Finite Group Databasebeta

← C3×#82274 #11808541

Subgroups — C3×#82274 #11808541

Order 1152.

1285 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#117C211yesmaximal, characteristic, complemented
2C2×C3×C4×#67C211yesmaximal, characteristic
3C3×C4×#217C211yesmaximal, characteristic, complemented
4C3×#4993C211yesmaximal, characteristic, complemented
5C3×#4992C211yesmaximal, characteristic, complemented
6C3×#4985C211yesmaximal, characteristic, complemented
7C3×#4983C211yesmaximal, characteristic, complemented
8C3×#4959C212yesmaximal
9C3×#4952C212yesmaximal, complemented
10C3×C4×(C2×D3)⋊C4C212yesmaximal, complemented
11C3×C4×Dic3⋊C4C212yesmaximal, complemented
12C3×#2395—31nomaximal
13#82274C311yesmaximal, characteristic, direct product factor
14C23×C32×C4C2211yescharacteristic
15C22×C3×#67C2211yescharacteristic
16C22×C3×C4×D3C2211yescharacteristic
17C32×#117—24no—
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C32×C4⋊C4C2211yescharacteristic
20C2×C32×C22⋊C4C2212yes—
21C2×C32×C42C2212yes—
22C2×C3×#217C2211yescharacteristic
23C2×C3×#217C2212yes—
24C3×C4×#67—24no—
25C2×C3×(C2×D3)⋊C4C2212yes—
26C2×C3×(C2×D3)⋊C4C2212yes—
27C2×C3×(C2×D3)⋊C4C2211yescharacteristic
28C2×C3×Dic3⋊C4C2212yes—
29C2×C3×Dic3⋊C4C2212yes—
30C2×C3×Dic3⋊C4C2211yescharacteristic
31C2×C3×C4×Dic3C2211yescharacteristic
32C2×C3×C4×Dic3C2212yes—
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2212yes—
35C3×C3⋊#95C2212yes—
36C3×C3⋊#95C2212yes—
37C3×C3⋊#95C2211yescharacteristic
38C3×C3⋊#95C2211yescharacteristic
39C2×C3×C4×D4—31no—
40C2×C3×#117—21no—
41C2×C3×#117D311yescharacteristic, complemented
42C2×C3×#117C2×C311yescharacteristic, complemented
43C2×C4×#67C2×C311yescharacteristic
44C3×#400—31no—
45C3×#399—31no—
46C3×#398—31no—
47C3×#397—31no—
48C3×#397—32no—
49C3×#394—32no—
50C3×C4×C4⋊C4—32no—
51C3×C4×C22⋊C4—31no—
52C3×C4×C22⋊C4—32no—
53C4×#217C2×C311yescharacteristic, complemented
54#4993C2×C311yescharacteristic, complemented
55#4992C2×C311yescharacteristic, complemented
56#4985C2×C311yescharacteristic, complemented
57#4983C2×C311yescharacteristic, complemented
58#4959C2×C312yes—
59#4952C2×C312yescomplemented
60C4×(C2×D3)⋊C4C2×C312yescomplemented
61C4×Dic3⋊C4C2×C312yescomplemented
62C24×C32C2311yescharacteristic
63C23×C3×D3C2311yescharacteristic
64C22×C32×C4—24no—
65C22×C32×C4D414yes—
66C22×C32×C4C2312yes—
67C22×C32×C4C2312yes—
68C22×C32×C4C2311yescharacteristic
69C22×C32×C4C2311yescharacteristic
70C2×C3×#67—22no—
71C2×C3×#67—22no—
72C2×C3×#67—22no—
73C22×C3×Dic3C2311yescharacteristic
74C22×C3×Dic3C2312yes—
75C22×C3×Dic3C2311yescharacteristic
76C22×C3×Dic3C2311yescharacteristic
77C22×C3×Dic3C2312yes—
78C2×C3×C4×D3—22no—
79C2×C3×C4×D3—24no—
80C32×C4⋊C4—22no—
81C32×C4⋊C4—22no—
82C32×C22⋊C4—24no—
83C32×C42—24no—
84C3×#217—22no—
85C3×#217—24no—
86C3×(C2×D3)⋊C4—24no—
87C3×(C2×D3)⋊C4—22no—
88C3×(C2×D3)⋊C4—24no—
89C3×Dic3⋊C4—22no—
90C3×Dic3⋊C4—24no—
91C3×Dic3⋊C4—24no—
92C3×C4×Dic3—22no—
93C3×C4×Dic3—24no—
94#2395—31noSylow
95C22×C3×D4—31no—
96C23×C3×C4—21no—
97C23×C3×C4—31no—
98C23×C3×C4C2×D311yescharacteristic
99C23×C3×C4C22×C311yescharacteristic
100C22×#67C22×C311yescharacteristic
101C22×C4×D3C22×C311yescharacteristic
102C3×C4×D4—64no—
103C3×#117—28no—
104C3×#117—24no—
105C3×#117—24no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—32no—
109C2×C3×C4⋊C4—32no—
110C2×C3×C4⋊C4—31no—
111C2×C3×C4⋊C4C2×D311yescharacteristic
112C2×C3×C4⋊C4C2×D311yescharacteristic
113C2×C3×C4⋊C4C22×C311yescharacteristic
114C2×C3×C4⋊C4C22×C311yescharacteristic
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4—32no—
117C2×C3×C22⋊C4—31no—
118C2×C3×C22⋊C4—32no—
119C2×C3×C22⋊C4—32no—
120C2×C3×C22⋊C4—31no—
121C2×C3×C22⋊C4C2×D312yes—
122C2×C3×C22⋊C4C22×C312yes—
123C2×C3×C42—22no—
124C2×C3×C42—32no—
125C2×C3×C42—31no—
126C2×C3×C42C2×D312yes—
127C2×C3×C42C22×C312yes—
128C2×#217C22×C311yescharacteristic
129C2×#217C22×C312yes—
130C4×#67—24no—
131C2×(C2×D3)⋊C4C22×C312yes—
132C2×(C2×D3)⋊C4C22×C312yes—
133C2×(C2×D3)⋊C4C22×C311yescharacteristic
134C2×Dic3⋊C4C22×C312yes—
135C2×Dic3⋊C4C22×C312yes—
136C2×Dic3⋊C4C22×C311yescharacteristic
137C2×C4×Dic3C22×C311yescharacteristic
138C2×C4×Dic3C22×C312yes—
139C3×#95—32no—
140C3×#95—31no—
141C3×#95—32no—
142C3×#95—31no—
143C3×#95—32no—
144C3×#95—32no—
145C3⋊#95C22×C312yes—
146C3⋊#95C22×C312yes—
147C3⋊#95C22×C312yes—
148C3⋊#95C22×C312yes—
149C3⋊#95C22×C311yescharacteristic
150C3⋊#95C22×C311yescharacteristic
151C23×C32—22no—
152C23×C32—22no—
153C23×C32—21no—
154C23×C32—21no—
155C23×C32C2×D412yes—
156C23×C32C2411yescharacteristic
157C22×C3×D3—21no—
158C22×C3×D3—22no—
159C22×C3×D3—22no—
160C22×C3×D3—21no—
161C22×C3×D3#4112yes—
162C2×C32×C4—22no—
163C2×C32×C4—28no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4—22no—
168C2×C32×C4—22no—
169C2×C32×C4—22no—
170C2×C32×C4C2×D412yes—
171C2×C32×C4C2×D412yes—
172C2×C32×C4#4114yes—
173C3×#67—28no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3#4112yes—
186C2×C3×Dic3#4114yes—
187C3×C4×D3—44no—
188C2×C4×D4—31no—
189C2×#117C3×D311yescharacteristic, complemented
190#400—31no—
191#399—31no—
192#398—31no—
193#397—31no—
194#397—32no—
195#394—32no—
196C4×C4⋊C4—32no—
197C4×C22⋊C4—31no—
198C4×C22⋊C4—32no—
199C24×C3—21no—
200C24×C3—31no—
201C24×C3C22×D311yescharacteristic
202C24×C3C23×C311yescharacteristic
203C23×D3C23×C311yescharacteristic
204C2×C3×D4—62no—
205C2×C3×D4—62no—
206C2×C3×D4—62no—
207C22×C3×C4—28no—
208C22×C3×C4—24no—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—22no—
214C22×C3×C4—22no—
215C22×C3×C4—32no—
216C22×C3×C4—31no—
217C22×C3×C4—32no—
218C22×C3×C4—31no—
219C22×C3×C4—31no—
220C22×C3×C4—64no—
221C22×C3×C4—62no—
222C22×C3×C4#6714yes—
223C22×C3×C4C3×D414yes—
224C22×C3×C4C22×D312yes—
225C22×C3×C4C22×D312yes—
226C22×C3×C4C22×D311yescharacteristic
227C22×C3×C4C22×D311yescharacteristic
228C22×C3×C4C23×C312yes—
229C22×C3×C4C23×C312yes—
230C22×C3×C4C23×C311yescharacteristic
231C22×C3×C4C23×C311yescharacteristic
232C2×#67—22no—
233C2×#67—22no—
234C2×#67—22no—
235C22×Dic3C23×C312yes—
236C22×Dic3C23×C311yescharacteristic
237C22×Dic3C23×C311yescharacteristic
238C22×Dic3C23×C311yescharacteristic
239C22×Dic3C23×C312yes—
240C2×C4×D3—22no—
241C2×C4×D3—24no—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—64no—
249C3×C4⋊C4—64no—
250C3×C4⋊C4—62no—
251C3×C22⋊C4—28no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—64no—
255C3×C22⋊C4—64no—
256C3×C22⋊C4—64no—
257C3×C22⋊C4—62no—
258C3×C22⋊C4—62no—
259C3×C42—28no—
260C3×C42—24no—
261C3×C42—24no—
262C3×C42—64no—
263C3×C42—62no—
264#217—22no—
265#217—24no—
266(C2×D3)⋊C4—24no—
267(C2×D3)⋊C4—22no—
268(C2×D3)⋊C4—24no—
269Dic3⋊C4—22no—
270Dic3⋊C4—24no—
271Dic3⋊C4—24no—
272C4×Dic3—22no—
273C4×Dic3—24no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—22no—
277C22×C32—42no—
278C22×C32—42no—
279C22×C32C22×D411yescharacteristic
280C22×C32C2×#4112yes—
281C22×C32C2×#4111yescharacteristic
282C22×C32C2×#4111yescharacteristic
283C22×C32C2×#4112yes—
284C2×C3×D3—22no—
285C2×C3×D3—24no—
286C2×C3×D3—41no—
287C2×C3×D3—42no—
288C2×C3×D3—41no—
289C32×C4—22no—
290C32×C4—42no—
291C32×C4—42no—
292C32×C4#12314yes—
293C3×Dic3—22no—
294C3×Dic3—24no—
295C3×Dic3—41no—
296C3×Dic3—42no—
297C3×Dic3—41no—
298C22×D4—31no—
299C23×C4—31no—
300C23×C4C2×C3×D311yescharacteristic
301C4×D4—64no—
302#117—24no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4C2×C3×D311yescharacteristic
307C2×C4⋊C4C2×C3×D311yescharacteristic
308C2×C22⋊C4—31no—
309C2×C22⋊C4—32no—
310C2×C22⋊C4—31no—
311C2×C22⋊C4—32no—
312C2×C22⋊C4—32no—
313C2×C22⋊C4C2×C3×D312yes—
314C2×C42—31no—
315C2×C42—32no—
316C2×C42C2×C3×D312yes—
317#95—32no—
318#95—31no—
319#95—32no—
320#95—31no—
321#95—32no—
322#95—32no—
323C23×C3—24no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—21no—
333C23×C3—22no—
334C23×C3—32no—
335C23×C3—42no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—62no—
339C23×C3—62no—
340C23×C3—61no—
341C23×C3—61no—
342C23×C3C2×#6712yes—
343C23×C3C2×C3×D412yes—
344C23×C3C23×D311yescharacteristic
345C23×C3C24×C311yescharacteristic
346C22×D3—21no—
347C22×D3—22no—
348C22×D3—22no—
349C22×D3—21no—
350C22×D3C3×#4112yes—
351C3×D4—68no—
352C2×C3×C4—24no—
353C2×C3×C4—216no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—28no—
357C2×C3×C4—28no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—24no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—34no—
376C2×C3×C4—32no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—62no—
382C2×C3×C4—62no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—62no—
386C2×C3×C4—62no—
387C2×C3×C4—62no—
388C2×C3×C4—62no—
389C2×C3×C4—62no—
390C2×C3×C4—61no—
391C2×C3×C4—61no—
392C2×C3×C4—124no—
393C2×C3×C4#23514yes—
394C2×C3×C4C2×#6712yes—
395C2×C3×C4C2×#6712yes—
396C2×C3×C4C2×C3×D412yes—
397C2×C3×C4C2×C3×D412yes—
398C2×C3×C4C3×#4114yes—
399#67—28no—
400C2×Dic3—21no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3C3×#4112yes—
412C2×Dic3C3×#4114yes—
413C4×D3—44no—
414C2×C32—42no—
415C2×C32C2×#12311yescharacteristic
416C2×C32#52511yescharacteristic
417C2×C32#52912yes—
418C2×C32#52912yes—
419C2×C32#53211yescharacteristic
420C3×D3—42no—
421C24—31no—
422C24C22×C3×D311yescharacteristic
423C2×D4—62no—
424C2×D4—62no—
425C2×D4—62no—
426C22×C4—24no—
427C22×C4—32no—
428C22×C4—31no—
429C22×C4—32no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—64no—
433C22×C4—62no—
434C22×C4C3×#6714yes—
435C22×C4C22×C3×D312yes—
436C22×C4C22×C3×D312yes—
437C22×C4C22×C3×D311yescharacteristic
438C22×C4C22×C3×D311yescharacteristic
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—64no—
442C4⋊C4—64no—
443C4⋊C4—62no—
444C22⋊C4—24no—
445C22⋊C4—64no—
446C22⋊C4—64no—
447C22⋊C4—64no—
448C22⋊C4—62no—
449C22⋊C4—62no—
450C42—24no—
451C42—64no—
452C42—62no—
453C22×C3—24no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—22no—
464C22×C3—21no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—64no—
474C22×C3—62no—
475C22×C3—122no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3C2×#23512yes—
479C22×C3D3×#4111yescharacteristic
480C22×C3D3×#4111yescharacteristic
481C22×C3D3×#4112yes—
482C22×C3C22×#6711yescharacteristic
483C22×C3C22×C3×D411yescharacteristic
484C22×C3C2×C3×#4112yes—
485C22×C3C2×C3×#4111yescharacteristic
486C22×C3C2×C3×#4111yescharacteristic
487C22×C3C2×C3×#4112yes—
488C2×D3—22no—
489C2×D3—24no—
490C2×D3—41no—
491C2×D3—42no—
492C2×D3—41no—
493C3×C4—24no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—24no—
497C3×C4—44no—
498C3×C4—44no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4—42no—
503C3×C4—64no—
504C3×C4—62no—
505C3×C4—122no—
506C3×C4—121no—
507C3×C4—121no—
508C3×C4#92914yes—
509C3×C4C3×#12314yes—
510Dic3—22no—
511Dic3—24no—
512Dic3—41no—
513Dic3—42no—
514Dic3—41no—
515C32#239511yescharacteristic, complemented, Sylow
516C23—22no—
517C23—22no—
518C23—21no—
519C23—21no—
520C23—32no—
521C23—62no—
522C23—62no—
523C23—61no—
524C23—61no—
525C23C2×C3×#6712yes—
526C23C23×C3×D311yescharacteristic
527D4—68no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—34no—
537C2×C4—32no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—124no—
550C2×C4C3×#23514yes—
551C2×C4C2×C3×#6712yes—
552C2×C4C2×C3×#6712yes—
553C2×C3—22no—
554C2×C3—22no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—44no—
559C2×C3—42no—
560C2×C3—42no—
561C2×C3—122no—
562C2×C3#552812yes—
563C2×C3#552812yes—
564C2×C3C2×#92911yescharacteristic
565C2×C3#582811yescharacteristic
566C2×C3#583211yescharacteristic
567C2×C3C2×C3×#12311yescharacteristic
568C2×C3C3×#52511yescharacteristic
569C2×C3C3×#52912yes—
570C2×C3C3×#52912yes—
571C2×C3C3×#53211yescharacteristic
572D3—42no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—42no—
577C22—42no—
578C22—64no—
579C22—62no—
580C22—122no—
581C22—121no—
582C22—121no—
583C22C2×C3×#23512yes—
584C22C3×D3×#4111yescharacteristic
585C22C3×D3×#4111yescharacteristic
586C22C3×D3×#4112yes—
587C22C22×C3×#6711yescharacteristic
588C4—22no—
589C4—42no—
590C4—42no—
591C4—64no—
592C4—62no—
593C4—122no—
594C4—121no—
595C4—121no—
596C4C3×#92914yes—
597C3—21no—
598C3#8227411yescharacteristic, direct product factor
599C3C3×#239511yescharacteristic, complemented
600C2—42no—
601C2—122no—
602C2C3×#552812yes—
603C2C3×#552812yes—
604C2C2×C3×#92911yescharacteristic
605C2C3×#582811yescharacteristic
606C2C3×#583211yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D3
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC2×C32×#117C2
socleC23×C32C24
prime radicalC22×C3×#67C22
hypercenterC2×C3×#117D3
hypocenterC3C3×#2395
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×#117

Complements

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

NH ≅ G/Ntype
1. C2×C32×#117C2semidirect
3. C3×C4×#217C2semidirect
4. C3×#4993C2semidirect
5. C3×#4992C2semidirect
6. C3×#4985C2semidirect
7. C3×#4983C2semidirect
9. C3×#4952C2semidirect
10. C3×C4×(C2×D3)⋊C4C2semidirect
11. C3×C4×Dic3⋊C4C2semidirect
13. #82274C3direct (N × H)
41. C2×C3×#117D3semidirect
42. C2×C3×#117C2×C3semidirect
53. C4×#217C2×C3semidirect
54. #4993C2×C3semidirect
55. #4992C2×C3semidirect
56. #4985C2×C3semidirect
57. #4983C2×C3semidirect
59. #4952C2×C3semidirect
60. C4×(C2×D3)⋊C4C2×C3semidirect
61. C4×Dic3⋊C4C2×C3semidirect
189. C2×#117C3×D3semidirect
515. C32#2395semidirect
598. C3#82274direct (N × H)
599. C3C3×#2395semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1172
C2×C3×C4×#672
C3×C4×#2172
C3×#49932
C3×#49922
C3×#49852
C3×#49832
C3×#49592
C3×#49522
C3×C4×(C2×D3)⋊C42
C3×C4×Dic3⋊C42
C3×#23953
#822743