—21no—
560C22—21no—
561C22—32no—
562C22—61no—
563C22—61no—
564C22—61no—
565C22—61no—
566C22—61no—
567C22—122no—
568C22—122no—
569C22C3×D3×C22⋊C411yescharacteristic
570C22C3×#67⋊C411yescharacteristic
571C22C3×C4×#6711yescharacteristic
572C4—42no—
573C4—61no—
574C4—61no—
575C4—122no—
576C4C3×#88311yescharacteristic
577C4C3×#88411yescharacteristic
578C4C3×#93711yescharacteristic
579C4C3×#93811yescharacteristic
580C3—21no—
581C3#7011511yescharacteristic, direct product factor
582C3C3×#190111yescharacteristic, complemented
583C2—21no—
584C2—21no—
585C2—62no—
586C2—121no—
587C2C3×D3×#10111yescharacteristic
588C2C3×#475111yescharacteristic
589C2C3×#495611yescharacteristic
subgroupG/H
centerC22×C3D3×C22⋊C4
commutatorC23×C3C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#101C2
socleC22×C32C2×C22⋊C4
prime radicalC23×C3×D3C2×C4
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#101D3
hypocenterC3C3×#1901
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#101

Complements

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

NH ≅ G/Ntype
3. C2×C32×#101C2semidirect
4. C2×C3×#836C2semidirect
6. C3×#4554C2semidirect
7. C3×#4554C2semidirect
9. #70115C3direct (N × H)
37. C2×C3×#101D3semidirect
38. C2×C3×#101C2×C3semidirect
40. C2×#836C2×C3semidirect
44. #4554C2×C3semidirect
45. #4554C2×C3semidirect
173. C2×#101C3×D3semidirect
281. C2×C3×D3#101semidirect
479. C2×D3C3×#101semidirect
501. C32#1901semidirect
581. C3#70115direct (N × H)
582. C3C3×#1901semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9462
C2×C3×D3×M162
C2×C32×#1012
C2×C3×#8362
C3×#51262
C3×#45542
C3×#19013
#701153