Finite Group Databasebeta

← #29686272

Subgroups — #29686272

Order 15616.

1092 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839966C211yesmaximal, characteristic
2C2×#16839864C211yesmaximal, characteristic
3C2×#16839860C211yesmaximal, characteristic
4C4×#14519775C211yesmaximal, characteristic, complemented
5C4×#14519739C211yesmaximal, characteristic
6#21446740C211yesmaximal, characteristic, complemented
7#21446576C211yesmaximal, characteristic, complemented
8#21446462C211yesmaximal, characteristic, complemented
9#21446458C211yesmaximal, characteristic, complemented
10#21446154C211yesmaximal, characteristic
11#21446150C211yesmaximal, characteristic
12#21445818C211yesmaximal, characteristic
13#21445673C211yesmaximal, characteristic, complemented
14#21445652C211yesmaximal, characteristic, complemented
15C61×#1763C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23C2×C42×D61—21no—
24#16839966C418yescomplemented
25#16839864—24no—
26#16839860—24no—
27C2×#14519711C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519697C2211yescharacteristic
37C2×#14519688C2211yescharacteristic
38#16839669C2211yescharacteristic
39#16839667C2211yescharacteristic
40#16839657—21no—
41#16839657—21no—
42#16839631C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×#11758196—21no—
46C4×#11758196—21no—
47C4×#11758198—21no—
48#16839609—21no—
49#16839609—21no—
50C42×Dic61C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#97×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C2×#11758190—22no—
68C2×#11758190—22no—
69C2×#11758190C2×C414yes—
70C2×#11758201—22no—
71C2×#11758201—22no—
72C2×#11758201C2×C414yes—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519775—22no—
76#14519775C2×C414yescomplemented
77C4×Dic122—22no—
78C4×Dic122—22no—
79C4×D244—22no—
80C4×D244D414yes—
81#14519739—22no—
82#14519739—22no—
83#14519739—22no—
84#14519739C2×C414yes—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61D411yescharacteristic
99C2×C4×Dic61D411yescharacteristic
100C2×C4×Dic61D411yescharacteristic
101C2×C4×Dic61D411yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104C42×D61—44no—
105#14519711—22no—
106#14519710—22no—
107#14519706—22no—
108#14519706C2×C414yescomplemented
109#14519702—22no—
110#14519701—22no—
111#14519700—22no—
112#14519700C2×C414yescomplemented
113#14519699—22no—
114#14519697—22no—
115#14519688—22no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122C22×Q8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C4×Q8×C61—22no—
126C4×Q8×C61—22no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61—21no—
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#103×C61—22no—
135#103×C61C2×C414yescomplemented
136#97×C61—22no—
137#95×C61—21no—
138C23×D61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61C22×C412yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C22×C412yes—
163C2×D244C2×D412yes—
164C2×D244#4112yes—
165C2×C4×D61—24no—
166C2×C4×D61—24no—
167C2×C4×D61—24no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122C22×C412yes—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C22×C412yes—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—42no—
191C4⋊C4×C61C22×C412yes—
192C4⋊C4×C61#4112yes—
193C42×C61—42no—
194C42×C61—42no—
195C42×C61C2×D412yes—
196C42×C61#4112yes—
197#11758201—28no—
198#11758198—24no—
199#11758198—24no—
200#11758198—24no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C2×D412yes—
205#11758196—24no—
206#11758196—24no—
207#11758196—42no—
208#11758196—42no—
209#11758196—42no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—41no—
216C4×Dic61—42no—
217C4×Dic61—42no—
218C4×Dic61—42no—
219C4×Dic61C22×C412yes—
220C4×Dic61#4112yes—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418C22×C412yes—
227#11758190—28no—
228C23×C61C22×D411yescharacteristic
229C22×D61—22no—
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—41no—
239Q8×C61—41no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61C4×D412yes—
281C2×Dic61C4×D412yes—
282C2×Dic61AGL(1,ℤ/8)12yes—
283C2×Dic61#13712yes—
284C2×Dic61#13712yes—
285D244—22no—
286D244—22no—
287D244—22no—
288D244C4×D414yes—
289C4×D61—48no—
290C4×D61—48no—
291C4×D61—48no—
292Dic122—22no—
293Dic122—22no—
294Dic122—41no—
295Dic122—41no—
296C8×C61—22no—
297C8×C61—41no—
298#92418—22no—
299#92418—41no—
300#21504—611nomaximal, Sylow
301C22×C61C2×C4×D411yescharacteristic
302C22×C61C2×#12111yescharacteristic
303C22×C61#52511yescharacteristic
304C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C61C2×#13711yescharacteristic
306C22×C61C2×#13711yescharacteristic
307C22×C61#57511yescharacteristic
308C2×D61—24no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—42no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61—42no—
320C4×C61—81no—
321C4×C61C2×C4×D411yescharacteristic
322C4×C61C4×#4111yescharacteristic
323C4×C61#52911yescharacteristic
324C4×C61#53211yescharacteristic
325Dic61—22no—
326Dic61—22no—
327Dic61—41no—
328Dic61—42no—
329Dic61—42no—
330Dic61—81no—
331Dic61#44014yescomplemented
332C2×#461—611no—
333C2×#440—611no—
334C2×#440—611no—
335C4×C4⋊Q8—611no—
336C42×D4—611no—
337#1981—611no—
338#1947—611no—
339#1928—611no—
340#1928—611no—
341#1877—611no—
342#1877—611no—
343#1802—611no—
344#1767—611no—
345#1763—611no—
346#1763D6111yescharacteristic, complemented
347C2×C61C4×#12111yescharacteristic
348C2×C61C2×#44011yescharacteristic
349C2×C61C4×#13711yescharacteristic
350C2×C61#303611yescharacteristic
351C2×C61#303911yescharacteristic
352C2×C61#311011yescharacteristic
353C2×C61#311611yescharacteristic
354D61—44no—
355C22×SD16—611no—
356C22×SD16—611no—
357C2×C4⋊Q8—611no—
358C2×C4×Q8—611no—
359C2×C4×Q8C2×D6111yescharacteristic
360C2×C4×D4—611no—
361C2×C4×D4—611no—
362C2×C4×D4—611no—
363C22×C42—1221no—
364#461—618no—
365#440—1224no—
366#440—1224no—
367C2×#107—611no—
368C2×#107—611no—
369C2×C4⋊C8—611no—
370C2×C4⋊C8—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103C2×D6111yescharacteristic
376C2×#103C2×D6111yescharacteristic
377C2×#102—611no—
378C2×#102—611no—
379C2×#97—611no—
380C2×#97C2×D6111yescharacteristic
381#391—611no—
382#391—611no—
383#389—1221no—
384#389—1221no—
385#384—611no—
386C4×C4⋊C4—611no—
387C4×C4⋊C4—611no—
388C4×C4⋊C4—1221no—
389C4×C4⋊C4—1221no—
390C4×C4⋊C4C2×D6111yescharacteristic
391C4×C22⋊C4—1221no—
392C4×C22⋊C4—1221no—
393C4×C22⋊C4—1221no—
394C43—611no—
395#340—611no—
396#340—611no—
397#340C2×D6111yescharacteristic
398#340C2×D6111yescharacteristic
399#336—611no—
400#336—611no—
401C61#2150411yescharacteristic, complemented, Sylow
402C22×Q8—611no—
403C22×Q8C22×D6111yescharacteristic
404C22×D4—611no—
405C23×C4—1221no—
406C23×C4—1221no—
407C23×C4—1221no—
408C2×SD16—614no—
409C2×SD16—614no—
410C2×SD16—1222no—
411C2×SD16—1222no—
412C2×SD16—1222no—
413C2×SD16—1222no—
414C22×C8—611no—
415C22×C8—611no—
416C22×C8C22×D6111yescharacteristic
417C22×C8C22×D6111yescharacteristic
418C4⋊Q8—614no—
419C4⋊Q8—1222no—
420C4×Q8—22no—
421C4×Q8—22no—
422C4×Q8—1222no—
423C4×Q8—1222no—
424C4×D4—614no—
425C4×D4—614no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C42—21no—
445C2×C42—21no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42—2444no—
456C2×C42C22×D6111yescharacteristic
457#107—1222no—
458#107—1222no—
459C4⋊C8—614no—
460C4⋊C8—1222no—
461#103—22no—
462#103—614no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103C4×D6114yescomplemented
467#102—1222no—
468#102—1222no—
469#97—22no—
470#97—1222no—
471#95—21no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478C24—1221no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—612no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8C2×C4×D6112yes—
491C2×D4—612no—
492C2×D4—612no—
493C2×D4—612no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—1224no—
506C22×C4—1224no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1224no—
510C22×C4—1221no—
511C22×C4—2442no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4D4×D6111yescharacteristic
518C22×C4D4×D6111yescharacteristic
519C22×C4C23×D6111yescharacteristic
520SD16—1228no—
521SD16—1228no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—612no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8C2×C4×D6112yes—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—42no—
539C4⋊C4—612no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C4⋊C4C2×C4×D6112yes—
549C4⋊C4#1175822212yes—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—1224no—
553C42—42no—
554C42—42no—
555C42—612no—
556C42—612no—
557C42—1222no—
558C42—1222no—
559C42—1222no—
560C42—1221no—
561C42—1221no—
562C42—2441no—
563C42—2442no—
564C42—2442no—
565C42—2442no—
566C42#1175822212yes—
567C42D4×D6112yes—
568C23—1222no—
569C23—1222no—
570C23—1222no—
571C23—2441no—
572C23—2441no—
573C23—2441no—
574C23—2441no—
575C23C2×D4×D6111yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580Q8—1222no—
581Q8—1222no—
582Q8—2441no—
583Q8—2441no—
584D4—614no—
585D4—1222no—
586D4—1222no—
587D4—1222no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—612no—
605C2×C4—612no—
606C2×C4—612no—
607C2×C4—612no—
608C2×C4—612no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—2448no—
619C2×C4—2448no—
620C2×C4—2448no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4#1451975012yes—
629C2×C4#1451975012yes—
630C2×C4C22×C4×D6111yescharacteristic
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C2×C4C2×#1175822211yescharacteristic
635C2×C4C2×#1175822211yescharacteristic
636C8—22no—
637C8—41no—
638C8—1222no—
639C8—2441no—
640C22—1224no—
641C22—2442no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22C4×D4×D6111yescharacteristic
648C22C2×#1451975011yescharacteristic
649C22#1684027111yescharacteristic
650C22AGL(1,ℤ/8)×D6111yescharacteristic
651C22#137×D6111yescharacteristic
652C22#137×D6111yescharacteristic
653C22#1684056111yescharacteristic
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—81no—
660C4—614no—
661C4—1222no—
662C4—1222no—
663C4—4881no—
664C4—2441no—
665C4—2442no—
666C4—2442no—
667C4C4×D4×D6111yescharacteristic
668C4C4×#1175822211yescharacteristic
669C4#1684029011yescharacteristic
670C4#1684030911yescharacteristic
671C2—2444no—
672C2C4×#1451975011yescharacteristic
673C2#440×D6111yescharacteristic
674C2#2145463511yescharacteristic
675C2#2145528211yescharacteristic
676C2#2145530511yescharacteristic
677C2#2145591611yescharacteristic
678C2#2145597911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1763C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1763D61
hypocenterC61#21504
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
4. C4×#14519775C2semidirect
6. #21446740C2semidirect
7. #21446576C2semidirect
8. #21446462C2semidirect
9. #21446458C2semidirect
13. #21445673C2semidirect
14. #21445652C2semidirect
15. C61×#1763C2semidirect
24. #16839966C4semidirect
76. #14519775C2×C4semidirect
108. #14519706C2×C4semidirect
112. #14519700C2×C4semidirect
135. #103×C61C2×C4semidirect
331. Dic61#440semidirect
346. #1763D61semidirect
401. C61#21504semidirect
466. #103C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399662
C2×#168398642
C2×#168398602
C4×#145197752
C4×#145197392
#214467402
#214465762
#214464622
#214464582
#214461542
#214461502
#214458182
#214456732
#214456522
C61×#17632
#2150461