Finite Group Databasebeta

← #29828816

Subgroups — #29828816

Order 15616.

1576 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D61C211yesmaximal, characteristic, complemented
2C2×#16839963C211yesmaximal, characteristic, complemented
3C2×#16839962C211yesmaximal, characteristic, complemented
4C2×#16839902C212yesmaximal, complemented
5#21454165C211yesmaximal, characteristic
6#21446175C212yesmaximal, complemented
7#21446172C212yesmaximal, complemented
8#21446135C212yesmaximal, complemented
9#21445821C211yesmaximal, characteristic, complemented
10#21445796C211yesmaximal, characteristic, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×#11758190C2212yes—
14C22×#11758201C2212yes—
15C2×#14519774C2211yescharacteristic
16C2×#14519752—21no—
17C2×#14519752—21no—
18C2×#14519752C2212yes—
19C2×Q8×Dic61C2211yescharacteristic
20C2×C4×Dic122—21no—
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519742—21no—
23C2×#14519742C2211yescharacteristic
24C4×Q8×D61—24no—
25C4×Q8×D61—24no—
26C2×C4⋊C4×D61—21no—
27C2×C4⋊C4×D61C2211yescharacteristic
28C2×C42×D61—21no—
29C2×C42×D61C2211yescharacteristic
30#16839963—24no—
31#16839962—24no—
32#16839902—28no—
33C2×C4⋊#92418C2211yescharacteristic
34C2×#14519700C2212yes—
35C2×#14519699C2212yes—
36C2×#14519697C2212yes—
37C2×#14519689C2212yes—
38#16839663—21no—
39#16839663—21no—
40#16839663C2212yes—
41#16839524C2212yes—
42#16839516C2212yes—
43C2×C4×Q8×C61C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic
45C2×#103×C61C2212yes—
46C61×#384C2211yescharacteristic
47C61×#340C2212yes—
48C22×Dic122—21no—
49C22×Dic122C2311yescharacteristic
50C2×Q8×D61—22no—
51C2×Q8×D61—22no—
52C2×Q8×D61—22no—
53C2×Q8×D61—22no—
54C2×Q8×D61—22no—
55C2×Q8×D61D414yescomplemented
56C22×D244—22no—
57C22×D244—21no—
58C22×D244—21no—
59C22×D244C2312yes—
60C22×C4×D61—21no—
61C22×C4×D61—21no—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C2×#11758190—24no—
66C2×#11758190—24no—
67C2×#11758190—24no—
68C2×#11758201—24no—
69C2×#11758201—24no—
70C2×#11758201—24no—
71C22×#92418C2312yes—
72#14519774—22no—
73#14519774—22no—
74#14519752—24no—
75#14519752—24no—
76#14519752—44no—
77#14519752—44no—
78Q8×Dic61—22no—
79Q8×Dic61—22no—
80C4×Dic122—28no—
81C4×Dic122—22no—
82C4×Dic122D414yescomplemented
83#14519742—28no—
84#14519742—22no—
85#14519742—22no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C4⋊C4×D61—28no—
92C4⋊C4×D61—22no—
93C4⋊C4×D61—22no—
94C2×#11758198—22no—
95C2×#11758198—22no—
96C2×#11758198—22no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61D411yescharacteristic
101C2×C4×Dic61D411yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103C42×D61—28no—
104C42×D61—22no—
105C42×D61—22no—
106C4⋊#92418—22no—
107#14519700—24no—
108#14519699—24no—
109#14519697—24no—
110#14519689—24no—
111C22×Q8×C61C2311yescharacteristic
112C22×C8×C61C2312yes—
113C4×Q8×C61—22no—
114C4×Q8×C61D414yescomplemented
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2312yes—
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122C4⋊C8×C61—22no—
123#103×C61—24no—
124C23×D61—22no—
125C23×D61C2×D411yescharacteristic
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61C2×D412yes—
130C2×Q8×C61C2×D412yes—
131C2×Q8×C61#4112yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143Q8×D61—28no—
144Q8×D61—28no—
145Q8×D61—28no—
146Q8×D61—28no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—22no—
153C2×D244—42no—
154C2×D244—42no—
155C2×D244—44no—
156C2×D244—44no—
157C2×D244—44no—
158C2×D244—42no—
159C2×D244—42no—
160C2×D244—42no—
161C2×D244—42no—
162C2×C4×D61—24no—
163C2×C4×D61—24no—
164C2×C4×D61—24no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—24no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—41no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×C4×D61SD1614yescomplemented
178C2×C4×D61C2×D412yes—
179C2×C4×D61#4112yes—
180C2×Dic122—24no—
181C2×Dic122—24no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—24no—
186C2×Dic122C2×D412yes—
187C2×Dic122C2×D412yes—
188C2×Dic122#4112yes—
189C2×C8×C61—22no—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C4⋊C4×C61—24no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—42no—
197C4⋊C4×C61—42no—
198C4⋊C4×C61C2×D412yes—
199C42×C61—24no—
200C42×C61C2×D412yes—
201C42×C61C2×D412yes—
202#11758201—48no—
203#11758198—44no—
204#11758198—44no—
205#11758198—44no—
206C4⋊Dic61—24no—
207C4⋊Dic61—21no—
208C4⋊Dic61C2×D412yes—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212C4×Dic61—24no—
213C4×Dic61—24no—
214C4×Dic61—24no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C2×#92418—22no—
220C2×#92418—22no—
221C2×#92418—22no—
222#11758190—48no—
223C23×C61C22×D411yescharacteristic
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—42no—
229C22×D61—42no—
230C22×D61—42no—
231C22×D61—42no—
232C22×D61—42no—
233C22×D61—42no—
234C22×D61—42no—
235C22×D61#12112yes—
236C22×D61C2×SD1612yes—
237C22×D61AGL(1,ℤ/8)12yes—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61#12114yescomplemented
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61#12012yes—
254C2×C4×C61#12012yes—
255C2×C4×C61C2×SD1612yes—
256C2×C4×C61#13712yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—22no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61#12112yes—
277C2×Dic61C2×SD1612yes—
278D244—42no—
279D244—42no—
280D244—42no—
281D244—42no—
282D244—84no—
283D244—88no—
284D244—84no—
285C4×D61—216no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—28no—
291C4×D61—44no—
292C4×D61—48no—
293Dic122—216no—
294Dic122—22no—
295Dic122—22no—
296Dic122—22no—
297Dic122#12114yescomplemented
298C8×C61—42no—
299#92418—42no—
300#32903—611nomaximal, Sylow
301C22×C61C2×#12011yescharacteristic
302C22×C61C2×#12111yescharacteristic
303C22×C61#52511yescharacteristic
304C22×C61C22×SD1611yescharacteristic
305C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C61C2×#13711yescharacteristic
307C22×C61#57511yescharacteristic
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—82no—
315C2×D61—82no—
316C2×D61—82no—
317C2×D61—82no—
318C2×D61—82no—
319C2×D61—82no—
320C2×D61—82no—
321C2×D61#46014yescomplemented
322C4×C61—24no—
323C4×C61—22no—
324C4×C61—42no—
325C4×C61—82no—
326C4×C61#44814yescomplemented
327C4×C61C2×#12111yescharacteristic
328C4×C61C2×#12111yescharacteristic
329C4×C61#52511yescharacteristic
330C4×C61#52511yescharacteristic
331Dic61—22no—
332Dic61—24no—
333Dic61—22no—
334Dic61—41no—
335Dic61—42no—
336C22×C4×Q8—611no—
337C2×#460—611no—
338C2×#460—611no—
339C2×#448—612no—
340#2856—611no—
341#1880—612no—
342#1880—612no—
343#1874—612no—
344#1803—611no—
345#1803D6111yescharacteristic, complemented
346#1797—611no—
347C2×C61#284711yescharacteristic
348C2×C61C2×#44811yescharacteristic
349C2×C61#300611yescharacteristic
350C2×C61C2×#46011yescharacteristic
351C2×C61#303711yescharacteristic
352C2×C61#310311yescharacteristic
353C2×C61#311211yescharacteristic
354D61—24no—
355D61—162no—
356C23×Q8—611no—
357C22×SD16—612no—
358C22×SD16—612no—
359C2×#127—611no—
360C2×#121—612no—
361C2×#121—1221no—
362C2×#121—1221no—
363C2×C4×Q8—611no—
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×C4×Q8—1221no—
367C2×C4×Q8—1221no—
368C2×C4×Q8—1224no—
369C2×C4×Q8—1224no—
370C2×C4×Q8C2×D6111yescharacteristic
371C22×C4⋊C4—611no—
372C22×C4⋊C4—1221no—
373C22×C42—611no—
374C22×C42—1221no—
375#460—1224no—
376#460—1224no—
377#448—1228no—
378C2×C4⋊C8—611no—
379C2×C4⋊C8C2×D6111yescharacteristic
380C2×#103—612no—
381C2×#103C2×D6112yes—
382C2×#102—612no—
383C2×#102—612no—
384C2×C22⋊C8—612no—
385#390—612no—
386#390—1221no—
387#390—1221no—
388#384C2×D6111yescharacteristic
389#340—612no—
390#340C2×D6112yes—
391#336—612no—
392C61#3290311yescharacteristic, complemented, Sylow
393C22×Q8—614no—
394C22×Q8—611no—
395C22×Q8—1221no—
396C22×Q8—1222no—
397C22×Q8—1222no—
398C22×Q8—1222no—
399C22×Q8—1222no—
400C22×Q8—1222no—
401C22×Q8C22×D6111yescharacteristic
402C22×D4—612no—
403C22×D4—1222no—
404C22×D4—1221no—
405C22×D4—1221no—
406C23×C4—611no—
407C23×C4—611no—
408C23×C4—611no—
409C23×C4—1221no—
410C23×C4—1221no—
411C2×SD16—1224no—
412C2×SD16—1224no—
413C2×SD16—1224no—
414C2×SD16—1224no—
415C2×SD16—1224no—
416C2×SD16—1224no—
417C22×C8—612no—
418C22×C8C22×D6112yes—
419#127—1222no—
420#127—1222no—
421#121—1224no—
422#121—1224no—
423#121—2444no—
424#121—2444no—
425C4×Q8—22no—
426C4×Q8—614no—
427C4×Q8—1228no—
428C4×Q8—1222no—
429C4×Q8—1228no—
430C4×Q8—1222no—
431C4×Q8—1222no—
432C4×Q8—1222no—
433C4×Q8—1222no—
434C4×Q8D24414yescomplemented
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—1222no—
440C2×C4⋊C4—1228no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1222no—
446C2×C4⋊C4C22×D6112yes—
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C22⋊C4—1222no—
449C2×C22⋊C4—1222no—
450C2×C22⋊C4—1222no—
451C2×C42—21no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—1228no—
456C2×C42—1221no—
457C2×C42—1221no—
458C2×C42—1221no—
459C2×C42—1222no—
460C2×C42—1222no—
461C2×C42C22×D6111yescharacteristic
462C4⋊C8—22no—
463C4⋊C8—1222no—
464#103—24no—
465#103—1224no—
466#102—1224no—
467#102—1224no—
468C22⋊C8—1224no—
469C24—611no—
470C24—1222no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—612no—
475C2×Q8—612no—
476C2×Q8—612no—
477C2×Q8—1224no—
478C2×Q8—1224no—
479C2×Q8—1224no—
480C2×Q8—1228no—
481C2×Q8—1228no—
482C2×Q8—1228no—
483C2×Q8—1228no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8C2×D24412yes—
488C2×Q8D4×D6112yes—
489C2×Q8#1175822612yes—
490C2×D4—1222no—
491C2×D4—1222no—
492C2×D4—1222no—
493C2×D4—1222no—
494C2×D4—1222no—
495C2×D4—1222no—
496C2×D4—2442no—
497C2×D4—2442no—
498C2×D4—2444no—
499C2×D4—2444no—
500C2×D4—2444no—
501C2×D4—2442no—
502C2×D4—2442no—
503C2×D4—2442no—
504C2×D4—2442no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—614no—
513C22×C4—611no—
514C22×C4—1224no—
515C22×C4—1224no—
516C22×C4—1224no—
517C22×C4—1222no—
518C22×C4—1224no—
519C22×C4—1221no—
520C22×C4—1222no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1222no—
526C22×C4—1221no—
527C22×C4—1222no—
528C22×C4—2441no—
529C22×C4—2442no—
530C22×C4—2442no—
531C22×C4D4×D6111yescharacteristic
532C22×C4D4×D6111yescharacteristic
533C22×C4C23×D6111yescharacteristic
534SD16—2448no—
535SD16—2448no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—1222no—
540C2×C8—1222no—
541C2×C8—1222no—
542C4⋊C4—21no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—612no—
549C4⋊C4—1224no—
550C4⋊C4—1224no—
551C4⋊C4—1224no—
552C4⋊C4—1221no—
553C4⋊C4—1224no—
554C4⋊C4C2×D24412yes—
555C22⋊C4—2444no—
556C22⋊C4—2444no—
557C22⋊C4—2444no—
558C42—24no—
559C42—1224no—
560C42—1224no—
561C42—1221no—
562C42—1221no—
563C42—1222no—
564C42—1222no—
565C42—1224no—
566C42C2×D24412yes—
567C42D4×D6112yes—
568C23—612no—
569C23—612no—
570C23—612no—
571C23—1221no—
572C23—1221no—
573C23—1221no—
574C23—1221no—
575C23—2442no—
576C23—2442no—
577C23—2442no—
578C23—2442no—
579C23—2442no—
580C23—2442no—
581C23—2442no—
582C23C2×D4×D6111yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—22no—
586Q8—614no—
587Q8—12216no—
588Q8—1222no—
589Q8—1222no—
590Q8—1222no—
591Q8#1451975214yescomplemented
592D4—4884no—
593D4—4888no—
594D4—4884no—
595D4—2442no—
596D4—2442no—
597D4—2442no—
598D4—2442no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—12216no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1221no—
618C2×C4—1222no—
619C2×C4—1221no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1228no—
625C2×C4—1222no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—2444no—
630C2×C4—2448no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4#1451974712yes—
634C2×C4#1451974712yes—
635C2×C4SD16×D6112yes—
636C2×C4#1451980912yes—
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4C2×D4×D6111yescharacteristic
639C2×C4C22×D24411yescharacteristic
640C2×C4#41×D6111yescharacteristic
641C2×C4#41×D6111yescharacteristic
642C2×C4C2×#1175822611yescharacteristic
643C8—42no—
644C8—2442no—
645C22—614no—
646C22—1222no—
647C22—1222no—
648C22—1222no—
649C22—1222no—
650C22—1222no—
651C22—1222no—
652C22—4882no—
653C22—4882no—
654C22—4882no—
655C22—4882no—
656C22—4882no—
657C22—4882no—
658C22—4882no—
659C22C2×#1451974711yescharacteristic
660C22#121×D6111yescharacteristic
661C22#1684027511yescharacteristic
662C22C2×SD16×D6111yescharacteristic
663C22AGL(1,ℤ/8)×D6111yescharacteristic
664C22C2×#1451980911yescharacteristic
665C22#1684056111yescharacteristic
666C4—24no—
667C4—22no—
668C4—42no—
669C4—82no—
670C4—1222no—
671C4—1224no—
672C4—1222no—
673C4—2441no—
674C4—2442no—
675C4#1683990214yescomplemented
676C4#121×D6111yescharacteristic
677C4C2×#1451975211yescharacteristic
678C4#1684027411yescharacteristic
679C4#1684027511yescharacteristic
680C2—1224no—
681C2—9762no—
682C2#2145410211yescharacteristic
683C2C2×#1683990211yescharacteristic
684C2#2145507211yescharacteristic
685C2#460×D6111yescharacteristic
686C2#2145529211yescharacteristic
687C2#2145585211yescharacteristic
688C2#2145594311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radical#21454165C2
hypercenter#1803D61
hypocenterC61#32903
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. C2×C4×Q8×D61C2semidirect
2. C2×#16839963C2semidirect
3. C2×#16839962C2semidirect
4. C2×#16839902C2semidirect
6. #21446175C2semidirect
7. #21446172C2semidirect
8. #21446135C2semidirect
9. #21445821C2semidirect
10. #21445796C2semidirect
11. C61×#1803C2semidirect
55. C2×Q8×D61D4semidirect
82. C4×Dic122D4semidirect
114. C4×Q8×C61D4semidirect
177. C2×C4×D61SD16semidirect
241. Q8×C61#121semidirect
297. Dic122#121semidirect
321. C2×D61#460semidirect
326. C4×C61#448semidirect
345. #1803D61semidirect
392. C61#32903semidirect
434. C4×Q8D244semidirect
591. Q8#14519752semidirect
675. C4#16839902semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D612
C2×#168399632
C2×#168399622
C2×#168399022
#214541652
#214461752
#214461722
#214461352
#214458212
#214457962
C61×#18032
#3290361