Finite Group Databasebeta

← #11803485

Subgroups — #11803485

Order 1152.

1261 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#634C211yesmaximal, characteristic, complemented
2C2×C3×#872C211yesmaximal, characteristic
3C2×#63857C211yesmaximal, characteristic
4C3×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
5#10591352C211yesmaximal, characteristic, complemented
6#10591334C212yesmaximal, complemented
7#10591330C212yesmaximal, complemented
8#10591313C211yesmaximal, characteristic
9#10591287C212yesmaximal
10#10591280C212yesmaximal
11#10591213C211yesmaximal, characteristic, complemented
12#82235—31nomaximal
13#73762—31nomaximal
14C22×#634C2211yescharacteristic
15C22×C3×C4×D3C2211yescharacteristic
16C2×C32×C42C2211yescharacteristic
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×#3750C2212yes—
19C2×C4×C3⋊Dic3C2212yes—
20C4×#634—24no—
21C2×C3×(C2×D3)⋊C4C2211yescharacteristic
22C2×C3×(C2×D3)⋊C4C2211yescharacteristic
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×Dic3⋊C4C2211yescharacteristic
26C2×C3×Dic3⋊C4C2211yescharacteristic
27C2×C3×C4×Dic3C2211yescharacteristic
28C2×C3×C4×Dic3C2212yes—
29C3×#872—24no—
30C2×Dic3⋊Dic3C2212yes—
31C2×Dic3⋊Dic3C2212yes—
32C2×(C2×D3)⋊Dic3C2212yes—
33C2×(C2×D3)⋊Dic3C2212yes—
34#63857C418yescomplemented
35C3×C3⋊#95C2212yes—
36#63579C2212yes—
37#63579C2212yes—
38#63579C2212yes—
39#63579C2212yes—
40C2×C3×#117—31no—
41C2×#924—31no—
42C2×#897—31no—
43C2×#872D311yescharacteristic
44C2×C4×Dic6—31no—
45C2×C4×Dic6—31no—
46C2×C4×Dic6—61no—
47C3×C4×C4⋊C4—31no—
48#5212—31no—
49#4987—31no—
50#4984—31no—
51#4978—32no—
52#4976—32no—
53#4973—31no—
54#4945—32no—
55#4944—32no—
56#4940—32no—
57C4×C4⋊Dic3D311yescharacteristic, complemented
58#4929—31no—
59#4928—31no—
60#4928—32no—
61#4670—32no—
62#4653—32no—
63C23×C3×D3C2311yescharacteristic
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C32×C4C2311yescharacteristic
67C22×C3⋊Dic3C2312yes—
68C22×C3⋊Dic3C2312yes—
69C2×#634—22no—
70C2×#634—22no—
71C2×#634C2×C414yes—
72C22×C3×Dic3C2311yescharacteristic
73C22×C3×Dic3C2311yescharacteristic
74C22×C3×Dic3C2312yes—
75C22×C3×Dic3C2312yes—
76C22×C3×Dic3C2311yescharacteristic
77C2×C3×C4×D3—22no—
78C2×C3×C4×D3—22no—
79C2×C3×C4×D3C2×C414yes—
80C32×C42D414yes—
81C4⋊(C3⋊Dic3)—22no—
82#3750—24no—
83C4×C3⋊Dic3—24no—
84C3×(C2×D3)⋊C4—22no—
85C3×(C2×D3)⋊C4—22no—
86C3×C4⋊Dic3—22no—
87C3×C4⋊Dic3C2×C414yes—
88C3×Dic3⋊C4—22no—
89C3×Dic3⋊C4—22no—
90C3×C4×Dic3—22no—
91C3×C4×Dic3—42no—
92Dic3⋊Dic3—42no—
93Dic3⋊Dic3C2×C418yes—
94(C2×D3)⋊Dic3—42no—
95(C2×D3)⋊Dic3C2×C418yes—
96#2323—91noSylow
97C23×C3×C4—31no—
98C22×C4×D3C2×D311yescharacteristic
99C22×Dic6—31no—
100C22×Dic6—31no—
101C22×Dic6—61no—
102C3×#117—64no—
103C2×C3×C4⋊C4—31no—
104C2×C3×C4⋊C4—31no—
105C2×C3×C4⋊C4—31no—
106C2×C3×C4⋊C4—31no—
107C2×C3×C22⋊C4—31no—
108C2×C3×C22⋊C4—31no—
109C2×C3×C42—21no—
110C2×C3×C42—31no—
111C2×C3×C42—32no—
112C2×C3×C42C2×D311yescharacteristic
113C2×C3×C42C2×D311yescharacteristic
114C2×#217—32no—
115C2×#217—32no—
116C2×(C2×D3)⋊C4—32no—
117C2×(C2×D3)⋊C4—32no—
118C2×(C2×D3)⋊C4C2×D311yescharacteristic
119C2×(C2×D3)⋊C4C2×D311yescharacteristic
120C2×C4⋊Dic3—32no—
121C2×C4⋊Dic3—31no—
122C2×C4⋊Dic3—31no—
123C2×C4⋊Dic3—32no—
124C2×C4⋊Dic3—61no—
125C2×C4⋊Dic3C2×D311yescharacteristic
126C2×C4⋊Dic3C2×D311yescharacteristic
127#924—38no—
128C2×Dic3⋊C4—32no—
129C2×Dic3⋊C4—32no—
130C2×Dic3⋊C4—32no—
131C2×Dic3⋊C4—32no—
132C2×Dic3⋊C4—62no—
133C2×Dic3⋊C4C2×D311yescharacteristic
134C2×Dic3⋊C4C2×D311yescharacteristic
135C2×C4×Dic3—32no—
136C2×C4×Dic3—32no—
137C2×C4×Dic3—62no—
138C2×C4×Dic3C2×D311yescharacteristic
139C2×C4×Dic3C2×D312yes—
140#897—38no—
141#872—24no—
142C4×Dic6—68no—
143C4×Dic6—64no—
144C4×Dic6—64no—
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—32no—
154C3⋊#95C2×D312yes—
155C23×C32C2411yescharacteristic
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—21no—
161C22×C3×D3—21no—
162C22×C3×D3C22×C412yes—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4C22×C412yes—
168C2×C32×C4C2×D412yes—
169C2×C32×C4C2×D412yes—
170C2×C32×C4C2×D412yes—
171C2×C32×C4#4112yes—
172C2×C3⋊Dic3—22no—
173C2×C3⋊Dic3—22no—
174C2×C3⋊Dic3—22no—
175C2×C3⋊Dic3—22no—
176C2×C3⋊Dic3—22no—
177C2×C3⋊Dic3C22×C414yes—
178#634—28no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3C22×C414yes—
193C2×C3×Dic3C22×C412yes—
194C3×C4×D3—28no—
195C2×C22⋊Q8—91no—
196C2×C4×Q8—91no—
197C2×#117—31no—
198#391—91no—
199#389—92no—
200#389—91no—
201#388—92no—
202#388—92no—
203#387—92no—
204#380—91no—
205C4×C4⋊C4—31no—
206C24×C3—31no—
207C23×D3C22×D311yescharacteristic
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—31no—
212C22×C3×C4—32no—
213C22×C3×C4—34no—
214C22×C3×C4—31no—
215C22×C3×C4—32no—
216C22×C3×C4—31no—
217C22×C3×C4—62no—
218C22×C3×C4—62no—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C22×D311yescharacteristic
221C22×C3×C4C22×D311yescharacteristic
222C22×C3×C4C22×D311yescharacteristic
223C22×C3×C4C22×D311yescharacteristic
224C22×C3×C4C22×D311yescharacteristic
225C22×Dic3—32no—
226C22×Dic3—32no—
227C22×Dic3—32no—
228C22×Dic3—32no—
229C22×Dic3—62no—
230C22×Dic3—62no—
231C22×Dic3C22×D311yescharacteristic
232C22×Dic3C22×D312yes—
233C22×Dic3C22×D311yescharacteristic
234C22×Dic3C22×D311yescharacteristic
235C22×Dic3C22×D312yes—
236C2×C4×D3—22no—
237C2×C4×D3—22no—
238C2×C4×D3C4×D314yes—
239C2×Dic6—34no—
240C2×Dic6—34no—
241C2×Dic6—64no—
242C2×Dic6—64no—
243C2×Dic6—64no—
244C2×Dic6—62no—
245C2×Dic6—62no—
246C2×Dic6—62no—
247C2×Dic6—62no—
248C3×C4⋊C4—34no—
249C3×C4⋊C4—62no—
250C3×C4⋊C4—62no—
251C3×C4⋊C4—62no—
252C3×C22⋊C4—62no—
253C3×C22⋊C4—62no—
254C3×C42—24no—
255C3×C42—62no—
256C3×C42—122no—
257C3×C42D1214yes—
258C3×C42#6714yes—
259#217—38no—
260#217—122no—
261(C2×D3)⋊C4—22no—
262(C2×D3)⋊C4—22no—
263(C2×D3)⋊C4—38no—
264(C2×D3)⋊C4—122no—
265C4⋊Dic3—22no—
266C4⋊Dic3—38no—
267C4⋊Dic3—64no—
268C4⋊Dic3—62no—
269C4⋊Dic3—62no—
270C4⋊Dic3—122no—
271C4⋊Dic3C4×D314yes—
272Dic3⋊C4—22no—
273Dic3⋊C4—22no—
274Dic3⋊C4—38no—
275Dic3⋊C4—68no—
276Dic3⋊C4—64no—
277Dic3⋊C4—64no—
278Dic3⋊C4—122no—
279C4×Dic3—22no—
280C4×Dic3—42no—
281C4×Dic3—68no—
282C4×Dic3—64no—
283C4×Dic3—64no—
284C22×C32C23×C411yescharacteristic
285C22×C32C22×D411yescharacteristic
286C22×C32C2×#4111yescharacteristic
287C22×C32H(5,𝔽2)11yescharacteristic
288C22×C322−1+411yescharacteristic
289C22×C322−1+411yescharacteristic
290C22×C322−1+411yescharacteristic
291C2×C3×D3—22no—
292C2×C3×D3—22no—
293C2×C3×D3—22no—
294C2×C3×D3—41no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—41no—
298C32×C4—22no—
299C32×C4—22no—
300C32×C4C4×D414yes—
301C3⋊Dic3—24no—
302C3⋊Dic3—42no—
303C3×Dic3—22no—
304C3×Dic3—24no—
305C3×Dic3—41no—
306C3×Dic3—41no—
307C3×Dic3—42no—
308C22×Q8—91no—
309C23×C4—31no—
310C22⋊Q8—98no—
311C4×Q8—184no—
312#117—64no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—92no—
318C2×C4⋊C4—91no—
319C2×C4⋊C4—92no—
320C2×C4⋊C4—92no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—92no—
324C2×C22⋊C4—92no—
325C2×C42—31no—
326C2×C42—32no—
327C2×C42—92no—
328C2×C42D3211yescharacteristic
329#95—32no—
330#95—92no—
331#95—92no—
332#95—92no—
333#95—92no—
334C23×C3—21no—
335C23×C3—32no—
336C23×C3—61no—
337C23×C3—61no—
338C23×C3—61no—
339C23×C3—61no—
340C23×C3—61no—
341C23×C3—61no—
342C23×C3C23×D311yescharacteristic
343C23×C3C23×D311yescharacteristic
344C22×D3—21no—
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3—21no—
350C22×D3C2×C4×D312yes—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—32no—
367C2×C3×C4—34no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—62no—
373C2×C3×C4—62no—
374C2×C3×C4—68no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—62no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—62no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—62no—
384C2×C3×C4C2×C4×D312yes—
385C2×C3×C4C2×C4×D312yes—
386C2×C3×C4C2×D1212yes—
387C2×C3×C4C2×D1212yes—
388C2×C3×C4C2×D1212yes—
389C2×C3×C4#23512yes—
390C2×C3×C4#23512yes—
391C2×C3×C4C2×#6712yes—
392C2×C3×C4C2×#6712yes—
393C2×C3×C4C2×#6712yes—
394C2×Dic3—21no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—34no—
408C2×Dic3—34no—
409C2×Dic3—64no—
410C2×Dic3—64no—
411C2×Dic3—64no—
412C2×Dic3—62no—
413C2×Dic3—62no—
414C2×Dic3—62no—
415C2×Dic3—62no—
416C2×Dic3—64no—
417C2×Dic3—62no—
418C2×Dic3—62no—
419C2×Dic3—62no—
420C2×Dic3—62no—
421C2×Dic3—62no—
422C2×Dic3—62no—
423C2×Dic3—122no—
424C2×Dic3—122no—
425C2×Dic3C2×C4×D314yes—
426C2×Dic3C2×C4×D312yes—
427C4×D3—28no—
428Dic6—616no—
429Dic6—68no—
430Dic6—68no—
431C2×C32C2×C4×D411yescharacteristic
432C2×C32#51911yescharacteristic
433C2×C32#52011yescharacteristic
434C2×C32#53611yescharacteristic
435C2×C32#53711yescharacteristic
436C2×C32#54111yescharacteristic
437C2×C32#54211yescharacteristic
438C3×D3—42no—
439C24—31no—
440C2×Q8—94no—
441C2×Q8—182no—
442C2×Q8—182no—
443C22×C4—32no—
444C22×C4—31no—
445C22×C4—34no—
446C22×C4—32no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—92no—
452C22×C4—92no—
453C22×C4C2×D3211yescharacteristic
454C22×C4C2×D3211yescharacteristic
455C22×C4C2×D3211yescharacteristic
456C4⋊C4—34no—
457C4⋊C4—62no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4—98no—
461C4⋊C4—184no—
462C4⋊C4—182no—
463C4⋊C4—362no—
464C22⋊C4—62no—
465C22⋊C4—62no—
466C22⋊C4—98no—
467C22⋊C4—362no—
468C42—62no—
469C42—122no—
470C42—184no—
471C42#62614yes—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—62no—
480C22×C3—62no—
481C22×C3—62no—
482C22×C3—121no—
483C22×C3—121no—
484C22×C3—121no—
485C22×C3—121no—
486C22×C3C22×C4×D311yescharacteristic
487C22×C3C22×C4×D311yescharacteristic
488C22×C3C22×D1211yescharacteristic
489C22×C3C2×#23511yescharacteristic
490C22×C3C2×#23511yescharacteristic
491C22×C3#100411yescharacteristic
492C22×C3#100711yescharacteristic
493C22×C3#100911yescharacteristic
494C22×C3#101011yescharacteristic
495C22×C3#101011yescharacteristic
496C22×C3#101011yescharacteristic
497C22×C3#101011yescharacteristic
498C22×C3#101011yescharacteristic
499C22×C3C22×#6711yescharacteristic
500C2×D3—22no—
501C2×D3—22no—
502C2×D3—22no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—41no—
506C2×D3—41no—
507C3×C4—24no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—24no—
513C3×C4—42no—
514C3×C4—62no—
515C3×C4—64no—
516C3×C4—121no—
517C3×C4—121no—
518C3×C4—122no—
519C3×C4C4×D1214yes—
520C3×C4C4×#6714yes—
521Dic3—22no—
522Dic3—24no—
523Dic3—42no—
524Dic3—41no—
525Dic3—41no—
526Dic3—68no—
527Dic3—64no—
528Dic3—64no—
529Dic3—124no—
530Dic3—122no—
531Dic3—122no—
532C32#232311yescharacteristic, complemented, Sylow
533C23—32no—
534C23—61no—
535C23—61no—
536C23—61no—
537C23—61no—
538C23—61no—
539C23—61no—
540C23C22×D3211yescharacteristic
541Q8—188no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—34no—
547C2×C4—32no—
548C2×C4—62no—
549C2×C4—61no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—68no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—94no—
563C2×C4—182no—
564C2×C4—182no—
565C2×C4—182no—
566C2×C4—182no—
567C2×C4—182no—
568C2×C4#379812yes—
569C2×C4C4×D3212yes—
570C2×C4C2×#62612yes—
571C2×C4C2×#62612yes—
572C2×C4C2×#62612yes—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—122no—
581C2×C3C2×C4×D1211yescharacteristic
582C2×C3#551011yescharacteristic
583C2×C3#551611yescharacteristic
584C2×C3#551811yescharacteristic
585C2×C3#552311yescharacteristic
586C2×C3#552611yescharacteristic
587C2×C3#553411yescharacteristic
588C2×C3#554611yescharacteristic
589C2×C3#556111yescharacteristic
590C2×C3#557211yescharacteristic
591C2×C3#558311yescharacteristic
592C2×C3C2×C4×#6711yescharacteristic
593C2×C3#582511yescharacteristic
594C2×C3#583311yescharacteristic
595D3—42no—
596C22—62no—
597C22—62no—
598C22—62no—
599C22—121no—
600C22—121no—
601C22—121no—
602C22—121no—
603C22#6429711yescharacteristic
604C22#6429811yescharacteristic
605C22#6429811yescharacteristic
606C22C2×#379811yescharacteristic
607C22C2×C4×D3211yescharacteristic
608C22#6430711yescharacteristic
609C22C22×#62611yescharacteristic
610C4—22no—
611C4—22no—
612C4—64no—
613C4—62no—
614C4—121no—
615C4—122no—
616C4—121no—
617C4—184no—
618C4—362no—
619C4C4×#62614yes—
620C3—21no—
621C3#7376211yescharacteristic, complemented
622C3#8223511yescharacteristic, complemented
623C2—122no—
624C2#1059388011yescharacteristic
625C2#1059388511yescharacteristic
626C2#1059388711yescharacteristic
627C2#1059398311yescharacteristic
628C2#1059485611yescharacteristic
629C2C2×C4×#62611yescharacteristic
630C2#1059504311yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C42C22
socle, cosocleC23×C32C24
prime radicalC23×C3×D3C23
hypercenterC2×C42D32
hypocenterC32#2323
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×#634C2semidirect
4. C3×C4×C4⋊Dic3C2semidirect
5. #10591352C2semidirect
6. #10591334C2semidirect
7. #10591330C2semidirect
11. #10591213C2semidirect
34. #63857C4semidirect
57. C4×C4⋊Dic3D3semidirect
532. C32#2323semidirect
621. C3#73762semidirect
622. C3#82235semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#6342
C2×C3×#8722
C2×#638572
C3×C4×C4⋊Dic32
#105913522
#105913342
#105913302
#105913132
#105912872
#105912802
#105912132
#822353
#737623