Finite Group Databasebeta

← #29585329

Subgroups — #29585329

Order 15616.

1434 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840236C211yesmaximal, characteristic
2C22×#14519693C211yesmaximal, characteristic
3#21448399C211yesmaximal, characteristic
4#21445528C211yesmaximal, characteristic, complemented
5C2×#16839519C211yesmaximal, characteristic, complemented
6C2×#16839518C211yesmaximal, characteristic, complemented
7C61×#1733C211yesmaximal, characteristic, complemented
8C23×D244C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10#16840236—22no—
11#16840236—22no—
12#16840236—22no—
13#16840236C414yescomplemented
14C2×C4×D244—22no—
15C2×#14519739—22no—
16C2×#14519736C2211yescharacteristic
17C2×#14519736C2211yescharacteristic
18C2×C22⋊C4×D61—22no—
19C2×#14519693—22no—
20C2×#14519693—22no—
21C2×#14519693C414yescomplemented
22#16839664—22no—
23#16839663C414yescomplemented
24#16839617—22no—
25#16839521—22no—
26#16839519—22no—
27#16839518—22no—
28C22×M16×C61C2211yescharacteristic
29C2×#117×C61C2211yescharacteristic
30C2×#117×C61C2211yescharacteristic
31C61×#339—22no—
32C24×D61D412yes—
33C22×D244—21no—
34C22×D244—21no—
35C22×D244—21no—
36C22×D244—21no—
37C22×D244—21no—
38C22×D244—21no—
39C22×D244—21no—
40C22×D244—21no—
41C22×D244C2×C412yes—
42C22×D244C2×C412yes—
43C22×D244C2×C412yes—
44C22×D244D412yes—
45C22×D244D412yes—
46C22×D244Q812yes—
47C22×C4×D61—24no—
48C2×#11758194—21no—
49C2×#11758194—21no—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194C2×C412yescomplemented
54C22×#92418—21no—
55C4×D244—28no—
56C4×D244—28no—
57#14519739—28no—
58#14519739—28no—
59#14519736—21no—
60#14519736—21no—
61#14519736—21no—
62#14519736—21no—
63#14519736—21no—
64#14519736—21no—
65#14519736C2×C412yes—
66#14519736D412yes—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C22⋊Dic61—21no—
70C2×C22⋊Dic61—21no—
71C2×#11758198—24no—
72C2×#11758198—28no—
73C22⋊C4×D61—28no—
74C22⋊C4×D61—28no—
75C2×C4×Dic61C2×C412yescomplemented
76C2×C4×Dic61D412yes—
77#14519693—28no—
78C23×C4×C61C2311yescharacteristic
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61—21no—
82C2×M16×C61—21no—
83C2×M16×C61—21no—
84C2×M16×C61C2×C412yescomplemented
85C22×C8×C61—21no—
86#117×C61—21no—
87#117×C61—21no—
88#117×C61—21no—
89#117×C61—21no—
90#117×C61—21no—
91#117×C61—21no—
92#117×C61C2×C412yes—
93#117×C61D412yes—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61C2×C412yescomplemented
96C2×C22⋊C4×C61—21no—
97C2×C22⋊C4×C61—21no—
98C2×C42×C61—21no—
99C2×C42×C61D412yes—
100C24×C61C2×D411yescharacteristic
101C23×D61—22no—
102C23×D61—22no—
103C23×D61—22no—
104C23×D61—22no—
105C23×D61—84no—
106C23×D61C22⋊C414yes—
107C23×D61C22⋊C414yes—
108C23×D61C4⋊C414yes—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61C22×C411yescharacteristic
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×Q811yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C22×Dic61—22no—
126C22×Dic61—22no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244C4214yes—
152C2×D244C22⋊C414yes—
153C2×D244C4⋊C414yes—
154C2×D244C4⋊C414yes—
155C2×C4×D61—216no—
156C2×C4×D61—88no—
157M16×C61—22no—
158M16×C61—22no—
159M16×C61—41no—
160M16×C61—41no—
161C2×C8×C61—22no—
162C2×C8×C61—41no—
163C2×C8×C61—41no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—41no—
171C22⋊C4×C61—41no—
172C42×C61—22no—
173C42×C61—41no—
174C42×C61—41no—
175C42×C61C22⋊C414yescomplemented
176C22⋊Dic61—22no—
177C22⋊Dic61—22no—
178C22⋊Dic61—41no—
179C22⋊Dic61—41no—
180#11758198—216no—
181#11758198—88no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61—41no—
185C4⋊Dic61—41no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189C4×Dic61C22⋊C414yescomplemented
190#11758194—22no—
191#11758194—22no—
192#11758194—41no—
193#11758194—41no—
194C2×#92418—22no—
195C2×#92418—41no—
196C2×#92418—41no—
197C23×C61—42no—
198C23×C61C2×C22⋊C411yescharacteristic
199C23×C61C2×C22⋊C411yescharacteristic
200C23×C61C2×C4⋊C411yescharacteristic
201C23×C61#12011yescharacteristic
202C23×C61#12011yescharacteristic
203C23×C61#12311yescharacteristic
204C23×C61#12311yescharacteristic
205C22×D61—24no—
206C22×D61—24no—
207C22×D61—24no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61—24no—
211C22×D61—48no—
212C22×D61—48no—
213C22×D61—84no—
214C22×D61—84no—
215C22×D61—84no—
216C22×D61—84no—
217C22×D61—84no—
218C22×D61#9518yes—
219C2×C4×C61—24no—
220C2×C4×C61—44no—
221C2×C4×C61—44no—
222C2×C4×C61—42no—
223C2×C4×C61C2×C4211yescharacteristic
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C4⋊C411yescharacteristic
228C2×C4×C61C2×C4⋊C411yescharacteristic
229C2×C4×C61#11711yescharacteristic
230C2×C4×C61#11711yescharacteristic
231C2×C4×C61C4×D411yescharacteristic
232C2×C4×C61C4×D411yescharacteristic
233C2×C4×C61C4×D411yescharacteristic
234C2×C4×C61C4×D411yescharacteristic
235C2×C4×C61C4×D411yescharacteristic
236C2×C4×C61C4×D411yescharacteristic
237C2×C4×C61C4×D411yescharacteristic
238C2×C4×C61C4×D411yescharacteristic
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61#12111yescharacteristic
242C2×C4×C61#12111yescharacteristic
243C2×C4×C61#12111yescharacteristic
244C2×C4×C61#12111yescharacteristic
245C2×C4×C61C22⋊Q811yescharacteristic
246C2×C4×C61C22⋊Q811yescharacteristic
247C2×C4×C61C22⋊Q811yescharacteristic
248C2×C4×C61C22⋊Q811yescharacteristic
249C2×C4×C61#12311yescharacteristic
250C2×C4×C61#12311yescharacteristic
251C2×Dic61—24no—
252C2×Dic61—44no—
253C2×Dic61—44no—
254C2×Dic61—42no—
255D244—24no—
256D244—24no—
257D244—24no—
258D244—24no—
259D244—24no—
260D244—24no—
261D244—24no—
262D244—24no—
263C4×D61—816no—
264C8×C61—42no—
265#92418—42no—
266#10740—611nomaximal, Sylow
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61C2×#9511yescharacteristic
275C22×C61#37911yescharacteristic
276C22×C61#38111yescharacteristic
277C22×C61#38511yescharacteristic
278C22×C61#38511yescharacteristic
279C22×C61#38611yescharacteristic
280C22×C61#38611yescharacteristic
281C2×D61—28no—
282C2×D61—48no—
283C2×D61—48no—
284C2×D61—48no—
285C2×D61—48no—
286C2×D61—84no—
287C2×D61—84no—
288C2×D61—84no—
289C4×C61—44no—
290C4×C61—82no—
291C4×C61C4×C22⋊C411yescharacteristic
292C4×C61C4×C22⋊C411yescharacteristic
293C4×C61#38011yescharacteristic
294C4×C61#38011yescharacteristic
295C4×C61#38511yescharacteristic
296C4×C61#38511yescharacteristic
297C4×C61#38611yescharacteristic
298C4×C61#38611yescharacteristic
299Dic61—44no—
300Dic61—82no—
301C2×#518—611no—
302C22×#100—611no—
303#2309—611no—
304#1733—611no—
305#1733D6111yescharacteristic, complemented
306C2×H(3,ℤ/4)—611no—
307C2×H(3,ℤ/4)—611no—
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61—21no—
313C2×C61—21no—
314C2×C61C22⋊#9511yescharacteristic
315C2×C61#191112yes—
316D61—48no—
317C23×D4—611no—
318C22×M16—611no—
319C22×M16C2×D6111yescharacteristic
320#518—614no—
321#518—1222no—
322#518—1222no—
323#518—1222no—
324C2×C4×D4—1222no—
325C2×C4×D4—1222no—
326C2×#117—611no—
327C2×#117—611no—
328C2×#117C2×D6111yescharacteristic
329C2×#117C2×D6111yescharacteristic
330C22×C22⋊C4—1222no—
331C2×#100—614no—
332C2×#100—1222no—
333C2×#100—1222no—
334#390—614no—
335#390—1222no—
336#379—1222no—
337#339—22no—
338#339—1222no—
339H(3,ℤ/4)—1222no—
340H(3,ℤ/4)—1222no—
341C61#1074011yescharacteristic, complemented, Sylow
342C25—612no—
343C22×D4—612no—
344C22×D4—612no—
345C22×D4—612no—
346C22×D4—612no—
347C22×D4—612no—
348C22×D4—612no—
349C22×D4—1221no—
350C22×D4—1221no—
351C22×D4—1221no—
352C22×D4—1221no—
353C22×D4—1221no—
354C22×D4—1221no—
355C22×D4—1221no—
356C22×D4—1221no—
357C23×C4—1224no—
358C23×C4C22×D6111yescharacteristic
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—612no—
365C2×M16—1221no—
366C2×M16—1221no—
367C2×M16—1221no—
368C2×M16—1221no—
369C2×M16—1221no—
370C2×M16C4×D6112yescomplemented
371C22×C8—21no—
372C22×C8—1221no—
373C4×D4—1228no—
374C4×D4—1228no—
375C4×D4—1228no—
376C4×D4—1228no—
377#117—21no—
378#117—21no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117—612no—
384#117—612no—
385#117—1221no—
386#117—1221no—
387#117—1221no—
388#117—1221no—
389#117—1221no—
390#117—1221no—
391#117C4×D6112yes—
392#117D24412yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4C4×D6112yescomplemented
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—1228no—
400C2×C22⋊C4—1228no—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1228no—
404C2×C22⋊C4—1221no—
405C2×C42—21no—
406C2×C42—612no—
407C2×C42—612no—
408C2×C42C22⋊D6112yes—
409#100—1228no—
410C24—614no—
411C24—614no—
412C24—614no—
413C24—1222no—
414C24—1222no—
415C24—1222no—
416C24—1222no—
417C24—4884no—
418C24D4×D6111yescharacteristic
419C2×D4—614no—
420C2×D4—614no—
421C2×D4—614no—
422C2×D4—614no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1222no—
436C2×D4—1222no—
437C2×D4—1222no—
438C2×D4—1222no—
439C2×D4—1222no—
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—1222no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—12216no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—4888no—
453C22×C4C2×C4×D6111yescharacteristic
454C22×C4C2×C4×D6111yescharacteristic
455C22×C4C2×C4×D6111yescharacteristic
456C22×C4C2×D24411yescharacteristic
457C22×C4#1175822211yescharacteristic
458C22×C4#1175822211yescharacteristic
459C22×C4D4×D6111yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4#1175822411yescharacteristic
464C22×C4Q8×D6111yescharacteristic
465C22×C4#1175822611yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467M16—22no—
468M16—22no—
469M16—41no—
470M16—41no—
471M16—1222no—
472M16—1222no—
473M16—2441no—
474M16—2441no—
475C2×C8—22no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—1222no—
479C2×C8—2441no—
480C2×C8—2441no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—2441no—
488C4⋊C4—2441no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—12216no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C22⋊C4—4888no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C42—22no—
500C42—41no—
501C42—41no—
502C42—614no—
503C42—1222no—
504C42—1222no—
505C42—1222no—
506C42#1175819814yescomplemented
507C23—42no—
508C23—618no—
509C23—1224no—
510C23—1224no—
511C23—1224no—
512C23—1224no—
513C23—1224no—
514C23—1224no—
515C23—4884no—
516C23—4884no—
517C23—4884no—
518C23—4884no—
519C23—4884no—
520C23—2448no—
521C23—2448no—
522C23C22⋊C4×D6111yescharacteristic
523C23C22⋊C4×D6111yescharacteristic
524C23C4⋊C4×D6111yescharacteristic
525C23#1451974511yescharacteristic
526C23#1451974711yescharacteristic
527C23#1451975911yescharacteristic
528C23#1451976211yescharacteristic
529D4—1224no—
530D4—1224no—
531D4—1224no—
532D4—1224no—
533D4—1224no—
534D4—1224no—
535D4—1224no—
536D4—1224no—
537C2×C4—24no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—1224no—
542C2×C4—48816no—
543C2×C4—2444no—
544C2×C4—2442no—
545C2×C4—2444no—
546C2×C4C42×D6111yescharacteristic
547C2×C4C22⋊C4×D6111yescharacteristic
548C2×C4C22⋊C4×D6111yescharacteristic
549C2×C4C2×#1175819811yescharacteristic
550C2×C4C4⋊C4×D6111yescharacteristic
551C2×C4C4⋊C4×D6111yescharacteristic
552C2×C4#1451973411yescharacteristic
553C2×C4#1451973511yescharacteristic
554C2×C4#1451973711yescharacteristic
555C2×C4#1451973711yescharacteristic
556C2×C4C4×C22⋊D6111yescharacteristic
557C2×C4C4×C22⋊D6111yescharacteristic
558C2×C4#1451973911yescharacteristic
559C2×C4#1451973911yescharacteristic
560C2×C4C4×D24411yescharacteristic
561C2×C4C4×D24411yescharacteristic
562C2×C4#1451974511yescharacteristic
563C2×C4#1451974711yescharacteristic
564C2×C4#1451974811yescharacteristic
565C2×C4#1451975011yescharacteristic
566C2×C4#1451975011yescharacteristic
567C2×C4#1451975211yescharacteristic
568C2×C4#1451975311yescharacteristic
569C2×C4#1451975511yescharacteristic
570C2×C4#1451975511yescharacteristic
571C2×C4#1451975711yescharacteristic
572C2×C4#1451975911yescharacteristic
573C2×C4#1451976211yescharacteristic
574C8—42no—
575C8—2442no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—1228no—
584C22—4884no—
585C22—4884no—
586C22—4884no—
587C22—2448no—
588C22—2448no—
589C22—2448no—
590C22—2448no—
591C22#95×D6111yescharacteristic
592C22#1683961711yescharacteristic
593C22#1683962411yescharacteristic
594C22#1683963311yescharacteristic
595C22#1683963611yescharacteristic
596C22#1683963811yescharacteristic
597C22#1683964011yescharacteristic
598C4—44no—
599C4—82no—
600C4—4882no—
601C4—2444no—
602C4#1683960911yescharacteristic
603C4C4×#1175819811yescharacteristic
604C4#1683961911yescharacteristic
605C4#1683962111yescharacteristic
606C4#1683963311yescharacteristic
607C4#1683963611yescharacteristic
608C4#1683963811yescharacteristic
609C4#1683964011yescharacteristic
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—21no—
615C2—21no—
616C2—2448no—
617C2#2144404011yescharacteristic
618C2#2144634812yes—
subgroupG/H
centerC22#16839638
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC61×#1733C2
socleC22×C61#386
prime radicalC23×D244C22
cosocleC23×C4×C61C23
hypercenter#1733D61
hypocenterC61#10740
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
4. #21445528C2semidirect
5. C2×#16839519C2semidirect
6. C2×#16839518C2semidirect
7. C61×#1733C2semidirect
13. #16840236C4semidirect
21. C2×#14519693C4semidirect
23. #16839663C4semidirect
53. C2×#11758194C2×C4semidirect
75. C2×C4×Dic61C2×C4semidirect
84. C2×M16×C61C2×C4semidirect
95. C2×C4⋊C4×C61C2×C4semidirect
175. C42×C61C22⋊C4semidirect
189. C4×Dic61C22⋊C4semidirect
305. #1733D61semidirect
341. C61#10740semidirect
370. C2×M16C4×D61semidirect
396. C2×C4⋊C4C4×D61semidirect
506. C42#11758198semidirect

Maximal subgroups

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

subgroupindex
C2×#168402362
C22×#145196932
#214483992
#214455282
C2×#168395192
C2×#168395182
C61×#17332
#1074061