Finite Group Databasebeta

← #11803488

Subgroups — #11803488

Order 1152.

1229 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#926C211yesmaximal, characteristic
2C2×C3×#924C211yesmaximal, characteristic
3C2×Dic3×Dic6C211yesmaximal, characteristic, complemented
4#10592911C211yesmaximal, characteristic
5#10591702C212yesmaximal
6#10591701C212yesmaximal
7#10591352C211yesmaximal, characteristic, complemented
8Dic3×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#10591265C212yesmaximal, complemented
10#10591262C212yesmaximal, complemented
11#10591220C211yesmaximal, characteristic, complemented
12#82744—31nomaximal
13#82235—31nomaximal
14C23×C32×C4C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C2×#3757C2212yes—
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×#217C2211yescharacteristic
20C2×C3×#217C2211yescharacteristic
21C2×C3×#217C2212yes—
22C3×#926—24no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C3×#924C418yescomplemented
27C2×C3×Dic3⋊C4C2212yes—
28C2×C3×C4×Dic3C2211yescharacteristic
29C2×C3×C4×Dic3C2211yescharacteristic
30C2×Dic3⋊Dic3C2212yes—
31C2×Dic3⋊Dic3C2212yes—
32C2×Dic32C2212yes—
33C2×Dic32C2212yes—
34Dic3×Dic6—24no—
35#63658C2212yes—
36#63579C2212yes—
37#63579C2212yes—
38#63579C2212yes—
39#63579C2212yes—
40C2×C3×C22⋊Q8—31no—
41C2×C3×#117—31no—
42C2×Q8×Dic3—31no—
43C2×#926D311yescharacteristic
44C2×#924D311yescharacteristic
45C2×C4×Dic6—31no—
46#5229—31no—
47#5212—31no—
48#5212—31no—
49#5212—61no—
50#4984—32no—
51#4984—31no—
52#4980—32no—
53#4978—32no—
54#4976—32no—
55Dic3×C4⋊C4—31no—
56#4951—32no—
57#4951—32no—
58#4945—32no—
59#4944—32no—
60C4×C4⋊Dic3—31no—
61#4929—31no—
62#4928—31no—
63C24×C32C2311yescharacteristic
64C22×C32×C4—22no—
65C22×C32×C4—22no—
66C22×C32×C4C2×C414yes—
67C22×C32×C4C2311yescharacteristic
68C22×C32×C4C2311yescharacteristic
69C22×C3⋊Dic3C2312yes—
70C22×C3⋊Dic3C2312yes—
71C22×C3×Dic3C2311yescharacteristic
72C22×C3×Dic3C2311yescharacteristic
73C22×C3×Dic3C2312yes—
74C22×C3×Dic3C2312yes—
75C22×C3×Dic3C2311yescharacteristic
76C22×C3×Dic3C2311yescharacteristic
77C2×C3×Dic6—22no—
78C2×C3×Dic6—22no—
79C2×C3×Dic6C2×C414yes—
80#3757—42no—
81C4⋊(C3⋊Dic3)—22no—
82C4⋊(C3⋊Dic3)—22no—
83C3×#217—22no—
84C3×#217—22no—
85C3×#217C2×C418yes—
86C3×C4⋊Dic3—22no—
87C3×C4⋊Dic3—22no—
88C3×C4⋊Dic3C2×C414yes—
89C3×Dic3⋊C4C2×C418yes—
90C3×C4×Dic3—22no—
91C3×C4×Dic3D414yes—
92Dic3⋊Dic3—24no—
93Dic3⋊Dic3—42no—
94Dic32—24no—
95Dic32—42no—
96#2323—91noSylow
97C22×C3×Q8—31no—
98C23×C3×C4—21no—
99C23×C3×C4C2×D311yescharacteristic
100C23×C3×C4C2×D311yescharacteristic
101C22×Dic6C2×D311yescharacteristic
102C3×C22⋊Q8—38no—
103C3×#117—64no—
104C2×C3×C4⋊C4—32no—
105C2×C3×C4⋊C4—31no—
106C2×C3×C4⋊C4—31no—
107C2×C3×C4⋊C4—31no—
108C2×C3×C22⋊C4—32no—
109C2×C3×C22⋊C4—31no—
110C2×C3×C22⋊C4—31no—
111C2×C3×C42—31no—
112C2×C3×C42—31no—
113C2×#217—32no—
114C2×#217—32no—
115C2×#217—62no—
116C2×#217C2×D311yescharacteristic
117C2×#217C2×D311yescharacteristic
118C2×#217C2×D312yes—
119Q8×Dic3—64no—
120#926—24no—
121C2×C4⋊Dic3—31no—
122C2×C4⋊Dic3—31no—
123C2×C4⋊Dic3—31no—
124C2×C4⋊Dic3—31no—
125C2×C4⋊Dic3—32no—
126C2×C4⋊Dic3—32no—
127C2×C4⋊Dic3—61no—
128C2×C4⋊Dic3—61no—
129C2×C4⋊Dic3C2×D311yescharacteristic
130C2×C4⋊Dic3C2×D311yescharacteristic
131C2×C4⋊Dic3C2×D311yescharacteristic
132#924Dic318yescomplemented
133C2×Dic3⋊C4—32no—
134C2×Dic3⋊C4—32no—
135C2×Dic3⋊C4C2×D312yes—
136C2×C4×Dic3—32no—
137C2×C4×Dic3—32no—
138C2×C4×Dic3—32no—
139C2×C4×Dic3—32no—
140C2×C4×Dic3C2×D311yescharacteristic
141C2×C4×Dic3C2×D311yescharacteristic
142C4×Dic6—64no—
143C3⋊#95—32no—
144C3⋊#95—32no—
145C3⋊#95—32no—
146C3⋊#95—32no—
147C3⋊#95—32no—
148C3⋊#95—32no—
149C3⋊#95—32no—
150C3⋊#95—32no—
151C3⋊#95—32no—
152C3⋊#95—32no—
153C3⋊#95—62no—
154C23×C32—21no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—21no—
159C23×C32—21no—
160C23×C32C22×C412yes—
161C23×C32C2411yescharacteristic
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—28no—
165C2×C32×C4C22×C412yes—
166C2×C32×C4C22×C412yes—
167C2×C32×C4C2×D412yes—
168C2×C32×C4#4112yes—
169C2×C3⋊Dic3—22no—
170C2×C3⋊Dic3—22no—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3—22no—
173C2×C3⋊Dic3—22no—
174C2×C3⋊Dic3—22no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3C22×C414yes—
190C2×C3×Dic3C22×C414yes—
191C2×C3×Dic3C2×D412yes—
192C2×C3×Dic3C2×D412yes—
193C3×Dic6—28no—
194C2×C22⋊Q8—31no—
195C2×C4×Q8—91no—
196C2×#117—31no—
197#391—91no—
198#389—91no—
199#389—92no—
200#388—92no—
201#388—92no—
202#387—92no—
203#380—91no—
204C4×C4⋊C4—91no—
205C24×C3—21no—
206C24×C3C22×D311yescharacteristic
207C24×C3C22×D311yescharacteristic
208C2×C3×Q8—34no—
209C2×C3×Q8—62no—
210C2×C3×Q8—62no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—22no—
214C22×C3×C4—22no—
215C22×C3×C4—24no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—31no—
219C22×C3×C4—32no—
220C22×C3×C4—32no—
221C22×C3×C4—31no—
222C22×C3×C4—31no—
223C22×C3×C4—31no—
224C22×C3×C4—42no—
225C22×C3×C4—42no—
226C22×C3×C4C4×D314yes—
227C22×C3×C4C2×Dic314yes—
228C22×C3×C4C22×D311yescharacteristic
229C22×C3×C4C22×D311yescharacteristic
230C22×C3×C4C22×D311yescharacteristic
231C22×C3×C4C22×D311yescharacteristic
232C22×Dic3—32no—
233C22×Dic3—32no—
234C22×Dic3—32no—
235C22×Dic3—32no—
236C22×Dic3—62no—
237C22×Dic3—62no—
238C22×Dic3C22×D311yescharacteristic
239C22×Dic3C22×D311yescharacteristic
240C22×Dic3C22×D312yes—
241C22×Dic3C22×D312yes—
242C22×Dic3C22×D311yescharacteristic
243C22×Dic3C22×D311yescharacteristic
244C2×Dic6—22no—
245C2×Dic6—22no—
246C2×Dic6C2×Dic314yes—
247C3×C4⋊C4—38no—
248C3×C4⋊C4—34no—
249C3×C4⋊C4—62no—
250C3×C4⋊C4—62no—
251C3×C22⋊C4—38no—
252C3×C22⋊C4—62no—
253C3×C22⋊C4—62no—
254C3×C42—34no—
255C3×C42—62no—
256#217—22no—
257#217—22no—
258#217—122no—
259#217—122no—
260#217—124no—
261#217C2×Dic318yes—
262C4⋊Dic3—22no—
263C4⋊Dic3—22no—
264C4⋊Dic3—62no—
265C4⋊Dic3—62no—
266C4⋊Dic3—62no—
267C4⋊Dic3—62no—
268C4⋊Dic3—64no—
269C4⋊Dic3—122no—
270C4⋊Dic3—122no—
271C4⋊Dic3—122no—
272C4⋊Dic3C2×Dic314yes—
273Dic3⋊C4—64no—
274Dic3⋊C4—122no—
275Dic3⋊C4C2×Dic318yes—
276C4×Dic3—22no—
277C4×Dic3—64no—
278C4×Dic3—64no—
279C4×Dic3—122no—
280C4×Dic3—122no—
281C4×Dic3#6714yes—
282C22×C32—22no—
283C22×C32—22no—
284C22×C32—22no—
285C22×C32—41no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—41no—
289C22×C32C23×C411yescharacteristic
290C22×C32C22×D411yescharacteristic
291C22×C32C2×#4111yescharacteristic
292C22×C32H(5,𝔽2)11yescharacteristic
293C22×C322−1+411yescharacteristic
294C22×C322−1+411yescharacteristic
295C22×C322−1+411yescharacteristic
296C32×C4—22no—
297C32×C4C4×D414yes—
298C3⋊Dic3—42no—
299C3⋊Dic3—42no—
300C3×Dic3—24no—
301C3×Dic3—24no—
302C3×Dic3—22no—
303C3×Dic3—22no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C22×Q8—31no—
307C23×C4D3211yescharacteristic
308C22⋊Q8—38no—
309C4×Q8—184no—
310#117—64no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—92no—
316C2×C4⋊C4—91no—
317C2×C4⋊C4—91no—
318C2×C4⋊C4—92no—
319C2×C22⋊C4—32no—
320C2×C22⋊C4—31no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—92no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—92no—
326C2×C42—92no—
327#95—92no—
328#95—92no—
329#95—92no—
330#95—92no—
331#95—92no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—22no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3C2×C4×D312yes—
353C23×C3C22×Dic312yes—
354C23×C3C23×D311yescharacteristic
355C23×C3C23×D311yescharacteristic
356C3×Q8—68no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—28no—
362C2×C3×C4—28no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—34no—
368C2×C3×C4—34no—
369C2×C3×C4—32no—
370C2×C3×C4—32no—
371C2×C3×C4—48no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—62no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4—61no—
381C2×C3×C4—62no—
382C2×C3×C4—61no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4C2×C4×D312yes—
389C2×C3×C4C2×C4×D312yes—
390C2×C3×C4#23512yes—
391C2×C3×C4D3×D412yes—
392C2×C3×C4#23712yes—
393C2×C3×C4C22×Dic312yes—
394C2×C3×C4C22×Dic312yes—
395C2×C3×C4C2×#6712yes—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—22no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—62no—
411C2×Dic3—62no—
412C2×Dic3—62no—
413C2×Dic3—62no—
414C2×Dic3—62no—
415C2×Dic3—62no—
416C2×Dic3—62no—
417C2×Dic3—62no—
418C2×Dic3—62no—
419C2×Dic3—62no—
420C2×Dic3—62no—
421C2×Dic3—62no—
422C2×Dic3—122no—
423C2×Dic3—122no—
424C2×Dic3—122no—
425C2×Dic3—122no—
426C2×Dic3—122no—
427C2×Dic3—122no—
428C2×Dic3C22×Dic314yes—
429C2×Dic3C22×Dic314yes—
430C2×Dic3C2×#6712yes—
431C2×Dic3C2×#6712yes—
432Dic6—28no—
433C2×C32—42no—
434C2×C32C2×C4×D411yescharacteristic
435C2×C32#51911yescharacteristic
436C2×C32#52011yescharacteristic
437C2×C32#53611yescharacteristic
438C2×C32#53711yescharacteristic
439C2×C32#54111yescharacteristic
440C2×C32#54211yescharacteristic
441C24C2×D3211yescharacteristic
442C2×Q8—34no—
443C2×Q8—62no—
444C2×Q8—62no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—31no—
448C22×C4—32no—
449C22×C4—32no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—92no—
454C22×C4—92no—
455C22×C4D3×Dic314yes—
456C22×C4C2×D3211yescharacteristic
457C22×C4C2×D3211yescharacteristic
458C4⋊C4—38no—
459C4⋊C4—34no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C4⋊C4—184no—
463C4⋊C4—182no—
464C4⋊C4—182no—
465C4⋊C4—362no—
466C22⋊C4—38no—
467C22⋊C4—62no—
468C22⋊C4—62no—
469C22⋊C4—362no—
470C42—34no—
471C42—62no—
472C42—184no—
473C42—362no—
474C22×C3—24no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3C22×C4×D311yescharacteristic
503C22×C3C2×#23511yescharacteristic
504C22×C3C2×D3×D411yescharacteristic
505C22×C3C2×#23711yescharacteristic
506C22×C3#100711yescharacteristic
507C22×C3#100711yescharacteristic
508C22×C3#100911yescharacteristic
509C22×C3#100911yescharacteristic
510C22×C3#101011yescharacteristic
511C22×C3#101011yescharacteristic
512C22×C3#101011yescharacteristic
513C22×C3#101011yescharacteristic
514C22×C3C23×Dic311yescharacteristic
515C22×C3C22×#6711yescharacteristic
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—24no—
519C3×C4—42no—
520C3×C4—64no—
521C3×C4—64no—
522C3×C4—62no—
523C3×C4—62no—
524C3×C4—121no—
525C3×C4—121no—
526C3×C4C4×#6714yes—
527C3×C4D4×Dic314yes—
528Dic3—24no—
529Dic3—22no—
530Dic3—24no—
531Dic3—22no—
532Dic3—41no—
533Dic3—41no—
534Dic3—122no—
535Dic3—122no—
536Dic3—122no—
537Dic3—122no—
538Dic3—242no—
539Dic3—242no—
540C32#232311yescharacteristic, complemented, Sylow
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23C2×D3×Dic312yes—
548C23C22×D3211yescharacteristic
549Q8—68no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—34no—
554C2×C4—34no—
555C2×C4—32no—
556C2×C4—32no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—62no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—182no—
572C2×C4—182no—
573C2×C4—182no—
574C2×C4—182no—
575C2×C4—182no—
576C2×C4—182no—
577C2×C4C2×D3×Dic312yes—
578C2×C4C2×D3×Dic312yes—
579C2×C4#380412yes—
580C2×C4D3×#6712yes—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—44no—
589C2×C3—42no—
590C2×C3—42no—
591C2×C3#551611yescharacteristic
592C2×C3#551811yescharacteristic
593C2×C3#552311yescharacteristic
594C2×C3#553411yescharacteristic
595C2×C3#562511yescharacteristic
596C2×C3#562811yescharacteristic
597C2×C3#562911yescharacteristic
598C2×C3#564311yescharacteristic
599C2×C3C2×C4×#6711yescharacteristic
600C2×C3C2×D4×Dic311yescharacteristic
601C2×C3#581211yescharacteristic
602C2×C3#582511yescharacteristic
603C2×C3#582711yescharacteristic
604C2×C3#583311yescharacteristic
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22#6429711yescharacteristic
613C22#6429711yescharacteristic
614C22#6429811yescharacteristic
615C22#6430811yescharacteristic
616C22C22×D3×Dic311yescharacteristic
617C22C2×#380411yescharacteristic
618C22C2×D3×#6711yescharacteristic
619C4—22no—
620C4—62no—
621C4—62no—
622C4—64no—
623C4—64no—
624C4—121no—
625C4—121no—
626C4—362no—
627C4—362no—
628C4Dic3×#6714yes—
629C3—21no—
630C3#8223511yescharacteristic, complemented
631C3#8274411yescharacteristic, complemented
632C2—42no—
633C2#1059481411yescharacteristic
634C2#1059481811yescharacteristic
635C2#1059482411yescharacteristic
636C2#1059488411yescharacteristic
637C2#1059489411yescharacteristic
638C2#1059499211yescharacteristic
639C2C2×Dic3×#6711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC23×C32×C4C22
socle, cosocleC23×C32C24
prime radicalC24×C32C23
hypercenterC23×C4D32
hypocenterC32#2323
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
3. C2×Dic3×Dic6C2semidirect
7. #10591352C2semidirect
8. Dic3×C4⋊Dic3C2semidirect
9. #10591265C2semidirect
10. #10591262C2semidirect
11. #10591220C2semidirect
26. C3×#924C4semidirect
132. #924Dic3semidirect
540. C32#2323semidirect
630. C3#82235semidirect
631. C3#82744semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9262
C2×C3×#9242
C2×Dic3×Dic62
#105929112
#105917022
#105917012
#105913522
Dic3×C4⋊Dic32
#105912652
#105912622
#105912202
#827443
#822353