Finite Group Databasebeta

← #11803401

Subgroups — #11803401

Order 1152.

1435 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64144C211yesmaximal, characteristic
2C2×C3×D4×Dic3C211yesmaximal, characteristic
3C22×#3724C211yesmaximal, characteristic, complemented
4C3×#5220C211yesmaximal, characteristic
5#10591703C212yesmaximal, complemented
6Dic3×#217C211yesmaximal, characteristic, complemented
7#10591697C211yesmaximal, characteristic, complemented
8#10591693C211yesmaximal, characteristic, complemented
9#10591693C212yesmaximal, complemented
10#10591264C212yesmaximal, complemented
11#10591197C212yesmaximal, complemented
12#82326—31nomaximal
13#74664—31nomaximal
14C22×C32×D4C2211yescharacteristic
15C23×C3⋊Dic3C2211yescharacteristic
16C23×C3×Dic3C2211yescharacteristic
17C23×C3×Dic3C2211yescharacteristic
18C23×C3×Dic3C2211yescharacteristic
19C2×#3757C2211yescharacteristic
20C2×#3757C2212yes—
21#64144C418yescomplemented
22C2×#3750C2212yes—
23C2×C3×#217C2211yescharacteristic
24C2×C3×#217C2211yescharacteristic
25C2×C3×#217C2211yescharacteristic
26C2×C3×#217C2212yes—
27C3×D4×Dic3—24no—
28C2×C3×C4⋊Dic3C2211yescharacteristic
29C2×C3×C4×Dic3C2211yescharacteristic
30C2×#3724—24no—
31C2×#3724C2212yes—
32C2×#3724C2212yes—
33C2×#3724C2212yes—
34C2×Dic32C2212yes—
35C3×C3⋊#95C2212yes—
36#63579C2212yes—
37#63579C2212yes—
38#63579C2212yes—
39#63579C2212yes—
40C2×C3×C4×D4—31no—
41C2×#935—31no—
42C2×#935—31no—
43C2×#935—61no—
44C2×D4×Dic3D311yescharacteristic
45C22×Dic3⋊C4—31no—
46C22×Dic3⋊C4—31no—
47C3×#386—31no—
48#5220D311yescharacteristic
49#5210—32no—
50#5210—31no—
51C4×#217—31no—
52#4976—32no—
53#4946—31no—
54#4946—32no—
55#4945—32no—
56#4944—32no—
57#4943—31no—
58#4943—31no—
59Dic3×C22⋊C4—31no—
60#4933—32no—
61#4652—32no—
62#4647—32no—
63C24×C32C2311yescharacteristic
64C24×C32C2311yescharacteristic
65C2×C32×D4—22no—
66C2×C32×D4—22no—
67C2×C32×D4C2×C414yes—
68C22×C32×C4C2311yescharacteristic
69C22×C3⋊Dic3—22no—
70C22×C3⋊Dic3—22no—
71C22×C3⋊Dic3C2×C414yescomplemented
72C22×C3⋊Dic3C2312yes—
73C22×C3⋊Dic3C2312yes—
74C22×C3×Dic3—22no—
75C22×C3×Dic3—22no—
76C22×C3×Dic3—42no—
77C22×C3×Dic3—42no—
78C22×C3×Dic3—42no—
79C22×C3×Dic3D414yes—
80C22×C3×Dic3C2311yescharacteristic
81C22×C3×Dic3C2311yescharacteristic
82C22×C3×Dic3C2312yes—
83C22×C3×Dic3C2312yes—
84C22×C3×Dic3C2311yescharacteristic
85C22×C3×Dic3C2311yescharacteristic
86#3757C2×C414yescomplemented
87#3757C2×C418yescomplemented
88#3750C2×C418yescomplemented
89C3×#217—22no—
90C3×#217—22no—
91C3×#217—22no—
92C3×#217—42no—
93C3×C4⋊Dic3—22no—
94C3×C4×Dic3—22no—
95#3724—24no—
96#3724—24no—
97#3724—42no—
98Dic32—42no—
99#2322—91noSylow
100C22×C3×D4—21no—
101C22×C3×D4C2×D311yescharacteristic
102C22×C3×D4C2×D311yescharacteristic
103C23×C3×C4—31no—
104C23×C3×C4—31no—
105C23×C3×C4—31no—
106C23×Dic3—31no—
107C23×Dic3—31no—
108C23×Dic3—61no—
109C23×Dic3C2×D311yescharacteristic
110C23×Dic3C2×D311yescharacteristic
111C23×Dic3C2×D311yescharacteristic
112C3×C4×D4—64no—
113C2×C3×C4⋊C4—31no—
114C2×C3×C22⋊C4—31no—
115C2×C3×C22⋊C4—31no—
116C2×C3×C22⋊C4—31no—
117C2×C3×C22⋊C4—32no—
118C2×C3×C42—31no—
119C2×#217—31no—
120C2×#217—32no—
121C2×#217—32no—
122C2×#217—31no—
123C2×#217—61no—
124C2×#217—62no—
125C2×#217C2×D311yescharacteristic
126C2×#217C2×D311yescharacteristic
127C2×#217C2×D311yescharacteristic
128C2×#217C2×D312yes—
129#935—38no—
130#935—38no—
131#935—68no—
132D4×Dic3—24no—
133C2×C4⋊Dic3C2×D311yescharacteristic
134C2×Dic3⋊C4—32no—
135C2×Dic3⋊C4—32no—
136C2×Dic3⋊C4—32no—
137C2×Dic3⋊C4—32no—
138C2×Dic3⋊C4—32no—
139C2×Dic3⋊C4—32no—
140C2×Dic3⋊C4—32no—
141C2×Dic3⋊C4—32no—
142C2×Dic3⋊C4—62no—
143C2×Dic3⋊C4—64no—
144C2×Dic3⋊C4—64no—
145C2×C4×Dic3—32no—
146C2×C4×Dic3—32no—
147C2×C4×Dic3C2×D311yescharacteristic
148C3×#95—32no—
149C3⋊#95—32no—
150C3⋊#95—32no—
151C3⋊#95—32no—
152C3⋊#95—32no—
153C3⋊#95—32no—
154C3⋊#95—32no—
155C3⋊#95—32no—
156C3⋊#95—32no—
157C3⋊#95C2×D312yes—
158C23×C32—22no—
159C23×C32—22no—
160C23×C32—21no—
161C23×C32—21no—
162C23×C32—22no—
163C23×C32—22no—
164C23×C32C22×C412yes—
165C23×C32C22×C412yes—
166C23×C32C2×D412yes—
167C23×C32#4112yes—
168C23×C32C2411yescharacteristic
169C32×D4—28no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4C22×C412yes—
173C2×C3⋊Dic3—28no—
174C2×C3⋊Dic3—22no—
175C2×C3⋊Dic3—22no—
176C2×C3⋊Dic3—22no—
177C2×C3⋊Dic3—22no—
178C2×C3⋊Dic3C22×C414yes—
179C2×C3⋊Dic3C22×C414yes—
180C2×C3×Dic3—22no—
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—22no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3—22no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—44no—
193C2×C3×Dic3—44no—
194C2×C3×Dic3—44no—
195C2×C3×Dic3C2×D412yes—
196C2×C3×Dic3C2×D412yes—
197C2×#123—91no—
198C2×C4×D4—31no—
199C22×C4⋊C4—91no—
200#389—92no—
201#388—92no—
202#387—92no—
203#386—31no—
204#385—92no—
205#385—91no—
206#381—91no—
207C4×C22⋊C4—91no—
208C24×C3—21no—
209C24×C3—21no—
210C24×C3C22×D311yescharacteristic
211C24×C3C22×D311yescharacteristic
212C24×C3C22×D311yescharacteristic
213C24×C3C22×D311yescharacteristic
214C2×C3×D4—22no—
215C2×C3×D4—22no—
216C2×C3×D4—22no—
217C2×C3×D4—22no—
218C2×C3×D4—24no—
219C2×C3×D4—42no—
220C2×C3×D4—42no—
221C2×C3×D4C4×D314yes—
222C2×C3×D4C4×D314yes—
223C22×C3×C4—21no—
224C22×C3×C4—31no—
225C22×C3×C4—31no—
226C22×C3×C4—34no—
227C22×C3×C4—31no—
228C22×C3×C4—32no—
229C22×C3×C4—31no—
230C22×C3×C4—32no—
231C22×C3×C4—62no—
232C22×C3×C4—62no—
233C22×C3×C4—122no—
234C22×C3×C4—122no—
235C22×C3×C4—122no—
236C22×C3×C4C22×D311yescharacteristic
237C22×C3×C4C22×D311yescharacteristic
238C22×Dic3—22no—
239C22×Dic3—22no—
240C22×Dic3—34no—
241C22×Dic3—34no—
242C22×Dic3—32no—
243C22×Dic3—32no—
244C22×Dic3—32no—
245C22×Dic3—32no—
246C22×Dic3—42no—
247C22×Dic3—42no—
248C22×Dic3—42no—
249C22×Dic3—62no—
250C22×Dic3—62no—
251C22×Dic3—64no—
252C22×Dic3—62no—
253C22×Dic3—62no—
254C22×Dic3—62no—
255C22×Dic3—62no—
256C22×Dic3—122no—
257C22×Dic3—122no—
258C22×Dic3#6714yes—
259C22×Dic3C22×D311yescharacteristic
260C22×Dic3C22×D312yes—
261C22×Dic3C22×D311yescharacteristic
262C22×Dic3C22×D312yes—
263C22×Dic3C22×D311yescharacteristic
264C22×Dic3C22×D311yescharacteristic
265C3×C4⋊C4—62no—
266C3×C22⋊C4—62no—
267C3×C22⋊C4—62no—
268C3×C22⋊C4—62no—
269C3×C22⋊C4—122no—
270C3×C42—62no—
271#217—22no—
272#217—22no—
273#217—22no—
274#217—34no—
275#217—34no—
276#217—38no—
277#217—38no—
278#217—42no—
279#217—64no—
280#217—68no—
281C4⋊Dic3—22no—
282Dic3⋊C4—38no—
283Dic3⋊C4—38no—
284Dic3⋊C4—68no—
285Dic3⋊C4—64no—
286Dic3⋊C4—64no—
287Dic3⋊C4—64no—
288Dic3⋊C4—64no—
289Dic3⋊C4—122no—
290Dic3⋊C4—122no—
291C4×Dic3—22no—
292C4×Dic3—122no—
293C4×Dic3—122no—
294C22×C32—24no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—24no—
298C22×C32—22no—
299C22×C32—24no—
300C22×C32—41no—
301C22×C32—41no—
302C22×C32—42no—
303C22×C32C4×D414yes—
304C22×C32C23×C411yescharacteristic
305C22×C32C22×D411yescharacteristic
306C22×C32C2×#4111yescharacteristic
307C22×C32H(5,𝔽2)12yes—
308C22×C322−1+412yes—
309C32×C4—22no—
310C3⋊Dic3—24no—
311C3⋊Dic3—24no—
312C3×Dic3—22no—
313C3×Dic3—22no—
314C3×Dic3—41no—
315C3×Dic3—42no—
316C3×Dic3—41no—
317C3×Dic3—42no—
318C22×D4D3211yescharacteristic
319C23×C4—31no—
320C23×C4—31no—
321C23×C4—31no—
322C23×C4—91no—
323#123—98no—
324C4×D4—64no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—92no—
327C2×C4⋊C4—92no—
328C2×C4⋊C4—92no—
329C2×C4⋊C4—92no—
330C2×C4⋊C4—184no—
331C2×C22⋊C4—31no—
332C2×C22⋊C4—31no—
333C2×C22⋊C4—31no—
334C2×C22⋊C4—32no—
335C2×C22⋊C4—91no—
336C2×C22⋊C4—92no—
337C2×C42—31no—
338C2×C42—92no—
339#95—32no—
340#95—92no—
341#95—92no—
342#95—92no—
343#95—92no—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C3—22no—
361C23×C3—42no—
362C23×C3—42no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—42no—
366C23×C3—42no—
367C23×C3C2×C4×D312yes—
368C23×C3C2×C4×D312yes—
369C23×C3C2×C4×D312yes—
370C23×C3C2×C4×D312yes—
371C23×C3#23512yes—
372C23×C3D3×D412yes—
373C23×C3#23712yes—
374C23×C3C2×#6712yes—
375C23×C3C23×D311yescharacteristic
376C23×C3C23×D311yescharacteristic
377C3×D4—28no—
378C3×D4—28no—
379C3×D4—48no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—32no—
386C2×C3×C4—32no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—62no—
390C2×C3×C4—62no—
391C2×C3×C4—62no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—62no—
395C2×C3×C4—62no—
396C2×C3×C4—62no—
397C2×C3×C4—62no—
398C2×C3×C4—62no—
399C2×C3×C4—62no—
400C2×C3×C4—62no—
401C2×C3×C4—124no—
402C2×C3×C4—124no—
403C2×C3×C4—124no—
404C2×C3×C4C2×C4×D312yes—
405C2×C3×C4C2×C4×D312yes—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—34no—
419C2×Dic3—34no—
420C2×Dic3—34no—
421C2×Dic3—34no—
422C2×Dic3—44no—
423C2×Dic3—44no—
424C2×Dic3—44no—
425C2×Dic3—64no—
426C2×Dic3—64no—
427C2×Dic3—68no—
428C2×Dic3—68no—
429C2×Dic3—62no—
430C2×Dic3—62no—
431C2×Dic3—62no—
432C2×Dic3—62no—
433C2×Dic3—62no—
434C2×Dic3—62no—
435C2×Dic3—62no—
436C2×Dic3—62no—
437C2×Dic3—122no—
438C2×Dic3—122no—
439C2×Dic3—122no—
440C2×Dic3—122no—
441C2×Dic3—128no—
442C2×Dic3C2×#6712yes—
443C2×Dic3C2×#6712yes—
444C2×C32—24no—
445C2×C32—42no—
446C2×C32C2×C4×D411yescharacteristic
447C2×C32#52012yes—
448C2×C32#53411yescharacteristic
449C2×C32#53611yescharacteristic
450C2×C32#53912yes—
451C24C2×D3211yescharacteristic
452C24C2×D3211yescharacteristic
453C2×D4—22no—
454C2×D4—22no—
455C2×D4#62414yes—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—34no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—92no—
466C22×C4—92no—
467C22×C4—94no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—182no—
472C22×C4—182no—
473C22×C4C2×D3211yescharacteristic
474C4⋊C4—62no—
475C4⋊C4—98no—
476C4⋊C4—184no—
477C4⋊C4—184no—
478C4⋊C4—362no—
479C22⋊C4—62no—
480C22⋊C4—62no—
481C22⋊C4—62no—
482C22⋊C4—94no—
483C22⋊C4—98no—
484C22⋊C4—122no—
485C42—62no—
486C42—362no—
487C22×C3—22no—
488C22×C3—24no—
489C22×C3—24no—
490C22×C3—22no—
491C22×C3—24no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—24no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—24no—
498C22×C3—24no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—24no—
505C22×C3—44no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—44no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—42no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—42no—
517C22×C3—82no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3#67⋊C414yes—
521C22×C3C4×#6714yes—
522C22×C3C22×C4×D311yescharacteristic
523C22×C3C22×C4×D311yescharacteristic
524C22×C3C2×#23511yescharacteristic
525C22×C3C2×D3×D411yescharacteristic
526C22×C3C2×#23711yescharacteristic
527C22×C3#100412yes—
528C22×C3#100412yes—
529C22×C3#101012yes—
530C22×C3#101012yes—
531C22×C3C22×#6711yescharacteristic
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—62no—
536C3×C4—62no—
537C3×C4—121no—
538C3×C4—122no—
539C3×C4—122no—
540C3×C4—121no—
541Dic3—22no—
542Dic3—22no—
543Dic3—41no—
544Dic3—41no—
545Dic3—42no—
546Dic3—42no—
547Dic3—64no—
548Dic3—64no—
549Dic3—64no—
550Dic3—64no—
551Dic3—124no—
552Dic3—124no—
553C32#232211yescharacteristic, complemented, Sylow
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—21no—
559C23—21no—
560C23#380412yes—
561C23C2×#62412yes—
562C23C2×#62412yes—
563C23D3×#6712yes—
564C23C22×D3211yescharacteristic
565D4—28no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—32no—
569C2×C4—32no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—62no—
582C2×C4—94no—
583C2×C4—94no—
584C2×C4—124no—
585C2×C4—124no—
586C2×C4—124no—
587C2×C4—182no—
588C2×C4—182no—
589C2×C4—182no—
590C2×C4—182no—
591C2×C4—188no—
592C2×C4C2×#62412yes—
593C2×C3—24no—
594C2×C3—24no—
595C2×C3—22no—
596C2×C3—21no—
597C2×C3—22no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—44no—
601C2×C3—42no—
602C2×C3—42no—
603C2×C3—82no—
604C2×C3C2×#67⋊C411yescharacteristic
605C2×C3#554612yes—
606C2×C3#554612yes—
607C2×C3#556412yes—
608C2×C3#558911yescharacteristic
609C2×C3#560212yes—
610C2×C3#564811yescharacteristic
611C2×C3C2×C4×#6711yescharacteristic
612C2×C3#580411yescharacteristic
613C2×C3#582511yescharacteristic
614C22—24no—
615C22—24no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—24no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22#6395214yes—
624C22#6430912yes—
625C22#6431312yes—
626C22C2×#380411yescharacteristic
627C22C22×#62411yescharacteristic
628C22C2×D3×#6711yescharacteristic
629C4—22no—
630C4—62no—
631C4—62no—
632C4—122no—
633C4—122no—
634C4—121no—
635C4—121no—
636C4—184no—
637C4—184no—
638C3—21no—
639C3#7466411yescharacteristic, complemented
640C3#8232611yescharacteristic, complemented
641C2—24no—
642C2—42no—
643C2#1059521612yes—
644C2#1059521912yes—
645C2#1059523011yescharacteristic
646C2#1059527011yescharacteristic
647C2C2×#6395211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
Fitting, prime radicalC22×C32×D4C22
socle, cosocleC23×C32C24
hypercenterC22×D4D32
hypocenterC32#2322
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
3. C22×#3724C2semidirect
5. #10591703C2semidirect
6. Dic3×#217C2semidirect
7. #10591697C2semidirect
8. #10591693C2semidirect
9. #10591693C2semidirect
10. #10591264C2semidirect
11. #10591197C2semidirect
21. #64144C4semidirect
71. C22×C3⋊Dic3C2×C4semidirect
86. #3757C2×C4semidirect
87. #3757C2×C4semidirect
88. #3750C2×C4semidirect
553. C32#2322semidirect
639. C3#74664semidirect
640. C3#82326semidirect

Maximal subgroups

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

subgroupindex
C2×#641442
C2×C3×D4×Dic32
C22×#37242
C3×#52202
#105917032
Dic3×#2172
#105916972
#105916932
#105912642
#105911972
#823263
#746643