Finite Group Databasebeta

← #11792557

Subgroups — #11792557

Order 1152.

1197 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64149C211yesmaximal, characteristic
2C22×C3×#214C211yesmaximal, characteristic, complemented
3C22×#3717C211yesmaximal, characteristic, complemented
4C2×C3×#832C211yesmaximal, characteristic, complemented
5#10591221C211yesmaximal, characteristic, complemented
6C2×#63564C211yesmaximal, characteristic, complemented
7#10589156C211yesmaximal, characteristic, complemented
8#72056—31nomaximal
9#70210—31nomaximal
10C22×C32×D4C2211yescharacteristic
11C22×#638—22no—
12C22×C4⋊(C3⋊D3)C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14#64149—22no—
15#64149—22no—
16C2×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×#214—22no—
18C2×C3×#214—22no—
19C2×C3×#214—22no—
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C22×C3×C3⋊C8C2211yescharacteristic
22C22×C3×C3⋊C8C2211yescharacteristic
23C2×#3717—22no—
24C2×#3717—22no—
25C2×#3717—22no—
26C3×#832—22no—
27#63579—22no—
28#63564—22no—
29C22×C3×SD16—31no—
30C22×#215—31no—
31C2×#940—31no—
32C2×#940—31no—
33C2×#940—61no—
34C22×#214D311yescharacteristic, complemented
35C22×#204—31no—
36C2×C3×#102—31no—
37#5229—31no—
38C2×#832D311yescharacteristic, complemented
39C2×#821—31no—
40#4973—31no—
41C2×#809—31no—
42#4550—31no—
43#4550—31no—
44C24×C32—21no—
45C23×C3⋊D3—21no—
46C2×C32×D4—21no—
47C2×C32×D4—21no—
48C2×C32×D4—21no—
49C2×C32×D4—21no—
50C2×C32×D4—21no—
51C2×C32×D4D412yes—
52C22×C32×C4C2311yescharacteristic
53C2×#638—28no—
54C2×#638—44no—
55C2×#638—44no—
56C22×C3⋊Dic3D412yes—
57C2×C4⋊(C3⋊D3)—21no—
58C2×C4⋊(C3⋊D3)—21no—
59C2×C4⋊(C3⋊D3)—21no—
60C2×C4⋊(C3⋊D3)—21no—
61C2×C4⋊(C3⋊D3)—21no—
62C2×C4⋊(C3⋊D3)D412yes—
63C22×C3×Dic3—21no—
64C22×C3×Dic3—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6D412yescomplemented
71C4×C3⋊Dic3—21no—
72C4×C3⋊Dic3—21no—
73C3×#214—28no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3—21no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8—21no—
80C2×C3×C3⋊C8D412yescomplemented
81C2×C3×C3⋊C8D412yescomplemented
82#3717—28no—
83#2014—91noSylow
84C22×C3×Q8—31no—
85C22×C3×D4—21no—
86C22×C3×D4C2×D311yescharacteristic
87C22×C3×D4C2×D311yescharacteristic
88C22×#67—62no—
89C22×#67—62no—
90C22×#67—64no—
91C22×D12—31no—
92C22×D12—31no—
93C22×D12—61no—
94C22×Dic6C2×D311yescharacteristic
95C2×C3×SD16—62no—
96C2×C3×SD16—62no—
97C2×C3×SD16—62no—
98C22×C3×C8—31no—
99C22×C3×C8—31no—
100C2×C3×C4⋊C4—31no—
101C2×#215—62no—
102C2×#215—62no—
103C2×#215—62no—
104#940—62no—
105#940—62no—
106#940—64no—
107#940—64no—
108#940—62no—
109#940—62no—
110C2×#214—22no—
111C2×#214—22no—
112C2×#214—22no—
113C2×C4⋊Dic3C2×D311yescharacteristic
114C2×C4×Dic3—31no—
115C2×C4×Dic3—31no—
116C2×C4×Dic3—61no—
117C22×C3⋊C8C2×D311yescharacteristic
118C22×C3⋊C8C2×D311yescharacteristic
119C2×#204—62no—
120C2×#204—62no—
121C2×#204—62no—
122C3×#102—62no—
123#832—22no—
124C3⋊#95—62no—
125C3⋊#95—62no—
126#821—62no—
127#809—62no—
128C23×C32—22no—
129C23×C32—41no—
130C23×C32—41no—
131C23×C32—42no—
132C23×C32—42no—
133C23×C32C2×D411yescharacteristic
134C22×C3⋊D3—22no—
135C22×C3⋊D3—41no—
136C22×C3⋊D3—41no—
137C22×C3⋊D3—42no—
138C22×C3⋊D3—42no—
139C32×D4—22no—
140C32×D4—22no—
141C32×D4—41no—
142C32×D4—41no—
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4#4111yescharacteristic
149#638—416no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3SD1614yescomplemented
153C4⋊(C3⋊D3)—22no—
154C4⋊(C3⋊D3)—22no—
155C4⋊(C3⋊D3)—41no—
156C4⋊(C3⋊D3)—41no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—41no—
159C2×C3×Dic3—41no—
160C2×C3×Dic3—41no—
161C2×C3×Dic3—42no—
162C3×Dic6—22no—
163C3×Dic6—22no—
164C3×Dic6—41no—
165C3×Dic6—41no—
166C3×C3⋊C8—22no—
167C3×C3⋊C8—22no—
168C22×SD16—31no—
169C22×SD16—91no—
170C2×#127—91no—
171C2×#102—31no—
172C2×#102—91no—
173#391—91no—
174#336—91no—
175C24×C3—22no—
176C24×C3—21no—
177C24×C3—21no—
178C23×D3—62no—
179C23×D3—61no—
180C23×D3—61no—
181C2×C3×Q8—32no—
182C2×C3×Q8—61no—
183C2×C3×Q8—61no—
184C2×C3×Q8—61no—
185C2×C3×Q8—61no—
186C2×C3×Q8—61no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4—41no—
200C2×C3×D4—41no—
201C2×C3×D4—41no—
202C2×C3×D4—41no—
203C2×C3×D4D1212yes—
204C2×C3×D4#6712yes—
205C22×C3×C4—21no—
206C22×C3×C4—61no—
207C22×C3×C4—61no—
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C22×D311yescharacteristic
210C2×#67—616no—
211C2×#67—68no—
212C2×#67—68no—
213C2×#67—124no—
214C2×#67—124no—
215C2×#67—128no—
216C2×#67—128no—
217C2×#67—124no—
218C2×#67—124no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—32no—
222C22×Dic3—32no—
223C22×Dic3—62no—
224C2×D12—32no—
225C2×D12—32no—
226C2×D12—62no—
227C2×D12—61no—
228C2×D12—61no—
229C2×D12—61no—
230C2×D12—61no—
231C2×D12—61no—
232C2×D12—61no—
233C2×D12—61no—
234C2×D12—61no—
235C2×D12—62no—
236C2×D12—61no—
237C2×D12—61no—
238C2×D12—121no—
239C2×D12—121no—
240C2×D12—121no—
241C2×D12—121no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6#6712yescomplemented
248C3×SD16—68no—
249C2×C3×C8—32no—
250C2×C3×C8—32no—
251C2×C3×C8—61no—
252C2×C3×C8—61no—
253C2×C3×C8—61no—
254C2×C3×C8—61no—
255C3×C4⋊C4—61no—
256C3×C4⋊C4—61no—
257#215—68no—
258#214—28no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4×Dic3—62no—
262C4×Dic3—62no—
263C4×Dic3—61no—
264C4×Dic3—61no—
265C4×Dic3—61no—
266C4×Dic3—61no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8#6712yescomplemented
272C2×C3⋊C8#6712yescomplemented
273#204—68no—
274C22×C32—42no—
275C22×C32—44no—
276C22×C32—82no—
277C22×C32—81no—
278C22×C32—81no—
279C22×C32#12011yescharacteristic
280C22×C32#12111yescharacteristic
281C22×C32#12711yescharacteristic
282C22×C32C2×SD1612yes—
283C22×C32AGL(1,ℤ/8)12yes—
284C2×C3⋊D3—42no—
285C2×C3⋊D3—44no—
286C2×C3⋊D3—82no—
287C2×C3⋊D3—81no—
288C2×C3⋊D3—81no—
289C32×C4#12011yescharacteristic
290C32×C4#12011yescharacteristic
291C32×C4#12111yescharacteristic
292C32×C4#12111yescharacteristic
293C3⋊Dic3—24no—
294C3×Dic3—42no—
295C3×Dic3—81no—
296C22×Q8—31no—
297C22×D4—91no—
298C22×D4—182no—
299C22×D4D3211yescharacteristic
300C2×SD16—62no—
301C2×SD16—62no—
302C2×SD16—62no—
303C2×SD16—182no—
304C2×SD16—182no—
305C2×SD16—182no—
306C22×C8—31no—
307C22×C8—31no—
308#127—182no—
309#127—182no—
310C2×C4⋊C4—31no—
311C2×C42—91no—
312#102—62no—
313#102—182no—
314#95—182no—
315C23×C3—24no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—21no—
319C23×C3—44no—
320C23×C3—44no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—41no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3D3×D411yescharacteristic
332C23×C3D3×D411yescharacteristic
333C22×D3—64no—
334C22×D3—62no—
335C22×D3—62no—
336C22×D3—122no—
337C22×D3—122no—
338C22×D3—124no—
339C22×D3—124no—
340C22×D3—121no—
341C22×D3—122no—
342C22×D3—122no—
343C22×D3—122no—
344C22×D3—122no—
345C22×D3—121no—
346C22×D3—121no—
347C22×D3—121no—
348C3×Q8—62no—
349C3×Q8—62no—
350C3×Q8—121no—
351C3×Q8—121no—
352C3×D4—22no—
353C3×D4—22no—
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—42no—
357C3×D4—42no—
358C3×D4—42no—
359C3×D4—42no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—62no—
371C2×C3×C4—121no—
372C2×C3×C4—121no—
373C2×C3×C4—122no—
374C2×C3×C4—121no—
375C2×C3×C4C2×D1211yescharacteristic
376C2×C3×C4#23511yescharacteristic
377C2×C3×C4D3×D411yescharacteristic
378C2×C3×C4D3×D411yescharacteristic
379C2×C3×C4D3×D411yescharacteristic
380C2×C3×C4D3×D411yescharacteristic
381C2×C3×C4D3×D411yescharacteristic
382C2×C3×C4#23711yescharacteristic
383C2×C3×C4C2×#6711yescharacteristic
384C2×C3×C4C2×#6711yescharacteristic
385C2×C3×C4C2×#6711yescharacteristic
386C2×C3×C4C2×#6711yescharacteristic
387#67—1232no—
388#67—1216no—
389#67—1216no—
390C2×Dic3—22no—
391C2×Dic3—34no—
392C2×Dic3—34no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—42no—
396C2×Dic3—41no—
397C2×Dic3—64no—
398C2×Dic3—64no—
399C2×Dic3—64no—
400C2×Dic3—62no—
401C2×Dic3—62no—
402C2×Dic3—62no—
403C2×Dic3—62no—
404D12—62no—
405D12—62no—
406D12—62no—
407D12—62no—
408D12—122no—
409D12—122no—
410D12—122no—
411D12—122no—
412D12—121no—
413D12—121no—
414D12—121no—
415D12—121no—
416Dic6—22no—
417Dic6—22no—
418Dic6—41no—
419Dic6—41no—
420C3×C8—62no—
421C3×C8—62no—
422C3⋊C8—22no—
423C3⋊C8—22no—
424C2×C32—82no—
425C2×C32#39211yescharacteristic
426C2×C32#45012yes—
427C2×C32#46012yes—
428C2×C32#49211yescharacteristic
429C2×C32#49611yescharacteristic
430C3⋊D3—82no—
431C24—21no—
432C24—181no—
433C2×Q8—32no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—92no—
445C2×D4—188no—
446C2×D4—181no—
447C2×D4—181no—
448C2×D4—181no—
449C2×D4—181no—
450C2×D4—181no—
451C2×D4—364no—
452C2×D4—364no—
453C2×D4#62612yes—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—92no—
457C22×C4C2×D3211yescharacteristic
458SD16—68no—
459SD16—188no—
460C2×C8—32no—
461C2×C8—32no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C42—181no—
469C42—181no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—48no—
476C22×C3—44no—
477C22×C3—44no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—44no—
481C22×C3—82no—
482C22×C3—84no—
483C22×C3—82no—
484C22×C3—81no—
485C22×C3—82no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—82no—
490C22×C3#88211yescharacteristic
491C22×C3#88411yescharacteristic
492C22×C3#91112yes—
493C22×C3D3×SD1612yes—
494C22×C3D3×SD1612yes—
495C22×C3#91412yes—
496C22×C3#93711yescharacteristic
497C22×C3#93911yescharacteristic
498C22×C3#94011yescharacteristic
499C22×C3#94011yescharacteristic
500C2×D3—124no—
501C2×D3—128no—
502C2×D3—124no—
503C2×D3—122no—
504C2×D3—122no—
505C2×D3—124no—
506C2×D3—242no—
507C2×D3—244no—
508C2×D3—242no—
509C2×D3—241no—
510C2×D3—242no—
511C2×D3—241no—
512C2×D3—241no—
513C2×D3—241no—
514C2×D3—242no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—122no—
520C3×C4—241no—
521C3×C4#88211yescharacteristic
522C3×C4#88411yescharacteristic
523C3×C4#93011yescharacteristic
524C3×C4#93711yescharacteristic
525C3×C4#93711yescharacteristic
526C3×C4#93811yescharacteristic
527C3×C4#93911yescharacteristic
528C3×C4#95311yescharacteristic
529Dic3—42no—
530Dic3—68no—
531Dic3—64no—
532Dic3—64no—
533Dic3—81no—
534C32#201411yescharacteristic, complemented, Sylow
535C23—22no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—182no—
541C23—362no—
542C23—362no—
543C23—361no—
544C23—361no—
545C23#381111yescharacteristic
546Q8—62no—
547Q8—62no—
548Q8—121no—
549Q8—121no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554D4—182no—
555D4—182no—
556D4—3616no—
557D4—361no—
558D4—361no—
559C2×C4—62no—
560C2×C4—94no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—122no—
564C2×C4—121no—
565C2×C4—182no—
566C2×C4—182no—
567C2×C4#380411yescharacteristic
568C2×C4C2×#62611yescharacteristic
569C2×C4D3×#6711yescharacteristic
570C2×C4D3×#6711yescharacteristic
571C2×C4D3×#6711yescharacteristic
572C2×C4#381111yescharacteristic
573C8—62no—
574C8—62no—
575C2×C3—21no—
576C2×C3—22no—
577C2×C3—22no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—84no—
581C2×C3—82no—
582C2×C3—82no—
583C2×C3#476212yes—
584C2×C3#477612yes—
585C2×C3#496011yescharacteristic
586C2×C3#515411yescharacteristic
587C2×C3#516312yes—
588C2×C3#516912yes—
589C2×C3#517511yescharacteristic
590C2×C3#517511yescharacteristic
591C2×C3#517611yescharacteristic
592C2×C3#522311yescharacteristic
593D3—244no—
594D3—242no—
595D3—242no—
596C22—44no—
597C22—42no—
598C22—81no—
599C22—82no—
600C22—81no—
601C22—364no—
602C22—362no—
603C22—721no—
604C22—722no—
605C22—721no—
606C22#6393012yes—
607C22#6393712yes—
608C22#6397111yescharacteristic
609C22#6397911yescharacteristic
610C22#6398111yescharacteristic
611C4—122no—
612C4—184no—
613C4—241no—
614C4#6395811yescharacteristic
615C4#6397711yescharacteristic
616C4#6397811yescharacteristic
617C4#6398311yescharacteristic
618C3—21no—
619C3#7021011yescharacteristic, complemented
620C3#7205611yescharacteristic, complemented
621C2—82no—
622C2—722no—
623C2#1059144412yes—
624C2#1059146712yes—
625C2#1059147511yescharacteristic
626C2#1059149211yescharacteristic
627C2#1059172411yescharacteristic
subgroupG/H
centerC23#3811
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×D4C22
socleC23×C32C2×D4
prime radicalC2×#64149C2
hypercenterC22×D4D32
hypocenterC32#2014
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C22×C3×#214C2semidirect
3. C22×#3717C2semidirect
4. C2×C3×#832C2semidirect
5. #10591221C2semidirect
6. C2×#63564C2semidirect
7. #10589156C2semidirect
34. C22×#214D3semidirect
38. C2×#832D3semidirect
70. C2×C3×Dic6D4semidirect
80. C2×C3×C3⋊C8D4semidirect
81. C2×C3×C3⋊C8D4semidirect
152. C2×C3⋊Dic3SD16semidirect
247. C2×Dic6#67semidirect
271. C2×C3⋊C8#67semidirect
272. C2×C3⋊C8#67semidirect
534. C32#2014semidirect
619. C3#70210semidirect
620. C3#72056semidirect

Maximal subgroups

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

subgroupindex
C2×#641492
C22×C3×#2142
C22×#37172
C2×C3×#8322
#105912212
C2×#635642
#105891562
#720563
#702103