Finite Group Databasebeta

← #29709212

Subgroups — #29709212

Order 15616.

880 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic244C211yesmaximal, characteristic, complemented
2C2×C4×#11758190C211yesmaximal, characteristic
3C4×#14519765C211yesmaximal, characteristic
4#21448296C211yesmaximal, characteristic, complemented
5#21447302C211yesmaximal, characteristic, complemented
6#21447239C211yesmaximal, characteristic, complemented
7#21447206C211yesmaximal, characteristic
8#21447199C211yesmaximal, characteristic, complemented
9#21447142C211yesmaximal, characteristic, complemented
10#21447137C211yesmaximal, characteristic
11#21447061C211yesmaximal, characteristic
12#21446916C211yesmaximal, characteristic
13C4×#14519702C211yesmaximal, characteristic, complemented
14C4×#14519699C211yesmaximal, characteristic
15C4×C4⋊C8×C61C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C4×Dic244—24no—
23C4×#11758190—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34#16839668C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839664C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839653—21no—
41#16839653—21no—
42#16839646—21no—
43#16839646—21no—
44#16839628C2211yescharacteristic
45C4×#11758196—21no—
46C4×C4⋊Dic61C2211yescharacteristic
47C4×#11758198—21no—
48C4×#11758198—21no—
49#16839608—21no—
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C43×C61C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×Dic244—22no—
66C2×Dic244—22no—
67C2×Dic244—22no—
68C2×#11758190—22no—
69C2×#11758190—22no—
70C2×#11758190—22no—
71#14519765—22no—
72#14519765—22no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C4×Dic122D414yes—
77C4×D244—22no—
78C4×D244—22no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96#14519711—22no—
97#14519709—22no—
98#14519702—22no—
99#14519702—22no—
100#14519702—22no—
101#14519702—22no—
102#14519702—22no—
103#14519699—22no—
104#14519699—22no—
105#14519699—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C42×C61D411yescharacteristic
122C2×C42×C61D411yescharacteristic
123C2×C42×C61D411yescharacteristic
124C2×C42×C61D411yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C4⋊C8×C61—22no—
129C4⋊C8×C61—22no—
130C4×C8×C61—22no—
131C4×C8×C61—22no—
132C23×D61—21no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244#4112yes—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122C2×D412yes—
167C2×Dic122#4112yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61#4112yes—
179C2×C8×C61#4112yes—
180C42×C61—22no—
181C42×C61—22no—
182C42×C61—22no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—41no—
188C42×C61C2×D412yes—
189C42×C61#4112yes—
190#11758198—24no—
191#11758198—24no—
192#11758198—42no—
193#11758198—42no—
194#11758198—42no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C2×D412yes—
205#11758196—24no—
206#11758196—42no—
207#11758196—42no—
208#11758196—42no—
209C4×Dic61—24no—
210C4×Dic61—42no—
211C4×Dic61—42no—
212Dic244—28no—
213#11758190—28no—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61#12112yes—
236C2×C4×C61#12112yes—
237C2×C4×C61#13512yes—
238C2×C4×C61#13512yes—
239C2×C4×C61#13712yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267D244—22no—
268D244—22no—
269D244—41no—
270D244—41no—
271C4×D61—84no—
272Dic122—22no—
273Dic122—22no—
274Dic122—22no—
275Dic122—22no—
276Dic122—41no—
277Dic122—41no—
278Dic122—41no—
279Dic122—41no—
280C8×C61—22no—
281C8×C61—22no—
282C8×C61—41no—
283C8×C61—41no—
284#23681—611nomaximal, Sylow
285C22×C61C2×#12111yescharacteristic
286C22×C61#52511yescharacteristic
287C22×C61#53211yescharacteristic
288C22×C61C2×#13511yescharacteristic
289C22×C61C2×#13511yescharacteristic
290C22×C61C2×#13711yescharacteristic
291C22×C61#57511yescharacteristic
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61#46414yes—
305C4×C61#52511yescharacteristic
306C4×C61#52511yescharacteristic
307C4×C61#52911yescharacteristic
308C4×C61#52911yescharacteristic
309Dic61—42no—
310Dic61—42no—
311Dic61—42no—
312Dic61—81no—
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316C2×C4×Q16—611no—
317C2×C4×SD16—611no—
318C4×#124—611no—
319#2295—611no—
320#2085—611no—
321#2074—611no—
322#2067—611no—
323#2066—611no—
324#2056—611no—
325#2055—611no—
326#2042—611no—
327#2015—611no—
328C4×C4⋊C8D6111yescharacteristic, complemented
329C4×#103—611no—
330C4×#102—611no—
331C2×C61#239211yescharacteristic
332C2×C61C2×#46411yescharacteristic
333C2×C61#303511yescharacteristic
334C2×C61#310211yescharacteristic
335C2×C61#310711yescharacteristic
336C2×C61#316811yescharacteristic
337C2×C61#317211yescharacteristic
338D61—82no—
339C22×Q16—611no—
340C22×SD16—611no—
341C2×#124—611no—
342C2×C4×Q8—611no—
343C2×C4×Q8—611no—
344C2×C4×D4—611no—
345C4×Q16—1224no—
346C4×SD16—1224no—
347C2×#107—611no—
348C2×#106—611no—
349C2×C4⋊C8C2×D6111yescharacteristic
350C2×C4⋊C8C2×D6111yescharacteristic
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×C4×C8C2×D6111yescharacteristic
360C2×C4×C8C2×D6111yescharacteristic
361#391—611no—
362#391—611no—
363#391—611no—
364#390—611no—
365#389—611no—
366#389—611no—
367#388—1221no—
368#388—1221no—
369#387—1221no—
370#387—1221no—
371#382—611no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—1221no—
374C4×C22⋊C4—1221no—
375C4×C22⋊C4—1221no—
376#376—1221no—
377C43C2×D6111yescharacteristic
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#336C2×D6111yescharacteristic
383#336C2×D6111yescharacteristic
384C61#2368111yescharacteristic, complemented, Sylow
385C22×Q8—611no—
386C22×Q8—611no—
387C22×D4—611no—
388C23×C4—1221no—
389C2×Q16—1222no—
390C2×Q16—1222no—
391C2×Q16—1222no—
392C2×SD16—1222no—
393C2×SD16—1222no—
394C2×SD16—1222no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399#124—1222no—
400#124—1222no—
401C4×Q8—614no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C42—1221no—
422C2×C42—1221no—
423C2×C42—1221no—
424C2×C42D24411yescharacteristic
425C2×C42D24411yescharacteristic
426C2×C42D24411yescharacteristic
427C2×C42D24411yescharacteristic
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430C2×C42C22×D6111yescharacteristic
431#107—1222no—
432#106—1222no—
433C4⋊C8—22no—
434C4⋊C8—22no—
435#103—1222no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#102—1222no—
441#102—1222no—
442#102—1222no—
443C4×C8—22no—
444C4×C8—22no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456C24—1221no—
457C2×Q8—612no—
458C2×Q8—612no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×D4—612no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—2442no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4C2×D24411yescharacteristic
486C22×C4C2×D24411yescharacteristic
487C22×C4C2×D24411yescharacteristic
488C22×C4C2×D24411yescharacteristic
489C22×C4C2×D24411yescharacteristic
490C22×C4C2×D24411yescharacteristic
491C22×C4C23×D6111yescharacteristic
492Q16—1228no—
493SD16—1228no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8#1175822212yes—
505C2×C8#1175822212yes—
506C4⋊C4—612no—
507C4⋊C4—1221no—
508C4⋊C4—1224no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C22⋊C4—1224no—
521C22⋊C4—1224no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C22⋊C4—2442no—
525C42—22no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—22no—
531C42—22no—
532C42—41no—
533C42—1224no—
534C42—2442no—
535C42—2442no—
536C42D4×D6112yes—
537C42#1175822612yes—
538C23—1222no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23C22×D24411yescharacteristic
546Q8—1222no—
547Q8—1222no—
548Q8—1222no—
549Q8—1222no—
550Q8—2441no—
551Q8—2441no—
552Q8—2441no—
553Q8—2441no—
554D4—1222no—
555D4—1222no—
556D4—2441no—
557D4—2441no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—4884no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4#1451975212yes—
594C2×C4#1451975212yes—
595C2×C4#1451979912yes—
596C2×C4#1451979912yes—
597C2×C4#1451981012yes—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4C2×#1175822211yescharacteristic
602C2×C4C2×#1175822211yescharacteristic
603C2×C4C2×#1175822611yescharacteristic
604C8—22no—
605C8—22no—
606C8—41no—
607C8—41no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—4881no—
612C22—2442no—
613C22—2442no—
614C22—2442no—
615C22C2×#1451975211yescharacteristic
616C22#1684027411yescharacteristic
617C22#1684031211yescharacteristic
618C22C2×#1451979911yescharacteristic
619C22C2×#1451979911yescharacteristic
620C22C2×#1451981011yescharacteristic
621C22#1684056211yescharacteristic
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—4881no—
628C4—4881no—
629C4—4881no—
630C4—4881no—
631C4—2442no—
632C4—2442no—
633C4—2442no—
634C4#1683998514yes—
635C4#1684027111yescharacteristic
636C4#1684027111yescharacteristic
637C4#1684029911yescharacteristic
638C4#1684029911yescharacteristic
639C2—4882no—
640C2#2144905911yescharacteristic
641C2C2×#1683998511yescharacteristic
642C2#2145527611yescharacteristic
643C2#2145583911yescharacteristic
644C2#2145589111yescharacteristic
645C2#2145638911yescharacteristic
646C2#2145642511yescharacteristic
subgroupG/H
centerC22×C4C2×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×C4⋊C8×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC4×C4⋊C8D61
hypocenterC61#23681
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic244C2semidirect
4. #21448296C2semidirect
5. #21447302C2semidirect
6. #21447239C2semidirect
8. #21447199C2semidirect
9. #21447142C2semidirect
13. C4×#14519702C2semidirect
15. C4×C4⋊C8×C61C2semidirect
328. C4×C4⋊C8D61semidirect
384. C61#23681semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2442
C2×C4×#117581902
C4×#145197652
#214482962
#214473022
#214472392
#214472062
#214471992
#214471422
#214471372
#214470612
#214469162
C4×#145197022
C4×#145196992
C4×C4⋊C8×C612
#2368161