Finite Group Databasebeta

← #29689750

Subgroups — #29689750

Order 15616.

1234 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840299C211yesmaximal, characteristic
2C2×#16839818C211yesmaximal, characteristic, complemented
3#384×D61C211yesmaximal, characteristic, complemented
4#21446786C212yesmaximal, complemented
5#21446632C212yesmaximal, complemented
6#21446135C212yesmaximal
7#21446123C212yesmaximal, complemented
8#21445824C211yesmaximal, characteristic, complemented
9#21445820C211yesmaximal, characteristic, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12#16840299—24no—
13#16840299C418yescomplemented
14C2×#14519772—21no—
15C2×#14519768C2211yescharacteristic
16C2×#14519766C2211yescharacteristic
17C2×#14519762—21no—
18C2×#14519753C2211yescharacteristic
19C2×#14519752C2211yescharacteristic
20C2×C4×D244—21no—
21C2×C4×D244C2211yescharacteristic
22C2×#14519734—21no—
23C2×C4⋊C4×D61—21no—
24C2×C4⋊C4×D61—21no—
25C2×C4⋊C4×D61C2212yes—
26C2×C42×D61C2211yescharacteristic
27#16839818—24no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519701C2212yes—
30C2×#14519699C2212yes—
31C2×#14519697C2212yes—
32C2×#14519689C2212yes—
33C2×#14519687C2212yes—
34#16839632C2211yescharacteristic
35#16839631—21no—
36#16839631—21no—
37#16839631C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#384C2211yescharacteristic
45C61×#340C2212yes—
46C22×D244—21no—
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C22×C4×D61—22no—
50C22×C4×D61D411yescharacteristic
51C22×C4×D61D411yescharacteristic
52C22×C4×D61C2311yescharacteristic
53C22×#92418C2312yes—
54#14519772—28no—
55#14519768—22no—
56#14519768C2×C414yescomplemented
57#14519766—22no—
58#14519766C2×C414yescomplemented
59#14519762—28no—
60#14519753—22no—
61#14519753C2×C414yes—
62#14519752—22no—
63#14519752C2×C414yes—
64C4×D244—28no—
65C4×D244—22no—
66C4×D244C2×C414yescomplemented
67#14519734—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C2×#11758196—22no—
78C4⋊C4×D61—42no—
79C4⋊C4×D61—42no—
80C4⋊C4×D61—44no—
81C4⋊C4×D61—44no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C42×D61C2×C414yes—
91C42×D61D414yes—
92C4⋊#92418—22no—
93#14519701—42no—
94#14519699—42no—
95#14519697—42no—
96#14519689—24no—
97#14519687—24no—
98C22×Q8×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100C4×Q8×C61—22no—
101C4×Q8×C61C2×C414yescomplemented
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61C2312yes—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C4⋊C8×C61—22no—
110#103×C61—42no—
111C23×D61—21no—
112C23×D61C2×D411yescharacteristic
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61C22×C412yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C22×Dic61—21no—
126C22×Dic61—22no—
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—24no—
135C2×D244—21no—
136C2×D244—42no—
137C2×D244—42no—
138C2×D244C22×C412yes—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—24no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61—44no—
151C2×C4×D61—44no—
152C2×C4×D61—44no—
153C2×C4×D61C22⋊C414yes—
154C2×C4×D61C22⋊C414yes—
155C2×C4×D61C22×C412yes—
156C2×C4×D61C2×D412yes—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61C22×C412yes—
167C42×C61—24no—
168C42×C61C22×C412yes—
169C42×C61C2×D412yes—
170#11758198—24no—
171#11758198—24no—
172#11758198—24no—
173#11758198—24no—
174#11758198—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61—24no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—42no—
180C4⋊Dic61—42no—
181C4⋊Dic61C22×C412yes—
182#11758196—24no—
183#11758196—24no—
184#11758196—44no—
185#11758196—44no—
186#11758196—44no—
187C4×Dic61C22⋊C414yes—
188C4×Dic61C22⋊C414yes—
189C4×Dic61C22×C412yes—
190C4×Dic61C2×D412yes—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C23×C61C22×D411yescharacteristic
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—22no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61C2×C22⋊C412yes—
207C22×D61AGL(1,ℤ/8)12yes—
208C22×D61#13712yes—
209Q8×C61—22no—
210Q8×C61—22no—
211Q8×C61—41no—
212Q8×C61—41no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61C2×C22⋊C412yes—
226C2×C4×C61AGL(1,ℤ/8)12yes—
227C2×C4×C61#13712yes—
228C2×C4×C61C23×C411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C2×C4×C612−1+411yescharacteristic
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×C22⋊C412yes—
249D244—22no—
250D244—22no—
251D244—48no—
252D244—41no—
253D244—41no—
254C4×D61—28no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—28no—
260C4×D61—48no—
261C4×D61—88no—
262C8×C61—42no—
263#92418—42no—
264#21867—611nomaximal, Sylow
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61#53511yescharacteristic
267C22×C61#53611yescharacteristic
268C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C61C2×#13711yescharacteristic
271C22×C61C2×#13711yescharacteristic
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61#41714yes—
286C4×C61—22no—
287C4×C61—42no—
288C4×C61—42no—
289C4×C61—82no—
290C4×C61#41714yes—
291C4×C61#52011yescharacteristic
292C4×C61#52011yescharacteristic
293C4×C61#53711yescharacteristic
294C4×C61#53711yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—22no—
298Dic61—42no—
299Dic61—82no—
300C2×#529—611no—
301C2×#432—611no—
302C2×#384—611no—
303#1992—612no—
304#1959—612no—
305#1874—612no—
306#1872—612no—
307#1803—611no—
308#1803D6111yescharacteristic, complemented
309#1802—611no—
310#1802—611no—
311C2×C61#231111yescharacteristic
312C2×C61C2×#41711yescharacteristic
313C2×C61C2×#41711yescharacteristic
314C2×C61#321711yescharacteristic
315C2×C61#321811yescharacteristic
316C2×C61#323811yescharacteristic
317C2×C61#323911yescharacteristic
318D61—24no—
319D61—82no—
320#529—618no—
321#529—1224no—
322C2×#126—1221no—
323C2×#125—611no—
324C2×#124—611no—
325C2×#123—1221no—
326C2×C22⋊Q8—611no—
327C2×#121—611no—
328C2×C4×Q8C2×D6111yescharacteristic
329C2×C4×D4—611no—
330C2×C4×D4—1221no—
331C2×#117—1221no—
332C22×C4⋊C4—612no—
333C22×C4⋊C4—1221no—
334C22×C4⋊C4—1221no—
335C22×C42—611no—
336#432—1224no—
337C2×C4⋊C8—611no—
338C2×C4⋊C8C2×D6111yescharacteristic
339C2×#103—612no—
340C2×#103C2×D6112yes—
341C2×#102—612no—
342C2×#102—612no—
343C2×C22⋊C8—612no—
344C2×#97—612no—
345#384—612no—
346#384—611no—
347#384—1221no—
348#384—1221no—
349#384C2×D6111yescharacteristic
350#340—612no—
351#340—612no—
352#340—612no—
353#340C2×D6112yes—
354C61#2186711yescharacteristic, complemented, Sylow
355C22×Q8C22×D6111yescharacteristic
356C22×D4—611no—
357C22×D4—1221no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—611no—
361C23×C4—1222no—
362C23×C4—1221no—
363C22×C8—612no—
364C22×C8C22×D6112yes—
365#126—1228no—
366#125—614no—
367#125—1222no—
368#124—614no—
369#124—1222no—
370#123—1228no—
371C22⋊Q8—614no—
372C22⋊Q8—1222no—
373#121—614no—
374#121—1222no—
375C4×Q8—22no—
376C4×Q8C4×D6114yescomplemented
377C4×D4—614no—
378C4×D4—1228no—
379C4×D4—1222no—
380#117—1228no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—2442no—
395C2×C4⋊C4—2442no—
396C2×C4⋊C4—2444no—
397C2×C4⋊C4—2444no—
398C2×C4⋊C4C22×D6112yes—
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C42—21no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—614no—
410C2×C42—614no—
411C2×C42C22×D6111yescharacteristic
412C4⋊C8—22no—
413C4⋊C8—1222no—
414#103—42no—
415#103—2442no—
416#102—2442no—
417#102—2442no—
418C22⋊C8—1224no—
419#97—1224no—
420C24—611no—
421C24—1221no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C2×C4×D6112yes—
428C2×D4—612no—
429C2×D4—1224no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—2442no—
436C2×D4—2442no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—614no—
446C22×C4—614no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1221no—
456C22×C4—1224no—
457C22×C4—1222no—
458C22×C4—2444no—
459C22×C4—2444no—
460C22×C4—2444no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1222no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—612no—
479C4⋊C4—1224no—
480C4⋊C4—1224no—
481C4⋊C4—1224no—
482C4⋊C4—1221no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—2442no—
486C4⋊C4—2444no—
487C4⋊C4—2444no—
488C4⋊C4—2444no—
489C4⋊C4—2442no—
490C4⋊C4C2×C4×D6112yes—
491C22⋊C4—1224no—
492C22⋊C4—1224no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C22⋊C4—1224no—
496C42—24no—
497C42—612no—
498C42—612no—
499C42—614no—
500C42—614no—
501C42C2×C4×D6112yes—
502C42D4×D6112yes—
503C23—612no—
504C23—612no—
505C23—612no—
506C23—1221no—
507C23—1221no—
508C23—1221no—
509C23—1221no—
510C23—1222no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23—2441no—
517C23C2×D4×D6111yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—1222no—
523D4—1222no—
524D4—2448no—
525D4—2441no—
526D4—2441no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—1228no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1228no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—4888no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2448no—
561C2×C4C22⋊C4×D6112yes—
562C2×C4C22⋊C4×D6112yes—
563C2×C4#1451980412yes—
564C2×C4#1451980912yes—
565C2×C4C22×C4×D6111yescharacteristic
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4#1451982211yescharacteristic
568C2×C4#1451982311yescharacteristic
569C2×C4#1451982411yescharacteristic
570C2×C4#1451982511yescharacteristic
571C8—42no—
572C8—2442no—
573C22—614no—
574C22—1222no—
575C22—1222no—
576C22—1222no—
577C22—1222no—
578C22—1222no—
579C22—1222no—
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22—4881no—
584C22—2442no—
585C22—2442no—
586C22—2442no—
587C22C2×C22⋊C4×D6111yescharacteristic
588C22#1684032811yescharacteristic
589C22#1684033511yescharacteristic
590C22AGL(1,ℤ/8)×D6111yescharacteristic
591C22C2×#1451980411yescharacteristic
592C22#137×D6111yescharacteristic
593C22C2×#1451980911yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—82no—
598C4—1222no—
599C4—1222no—
600C4—1222no—
601C4—4882no—
602C4—2442no—
603C4#1683976214yes—
604C4#1684024211yescharacteristic
605C4#1684024311yescharacteristic
606C4#1684034111yescharacteristic
607C4#1684034211yescharacteristic
608C2—1224no—
609C2—4882no—
610C2#2144840811yescharacteristic
611C2#417×D6111yescharacteristic
612C2C2×#1683976211yescharacteristic
613C2#2145684111yescharacteristic
614C2#2145685011yescharacteristic
615C2#2145704411yescharacteristic
616C2#2145705311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#21867
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×#16839818C2semidirect
3. #384×D61C2semidirect
4. #21446786C2semidirect
5. #21446632C2semidirect
7. #21446123C2semidirect
8. #21445824C2semidirect
9. #21445820C2semidirect
10. #21445816C2semidirect
11. C61×#1803C2semidirect
13. #16840299C4semidirect
56. #14519768C2×C4semidirect
58. #14519766C2×C4semidirect
66. C4×D244C2×C4semidirect
101. C4×Q8×C61C2×C4semidirect
308. #1803D61semidirect
354. C61#21867semidirect
376. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402992
C2×#168398182
#384×D612
#214467862
#214466322
#214461352
#214461232
#214458242
#214458202
#214458162
C61×#18032
#2186761