Finite Group Databasebeta

← C3×#75981 #11805550

Subgroups — C3×#75981 #11805550

Order 1152.

1221 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#887C211yesmaximal, characteristic
2C2×C3×#887C211yesmaximal, characteristic
3C2×C3×C4×Dic6C211yesmaximal, characteristic
4C32×#389C211yesmaximal, characteristic
5C3×#4984C211yesmaximal, characteristic
6C3×#4976C212yesmaximal
7C3×#4975C212yesmaximal
8C3×#4975C211yesmaximal, characteristic
9C3×C4×Dic3⋊C4C211yesmaximal, characteristic
10C3×#4649C212yesmaximal
11C3×#4647C212yesmaximal
12C3×Q8×C4⋊C4—31nomaximal
13#75981C311yesmaximal, characteristic, direct product factor
14C22×C3×Dic6C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C4⋊C4C2212yes—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C42C2211yescharacteristic
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×C4⋊Dic3C2212yes—
21C2×C3×Dic3⋊C4C2212yes—
22C2×C3×Dic3⋊C4C2212yes—
23C2×C3×Dic3⋊C4C2211yescharacteristic
24C2×C3×Dic3⋊C4C2212yes—
25C2×C3×Dic3⋊C4C2212yes—
26C2×C3×Dic3⋊C4C2212yes—
27C2×C3×Dic3⋊C4C2211yescharacteristic
28C2×C3×C4×Dic3C2212yes—
29C2×C3×C4×Dic3C2212yes—
30C2×C3×C4×Dic3C2211yescharacteristic
31C2×C3×C4×Dic3C2212yes—
32C2×C3×C4×Dic3C2211yescharacteristic
33C3×#887—24no—
34C3×#887C418yescomplemented
35C3×C4×Dic6—24no—
36C32×#95C2212yes—
37C3×C3⋊#95C2211yescharacteristic
38C3×C3⋊#95C2212yes—
39C3×C3⋊#95C2211yescharacteristic
40C2×C3×C4×Q8—31no—
41C2×C3×C4×Q8—31no—
42C2×C3×C4×Q8—31no—
43C2×#887C2×C311yescharacteristic
44C2×#887C2×C311yescharacteristic
45C2×C4×Dic6C2×C311yescharacteristic
46C3×#389—21no—
47C3×#389—32no—
48C3×#389—31no—
49C3×#389—32no—
50C3×#389D311yescharacteristic
51C3×#389C2×C311yescharacteristic
52C3×#384—32no—
53C3×#384—31no—
54C3×C4×C4⋊C4—31no—
55C3×C4×C4⋊C4—32no—
56#4984C2×C311yescharacteristic
57#4976C2×C312yes—
58#4975C2×C312yes—
59#4975C2×C311yescharacteristic
60C4×Dic3⋊C4C2×C311yescharacteristic
61#4649C2×C312yes—
62#4647C2×C312yes—
63C22×C32×C4C2312yes—
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2312yes—
66C22×C32×C4C2311yescharacteristic
67C22×C32×C4C2311yescharacteristic
68C22×C3×Dic3C2311yescharacteristic
69C22×C3×Dic3C2312yes—
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2311yescharacteristic
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2311yescharacteristic
74C2×C3×Dic6C2×C414yes—
75C2×C3×Dic6D414yes—
76C2×C3×Dic6Q814yes—
77C32×C4⋊C4—22no—
78C32×C4⋊C4—42no—
79C32×C4⋊C4C2×C414yes—
80C32×C42—22no—
81C3×C4⋊Dic3—22no—
82C3×C4⋊Dic3—42no—
83C3×Dic3⋊C4—24no—
84C3×Dic3⋊C4—24no—
85C3×Dic3⋊C4—24no—
86C3×Dic3⋊C4—22no—
87C3×Dic3⋊C4—42no—
88C3×Dic3⋊C4C2×C418yes—
89C3×Dic3⋊C4Q814yescomplemented
90C3×C4×Dic3—22no—
91C3×C4×Dic3—24no—
92C3×C4×Dic3—24no—
93C3×C4×Dic3C2×C414yes—
94C3×C4×Dic3C2×C418yes—
95Q8×C4⋊C4—31noSylow
96C22×C3×Q8—31no—
97C22×Dic6C22×C311yescharacteristic
98C3×C4×Q8—38no—
99C3×C4×Q8—64no—
100C3×C4×Q8—64no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4—31no—
105C2×C3×C4⋊C4—32no—
106C2×C3×C4⋊C4—31no—
107C2×C3×C4⋊C4—32no—
108C2×C3×C4⋊C4—32no—
109C2×C3×C4⋊C4—32no—
110C2×C3×C4⋊C4—31no—
111C2×C3×C4⋊C4—32no—
112C2×C3×C4⋊C4—32no—
113C2×C3×C4⋊C4C2×D311yescharacteristic
114C2×C3×C4⋊C4C2×D312yes—
115C2×C3×C4⋊C4C2×D311yescharacteristic
116C2×C3×C4⋊C4C22×C311yescharacteristic
117C2×C3×C4⋊C4C22×C312yes—
118C2×C3×C4⋊C4C22×C311yescharacteristic
119C2×C3×C42—21no—
120C2×C3×C42—32no—
121C2×C3×C42—31no—
122C2×C3×C42—32no—
123C2×C3×C42—32no—
124C2×C3×C42—31no—
125C2×C3×C42C2×D311yescharacteristic
126C2×C3×C42C22×C311yescharacteristic
127C2×C4⋊Dic3C22×C311yescharacteristic
128C2×C4⋊Dic3C22×C312yes—
129C2×Dic3⋊C4C22×C312yes—
130C2×Dic3⋊C4C22×C312yes—
131C2×Dic3⋊C4C22×C311yescharacteristic
132C2×Dic3⋊C4C22×C312yes—
133C2×Dic3⋊C4C22×C312yes—
134C2×Dic3⋊C4C22×C312yes—
135C2×Dic3⋊C4C22×C311yescharacteristic
136C2×C4×Dic3C22×C312yes—
137C2×C4×Dic3C22×C312yes—
138C2×C4×Dic3C22×C311yescharacteristic
139C2×C4×Dic3C22×C312yes—
140C2×C4×Dic3C22×C311yescharacteristic
141#887—24no—
142#887C3×C418yescomplemented
143C4×Dic6—24no—
144C3×#95—22no—
145C3×#95—31no—
146C3×#95—32no—
147C3×#95—31no—
148C3×#95C2×D312yes—
149C3×#95C22×C312yes—
150C3⋊#95C22×C311yescharacteristic
151C3⋊#95C22×C312yes—
152C3⋊#95C22×C311yescharacteristic
153C23×C32C2411yescharacteristic
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—21no—
159C2×C32×C4—22no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—22no—
163C2×C32×C4—22no—
164C2×C32×C4C22×C414yes—
165C2×C32×C4C22×C412yes—
166C2×C32×C4C2×D412yes—
167C2×C32×C4C2×Q812yes—
168C2×C32×C4C2×Q812yes—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3C22×C412yes—
180C2×C3×Dic3C22×C414yes—
181C2×C3×Dic3C22×C412yes—
182C2×C3×Dic3C2×D414yes—
183C2×C3×Dic3C2×Q812yes—
184C2×C3×Dic3C2×Q814yes—
185C2×C3×Dic3C2×Q812yes—
186C2×C3×Dic3#4112yes—
187C3×Dic6C4⋊C4116yescomplemented
188C2×C4×Q8—31no—
189C2×C4×Q8—31no—
190C2×C4×Q8—31no—
191#389—32no—
192#389—31no—
193#389—32no—
194#389C3×D311yescharacteristic
195#384—32no—
196#384—31no—
197C4×C4⋊C4—31no—
198C4×C4⋊C4—32no—
199C2×C3×Q8—34no—
200C2×C3×Q8—34no—
201C2×C3×Q8—34no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4—31no—
208C22×C3×C4—32no—
209C22×C3×C4—31no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—32no—
213C22×C3×C4C22×D311yescharacteristic
214C22×C3×C4C22×D312yes—
215C22×C3×C4C22×D312yes—
216C22×C3×C4C22×D311yescharacteristic
217C22×C3×C4C22×D311yescharacteristic
218C22×C3×C4C23×C311yescharacteristic
219C22×C3×C4C23×C312yes—
220C22×C3×C4C23×C312yes—
221C22×C3×C4C23×C311yescharacteristic
222C22×C3×C4C23×C311yescharacteristic
223C22×Dic3C23×C311yescharacteristic
224C22×Dic3C23×C312yes—
225C22×Dic3C23×C311yescharacteristic
226C22×Dic3C23×C311yescharacteristic
227C22×Dic3C23×C312yes—
228C22×Dic3C23×C311yescharacteristic
229C2×Dic6C2×C3×C414yes—
230C2×Dic6C3×D414yes—
231C2×Dic6C3×Q814yes—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—38no—
236C3×C4⋊C4—34no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—44no—
241C3×C4⋊C4—62no—
242C3×C4⋊C4—64no—
243C3×C4⋊C4—64no—
244C3×C4⋊C4—64no—
245C3×C4⋊C4—62no—
246C3×C4⋊C4—122no—
247C3×C4⋊C4—122no—
248C3×C4⋊C4C4×D314yes—
249C3×C4⋊C4C2×C3×C414yes—
250C3×C42—22no—
251C3×C42—22no—
252C3×C42—38no—
253C3×C42—34no—
254C3×C42—42no—
255C3×C42—64no—
256C3×C42—64no—
257C3×C42—62no—
258C4⋊Dic3—22no—
259C4⋊Dic3—42no—
260Dic3⋊C4—24no—
261Dic3⋊C4—24no—
262Dic3⋊C4—24no—
263Dic3⋊C4—22no—
264Dic3⋊C4—42no—
265Dic3⋊C4C2×C3×C418yes—
266Dic3⋊C4C3×Q814yescomplemented
267C4×Dic3—22no—
268C4×Dic3—24no—
269C4×Dic3—24no—
270C4×Dic3C2×C3×C414yes—
271C4×Dic3C2×C3×C418yes—
272C22×C32C23×C411yescharacteristic
273C22×C32C22×D411yescharacteristic
274C22×C32C22×Q811yescharacteristic
275C22×C32C22×Q811yescharacteristic
276C22×C32C2×#4111yescharacteristic
277C22×C32H(5,𝔽2)11yescharacteristic
278C22×C322−1+411yescharacteristic
279C32×C4—24no—
280C32×C4—22no—
281C32×C4—41no—
282C32×C4—42no—
283C32×C4C2×C4⋊C414yes—
284C3×Dic3—22no—
285C3×Dic3—22no—
286C3×Dic3—41no—
287C3×Dic3—42no—
288C3×Dic3C2×C4⋊C418yes—
289C3×Dic3C4×Q814yescomplemented
290C22×Q8—31no—
291C4×Q8—38no—
292C4×Q8—64no—
293C4×Q8—64no—
294C2×C4⋊C4—32no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—32no—
297C2×C4⋊C4—32no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4C2×C3×D311yescharacteristic
304C2×C4⋊C4C2×C3×D312yes—
305C2×C4⋊C4C2×C3×D311yescharacteristic
306C2×C42—32no—
307C2×C42—31no—
308C2×C42—32no—
309C2×C42—32no—
310C2×C42—31no—
311C2×C42C2×C3×D311yescharacteristic
312#95—31no—
313#95—32no—
314#95—31no—
315#95C2×C3×D312yes—
316C23×C3—21no—
317C23×C3C23×D311yescharacteristic
318C23×C3C24×C311yescharacteristic
319C3×Q8—316no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—32no—
347C2×C3×C4—34no—
348C2×C3×C4—34no—
349C2×C3×C4—32no—
350C2×C3×C4—34no—
351C2×C3×C4—32no—
352C2×C3×C4—32no—
353C2×C3×C4—32no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—62no—
369C2×C3×C4—62no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—62no—
373C2×C3×C4C2×C4×D314yes—
374C2×C3×C4C2×C4×D312yes—
375C2×C3×C4D3×D412yes—
376C2×C3×C4D3×Q812yes—
377C2×C3×C4D3×Q812yes—
378C2×C3×C4C22×C3×C414yes—
379C2×C3×C4C22×C3×C412yes—
380C2×C3×C4C2×C3×D412yes—
381C2×C3×C4C2×C3×Q812yes—
382C2×C3×C4C2×C3×Q812yes—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—21no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—21no—
393C2×Dic3C22×C3×C412yes—
394C2×Dic3C22×C3×C414yes—
395C2×Dic3C22×C3×C412yes—
396C2×Dic3C2×C3×D414yes—
397C2×Dic3C2×C3×Q812yes—
398C2×Dic3C2×C3×Q814yes—
399C2×Dic3C2×C3×Q812yes—
400C2×Dic3C3×#4112yes—
401Dic6C3×C4⋊C4116yescomplemented
402C2×C32C22×C4⋊C411yescharacteristic
403C2×C32C2×C4×Q811yescharacteristic
404C2×C32#52011yescharacteristic
405C2×C32D4×Q811yescharacteristic
406C2×C32#55011yescharacteristic
407C2×C32#55711yescharacteristic
408C2×C32Q8211yescharacteristic
409C2×Q8—34no—
410C2×Q8—34no—
411C2×Q8—34no—
412C22×C4—31no—
413C22×C4—32no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—32no—
418C22×C4C22×C3×D311yescharacteristic
419C22×C4C22×C3×D312yes—
420C22×C4C22×C3×D312yes—
421C22×C4C22×C3×D311yescharacteristic
422C22×C4C22×C3×D311yescharacteristic
423C4⋊C4—22no—
424C4⋊C4—38no—
425C4⋊C4—34no—
426C4⋊C4—42no—
427C4⋊C4—64no—
428C4⋊C4—64no—
429C4⋊C4—62no—
430C4⋊C4—64no—
431C4⋊C4—62no—
432C4⋊C4—122no—
433C4⋊C4—122no—
434C4⋊C4C3×C4×D314yes—
435C42—22no—
436C42—38no—
437C42—34no—
438C42—64no—
439C42—64no—
440C42—62no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3C22×C4×D311yescharacteristic
449C22×C3C2×D3×D411yescharacteristic
450C22×C3#100411yescharacteristic
451C22×C3C2×D3×Q811yescharacteristic
452C22×C3C2×D3×Q811yescharacteristic
453C22×C3D3×#4111yescharacteristic
454C22×C3#101011yescharacteristic
455C22×C3C23×C3×C411yescharacteristic
456C22×C3C22×C3×D411yescharacteristic
457C22×C3C22×C3×Q811yescharacteristic
458C22×C3C22×C3×Q811yescharacteristic
459C22×C3C2×C3×#4111yescharacteristic
460C22×C3C3×H(5,𝔽2)11yescharacteristic
461C22×C3C3×2−1+411yescharacteristic
462C3×C4—22no—
463C3×C4—24no—
464C3×C4—24no—
465C3×C4—22no—
466C3×C4—24no—
467C3×C4—38no—
468C3×C4—34no—
469C3×C4—44no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—41no—
473C3×C4—42no—
474C3×C4—41no—
475C3×C4—42no—
476C3×C4—62no—
477C3×C4—62no—
478C3×C4—82no—
479C3×C4—121no—
480C3×C4—122no—
481C3×C4D3×C4⋊C414yes—
482C3×C4C2×C3×C4⋊C414yes—
483Dic3—22no—
484Dic3—22no—
485Dic3—41no—
486Dic3—42no—
487Dic3C2×C3×C4⋊C418yes—
488Dic3C3×C4×Q814yescomplemented
489C32Q8×C4⋊C411yescharacteristic, complemented, Sylow
490C23C23×C3×D311yescharacteristic
491Q8—316no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—32no—
503C2×C4—34no—
504C2×C4—34no—
505C2×C4—32no—
506C2×C4—34no—
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—32no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—62no—
515C2×C4—61no—
516C2×C4—62no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—62no—
520C2×C4C2×C3×C4×D314yes—
521C2×C4C2×C3×C4×D312yes—
522C2×C4C3×D3×D412yes—
523C2×C4C3×D3×Q812yes—
524C2×C4C3×D3×Q812yes—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3C2×D3×C4⋊C411yescharacteristic
533C2×C3#554611yescharacteristic
534C2×C3C4×D3×Q811yescharacteristic
535C2×C3#559911yescharacteristic
536C2×C3#563211yescharacteristic
537C2×C3#568511yescharacteristic
538C2×C3#572211yescharacteristic
539C2×C3C22×C3×C4⋊C411yescharacteristic
540C2×C3C2×C3×C4×Q811yescharacteristic
541C2×C3C3×#52011yescharacteristic
542C2×C3C3×D4×Q811yescharacteristic
543C2×C3C3×#55011yescharacteristic
544C2×C3C3×#55711yescharacteristic
545C2×C3C3×Q8211yescharacteristic
546C22C22×C3×C4×D311yescharacteristic
547C22C2×C3×D3×D411yescharacteristic
548C22C3×#100411yescharacteristic
549C22C2×C3×D3×Q811yescharacteristic
550C22C2×C3×D3×Q811yescharacteristic
551C22C3×D3×#4111yescharacteristic
552C22C3×#101011yescharacteristic
553C4—24no—
554C4—22no—
555C4—38no—
556C4—34no—
557C4—41no—
558C4—42no—
559C4—62no—
560C4—62no—
561C4—122no—
562C4—121no—
563C4C3×D3×C4⋊C414yes—
564C3—21no—
565C3#7598111yescharacteristic, direct product factor
566C3C3×Q8×C4⋊C411yescharacteristic, complemented
567C2C2×C3×D3×C4⋊C411yescharacteristic
568C2C3×#554611yescharacteristic
569C2C3×C4×D3×Q811yescharacteristic
570C2C3×#559911yescharacteristic
571C2C3×#563211yescharacteristic
572C2C3×#568511yescharacteristic
573C2C3×#572211yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#389C2
socle, prime radicalC23×C32C24
cosocleC23×C3C24×C3
hypercenterC3×#389D3
hypocenterC3C3×Q8×C4⋊C4
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#389

Complements

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

NH ≅ G/Ntype
13. #75981C3direct (N × H)
34. C3×#887C4semidirect
89. C3×Dic3⋊C4Q8semidirect
142. #887C3×C4semidirect
187. C3×Dic6C4⋊C4semidirect
266. Dic3⋊C4C3×Q8semidirect
289. C3×Dic3C4×Q8semidirect
401. Dic6C3×C4⋊C4semidirect
488. Dic3C3×C4×Q8semidirect
489. C32Q8×C4⋊C4semidirect
565. C3#75981direct (N × H)
566. C3C3×Q8×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8872
C2×C3×C4×Dic62
C32×#3892
C3×#49842
C3×#49762
C3×#49752
C3×C4×Dic3⋊C42
C3×#46492
C3×#46472
C3×Q8×C4⋊C43
#759813