Finite Group Databasebeta

← C3×#82491 #11808646

Subgroups — C3×#82491 #11808646

Order 1152.

1205 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#872C211yesmaximal, characteristic
3C3×#5231C211yesmaximal, characteristic
4C3×#5230C211yesmaximal, characteristic, complemented
5C3×#4994C212yesmaximal
6C3×#4993C211yesmaximal, characteristic
7C3×#4985C211yesmaximal, characteristic, complemented
8C3×#4982C212yesmaximal, complemented
9C3×C4×(C2×D3)⋊C4C212yesmaximal
10C3×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
11C3×C4×Dic3⋊C4C212yesmaximal, complemented
12C3×#2396—31nomaximal
13#82491C311yesmaximal, characteristic, direct product factor
14C22×C32×Q8C2211yescharacteristic
15C22×C3×C4×D3C2211yescharacteristic
16C32×C4×Q8—24no—
17C2×C32×C4⋊C4C2212yes—
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C32×C42C2212yes—
20C2×C32×C42C2211yescharacteristic
21C2×C3×(C2×D3)⋊C4C2212yes—
22C2×C3×(C2×D3)⋊C4C2212yes—
23C2×C3×(C2×D3)⋊C4C2211yescharacteristic
24C2×C3×(C2×D3)⋊C4C2211yescharacteristic
25C2×C3×C4⋊Dic3C2212yes—
26C2×C3×Dic3⋊C4C2212yes—
27C2×C3×Dic3⋊C4C2212yes—
28C2×C3×Dic3⋊C4C2211yescharacteristic
29C2×C3×Dic3⋊C4C2211yescharacteristic
30C2×C3×C4×Dic3C2211yescharacteristic
31C2×C3×C4×Dic3C2212yes—
32C3×#872—24no—
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2212yes—
35C3×C3⋊#95C2211yescharacteristic
36C3×C3⋊#95C2211yescharacteristic
37C3×C3⋊#95C2212yes—
38C3×C3⋊#95C2212yes—
39C2×C3×C4×Q8—21no—
40C2×C3×C4×Q8D311yescharacteristic, complemented
41C2×C3×C4×Q8C2×C311yescharacteristic, complemented
42C2×C3×#117—31no—
43C2×#872C2×C311yescharacteristic
44C3×#400—31no—
45C3×#399—31no—
46C3×#398—32no—
47C3×#396—32no—
48C3×#395—31no—
49C3×#393—31no—
50C3×C4×C4⋊C4—32no—
51C3×C4×C4⋊C4—31no—
52C3×C4×C22⋊C4—32no—
53#5231C2×C311yescharacteristic
54#5230C2×C311yescharacteristic, complemented
55#4994C2×C312yes—
56#4993C2×C311yescharacteristic
57#4985C2×C311yescharacteristic, complemented
58#4982C2×C312yescomplemented
59C4×(C2×D3)⋊C4C2×C312yes—
60C4×C4⋊Dic3C2×C311yescharacteristic, complemented
61C4×Dic3⋊C4C2×C312yescomplemented
62C23×C3×D3C2311yescharacteristic
63C2×C32×Q8—22no—
64C2×C32×Q8—24no—
65C22×C32×C4C2312yes—
66C22×C32×C4C2311yescharacteristic
67C22×C32×C4C2311yescharacteristic
68C22×C32×C4C2312yes—
69C22×C32×C4C2311yescharacteristic
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2311yescharacteristic
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2311yescharacteristic
74C22×C3×Dic3C2312yes—
75C2×C3×C4×D3—22no—
76C2×C3×C4×D3—22no—
77C2×C3×C4×D3Q814yescomplemented
78C32×C4⋊C4—24no—
79C32×C4⋊C4—22no—
80C32×C42—24no—
81C32×C42D414yes—
82C3×(C2×D3)⋊C4—22no—
83C3×(C2×D3)⋊C4—22no—
84C3×(C2×D3)⋊C4—24no—
85C3×(C2×D3)⋊C4—24no—
86C3×C4⋊Dic3—24no—
87C3×Dic3⋊C4—22no—
88C3×Dic3⋊C4—22no—
89C3×Dic3⋊C4—24no—
90C3×Dic3⋊C4—24no—
91C3×C4×Dic3—22no—
92C3×C4×Dic3—24no—
93#2396—31noSylow
94C22×C3×Q8—21no—
95C22×C3×Q8C2×D311yescharacteristic
96C22×C3×Q8C22×C311yescharacteristic
97C23×C3×C4—31no—
98C22×C4×D3C22×C311yescharacteristic
99C3×C4×Q8—28no—
100C3×C4×Q8—24no—
101C3×C4×Q8—24no—
102C3×#117—64no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—32no—
106C2×C3×C4⋊C4—32no—
107C2×C3×C4⋊C4—31no—
108C2×C3×C4⋊C4—31no—
109C2×C3×C4⋊C4—32no—
110C2×C3×C4⋊C4C2×D312yes—
111C2×C3×C4⋊C4C2×D311yescharacteristic
112C2×C3×C4⋊C4C22×C312yes—
113C2×C3×C4⋊C4C22×C311yescharacteristic
114C2×C3×C22⋊C4—32no—
115C2×C3×C22⋊C4—32no—
116C2×C3×C22⋊C4—31no—
117C2×C3×C22⋊C4—31no—
118C2×C3×C42—22no—
119C2×C3×C42—21no—
120C2×C3×C42—32no—
121C2×C3×C42—31no—
122C2×C3×C42C2×D312yes—
123C2×C3×C42C2×D311yescharacteristic
124C2×C3×C42C22×C312yes—
125C2×C3×C42C22×C311yescharacteristic
126C2×(C2×D3)⋊C4C22×C312yes—
127C2×(C2×D3)⋊C4C22×C312yes—
128C2×(C2×D3)⋊C4C22×C311yescharacteristic
129C2×(C2×D3)⋊C4C22×C311yescharacteristic
130C2×C4⋊Dic3C22×C312yes—
131C2×Dic3⋊C4C22×C312yes—
132C2×Dic3⋊C4C22×C312yes—
133C2×Dic3⋊C4C22×C311yescharacteristic
134C2×Dic3⋊C4C22×C311yescharacteristic
135C2×C4×Dic3C22×C311yescharacteristic
136C2×C4×Dic3C22×C312yes—
137#872—24no—
138C3×#95—32no—
139C3×#95—32no—
140C3×#95—32no—
141C3×#95—31no—
142C3×#95—31no—
143C3×#95—32no—
144C3⋊#95C22×C312yes—
145C3⋊#95C22×C312yes—
146C3⋊#95C22×C311yescharacteristic
147C3⋊#95C22×C311yescharacteristic
148C3⋊#95C22×C312yes—
149C3⋊#95C22×C312yes—
150C23×C32C2411yescharacteristic
151C22×C3×D3—21no—
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3—21no—
155C22×C3×D3C2×Q812yes—
156C32×Q8—28no—
157C2×C32×C4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—22no—
160C2×C32×C4—22no—
161C2×C32×C4—22no—
162C2×C32×C4—22no—
163C2×C32×C4—22no—
164C2×C32×C4C2×D412yes—
165C2×C32×C4C2×D412yes—
166C2×C32×C4C2×D412yes—
167C2×C32×C4C2×Q812yes—
168C2×C32×C4#4114yes—
169C2×C32×C4#4112yes—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3C2×Q812yes—
182C2×C3×Dic3#4114yes—
183C3×C4×D3—28no—
184C2×C4×Q8C3×D311yescharacteristic, complemented
185C2×#117—31no—
186#400—31no—
187#399—31no—
188#398—32no—
189#396—32no—
190#395—31no—
191#393—31no—
192C4×C4⋊C4—32no—
193C4×C4⋊C4—31no—
194C4×C22⋊C4—32no—
195C24×C3—31no—
196C23×D3C23×C311yescharacteristic
197C2×C3×Q8—24no—
198C2×C3×Q8—28no—
199C2×C3×Q8—24no—
200C2×C3×Q8—22no—
201C2×C3×Q8—22no—
202C2×C3×Q8—24no—
203C22×C3×C4—21no—
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4—21no—
208C22×C3×C4—31no—
209C22×C3×C4—32no—
210C22×C3×C4—34no—
211C22×C3×C4—32no—
212C22×C3×C4—31no—
213C22×C3×C4—31no—
214C22×C3×C4—62no—
215C22×C3×C4—62no—
216C22×C3×C4C22×D312yes—
217C22×C3×C4C22×D312yes—
218C22×C3×C4C22×D311yescharacteristic
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C22×D311yescharacteristic
221C22×C3×C4C23×C312yes—
222C22×C3×C4C23×C311yescharacteristic
223C22×C3×C4C23×C311yescharacteristic
224C22×C3×C4C23×C312yes—
225C22×C3×C4C23×C311yescharacteristic
226C22×Dic3C23×C312yes—
227C22×Dic3C23×C311yescharacteristic
228C22×Dic3C23×C311yescharacteristic
229C22×Dic3C23×C311yescharacteristic
230C22×Dic3C23×C312yes—
231C2×C4×D3—22no—
232C2×C4×D3—22no—
233C2×C4×D3C3×Q814yescomplemented
234C3×C4⋊C4—28no—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—64no—
241C3×C4⋊C4—64no—
242C3×C4⋊C4—64no—
243C3×C4⋊C4—62no—
244C3×C4⋊C4—62no—
245C3×C22⋊C4—64no—
246C3×C22⋊C4—64no—
247C3×C22⋊C4—62no—
248C3×C22⋊C4—62no—
249C3×C42—28no—
250C3×C42—24no—
251C3×C42—24no—
252C3×C42—24no—
253C3×C42—64no—
254C3×C42—62no—
255C3×C42#6714yes—
256C3×C42C3×D414yes—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—24no—
260(C2×D3)⋊C4—24no—
261C4⋊Dic3—24no—
262Dic3⋊C4—22no—
263Dic3⋊C4—22no—
264Dic3⋊C4—24no—
265Dic3⋊C4—24no—
266C4×Dic3—22no—
267C4×Dic3—24no—
268C22×C32C22×D411yescharacteristic
269C22×C32C22×Q811yescharacteristic
270C22×C32C2×#4112yes—
271C22×C32C2×#4111yescharacteristic
272C22×C32C2×#4112yes—
273C2×C3×D3—22no—
274C2×C3×D3—24no—
275C2×C3×D3—41no—
276C2×C3×D3—42no—
277C2×C3×D3—41no—
278C32×C4—22no—
279C32×C4—24no—
280C32×C4—22no—
281C32×C4—42no—
282C32×C4C22⋊Q814yes—
283C3×Dic3—22no—
284C3×Dic3—24no—
285C3×Dic3—41no—
286C3×Dic3—42no—
287C3×Dic3—41no—
288C22×Q8C2×C3×D311yescharacteristic
289C23×C4—31no—
290C4×Q8—24no—
291#117—64no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—32no—
295C2×C4⋊C4—32no—
296C2×C4⋊C4—32no—
297C2×C4⋊C4C2×C3×D312yes—
298C2×C4⋊C4C2×C3×D311yescharacteristic
299C2×C22⋊C4—31no—
300C2×C22⋊C4—31no—
301C2×C22⋊C4—32no—
302C2×C22⋊C4—32no—
303C2×C42—31no—
304C2×C42—32no—
305C2×C42C2×C3×D312yes—
306C2×C42C2×C3×D311yescharacteristic
307#95—32no—
308#95—32no—
309#95—32no—
310#95—31no—
311#95—31no—
312#95—32no—
313C23×C3—21no—
314C23×C3—32no—
315C23×C3—62no—
316C23×C3—61no—
317C23×C3—62no—
318C23×C3—61no—
319C23×C3C23×D311yescharacteristic
320C23×C3C24×C311yescharacteristic
321C22×D3—21no—
322C22×D3—22no—
323C22×D3—22no—
324C22×D3—21no—
325C22×D3C2×C3×Q812yes—
326C3×Q8—216no—
327C3×Q8—28no—
328C3×Q8—28no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—32no—
354C2×C3×C4—34no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—68no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—62no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4#23514yes—
371C2×C3×C4D3×Q812yes—
372C2×C3×C4#23912yes—
373C2×C3×C4C2×#6712yes—
374C2×C3×C4C2×#6712yes—
375C2×C3×C4C2×#6712yes—
376C2×C3×C4C2×C3×D412yes—
377C2×C3×C4C2×C3×D412yes—
378C2×C3×C4C2×C3×D412yes—
379C2×C3×C4C2×C3×Q812yes—
380C2×C3×C4C3×#4114yes—
381C2×C3×C4C3×#4112yes—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3C2×C3×Q812yes—
394C2×Dic3C3×#4114yes—
395C4×D3—28no—
396C2×C32C2×C22⋊Q811yescharacteristic
397C2×C32#52511yescharacteristic
398C2×C32#52912yes—
399C2×C32#53211yescharacteristic
400C2×C32#53312yes—
401C3×D3—42no—
402C24—31no—
403C2×Q8—24no—
404C2×Q8—22no—
405C22×C4—31no—
406C22×C4—32no—
407C22×C4—34no—
408C22×C4—32no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4C22×C3×D312yes—
414C22×C4C22×C3×D312yes—
415C22×C4C22×C3×D311yescharacteristic
416C22×C4C22×C3×D311yescharacteristic
417C22×C4C22×C3×D311yescharacteristic
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—64no—
421C4⋊C4—64no—
422C4⋊C4—64no—
423C4⋊C4—62no—
424C4⋊C4—62no—
425C22⋊C4—64no—
426C22⋊C4—64no—
427C22⋊C4—62no—
428C22⋊C4—62no—
429C42—24no—
430C42—64no—
431C42—62no—
432C42C3×#6714yes—
433C22×C3—22no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—22no—
437C22×C3—21no—
438C22×C3—62no—
439C22×C3—64no—
440C22×C3—121no—
441C22×C3—121no—
442C22×C3—122no—
443C22×C3C2×#23512yes—
444C22×C3C2×D3×Q811yescharacteristic
445C22×C3C2×#23911yescharacteristic
446C22×C3D3×#4112yes—
447C22×C3C22×#6711yescharacteristic
448C22×C3C22×C3×D411yescharacteristic
449C22×C3C22×C3×Q811yescharacteristic
450C22×C3C2×C3×#4112yes—
451C22×C3C2×C3×#4111yescharacteristic
452C22×C3C2×C3×#4112yes—
453C2×D3—22no—
454C2×D3—24no—
455C2×D3—41no—
456C2×D3—42no—
457C2×D3—41no—
458C3×C4—28no—
459C3×C4—24no—
460C3×C4—24no—
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—24no—
465C3×C4—22no—
466C3×C4—24no—
467C3×C4—42no—
468C3×C4—44no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—62no—
472C3×C4—64no—
473C3×C4—121no—
474C3×C4—121no—
475C3×C4—122no—
476C3×C4#94614yes—
477C3×C4C3×C22⋊Q814yes—
478Dic3—22no—
479Dic3—24no—
480Dic3—41no—
481Dic3—42no—
482Dic3—41no—
483C32#239611yescharacteristic, complemented, Sylow
484C23—32no—
485C23—62no—
486C23—61no—
487C23—62no—
488C23—61no—
489C23C23×C3×D311yescharacteristic
490Q8—28no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—32no—
499C2×C4—34no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—68no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4C3×#23514yes—
513C2×C4C3×D3×Q812yes—
514C2×C4C3×#23912yes—
515C2×C4C2×C3×#6712yes—
516C2×C4C2×C3×#6712yes—
517C2×C4C2×C3×#6712yes—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—122no—
524C2×C3#557812yes—
525C2×C3#558012yes—
526C2×C3#579111yescharacteristic
527C2×C3C2×#94611yescharacteristic
528C2×C3#583211yescharacteristic
529C2×C3C2×C3×C22⋊Q811yescharacteristic
530C2×C3C3×#52511yescharacteristic
531C2×C3C3×#52912yes—
532C2×C3C3×#53211yescharacteristic
533C2×C3C3×#53312yes—
534D3—42no—
535C22—62no—
536C22—64no—
537C22—121no—
538C22—121no—
539C22—122no—
540C22C2×C3×#23512yes—
541C22C2×C3×D3×Q811yescharacteristic
542C22C2×C3×#23911yescharacteristic
543C22C3×D3×#4112yes—
544C22C22×C3×#6711yescharacteristic
545C4—24no—
546C4—22no—
547C4—22no—
548C4—42no—
549C4—62no—
550C4—64no—
551C4—121no—
552C4—121no—
553C4—122no—
554C4C3×#94614yes—
555C3—21no—
556C3#8249111yescharacteristic, direct product factor
557C3C3×#239611yescharacteristic, complemented
558C2—122no—
559C2C3×#557812yes—
560C2C3×#558012yes—
561C2C3×#579111yescharacteristic
562C2C2×C3×#94611yescharacteristic
563C2C3×#583211yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D3
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC2×C32×C4×Q8C2
socleC23×C32C24
prime radicalC23×C3×D3C23
hypercenterC2×C3×C4×Q8D3
hypocenterC3C3×#2396
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×Q8C2semidirect
4. C3×#5230C2semidirect
7. C3×#4985C2semidirect
8. C3×#4982C2semidirect
10. C3×C4×C4⋊Dic3C2semidirect
11. C3×C4×Dic3⋊C4C2semidirect
13. #82491C3direct (N × H)
40. C2×C3×C4×Q8D3semidirect
41. C2×C3×C4×Q8C2×C3semidirect
54. #5230C2×C3semidirect
57. #4985C2×C3semidirect
58. #4982C2×C3semidirect
60. C4×C4⋊Dic3C2×C3semidirect
61. C4×Dic3⋊C4C2×C3semidirect
77. C2×C3×C4×D3Q8semidirect
184. C2×C4×Q8C3×D3semidirect
233. C2×C4×D3C3×Q8semidirect
483. C32#2396semidirect
556. C3#82491direct (N × H)
557. C3C3×#2396semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Q82
C2×C3×#8722
C3×#52312
C3×#52302
C3×#49942
C3×#49932
C3×#49852
C3×#49822
C3×C4×(C2×D3)⋊C42
C3×C4×C4⋊Dic32
C3×C4×Dic3⋊C42
C3×#23963
#824913