Finite Group Databasebeta

← #29679505

Subgroups — #29679505

Order 15616.

1598 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758226C211yesmaximal, characteristic
2C2×#16839784C211yesmaximal, characteristic, complemented
3C2×#16839762C212yesmaximal
4#384×D61C211yesmaximal, characteristic, complemented
5#21445824C211yesmaximal, characteristic, complemented
6#21445822C212yesmaximal, complemented
7#21445820C211yesmaximal, characteristic, complemented
8#21445818C212yesmaximal, complemented
9#21445816C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C22×#11758189C2212yes—
14C4×#11758226—24no—
15C4×#11758226C418yescomplemented
16C2×Q8×Dic61C2211yescharacteristic
17C2×C4×D244—21no—
18C2×C4×D244C2211yescharacteristic
19C2×#14519739—21no—
20C2×#14519739C2211yescharacteristic
21C2×#14519735—21no—
22C2×#14519735C2211yescharacteristic
23C2×C4⋊C4×D61—21no—
24C2×C4⋊C4×D61—21no—
25C2×C4⋊C4×D61C2212yes—
26C2×C42×D61—21no—
27C2×C42×D61C2211yescharacteristic
28#16839784—24no—
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519706C2212yes—
31#16839762—28no—
32C2×#14519701C2212yes—
33C2×#14519699C2212yes—
34C2×#14519697C2212yes—
35#16839632C2211yescharacteristic
36#16839631—21no—
37#16839631—21no—
38#16839631C2212yes—
39#16839525C2212yes—
40#16839524C2212yes—
41#16839524C2212yes—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C61×#384C2211yescharacteristic
46C61×#340C2212yes—
47C2×#11758226—22no—
48C2×#11758226—22no—
49C2×#11758226—22no—
50C2×#11758226C2×C414yes—
51C2×#11758226D414yes—
52C2×#11758226Q814yes—
53C22×D244—21no—
54C22×D244C2311yescharacteristic
55C22×C4×D61—21no—
56C22×C4×D61—21no—
57C22×C4×D61—21no—
58C22×C4×D61—22no—
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C2×#11758189—24no—
63C2×#11758189—24no—
64C2×#11758189—24no—
65C22×#92418C2312yes—
66Q8×Dic61—22no—
67Q8×Dic61C2×C414yescomplemented
68C4×D244—28no—
69C4×D244—22no—
70C4×D244C2×C414yescomplemented
71#14519739—28no—
72#14519739—22no—
73#14519739C2×C414yescomplemented
74#14519735—22no—
75#14519735—28no—
76#14519735C2×C414yes—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2312yes—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—22no—
83C2×#11758196—22no—
84C2×#11758196—22no—
85C4⋊C4×D61—24no—
86C4⋊C4×D61—24no—
87C4⋊C4×D61—44no—
88C4⋊C4×D61—44no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61D411yescharacteristic
95C2×C4×Dic61D411yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97C42×D61—28no—
98C42×D61C2×C414yes—
99C42×D61D414yes—
100C4⋊#92418—22no—
101#14519706—24no—
102#14519701—24no—
103#14519699—24no—
104#14519697—24no—
105C22×Q8×C61C2311yescharacteristic
106C22×C8×C61C2312yes—
107C4×Q8×C61—22no—
108C4×Q8×C61C2×C414yescomplemented
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2312yes—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116C4⋊C8×C61—22no—
117#103×C61—24no—
118C23×D61—21no—
119C23×D61C2×D411yescharacteristic
120C2×Q8×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61C22×C412yes—
124C2×Q8×C61C2×D412yes—
125C2×Q8×C61C2×Q812yes—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137#11758226—28no—
138#11758226—28no—
139#11758226—28no—
140#11758226C4⋊C4116yescomplemented
141C2×D244—24no—
142C2×D244—24no—
143C2×D244—24no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244C22×C412yes—
148C2×D244C2×D412yes—
149C2×D244C2×Q812yes—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—24no—
153C2×C4×D61—24no—
154C2×C4×D61—24no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—44no—
164C2×C4×D61—44no—
165C2×C4×D61—44no—
166C2×C4×D61C22⋊C414yes—
167C2×C4×D61C22⋊C414yes—
168C2×C4×D61C22×C412yes—
169C2×C4×D61C2×D412yes—
170C2×C4×D61C2×D412yes—
171C2×C4×D61C2×Q812yes—
172C2×C4×D61#4112yes—
173C2×C4×D61#4112yes—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—42no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61C22×C412yes—
184C42×C61—24no—
185C42×C61C22×C412yes—
186C42×C61C2×D412yes—
187#11758198—24no—
188#11758198—24no—
189#11758198—24no—
190C4⋊Dic61—21no—
191C4⋊Dic61—24no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—42no—
195C4⋊Dic61—42no—
196C4⋊Dic61C22×C412yes—
197#11758196—44no—
198#11758196—44no—
199#11758196—44no—
200C4×Dic61—24no—
201C4×Dic61—24no—
202C4×Dic61C22⋊C414yes—
203C4×Dic61C22⋊C414yes—
204C4×Dic61C22×C412yes—
205C4×Dic61C2×D412yes—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—22no—
209#11758189—48no—
210C23×C61C22×D411yescharacteristic
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61C2×C22⋊C412yes—
223C22×D61#12112yes—
224C22×D61C22⋊Q812yes—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—22no—
228Q8×C61C2×C4⋊C414yescomplemented
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61AGL(1,ℤ/8)12yes—
243C2×C4×C61#13712yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61C2×C22⋊C412yes—
260C2×Dic61C2×C22⋊C412yes—
261C2×Dic61#12112yes—
262C2×Dic61C22⋊Q812yes—
263C2×Dic61AGL(1,ℤ/8)12yes—
264C2×Dic61#13712yes—
265D244—216no—
266D244—22no—
267D244—22no—
268D244—22no—
269D244C2×C4⋊C414yescomplemented
270C4×D61—28no—
271C4×D61—216no—
272C4×D61—28no—
273C4×D61—48no—
274C4×D61—48no—
275C4×D61—88no—
276C4×D61C2×C4⋊C414yes—
277C4×D61#11714yes—
278C4×D61#12114yes—
279C4×D61C22⋊Q814yes—
280C8×C61—42no—
281#92418—42no—
282#20648—611nomaximal, Sylow
283C22×C61C22×C22⋊C411yescharacteristic
284C22×C61C2×#12111yescharacteristic
285C22×C61C2×C22⋊Q811yescharacteristic
286C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C22×C61C2×#13711yescharacteristic
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—24no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61#38014yes—
304C4×C61—22no—
305C4×C61—24no—
306C4×C61—42no—
307C4×C61—82no—
308C4×C61#41714yescomplemented
309C4×C61C22×C4⋊C411yescharacteristic
310C4×C61C2×#11711yescharacteristic
311C4×C61C2×#12111yescharacteristic
312C4×C61C2×C22⋊Q811yescharacteristic
313Dic61—22no—
314Dic61—42no—
315Dic61—82no—
316Dic61#38014yes—
317Dic61#41714yescomplemented
318C2×C4×#41—611no—
319C2×#423—611no—
320C2×#417—612no—
321C2×#384—611no—
322#1803—612no—
323#1803—611no—
324#1803D6111yescharacteristic, complemented
325#1802—611no—
326#1802—611no—
327#1802—612no—
328#1736—612no—
329C2×C61C2×#38011yescharacteristic
330C2×C61C2×#41711yescharacteristic
331C2×C61C2×#41711yescharacteristic
332C2×C61#303811yescharacteristic
333C2×C61#303911yescharacteristic
334C2×C61#307911yescharacteristic
335C2×C61#307911yescharacteristic
336D61—24no—
337D61—44no—
338C22×#41—611no—
339C22×M16—612no—
340C4×#41—618no—
341C4×#41—1224no—
342C2×C4×Q8—611no—
343C2×C4×Q8C2×D6111yescharacteristic
344C2×C4×D4—611no—
345C2×C4×D4—611no—
346C2×C4×D4—1221no—
347C2×C4×D4—1221no—
348C2×#117—611no—
349C2×#117—1221no—
350C22×C4⋊C4—612no—
351C22×C4⋊C4—1221no—
352C22×C4⋊C4—1221no—
353C22×C42—611no—
354C22×C42—1221no—
355#423—1224no—
356C2×C4⋊C8—612no—
357C2×C4⋊C8—611no—
358C2×C4⋊C8C2×D6111yescharacteristic
359#417—1228no—
360C2×#103—612no—
361C2×#103C2×D6112yes—
362C2×#102—612no—
363C2×#102—612no—
364#384—612no—
365#384—611no—
366#384—1221no—
367#384—1221no—
368#384C2×D6111yescharacteristic
369#340—612no—
370#340—612no—
371#340—612no—
372#340C2×D6112yes—
373C61#2064811yescharacteristic, complemented, Sylow
374C2×#41—614no—
375C2×#41—614no—
376C2×#41—614no—
377C2×#41—1222no—
378C2×#41—1222no—
379C2×#41—1222no—
380C22×Q8C22×D6111yescharacteristic
381C22×D4—611no—
382C22×D4—1221no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—1221no—
387C23×C4—1221no—
388C23×C4—1222no—
389C23×C4—1221no—
390C2×M16—1224no—
391C2×M16—1224no—
392C2×M16—1224no—
393C22×C8—612no—
394C22×C8C22×D6112yes—
395C4×Q8—22no—
396C4×Q8—614no—
397C4×Q8—1222no—
398C4×Q8C4×D6114yescomplemented
399C4×D4—614no—
400C4×D4—614no—
401C4×D4—1228no—
402C4×D4—1222no—
403C4×D4—1228no—
404C4×D4—1222no—
405#117—614no—
406#117—1228no—
407#117—1222no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—1224no—
414C2×C4⋊C4—1224no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—2444no—
422C2×C4⋊C4—2444no—
423C2×C4⋊C4C22×D6112yes—
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C42—21no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—614no—
433C2×C42—614no—
434C2×C42—1228no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42C22×D6111yescharacteristic
438C4⋊C8—22no—
439C4⋊C8—1224no—
440C4⋊C8—1222no—
441#103—24no—
442#103—1224no—
443#102—1224no—
444#102—1224no—
445C24—611no—
446C24—1221no—
447#41—6116no—
448#41—1228no—
449#41—1228no—
450#41—1228no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8C2×C4×D6112yes—
455C2×Q8D4×D6112yes—
456C2×Q8Q8×D6112yes—
457C2×D4—612no—
458C2×D4—612no—
459C2×D4—612no—
460C2×D4—1224no—
461C2×D4—1224no—
462C2×D4—1224no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—612no—
472C22×C4—612no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—611no—
478C22×C4—614no—
479C22×C4—614no—
480C22×C4—1224no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1224no—
484C22×C4—1224no—
485C22×C4—1224no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1221no—
489C22×C4—1224no—
490C22×C4—1222no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2444no—
494C22×C4—2444no—
495C22×C4—2444no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4C23×D6111yescharacteristic
501M16—2448no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C4⋊C4—21no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—612no—
515C4⋊C4—1224no—
516C4⋊C4—1221no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—2442no—
520C4⋊C4—2444no—
521C4⋊C4—2444no—
522C4⋊C4—2444no—
523C4⋊C4—2442no—
524C4⋊C4C2×C4×D6112yes—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C22⋊C4—1224no—
528C42—24no—
529C42—612no—
530C42—612no—
531C42—614no—
532C42—614no—
533C42—1224no—
534C42—1224no—
535C42C2×C4×D6112yes—
536C42D4×D6112yes—
537C23—612no—
538C23—612no—
539C23—612no—
540C23—1221no—
541C23—1221no—
542C23—1221no—
543C23—1221no—
544C23—1222no—
545C23—1222no—
546C23—1222no—
547C23—2441no—
548C23—2441no—
549C23—2441no—
550C23—2441no—
551C23C2×D4×D6111yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8C4⋊C4×D6114yescomplemented
556D4—614no—
557D4—12216no—
558D4—1222no—
559D4—1222no—
560D4—1222no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—614no—
579C2×C4—614no—
580C2×C4—614no—
581C2×C4—614no—
582C2×C4—612no—
583C2×C4—1228no—
584C2×C4—12216no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1228no—
588C2×C4—1222no—
589C2×C4—4888no—
590C2×C4—2448no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2448no—
597C2×C4C22⋊C4×D6112yes—
598C2×C4C22⋊C4×D6112yes—
599C2×C4#1451980412yes—
600C2×C4#1451980912yes—
601C2×C4C22×C4×D6111yescharacteristic
602C2×C4C2×D4×D6111yescharacteristic
603C2×C4C2×D4×D6111yescharacteristic
604C2×C4C2×Q8×D6111yescharacteristic
605C2×C4#41×D6111yescharacteristic
606C2×C4#41×D6111yescharacteristic
607C8—42no—
608C8—2442no—
609C22—614no—
610C22—1222no—
611C22—1222no—
612C22—1222no—
613C22—1222no—
614C22—1222no—
615C22—1222no—
616C22—1224no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22—2442no—
622C22—2442no—
623C22C2×C22⋊C4×D6111yescharacteristic
624C22#121×D6111yescharacteristic
625C22C22⋊Q8×D6111yescharacteristic
626C22AGL(1,ℤ/8)×D6111yescharacteristic
627C22C2×#1451980411yescharacteristic
628C22#137×D6111yescharacteristic
629C22C2×#1451980911yescharacteristic
630C4—22no—
631C4—24no—
632C4—42no—
633C4—82no—
634C4—614no—
635C4—614no—
636C4—1222no—
637C4—4882no—
638C4—2442no—
639C4#1683976214yescomplemented
640C4C2×C4⋊C4×D6111yescharacteristic
641C4#117×D6111yescharacteristic
642C4#121×D6111yescharacteristic
643C4C22⋊Q8×D6111yescharacteristic
644C2—1224no—
645C2—2444no—
646C2#380×D6111yescharacteristic
647C2#417×D6111yescharacteristic
648C2C2×#1683976211yescharacteristic
649C2#2145529611yescharacteristic
650C2#2145530311yescharacteristic
651C2#2145562911yescharacteristic
652C2#2145563311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#20648
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#16839784C2semidirect
4. #384×D61C2semidirect
5. #21445824C2semidirect
6. #21445822C2semidirect
7. #21445820C2semidirect
8. #21445818C2semidirect
9. #21445816C2semidirect
10. #21445537C2semidirect
11. C61×#1803C2semidirect
15. C4×#11758226C4semidirect
67. Q8×Dic61C2×C4semidirect
70. C4×D244C2×C4semidirect
73. #14519739C2×C4semidirect
108. C4×Q8×C61C2×C4semidirect
140. #11758226C4⋊C4semidirect
228. Q8×C61C2×C4⋊C4semidirect
269. D244C2×C4⋊C4semidirect
308. C4×C61#417semidirect
317. Dic61#417semidirect
324. #1803D61semidirect
373. C61#20648semidirect
398. C4×Q8C4×D61semidirect
555. Q8C4⋊C4×D61semidirect
639. C4#16839762semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582262
C2×#168397842
C2×#168397622
#384×D612
#214458242
#214458222
#214458202
#214458182
#214458162
#214455372
C61×#18032
#2064861