Finite Group Databasebeta

← #29681545

Subgroups — #29681545

Order 15616.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519802C211yesmaximal, characteristic
2C2×#16839779C211yesmaximal, characteristic, complemented
3C2×#16839766C211yesmaximal, characteristic, complemented
4#21448399C211yesmaximal, characteristic, complemented
5#21446195C218yesmaximal, complemented
6#21445558C211yesmaximal, characteristic, complemented
7#21445528C211yesmaximal, characteristic, complemented
8C61×#3428C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C2×#14519802—24no—
11C2×#14519802—22no—
12C2×#14519802C418yescomplemented
13C2×#14519802C2214yes—
14C22×#11758201—21no—
15C22×#11758199—21no—
16C22×#11758194C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19#16839779—24no—
20C2×#14519704C2214yescomplemented
21#16839766—22no—
22#16839766C2214yescomplemented
23C2×#14519697—22no—
24#16839664—22no—
25#16839663C2214yescomplemented
26#16839617—22no—
27#16839531C2214yescomplemented
28#16839521C2214yescomplemented
29C22×#41×C61C2211yescharacteristic
30C61×#520—22no—
31C61×#520C2214yescomplemented
32C61×#519—21no—
33C61×#518—21no—
34C61×#518—22no—
35C4×#41×C61—22no—
36C4×#41×C61—22no—
37C2×#117×C61—22no—
38C2×#117×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C24×D61—21no—
41C22×D244—21no—
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—22no—
45C22×D244—21no—
46C22×D244—22no—
47C22×D244—21no—
48C22×D244—22no—
49C22×D244C2×C414yescomplemented
50C22×D244C2312yes—
51#14519802—216no—
52#14519802—28no—
53#14519802C2×C4116yescomplemented
54C2×#11758201—28no—
55C2×#11758201—24no—
56C2×#11758199—28no—
57C2×#11758199—24no—
58C2×#11758194—22no—
59C2×#11758194—21no—
60C2×#11758194—21no—
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194C2×C414yescomplemented
64C2×#11758194C2312yes—
65C2×#11758194C2312yes—
66C22×#92418—21no—
67C22×#92418—21no—
68#14519736—22no—
69#14519736—22no—
70C2×C4⋊Dic61—21no—
71C2×C22⋊Dic61—21no—
72C2×#11758198—28no—
73C2×C4×Dic61C2312yes—
74#14519704—28no—
75#14519697—28no—
76C2×#41×C61—21no—
77C2×#41×C61—21no—
78C2×#41×C61—22no—
79C2×#41×C61—22no—
80C2×#41×C61C2×C414yes—
81C2×#41×C61D414yes—
82C2×#41×C61D412yes—
83C2×#41×C61C2312yes—
84C22×Q8×C61D411yescharacteristic
85C22×D4×C61—21no—
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C23×C4×C61C2311yescharacteristic
89C4×Q8×C61—24no—
90C4×Q8×C61—24no—
91C4×D4×C61—28no—
92C4×D4×C61—24no—
93C4×D4×C61—24no—
94C4×D4×C61—28no—
95#117×C61—24no—
96#117×C61—24no—
97#117×C61—21no—
98#117×C61—21no—
99#117×C61—21no—
100#117×C61—21no—
101#117×C61—21no—
102#117×C61—21no—
103#117×C61—24no—
104#117×C61D412yes—
105#117×C61C2312yes—
106C2×C4⋊C4×C61—24no—
107C2×C4⋊C4×C61D412yes—
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—24no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61D412yes—
112C2×C42×C61—24no—
113C2×C42×C61—21no—
114C2×C42×C61D412yes—
115C24×C61—21no—
116C24×C61C2×D411yescharacteristic
117C23×D61—22no—
118C23×D61—24no—
119C23×D61—41no—
120C23×D61—41no—
121C23×D61—42no—
122C23×D61—84no—
123#41×C61—24no—
124#41×C61—24no—
125#41×C61—24no—
126#41×C61—24no—
127#41×C61—24no—
128#41×C61—22no—
129#41×C61—22no—
130#41×C61C22⋊C418yescomplemented
131#41×C61C22×C414yes—
132#41×C61C2×D414yes—
133C2×Q8×C61—21no—
134C2×Q8×C61—22no—
135C2×Q8×C61C22⋊C414yescomplemented
136C2×Q8×C61C2×D412yes—
137C2×D4×C61—24no—
138C2×D4×C61—22no—
139C2×D4×C61—24no—
140C2×D4×C61—22no—
141C2×D4×C61—21no—
142C2×D4×C61—22no—
143C2×D4×C61C22⋊C414yescomplemented
144C2×D4×C61C2×D412yes—
145C22×C4×C61—21no—
146C22×C4×C61—24no—
147C22×C4×C61—22no—
148C22×C4×C61—24no—
149C22×C4×C61—22no—
150C22×C4×C61—28no—
151C22×C4×C61—22no—
152C22×C4×C61C22×C412yes—
153C22×C4×C61C2×D414yes—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61#4112yes—
161C22×C4×C61#4112yes—
162C22×C4×C61C2411yescharacteristic
163C22×Dic61—22no—
164C2×D244—24no—
165C2×D244—24no—
166C2×D244—24no—
167C2×D244—24no—
168C2×D244—22no—
169C2×D244—22no—
170C2×D244—22no—
171C2×D244—22no—
172C2×D244—24no—
173C2×D244—22no—
174C2×D244—41no—
175C2×D244—41no—
176C2×D244—42no—
177C2×D244—42no—
178C2×D244—42no—
179C2×D244—42no—
180C2×D244—42no—
181C2×D244C22×C414yes—
182C4⋊C4×C61—28no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61C2×D414yes—
187C22⋊C4×C61—28no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C42×C61—28no—
193C42×C61—22no—
194C42×C61—22no—
195C42×C61—22no—
196C42×C61—22no—
197C22⋊Dic61—42no—
198#11758201—216no—
199#11758199—216no—
200#11758198—88no—
201C4⋊Dic61—42no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204#11758194—24no—
205#11758194—22no—
206#11758194—22no—
207#11758194—22no—
208#11758194—42no—
209#11758194C22×C414yes—
210C2×#92418—24no—
211C2×#92418—22no—
212C2×#92418—22no—
213C2×#92418—42no—
214C23×C61—22no—
215C23×C61—24no—
216C23×C61—42no—
217C23×C61—82no—
218C23×C61C4×D412yes—
219C23×C61#12011yescharacteristic
220C23×C61#12011yescharacteristic
221C23×C61#12112yes—
222C23×C61C22×D411yescharacteristic
223C22×D61—24no—
224C22×D61—48no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61—44no—
228C22×D61—44no—
229C22×D61—88no—
230C22×D61—84no—
231C22×D61—84no—
232C22×D61—88no—
233Q8×C61—22no—
234Q8×C61—24no—
235Q8×C61C2×C22⋊C414yes—
236D4×C61—24no—
237D4×C61—24no—
238D4×C61—28no—
239D4×C61—22no—
240D4×C61—24no—
241D4×C61C2×C22⋊C414yes—
242C2×C4×C61—22no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—24no—
246C2×C4×C61—28no—
247C2×C4×C61—24no—
248C2×C4×C61—24no—
249C2×C4×C61—44no—
250C2×C4×C61—82no—
251C2×C4×C61—88no—
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C4×D412yes—
255C2×C4×C61C4×D412yes—
256C2×C4×C61C4×D412yes—
257C2×C4×C61#12011yescharacteristic
258C2×C4×C61#12011yescharacteristic
259C2×C4×C61#12112yes—
260C2×C4×C61#12112yes—
261C2×C4×C61#12112yes—
262C2×C4×C61#12312yes—
263C2×C4×C61#12312yes—
264C2×C4×C61C23×C411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×Dic61—44no—
271C2×Dic61—42no—
272D244—24no—
273D244—24no—
274D244—42no—
275D244—42no—
276D244—44no—
277D244—44no—
278#92418—24no—
279#92418—42no—
280#20945—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—24no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—44no—
288C22×C61—84no—
289C22×C61#38612yes—
290C22×C61#38612yes—
291C22×C61C2×C4×D411yescharacteristic
292C22×C61C2×#12011yescharacteristic
293C22×C61C2×#12111yescharacteristic
294C2×D61—48no—
295C2×D61—44no—
296C2×D61—84no—
297C2×D61—88no—
298C2×D61—88no—
299C2×D61—84no—
300C4×C61—24no—
301C4×C61—44no—
302C4×C61—44no—
303C4×C61#38612yes—
304C4×C61#38612yes—
305C4×C61C22×C22⋊C411yescharacteristic
306C4×C61C2×C4×D411yescharacteristic
307C4×C61C2×#12111yescharacteristic
308C4×C61C2×#12311yescharacteristic
309Dic61—82no—
310C22×AGL(1,ℤ/8)—611no—
311#3428D6111yescharacteristic, complemented
312C2×#421—611no—
313C2×#418—611no—
314#2309—611no—
315#1883—618no—
316#1742—611no—
317#1733—611no—
318C2×C61—21no—
319C2×C61—21no—
320C2×C61—22no—
321C2×C61—22no—
322C2×C61—44no—
323C2×C61C2×#38611yescharacteristic
324C2×C61#302112yes—
325D61—84no—
326C22×#41C2×D6111yescharacteristic
327C23×D4—611no—
328C2×AGL(1,ℤ/8)—618no—
329C2×AGL(1,ℤ/8)—614no—
330C2×AGL(1,ℤ/8)—1224no—
331C2×AGL(1,ℤ/8)—1222no—
332C22×SD16—1221no—
333C22×D8—1221no—
334C22×M16—611no—
335C22×M16—611no—
336#520—22no—
337#520C2×D6114yescomplemented
338#519—21no—
339#518—21no—
340#518—22no—
341C4×#41—22no—
342C4×#41—22no—
343C2×#117—22no—
344C2×#117—611no—
345C2×#117C2×D6111yescharacteristic
346C2×#117C2×D6111yescharacteristic
347#421—1224no—
348C2×#104—614no—
349#418—614no—
350#418—1222no—
351C2×#102—1222no—
352#390—614no—
353#390—1222no—
354#379—1222no—
355#343—614no—
356#339—614no—
357C61#2094511yescharacteristic, complemented, Sylow
358C25—1221no—
359C2×#41—22no—
360C2×#41—22no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41C4×D6114yes—
364C2×#41D24414yes—
365C2×#41C22⋊D6112yes—
366C2×#41C22×D6112yes—
367C22×Q8C22⋊D6111yescharacteristic
368C22×D4—21no—
369C22×D4—614no—
370C22×D4—612no—
371C22×D4—1222no—
372C22×D4—1222no—
373C22×D4—1222no—
374C22×D4—1221no—
375C22×D4—1221no—
376C22×D4—1221no—
377C22×D4—1221no—
378C22×D4—1221no—
379C22×D4C22⋊D6111yescharacteristic
380C23×C4—21no—
381C23×C4C22×D6111yescharacteristic
382AGL(1,ℤ/8)—6116no—
383AGL(1,ℤ/8)—12216no—
384AGL(1,ℤ/8)—1228no—
385C2×SD16—1228no—
386C2×SD16—1224no—
387C2×D8—1228no—
388C2×D8—1224no—
389C2×M16—614no—
390C2×M16—612no—
391C2×M16—612no—
392C2×M16—1222no—
393C2×M16—1221no—
394C2×M16—1221no—
395C2×M16—1222no—
396C2×M16—1222no—
397C22×C8—1221no—
398C22×C8—1221no—
399C4×Q8—24no—
400C4×Q8—24no—
401C4×D4—28no—
402C4×D4—24no—
403C4×D4—24no—
404C4×D4—28no—
405#117—24no—
406#117—24no—
407#117—21no—
408#117—21no—
409#117—21no—
410#117—21no—
411#117—21no—
412#117—21no—
413#117—24no—
414#117—1222no—
415#117—1222no—
416#117C22⋊D6112yes—
417#117C22×D6112yes—
418C2×C4⋊C4—24no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4C22⋊D6112yes—
421C2×C4⋊C4C22×D6112yes—
422C2×C22⋊C4—24no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1228no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4C22⋊D6112yes—
427C2×C42—24no—
428C2×C42—21no—
429C2×C42—612no—
430C2×C42C22⋊D6112yes—
431#104—1228no—
432#102—1228no—
433C24—21no—
434C24—1224no—
435C24—1222no—
436C24—4884no—
437C24—2441no—
438C24—2441no—
439C24—2442no—
440C24C2×C22⋊D6111yescharacteristic
441#41—22no—
442#41—22no—
443#41—24no—
444#41—24no—
445#41—24no—
446#41—24no—
447#41—24no—
448#41#1175819818yescomplemented
449#41C2×C4×D6114yes—
450#41C2×D24414yes—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8#1175819814yescomplemented
454C2×Q8C2×C22⋊D6112yes—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—614no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4—1224no—
468C2×D4—1224no—
469C2×D4—1224no—
470C2×D4—1224no—
471C2×D4—1224no—
472C2×D4—2441no—
473C2×D4—2441no—
474C2×D4—2442no—
475C2×D4—2442no—
476C2×D4—2442no—
477C2×D4—2442no—
478C2×D4—2442no—
479C2×D4#1175819814yescomplemented
480C2×D4C2×C22⋊D6112yes—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—28no—
487C22×C4—22no—
488C22×C4—1222no—
489C22×C4C2×C4×D6112yes—
490C22×C4C2×D24412yes—
491C22×C4#1175822212yes—
492C22×C4#1175822212yes—
493C22×C4C2×C22⋊D6114yes—
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C23×D6111yescharacteristic
500SD16—12216no—
501D8—12216no—
502M16—614no—
503M16—1222no—
504M16—1224no—
505M16—1222no—
506M16—1222no—
507M16—2442no—
508C2×C8—1222no—
509C2×C8—1224no—
510C2×C8—1222no—
511C2×C8—2442no—
512C4⋊C4—28no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—2442no—
517C4⋊C4C2×C22⋊D6114yes—
518C22⋊C4—28no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—4888no—
524C22⋊C4—2442no—
525C42—28no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—1222no—
531C42—1222no—
532C23—22no—
533C23—24no—
534C23—42no—
535C23—82no—
536C23—1224no—
537C23—4884no—
538C23—4884no—
539C23—4888no—
540C23—4888no—
541C23—2448no—
542C23—2442no—
543C23—2442no—
544C23—2444no—
545C23—2444no—
546C23C4×C22⋊D6112yes—
547C23#1451974611yescharacteristic
548C23#1451974611yescharacteristic
549C23#1451975112yes—
550C23C22×C22⋊D6111yescharacteristic
551Q8—22no—
552Q8—24no—
553Q8C2×#1175819814yes—
554D4—24no—
555D4—24no—
556D4—28no—
557D4—22no—
558D4—24no—
559D4—1224no—
560D4—1224no—
561D4—2442no—
562D4—2442no—
563D4—2444no—
564D4—2444no—
565D4C2×#1175819814yes—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—44no—
574C2×C4—82no—
575C2×C4—88no—
576C2×C4—2444no—
577C2×C4—2442no—
578C2×C4C2×#1175819812yes—
579C2×C4C2×#1175819812yes—
580C2×C4C4×C22⋊D6112yes—
581C2×C4C4×C22⋊D6112yes—
582C2×C4C4×C22⋊D6112yes—
583C2×C4#1451974611yescharacteristic
584C2×C4#1451974611yescharacteristic
585C2×C4#1451975112yes—
586C2×C4#1451975112yes—
587C2×C4#1451975112yes—
588C2×C4#1451976012yes—
589C2×C4#1451976012yes—
590C2×C4C22×C4×D6111yescharacteristic
591C2×C4C22×C22⋊D6111yescharacteristic
592C2×C4C22×C22⋊D6111yescharacteristic
593C2×C4C22×D24411yescharacteristic
594C2×C4C2×#1175822211yescharacteristic
595C2×C4C2×#1175822211yescharacteristic
596C8—1224no—
597C8—2442no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—84no—
606C22—4888no—
607C22—4888no—
608C22—4884no—
609C22—4884no—
610C22—2448no—
611C22—2444no—
612C22#1683964112yes—
613C22#1683964112yes—
614C22C2×C4×C22⋊D6111yescharacteristic
615C22C2×#1451974611yescharacteristic
616C22C2×#1451975111yescharacteristic
617C4—24no—
618C4—44no—
619C4—44no—
620C4—4882no—
621C4#1683964112yes—
622C4#1683964112yes—
623C4C22×#1175819811yescharacteristic
624C4C2×C4×C22⋊D6111yescharacteristic
625C4C2×#1451975111yescharacteristic
626C4C2×#1451976011yescharacteristic
627C2—21no—
628C2—22no—
629C2—22no—
630C2—21no—
631C2—44no—
632C2—4884no—
633C2C2×#1683964111yescharacteristic
634C2#2145519112yes—
subgroupG/H
centerC22C2×#14519746
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#3428C2
socleC22×C61C2×#120
prime radicalC22×#14519802C2
cosocleC22×C4×C61C24
hypercenter#3428D61
hypocenterC61#20945
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3428

Complements

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

NH ≅ G/Ntype
2. C2×#16839779C2semidirect
3. C2×#16839766C2semidirect
4. #21448399C2semidirect
5. #21446195C2semidirect
6. #21445558C2semidirect
7. #21445528C2semidirect
8. C61×#3428C2semidirect
12. C2×#14519802C4semidirect
20. C2×#14519704C22semidirect
22. #16839766C22semidirect
25. #16839663C22semidirect
27. #16839531C22semidirect
28. #16839521C22semidirect
31. C61×#520C22semidirect
49. C22×D244C2×C4semidirect
53. #14519802C2×C4semidirect
63. C2×#11758194C2×C4semidirect
130. #41×C61C22⋊C4semidirect
135. C2×Q8×C61C22⋊C4semidirect
143. C2×D4×C61C22⋊C4semidirect
311. #3428D61semidirect
337. #520C2×D61semidirect
357. C61#20945semidirect
448. #41#11758198semidirect
453. C2×Q8#11758198semidirect
479. C2×D4#11758198semidirect

Maximal subgroups

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

subgroupindex
C22×#145198022
C2×#168397792
C2×#168397662
#214483992
#214461952
#214455582
#214455282
C61×#34282
#2094561