Finite Group Databasebeta

← #11788142

Subgroups — #11788142

Order 1152.

881 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
2C2×#63908C211yesmaximal, characteristic
3C2×C3×#821C211yesmaximal, characteristic, complemented
4#10591313C211yesmaximal, characteristic, complemented
5C2×#63563C211yesmaximal, characteristic, complemented
6#10589417C211yesmaximal, characteristic, complemented
7#10589164C211yesmaximal, characteristic, complemented
8#71990—31nomaximal
9#70203—31nomaximal
10C22×C3×D12C2211yescharacteristic
11C22×C3×C4×D3C2211yescharacteristic
12C22×#625—21no—
13C22×C32×C8C2211yescharacteristic
14C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C4⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊C8C2211yescharacteristic
18C2×C3×ΓL(1,𝔽25)—22no—
19C2×C3×ΓL(1,𝔽25)—22no—
20C2×C3×ΓL(1,𝔽25)—22no—
21C2×(C2×D3)⋊Dic3—21no—
22C2×(C2×D3)⋊Dic3—21no—
23#63908—22no—
24#63908C414yescomplemented
25C3×#821—22no—
26#63579—21no—
27#63563—22no—
28C22×C3×M16—31no—
29C2×#938—31no—
30C2×#930—31no—
31C22×ΓL(1,𝔽25)D311yescharacteristic, complemented
32C2×C3×#102—31no—
33C2×#832—31no—
34#5212—31no—
35C2×#821—31no—
36C2×#821D311yescharacteristic, complemented
37#4987—31no—
38#4551—31no—
39#4551—31no—
40#4551—31no—
41#4551—61no—
42#4529—31no—
43C23×C3×D3—21no—
44C23×C3×D3D411yescharacteristic
45C22×C32×C4C2311yescharacteristic
46C22×C3⋊Dic3—21no—
47C22×C3⋊Dic3—21no—
48C22×C3×Dic3—21no—
49C22×C3×Dic3D411yescharacteristic
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12—21no—
53C2×C3×D12—21no—
54C2×C3×D12—21no—
55C2×C3×D12C2×C412yes—
56C2×C3×C4×D3—21no—
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3—21no—
60C2×C3×C4×D3C2×C412yes—
61C2×C3×C4×D3D412yes—
62C2×#625—24no—
63C2×#625—42no—
64C2×#625—42no—
65C2×C32×C8—21no—
66C2×C32×C8—21no—
67C2×C32×C8D412yescomplemented
68C4⋊(C3⋊Dic3)—21no—
69C4⋊(C3⋊Dic3)—21no—
70C4⋊(C3⋊Dic3)—21no—
71C4⋊(C3⋊Dic3)C2×C412yes—
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8D412yescomplemented
77C3×ΓL(1,𝔽25)—28no—
78(C2×D3)⋊Dic3—24no—
79(C2×D3)⋊Dic3—42no—
80#1895—91noSylow
81C22×C3×D4—31no—
82C23×C3×C4—31no—
83C22×#67—61no—
84C22×#67—61no—
85C22×D12C2×D311yescharacteristic
86C22×C4×D3C2×D311yescharacteristic
87C2×C3×M16—62no—
88C2×C3×M16—62no—
89C2×C3×M16—62no—
90C22×C3×C8—21no—
91C22×C3×C8—31no—
92C22×C3×C8C2×D311yescharacteristic
93C22×C3×C8C2×D311yescharacteristic
94C2×C3×C4⋊C4—31no—
95C2×#217—61no—
96C2×#217—61no—
97#938—34no—
98#938—62no—
99#930—34no—
100#930—62no—
101C2×(C2×D3)⋊C4—61no—
102C2×(C2×D3)⋊C4—61no—
103C2×C4⋊Dic3—31no—
104C2×C4⋊Dic3—31no—
105C2×C4⋊Dic3—31no—
106C2×C4⋊Dic3—31no—
107C2×C4⋊Dic3—61no—
108C2×C4⋊Dic3—61no—
109C2×C4⋊Dic3C2×D311yescharacteristic
110C22×C3⋊C8C2×D311yescharacteristic
111C2×ΓL(1,𝔽25)—22no—
112C2×ΓL(1,𝔽25)—22no—
113C2×ΓL(1,𝔽25)—22no—
114C3×#102—62no—
115#832—62no—
116C3⋊#95—61no—
117C3⋊#95—61no—
118#821—22no—
119#821—62no—
120C23×C32C2×D411yescharacteristic
121C22×C3×D3—21no—
122C22×C3×D3—21no—
123C22×C3×D3—21no—
124C22×C3×D3—21no—
125C22×C3×D3—21no—
126C22×C3×D3—21no—
127C22×C3×D3—22no—
128C22×C3×D3—41no—
129C22×C3×D3—41no—
130C22×C3×D3—41no—
131C22×C3×D3—41no—
132C22×C3×D3—41no—
133C22×C3×D3—41no—
134C22×C3×D3C22⋊C412yes—
135C2×C32×C4C22×C411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4#4111yescharacteristic
140C2×C32×C4#4111yescharacteristic
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—41no—
143C2×C3⋊Dic3—41no—
144C2×C3⋊Dic3—41no—
145C2×C3⋊Dic3—41no—
146C2×C3⋊Dic3—41no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—41no—
150C2×C3×Dic3—41no—
151C2×C3×Dic3—41no—
152C2×C3×Dic3C22⋊C412yes—
153C3×D12—22no—
154C3×D12—22no—
155C3×D12—41no—
156C3×D12—41no—
157C3×C4×D3—22no—
158C3×C4×D3—22no—
159C3×C4×D3—22no—
160C3×C4×D3—22no—
161#625—48no—
162C32×C8—22no—
163C3×C3⋊C8—22no—
164C22×M16—31no—
165C2×#121—91no—
166C2×#102—31no—
167C2×#102—91no—
168#380—91no—
169#340—91no—
170#340—91no—
171C24×C3—31no—
172C24×C3—61no—
173C23×D3—21no—
174C23×D3D1211yescharacteristic
175C2×C3×D4—32no—
176C2×C3×D4—61no—
177C2×C3×D4—61no—
178C2×C3×D4—61no—
179C2×C3×D4—61no—
180C2×C3×D4—61no—
181C22×C3×C4—21no—
182C22×C3×C4—31no—
183C22×C3×C4—32no—
184C22×C3×C4—32no—
185C22×C3×C4—61no—
186C22×C3×C4—61no—
187C22×C3×C4—61no—
188C22×C3×C4—61no—
189C22×C3×C4—61no—
190C22×C3×C4C22×D311yescharacteristic
191C22×C3×C4C22×D311yescharacteristic
192C2×#67—64no—
193C2×#67—64no—
194C2×#67—122no—
195C2×#67—122no—
196C2×#67—122no—
197C2×#67—122no—
198C22×Dic3—21no—
199C22×Dic3—61no—
200C22×Dic3—61no—
201C22×Dic3—61no—
202C22×Dic3—61no—
203C22×Dic3—121no—
204C22×Dic3—121no—
205C22×Dic3D1211yescharacteristic
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12C4×D312yes—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3C4×D312yes—
217C2×C4×D3#6712yes—
218C3×M16—68no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—22no—
224C2×C3×C8—32no—
225C2×C3×C8—41no—
226C2×C3×C8—41no—
227C2×C3×C8—61no—
228C2×C3×C8—61no—
229C2×C3×C8#6712yescomplemented
230C2×C3×C8#6712yescomplemented
231C3×C4⋊C4—61no—
232C3×C4⋊C4—61no—
233#217—64no—
234#217—122no—
235(C2×D3)⋊C4—64no—
236(C2×D3)⋊C4—122no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—32no—
240C4⋊Dic3—32no—
241C4⋊Dic3—62no—
242C4⋊Dic3—61no—
243C4⋊Dic3—61no—
244C4⋊Dic3—62no—
245C4⋊Dic3—61no—
246C4⋊Dic3—61no—
247C4⋊Dic3—61no—
248C4⋊Dic3—61no—
249C4⋊Dic3—121no—
250C4⋊Dic3—121no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8#6712yescomplemented
254ΓL(1,𝔽25)—28no—
255C22×C32C2×C22⋊C411yescharacteristic
256C22×C32#12111yescharacteristic
257C22×C32#12111yescharacteristic
258C22×C32AGL(1,ℤ/8)11yescharacteristic
259C22×C32AGL(1,ℤ/8)11yescharacteristic
260C22×C32AGL(1,ℤ/8)11yescharacteristic
261C22×C32#13711yescharacteristic
262C2×C3×D3—22no—
263C2×C3×D3—22no—
264C2×C3×D3—22no—
265C2×C3×D3—41no—
266C2×C3×D3—41no—
267C2×C3×D3—41no—
268C2×C3×D3—41no—
269C2×C3×D3—42no—
270C2×C3×D3—42no—
271C2×C3×D3—42no—
272C2×C3×D3—81no—
273C2×C3×D3—81no—
274C2×C3×D3—81no—
275C2×C3×D3—81no—
276C32×C4C4×D411yescharacteristic
277C32×C4C4×D411yescharacteristic
278C32×C4#12011yescharacteristic
279C32×C4#12311yescharacteristic
280C3⋊Dic3—42no—
281C3⋊Dic3—81no—
282C3×Dic3—22no—
283C3×Dic3—81no—
284C22×D4—31no—
285C22×D4—181no—
286C23×C4—31no—
287C2×M16—62no—
288C2×M16—62no—
289C2×M16—62no—
290C22×C8—31no—
291C22×C8D3211yescharacteristic
292#121—94no—
293#121—182no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—91no—
296C2×C4⋊C4—91no—
297C2×C22⋊C4—181no—
298C2×C22⋊C4—181no—
299#102—62no—
300#102—182no—
301#95—181no—
302C23×C3—21no—
303C23×C3—32no—
304C23×C3—61no—
305C23×C3—61no—
306C23×C3—61no—
307C23×C3—61no—
308C23×C3—61no—
309C23×C3—61no—
310C23×C3—62no—
311C23×C3—121no—
312C23×C3—121no—
313C23×C3—121no—
314C23×C3—121no—
315C23×C3—121no—
316C23×C3—121no—
317C23×C3C2×D1211yescharacteristic
318C23×C3D3×D411yescharacteristic
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3—21no—
323C22×D3—21no—
324C22×D3—21no—
325C22×D3—22no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3(C2×D3)⋊C412yes—
333C3×D4—62no—
334C3×D4—62no—
335C3×D4—121no—
336C3×D4—121no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—32no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—62no—
349C2×C3×C4—62no—
350C2×C3×C4—121no—
351C2×C3×C4—121no—
352C2×C3×C4—121no—
353C2×C3×C4C2×C4×D311yescharacteristic
354C2×C3×C4C2×C4×D311yescharacteristic
355C2×C3×C4#23511yescharacteristic
356C2×C3×C4#23511yescharacteristic
357C2×C3×C4#23511yescharacteristic
358C2×C3×C4D3×D411yescharacteristic
359C2×C3×C4D3×D411yescharacteristic
360C2×C3×C4#23711yescharacteristic
361C2×C3×C4C2×#6711yescharacteristic
362C2×C3×C4C2×#6711yescharacteristic
363C2×C3×C4C2×#6711yescharacteristic
364C2×C3×C4C2×#6711yescharacteristic
365#67—128no—
366#67—128no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—62no—
373C2×Dic3—62no—
374C2×Dic3—122no—
375C2×Dic3—121no—
376C2×Dic3—121no—
377C2×Dic3—121no—
378C2×Dic3—121no—
379C2×Dic3—122no—
380C2×Dic3—121no—
381C2×Dic3—121no—
382C2×Dic3—121no—
383C2×Dic3—121no—
384C2×Dic3—121no—
385C2×Dic3—121no—
386C2×Dic3—241no—
387C2×Dic3—241no—
388C2×Dic3—241no—
389C2×Dic3—241no—
390C2×Dic3(C2×D3)⋊C412yes—
391D12—22no—
392D12—22no—
393D12—41no—
394D12—41no—
395C4×D3—22no—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C3×C8—22no—
400C3×C8—22no—
401C3×C8—42no—
402C3×C8—62no—
403C3⋊C8—22no—
404C2×C32#38611yescharacteristic
405C2×C32#41711yescharacteristic
406C2×C32#41811yescharacteristic
407C2×C32#46811yescharacteristic
408C2×C32#46811yescharacteristic
409C2×C32#46911yescharacteristic
410C2×C32#46911yescharacteristic
411C3×D3—42no—
412C3×D3—82no—
413C24—31no—
414C24—61no—
415C2×D4—32no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—184no—
422C2×D4—362no—
423C2×D4—362no—
424C22×C4—31no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—181no—
433C22×C4—181no—
434C22×C4C2×D3211yescharacteristic
435M16—68no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—32no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8#62512yescomplemented
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—92no—
445C4⋊C4—181no—
446C4⋊C4—181no—
447C4⋊C4—181no—
448C22⋊C4—184no—
449C22⋊C4—362no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—62no—
458C22×C3—62no—
459C22×C3—62no—
460C22×C3—121no—
461C22×C3—121no—
462C22×C3—121no—
463C22×C3—121no—
464C22×C3—122no—
465C22×C3—122no—
466C22×C3—122no—
467C22×C3—241no—
468C22×C3—241no—
469C22×C3—241no—
470C22×C3—241no—
471C22×C3D3×C22⋊C411yescharacteristic
472C22×C3#88411yescharacteristic
473C22×C3#90811yescharacteristic
474C22×C3#90811yescharacteristic
475C22×C3#90811yescharacteristic
476C22×C3#90911yescharacteristic
477C22×C3#91111yescharacteristic
478C22×C3#91111yescharacteristic
479C22×C3#91411yescharacteristic
480C22×C3#91511yescharacteristic
481C22×C3C2×(C2×D3)⋊C411yescharacteristic
482C22×C3#93011yescharacteristic
483C22×C3#93011yescharacteristic
484C22×C3#93811yescharacteristic
485C2×D3—22no—
486C2×D3—22no—
487C2×D3—22no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—81no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—62no—
504C3×C4—241no—
505C3×C4#67⋊C411yescharacteristic
506C3×C4#88311yescharacteristic
507C3×C4C4×#6711yescharacteristic
508C3×C4C4×#6711yescharacteristic
509C3×C4C4×#6711yescharacteristic
510C3×C4#92911yescharacteristic
511C3×C4#93711yescharacteristic
512C3×C4#95311yescharacteristic
513Dic3—22no—
514Dic3—81no—
515Dic3—122no—
516Dic3—122no—
517Dic3—242no—
518Dic3—242no—
519Dic3—241no—
520Dic3—241no—
521C32#189511yescharacteristic, complemented, Sylow
522C23—32no—
523C23—61no—
524C23—61no—
525C23—61no—
526C23—61no—
527C23—61no—
528C23—61no—
529C23—62no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23D3×D1211yescharacteristic
537D4—62no—
538D4—62no—
539D4—121no—
540D4—121no—
541D4—368no—
542C2×C4—32no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—182no—
553C2×C4—361no—
554C2×C4—361no—
555C2×C4—361no—
556C2×C4—361no—
557C2×C4—361no—
558C2×C4#379511yescharacteristic
559C2×C4#379811yescharacteristic
560C2×C4C4×D3211yescharacteristic
561C2×C4C2×#62511yescharacteristic
562C2×C4D3×#6711yescharacteristic
563C2×C4D3×#6711yescharacteristic
564C8—22no—
565C8—62no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—122no—
574C2×C3—242no—
575C2×C3#477011yescharacteristic
576C2×C3#477111yescharacteristic
577C2×C3#478011yescharacteristic
578C2×C3#478211yescharacteristic
579C2×C3#495611yescharacteristic
580C2×C3#513111yescharacteristic
581C2×C3#513311yescharacteristic
582C2×C3#513411yescharacteristic
583C2×C3#513411yescharacteristic
584C2×C3#513511yescharacteristic
585C2×C3#513511yescharacteristic
586C2×C3#515911yescharacteristic
587C2×C3#517411yescharacteristic
588C2×C3#521311yescharacteristic
589D3—42no—
590D3—82no—
591C22—62no—
592C22—62no—
593C22—62no—
594C22—121no—
595C22—121no—
596C22—121no—
597C22—121no—
598C22—122no—
599C22—122no—
600C22—122no—
601C22—241no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22#6379411yescharacteristic
606C22#6379511yescharacteristic
607C22#6379511yescharacteristic
608C22#6380011yescharacteristic
609C22#6390611yescharacteristic
610C22#6390811yescharacteristic
611C22D3×(C2×D3)⋊C411yescharacteristic
612C4—62no—
613C4—241no—
614C4—362no—
615C4—721no—
616C4#6387911yescharacteristic
617C4C4×#62511yescharacteristic
618C4#6390511yescharacteristic
619C4#6397611yescharacteristic
620C3—21no—
621C3#7020311yescharacteristic, complemented
622C3#7199011yescharacteristic, complemented
623C2—122no—
624C2—242no—
625C2#1059067011yescharacteristic
626C2#1059082711yescharacteristic
627C2#1059082811yescharacteristic
628C2#1059083311yescharacteristic
629C2#1059083611yescharacteristic
630C2#1059093911yescharacteristic
631C2#1059138111yescharacteristic
subgroupG/H
centerC23D3×D12
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×C8C22
socleC23×C32C2×D4
prime radicalC2×#63908C2
cosocleC22×C32×C4C23
hypercenterC22×C8D32
hypocenterC32#1895
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C22×C3×ΓL(1,𝔽25)C2semidirect
3. C2×C3×#821C2semidirect
4. #10591313C2semidirect
5. C2×#63563C2semidirect
6. #10589417C2semidirect
7. #10589164C2semidirect
24. #63908C4semidirect
31. C22×ΓL(1,𝔽25)D3semidirect
36. C2×#821D3semidirect
67. C2×C32×C8D4semidirect
76. C2×C3×C3⋊C8D4semidirect
229. C2×C3×C8#67semidirect
230. C2×C3×C8#67semidirect
253. C2×C3⋊C8#67semidirect
441. C2×C8#625semidirect
521. C32#1895semidirect
621. C3#70203semidirect
622. C3#71990semidirect

Maximal subgroups

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

subgroupindex
C22×C3×ΓL(1,𝔽25)2
C2×#639082
C2×C3×#8212
#105913132
C2×#635632
#105894172
#105891642
#719903
#702033