Finite Group Databasebeta

← #11788312

Subgroups — #11788312

Order 1152.

829 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×M16C211yesmaximal, characteristic, complemented
2C2×#63908C211yesmaximal, characteristic
3C2×C3×#823C211yesmaximal, characteristic, complemented
4C2×#63558C211yesmaximal, characteristic, complemented
5#10591069C211yesmaximal, characteristic, complemented
6#10589420C211yesmaximal, characteristic, complemented
7#10589168C211yesmaximal, characteristic, complemented
8#72002—31nomaximal
9#70081—31nomaximal
10C22×C3×D12C2211yescharacteristic
11C22×C3×C4×D3C2211yescharacteristic
12C22×#625C412yes—
13C2×C32×M16C2211yescharacteristic
14C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×#3748C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C2×C3×#208C2211yescharacteristic
18C3×D3×M16—22no—
19C3×D3×M16—22no—
20C2×C3×ΓL(1,𝔽25)—21no—
21C2×C3×D3×C8—21no—
22C2×(C2×D3)⋊Dic3C412yes—
23#63908—21no—
24#63908—21no—
25#63908—21no—
26#63908—21no—
27C3×#823—22no—
28#63558—22no—
29(C2×D3)⋊(C3⋊C8)—21no—
30(C2×D3)⋊(C3⋊C8)—21no—
31C22×C3×M16—31no—
32C2×#938—31no—
33C2×#930—31no—
34C2×D3×M16D311yescharacteristic, complemented
35C2×C3×#100—31no—
36C2×#833—31no—
37#5207—31no—
38C2×#823—31no—
39C2×#823D311yescharacteristic, complemented
40#5126—31no—
41#4554—31no—
42#4554—31no—
43#4554—31no—
44#4554—31no—
45#4554—61no—
46C23×C3×D3C2×C411yescharacteristic
47C23×C3×D3C2×C412yes—
48C22×C32×C4C2311yescharacteristic
49C22×C3⋊Dic3C2×C412yes—
50C22×C3×Dic3C2×C411yescharacteristic
51C2×C3×D12—21no—
52C2×C3×D12—21no—
53C2×C3×D12—21no—
54C2×C3×D12—21no—
55C2×C3×D12—21no—
56C2×C3×D12—21no—
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3—21no—
60C2×C3×C4×D3—21no—
61C2×C3×C4×D3D411yescharacteristic
62C2×C3×C4×D3D411yescharacteristic
63C2×C3×C4×D3D411yescharacteristic
64C2×C3×C4×D3D411yescharacteristic
65C2×#625—22no—
66C2×#625—22no—
67C2×#625—44no—
68C32×M16—21no—
69C32×M16D412yescomplemented
70C2×C32×C8—21no—
71C4⋊(C3⋊Dic3)—21no—
72C4⋊(C3⋊Dic3)—21no—
73#3748—21no—
74#3748—21no—
75C2×C3⋊(C3⋊C8)—21no—
76C3×#208—21no—
77C3×#208—21no—
78C3×#208—21no—
79C3×#208D412yescomplemented
80C2×C3×C3⋊C8—21no—
81C2×C3×C3⋊C8—21no—
82C3×ΓL(1,𝔽25)—24no—
83C3×D3×C8—24no—
84(C2×D3)⋊Dic3—22no—
85(C2×D3)⋊Dic3—22no—
86#1900—91noSylow
87C22×C3×D4—31no—
88C23×C3×C4—31no—
89C22×#67—32no—
90C22×#67—32no—
91C22×D12C2×D311yescharacteristic
92C22×C4×D3C2×D311yescharacteristic
93C2×C3×M16—21no—
94C2×C3×M16—31no—
95C2×C3×M16—31no—
96C2×C3×M16—61no—
97C2×C3×M16—62no—
98C2×C3×M16—62no—
99C2×C3×M16C2×D311yescharacteristic
100C2×C3×M16C2×D311yescharacteristic
101C22×C3×C8—61no—
102C2×#217—32no—
103#938—61no—
104#938—61no—
105#938—61no—
106#938—61no—
107#930—61no—
108#930—61no—
109#930—61no—
110#930—61no—
111C2×(C2×D3)⋊C4—32no—
112C2×C4⋊Dic3—31no—
113C2×C4⋊Dic3—31no—
114C2×C4⋊Dic3—61no—
115C2×#208—31no—
116C2×#208—31no—
117C2×#208—61no—
118C2×#208C2×D311yescharacteristic
119C2×#208C2×D311yescharacteristic
120D3×M16—22no—
121D3×M16—22no—
122C2×ΓL(1,𝔽25)—21no—
123C2×D3×C8—21no—
124C3×#100—62no—
125#833—62no—
126#830—61no—
127#830—61no—
128#823—22no—
129#823—62no—
130(C2×D3)⋊C8—61no—
131(C2×D3)⋊C8—61no—
132C23×C32C22×C411yescharacteristic
133C22×C3×D3—21no—
134C22×C3×D3—21no—
135C22×C3×D3—22no—
136C22×C3×D3—22no—
137C22×C3×D3—22no—
138C22×C3×D3—22no—
139C22×C3×D3—41no—
140C22×C3×D3—42no—
141C22×C3×D3—42no—
142C22×C3×D3C22⋊C411yescharacteristic
143C22×C3×D3C22⋊C411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4#4111yescharacteristic
149C2×C32×C4#4111yescharacteristic
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—42no—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3C22⋊C411yescharacteristic
155C2×C3×Dic3C22⋊C411yescharacteristic
156C3×D12—41no—
157C3×D12—41no—
158C3×D12—41no—
159C3×D12—41no—
160C3×C4×D3—21no—
161C3×C4×D3—21no—
162C3×C4×D3—21no—
163C3×C4×D3—21no—
164C3×C4×D3—21no—
165C3×C4×D3—21no—
166C3×C4×D3—21no—
167C3×C4×D3—21no—
168#625—44no—
169#625—44no—
170C32×C8—22no—
171C3⋊(C3⋊C8)—41no—
172C3×C3⋊C8—22no—
173C3×C3⋊C8—41no—
174C22×M16—31no—
175C2×#121—91no—
176C2×#100—31no—
177C2×#100—91no—
178#407—91no—
179#343—91no—
180#343—91no—
181C24×C3—31no—
182C24×C3—32no—
183C23×D3C4×D311yescharacteristic
184C23×D3C4×D312yes—
185C2×C3×D4—61no—
186C2×C3×D4—61no—
187C2×C3×D4—61no—
188C2×C3×D4—61no—
189C2×C3×D4—61no—
190C2×C3×D4—61no—
191C22×C3×C4—21no—
192C22×C3×C4—31no—
193C22×C3×C4—31no—
194C22×C3×C4—31no—
195C22×C3×C4—31no—
196C22×C3×C4—31no—
197C22×C3×C4—61no—
198C22×C3×C4—61no—
199C22×C3×C4—61no—
200C22×C3×C4—61no—
201C22×C3×C4C22×D311yescharacteristic
202C22×C3×C4C22×D311yescharacteristic
203C2×#67—62no—
204C2×#67—62no—
205C2×#67—62no—
206C2×#67—62no—
207C2×#67—124no—
208C2×#67—124no—
209C22×Dic3—32no—
210C22×Dic3—32no—
211C22×Dic3—62no—
212C22×Dic3C4×D311yescharacteristic
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3D1211yescharacteristic
224C2×C4×D3D1211yescharacteristic
225C2×C4×D3#6711yescharacteristic
226C2×C4×D3#6711yescharacteristic
227C3×M16—21no—
228C3×M16—21no—
229C3×M16—22no—
230C3×M16—32no—
231C3×M16—41no—
232C3×M16—64no—
233C3×M16—61no—
234C3×M16—61no—
235C3×M16—61no—
236C3×M16#6712yescomplemented
237C3×M16#6712yescomplemented
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—41no—
241C2×C3×C8—64no—
242C2×C3×C8—61no—
243C2×C3×C8—61no—
244#217—62no—
245#217—62no—
246(C2×D3)⋊C4—62no—
247(C2×D3)⋊C4—62no—
248C4⋊Dic3—61no—
249C4⋊Dic3—61no—
250C4⋊Dic3—61no—
251C4⋊Dic3—61no—
252C4⋊Dic3—121no—
253C4⋊Dic3—121no—
254#208—21no—
255#208—21no—
256#208—21no—
257#208—62no—
258#208—61no—
259#208—61no—
260#208—61no—
261#208—61no—
262#208—121no—
263#208#6712yescomplemented
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—61no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8—121no—
269ΓL(1,𝔽25)—24no—
270D3×C8—24no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32C2×C22⋊C411yescharacteristic
274C22×C32C4×D411yescharacteristic
275C22×C32C4×D411yescharacteristic
276C2×C3×D3—21no—
277C2×C3×D3—21no—
278C2×C3×D3—21no—
279C2×C3×D3—21no—
280C2×C3×D3—21no—
281C2×C3×D3—42no—
282C2×C3×D3—42no—
283C2×C3×D3—42no—
284C2×C3×D3—42no—
285C2×C3×D3—42no—
286C2×C3×D3—82no—
287C2×C3×D3—82no—
288C2×C3×D3#10012yescomplemented
289C32×C4#12011yescharacteristic
290C32×C4#12111yescharacteristic
291C32×C4#12111yescharacteristic
292C32×C4#12311yescharacteristic
293C3⋊Dic3—42no—
294C3×Dic3—21no—
295C3×Dic3—21no—
296C22×D4—31no—
297C22×D4—92no—
298C23×C4—31no—
299C2×M16—31no—
300C2×M16—31no—
301C2×M16—61no—
302C2×M16—62no—
303C2×M16—62no—
304C2×M16—91no—
305C2×M16D3211yescharacteristic
306C22×C8—61no—
307#121—181no—
308#121—181no—
309#121—181no—
310#121—181no—
311C2×C4⋊C4—91no—
312C2×C22⋊C4—92no—
313#100—62no—
314#100—182no—
315C22⋊C8—181no—
316C22⋊C8—181no—
317C23×C3—21no—
318C23×C3—31no—
319C23×C3—31no—
320C23×C3—62no—
321C23×C3—61no—
322C23×C3—61no—
323C23×C3—62no—
324C23×C3—62no—
325C23×C3—62no—
326C23×C3—121no—
327C23×C3—122no—
328C23×C3—122no—
329C23×C3C2×C4×D311yescharacteristic
330C23×C3C2×C4×D311yescharacteristic
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—22no—
334C22×D3—22no—
335C22×D3—22no—
336C22×D3—22no—
337C22×D3—41no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3(C2×D3)⋊C411yescharacteristic
341C22×D3(C2×D3)⋊C411yescharacteristic
342C3×D4—121no—
343C3×D4—121no—
344C3×D4—121no—
345C3×D4—121no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4C2×D1211yescharacteristic
365C2×C3×C4#23511yescharacteristic
366C2×C3×C4#23511yescharacteristic
367C2×C3×C4#23511yescharacteristic
368C2×C3×C4D3×D411yescharacteristic
369C2×C3×C4D3×D411yescharacteristic
370C2×C3×C4D3×D411yescharacteristic
371C2×C3×C4#23711yescharacteristic
372C2×C3×C4C2×#6711yescharacteristic
373C2×C3×C4C2×#6711yescharacteristic
374C2×C3×C4C2×#6711yescharacteristic
375C2×C3×C4C2×#6711yescharacteristic
376#67—124no—
377#67—124no—
378#67—124no—
379#67—124no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—122no—
385C2×Dic3—122no—
386C2×Dic3—122no—
387C2×Dic3—242no—
388C2×Dic3(C2×D3)⋊C411yescharacteristic
389C2×Dic3(C2×D3)⋊C411yescharacteristic
390D12—41no—
391D12—41no—
392D12—41no—
393D12—41no—
394C4×D3—21no—
395C4×D3—21no—
396C4×D3—21no—
397C4×D3—21no—
398C4×D3—21no—
399C4×D3—21no—
400C4×D3—21no—
401C4×D3—21no—
402C3×C8—22no—
403C3×C8—22no—
404C3×C8—42no—
405C3×C8—62no—
406C3×C8—121no—
407C3⋊C8—22no—
408C3⋊C8—41no—
409C3⋊C8—122no—
410C3⋊C8—121no—
411C3⋊C8—121no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32#38611yescharacteristic
415C2×C32C2×#10011yescharacteristic
416C2×C32#41311yescharacteristic
417C3×D3—22no—
418C3×D3—41no—
419C3×D3—82no—
420C24—31no—
421C24—32no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—182no—
429C2×D4—182no—
430C2×D4—364no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—92no—
441C22×C4C2×D3211yescharacteristic
442M16—21no—
443M16—32no—
444M16—64no—
445M16—61no—
446M16—61no—
447M16—61no—
448M16—181no—
449M16—181no—
450M16#62512yescomplemented
451C2×C8—21no—
452C2×C8—64no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—181no—
456C4⋊C4—181no—
457C4⋊C4—181no—
458C22⋊C4—182no—
459C22⋊C4—182no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—32no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—61no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—61no—
474C22×C3—61no—
475C22×C3—122no—
476C22×C3—122no—
477C22×C3—122no—
478C22×C3—122no—
479C22×C3—122no—
480C22×C3—242no—
481C22×C3—242no—
482C22×C3D3×C22⋊C411yescharacteristic
483C22×C3#67⋊C411yescharacteristic
484C22×C3C2×(C2×D3)⋊C411yescharacteristic
485C22×C3C4×#6711yescharacteristic
486C22×C3C4×#6711yescharacteristic
487C22×C3C4×#6711yescharacteristic
488C2×D3—21no—
489C2×D3—21no—
490C2×D3—21no—
491C2×D3—21no—
492C2×D3—21no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—42no—
497C2×D3—42no—
498C2×D3—82no—
499C2×D3—82no—
500C2×D3#82312yescomplemented
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—61no—
506C3×C4—61no—
507C3×C4#88311yescharacteristic
508C3×C4#88411yescharacteristic
509C3×C4#92911yescharacteristic
510C3×C4#93011yescharacteristic
511C3×C4#93011yescharacteristic
512C3×C4#93711yescharacteristic
513C3×C4#93811yescharacteristic
514C3×C4#95311yescharacteristic
515Dic3—21no—
516Dic3—21no—
517Dic3—122no—
518Dic3—122no—
519Dic3—242no—
520C32#190011yescharacteristic, complemented, Sylow
521C23—31no—
522C23—31no—
523C23—62no—
524C23—61no—
525C23—61no—
526C23—62no—
527C23—62no—
528C23—62no—
529C23—121no—
530C23—122no—
531C23—122no—
532C23C4×D3211yescharacteristic
533D4—121no—
534D4—121no—
535D4—121no—
536D4—121no—
537D4—364no—
538D4—364no—
539C2×C4—31no—
540C2×C4—31no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—182no—
552C2×C4—362no—
553C2×C4#379511yescharacteristic
554C2×C4#379811yescharacteristic
555C2×C4D3×D1211yescharacteristic
556C2×C4C2×#62511yescharacteristic
557C2×C4D3×#6711yescharacteristic
558C2×C4D3×#6711yescharacteristic
559C8—22no—
560C8—62no—
561C8—121no—
562C8—361no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—41no—
571C2×C3—41no—
572C2×C3—62no—
573C2×C3—121no—
574C2×C3—242no—
575C2×C3D3×#10011yescharacteristic
576C2×C3#474511yescharacteristic
577C2×C3#495611yescharacteristic
578C2×C3C2×#82311yescharacteristic
579C2×C3#513211yescharacteristic
580C2×C3#521311yescharacteristic
581D3—22no—
582D3—41no—
583D3—82no—
584C22—21no—
585C22—21no—
586C22—32no—
587C22—61no—
588C22—61no—
589C22—61no—
590C22—61no—
591C22—61no—
592C22—122no—
593C22—122no—
594C22—122no—
595C22—122no—
596C22—122no—
597C22—242no—
598C22—242no—
599C22#6387911yescharacteristic
600C22C4×#62511yescharacteristic
601C22D3×(C2×D3)⋊C411yescharacteristic
602C4—61no—
603C4—61no—
604C4—362no—
605C4#6390511yescharacteristic
606C4#6390611yescharacteristic
607C4#6390811yescharacteristic
608C4#6397611yescharacteristic
609C3—21no—
610C3#7008111yescharacteristic, complemented
611C3#7200211yescharacteristic, complemented
612C2—21no—
613C2—21no—
614C2—62no—
615C2—121no—
616C2—242no—
617C2D3×#82311yescharacteristic
618C2#1059100611yescharacteristic
619C2#1059138111yescharacteristic
subgroupG/H
centerC22D3×(C2×D3)⋊C4
commutatorC23×C32C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32C2×C22⋊C4
prime radicalC2×#63908C2
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1900
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×M16C2semidirect
3. C2×C3×#823C2semidirect
4. C2×#63558C2semidirect
5. #10591069C2semidirect
6. #10589420C2semidirect
7. #10589168C2semidirect
34. C2×D3×M16D3semidirect
39. C2×#823D3semidirect
69. C32×M16D4semidirect
79. C3×#208D4semidirect
236. C3×M16#67semidirect
237. C3×M16#67semidirect
263. #208#67semidirect
288. C2×C3×D3#100semidirect
450. M16#625semidirect
500. C2×D3#823semidirect
520. C32#1900semidirect
610. C3#70081semidirect
611. C3#72002semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×M162
C2×#639082
C2×C3×#8232
C2×#635582
#105910692
#105894202
#105891682
#720023
#700813