Finite Group Databasebeta

← #29684356

Subgroups — #29684356

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448297C211yesmaximal, characteristic, complemented
2#21448284C211yesmaximal, characteristic
3#21446702C211yesmaximal, characteristic
4#21446675C211yesmaximal, characteristic
5#21446558C211yesmaximal, characteristic, complemented
6#21446553C211yesmaximal, characteristic, complemented
7#21446518C211yesmaximal, characteristic
8#21446511C211yesmaximal, characteristic, complemented
9#21446504C211yesmaximal, characteristic
10#21446497C211yesmaximal, characteristic, complemented
11#21445716C211yesmaximal, characteristic, complemented
12#21445649C211yesmaximal, characteristic, complemented
13#21445644C211yesmaximal, characteristic
14#97×Dic61C211yesmaximal, characteristic, complemented
15C2×C61×#440C211yesmaximal, characteristic
16C2×#14519736—21no—
17C2×#14519711C2211yescharacteristic
18C2×#14519709C2211yescharacteristic
19C2×#14519701C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519688C2211yescharacteristic
24C2×#14519687C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27C2×C8×Dic61C2211yescharacteristic
28#16839669C2211yescharacteristic
29#16839669C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839655—21no—
37#16839655—21no—
38#16839648—21no—
39#16839648—21no—
40C4×C4⋊Dic61—21no—
41C4×C4⋊Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×C22⋊Dic61—21no—
44#16839608—21no—
45C42×Dic61C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C61×#440C418yescomplemented
56C2×#107×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519736—44no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61D411yescharacteristic
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83#14519711—22no—
84#14519709—22no—
85#14519701—22no—
86#14519701—22no—
87#14519698—22no—
88#14519698—22no—
89#14519688—22no—
90#14519687—22no—
91#14519687—22no—
92C8×Dic61—22no—
93C8×Dic61—22no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104C22×Q8×C61C2311yescharacteristic
105C22×D4×C61C2311yescharacteristic
106C23×C4×C61—21no—
107C2×SD16×C61—22no—
108C2×SD16×C61—22no—
109C2×SD16×C61C2×C414yes—
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C4×Q8×C61—22no—
113C4×Q8×C61C2×C414yes—
114C4×D4×C61—22no—
115C4×D4×C61C2×C414yes—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#107×C61C2×C414yes—
123#103×C61C2×C414yes—
124#102×C61C2×C414yes—
125#97×C61C2×C414yes—
126C24×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61C22×C412yes—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61C22×C412yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—24no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156SD16×C61—28no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61C22×C412yes—
160C2×C8×C61C22×C412yes—
161C2×C8×C61#4112yes—
162C2×C8×C61#4112yes—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C22×C412yes—
167C4⋊C4×C61C22×C412yes—
168C22⋊C4×C61—24no—
169C42×C61—24no—
170C42×C61C22×C412yes—
171C42×C61#4112yes—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174C22⋊Dic61—42no—
175C22⋊Dic61—42no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—42no—
185C4⋊Dic61—42no—
186C4⋊Dic61—42no—
187C4⋊Dic61—42no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—41no—
195C4×Dic61—41no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418#4112yes—
203C2×#92418#4112yes—
204C23×C61—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C22×D411yescharacteristic
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216D4×C61—22no—
217D4×C61—22no—
218D4×C61—41no—
219D4×C61—41no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—48no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61C4×D412yes—
235C2×C4×C61C4×D412yes—
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
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×Dic61AGL(1,ℤ/8)12yes—
265C2×Dic61#13712yes—
266C8×C61—22no—
267C8×C61#11714yescomplemented
268#92418—22no—
269#92418—22no—
270#21288—611nomaximal, Sylow
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61C2×C4×D411yescharacteristic
279C22×C61#53211yescharacteristic
280C22×C61#53211yescharacteristic
281C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C61C2×#13711yescharacteristic
283C22×C61#57511yescharacteristic
284C22×C61#57511yescharacteristic
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—42no—
288C4×C61—42no—
289C4×C61—42no—
290C4×C61C2×#11711yescharacteristic
291C4×C61C4×#4111yescharacteristic
292C4×C61#52911yescharacteristic
293C4×C61#52911yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—41no—
297Dic61—41no—
298Dic61—81no—
299Dic61—81no—
300Dic61—81no—
301Dic61—81no—
302C2×#440D6111yescharacteristic
303#2295—611no—
304#2293—611no—
305#1973—611no—
306#1968—611no—
307#1943—611no—
308#1942—611no—
309#1936—611no—
310#1935—611no—
311#1934—611no—
312#1933—611no—
313#1776—611no—
314#1763—611no—
315#1762—611no—
316C4×#97—611no—
317C2×C61—82no—
318C2×C61#229311yescharacteristic
319C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
320C2×C61C4×#13711yescharacteristic
321C2×C61#318011yescharacteristic
322C2×C61#318111yescharacteristic
323C2×C61#318211yescharacteristic
324C2×C61#318311yescharacteristic
325C22×SD16C2×D6111yescharacteristic
326C2×C4×Q8C2×D6111yescharacteristic
327C2×C4×D4C2×D6111yescharacteristic
328C2×#117—1221no—
329#440Dic6118yescomplemented
330C2×#107—611no—
331C2×#107C2×D6111yescharacteristic
332C2×#106—611no—
333C2×#103—611no—
334C2×#103—611no—
335C2×#103C2×D6111yescharacteristic
336C2×#102—611no—
337C2×#102—611no—
338C2×#102C2×D6111yescharacteristic
339C2×#97—611no—
340C2×#97—611no—
341C2×#97—611no—
342C2×#97C2×D6111yescharacteristic
343C2×C4×C8—611no—
344C2×C4×C8—611no—
345#391—611no—
346#391—611no—
347#390—611no—
348#390—611no—
349#389—611no—
350#389—611no—
351#389—611no—
352#389—611no—
353#388—1221no—
354#388—1221no—
355#387—1221no—
356#387—1221no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—1221no—
360C4×C22⋊C4—1221no—
361#376—1221no—
362C43—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369C61#2128811yescharacteristic, complemented, Sylow
370C22×Q8C22×D6111yescharacteristic
371C22×D4C22×D6111yescharacteristic
372C23×C4—21no—
373C2×SD16—22no—
374C2×SD16—22no—
375C2×SD16C2×Dic6114yes—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8C22×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380C4×Q8—22no—
381C4×Q8C2×Dic6114yes—
382C4×D4—22no—
383C4×D4C2×Dic6114yes—
384#117—2444no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—21no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42C22×D6111yescharacteristic
411#107—1222no—
412#107C2×Dic6114yes—
413#106—1222no—
414#103—1222no—
415#103—1222no—
416#103C2×Dic6114yes—
417#102—1222no—
418#102—1222no—
419#102C2×Dic6114yes—
420#97—1222no—
421#97—1222no—
422#97—1222no—
423#97C2×Dic6114yes—
424C4×C8—1222no—
425C4×C8—1222no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C22×Dic6112yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C22×Dic6112yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—612no—
470C2×C8—612no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8#1175822212yes—
476C2×C8#1175822212yes—
477C2×C8C22×Dic6112yes—
478C2×C8C22×Dic6112yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—2442no—
491C4⋊C4—2442no—
492C4⋊C4—2442no—
493C4⋊C4—2442no—
494C4⋊C4C22×Dic6112yes—
495C4⋊C4C22×Dic6112yes—
496C22⋊C4—24no—
497C22⋊C4—2442no—
498C22⋊C4—2442no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C42—24no—
502C42—1222no—
503C42—1222no—
504C42—1221no—
505C42—1221no—
506C42—1221no—
507C42—1221no—
508C42—2441no—
509C42—2441no—
510C42—2442no—
511C42—2442no—
512C42#1175822412yes—
513C42C22×Dic6112yes—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D6111yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4D4×Dic6112yes—
569C2×C4D4×Dic6112yes—
570C2×C4C23×Dic6111yescharacteristic
571C2×C4#41×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C2×C4C2×#1175822411yescharacteristic
574C2×C4C2×#1175822211yescharacteristic
575C2×C4C2×#1175822211yescharacteristic
576C8—22no—
577C8—1222no—
578C8—1222no—
579C8#1451973614yescomplemented
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×D4×Dic6111yescharacteristic
588C22#1684030911yescharacteristic
589C22#1684030911yescharacteristic
590C22AGL(1,ℤ/8)×D6111yescharacteristic
591C22#137×D6111yescharacteristic
592C22#1684055811yescharacteristic
593C22#1684056111yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—1222no—
600C4—1222no—
601C4—4881no—
602C4—4881no—
603C4—4881no—
604C4—4881no—
605C4—2441no—
606C4—2441no—
607C4C2×#1451973611yescharacteristic
608C4#41×Dic6111yescharacteristic
609C4#1684029411yescharacteristic
610C4#1684029411yescharacteristic
611C2—82no—
612C2#2144828411yescharacteristic
613C2AGL(1,ℤ/8)×Dic6111yescharacteristic
614C2#137×Dic6111yescharacteristic
615C2#2145651311yescharacteristic
616C2#2145652411yescharacteristic
617C2#2145653411yescharacteristic
618C2#2145654411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#440C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC2×#440D61
hypocenterC61#21288
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #21448297C2semidirect
5. #21446558C2semidirect
6. #21446553C2semidirect
8. #21446511C2semidirect
10. #21446497C2semidirect
11. #21445716C2semidirect
12. #21445649C2semidirect
14. #97×Dic61C2semidirect
55. C61×#440C4semidirect
267. C8×C61#117semidirect
329. #440Dic61semidirect
369. C61#21288semidirect
579. C8#14519736semidirect

Maximal subgroups

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

subgroupindex
#214482972
#214482842
#214467022
#214466752
#214465582
#214465532
#214465182
#214465112
#214465042
#214464972
#214457162
#214456492
#214456442
#97×Dic612
C2×C61×#4402
#2128861