Finite Group Databasebeta

← #29696936

Subgroups — #29696936

Order 15616.

1212 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839815C211yesmaximal, characteristic, complemented
2C2×#16839724C211yesmaximal, characteristic
3C2×C4×#11758198C211yesmaximal, characteristic
4#21446433C212yesmaximal, complemented
5#21446423C212yesmaximal, complemented
6#21446003C212yesmaximal, complemented
7#21445797C211yesmaximal, characteristic, complemented
8#21445728C211yesmaximal, characteristic
9#21445599C212yesmaximal, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1817C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×#11758189C2212yes—
14C22×#11758198C411yescharacteristic
15C22×#11758198C411yescharacteristic
16C22×C4×Dic61C2211yescharacteristic
17#16839815—24no—
18C2×C4⋊#92418C2212yescomplemented
19C2×#14519706C2212yescomplemented
20#16839724—24no—
21C2×C22⋊#92418C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23C2×#14519689C2212yes—
24C2×#14519689C2212yes—
25C2×#14519689C2212yes—
26C2×#14519687C2212yescomplemented
27C2×C4×#92418C2212yescomplemented
28C4×#11758198C411yescharacteristic
29C4×#11758198C411yescharacteristic
30C4×#11758198C411yescharacteristic
31C4×#11758198C411yescharacteristic
32C4×#11758198C2211yescharacteristic
33C4×#11758198C2211yescharacteristic
34C4×#11758198C2211yescharacteristic
35C4×#11758198C2211yescharacteristic
36C2×#14519684C411yescharacteristic
37C2×#14519684C411yescharacteristic
38#16839516C2212yescomplemented
39#16839516C2212yescomplemented
40C22×C42×C61C2211yescharacteristic
41C2×C22⋊C8×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2211yescharacteristic
43C61×#336C2212yescomplemented
44C61×#336C2212yescomplemented
45C24×D61C2×C411yescharacteristic
46C23×Dic61C2×C411yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C23×Dic61C2×C411yescharacteristic
49C22×C4×D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61—24no—
52C22×C4×D61—42no—
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C2×#11758189—24no—
58C2×#11758189—24no—
59C2×#11758189—42no—
60C22×#92418C2312yes—
61C22×#92418C2312yes—
62C2×#11758198—24no—
63C2×#11758198—24no—
64C2×#11758198C2×C411yescharacteristic
65C2×#11758198C2×C411yescharacteristic
66C2×#11758198C2×C411yescharacteristic
67C2×#11758198C2×C411yescharacteristic
68C2×#11758198C2×C411yescharacteristic
69C2×#11758198C2×C411yescharacteristic
70C2×#11758198C2×C411yescharacteristic
71C2×#11758198C2×C411yescharacteristic
72C2×C4×Dic61—42no—
73C2×C4×Dic61C2×C411yescharacteristic
74C2×C4×Dic61C2×C411yescharacteristic
75C2×C4×Dic61C2311yescharacteristic
76C2×C4×Dic61C2311yescharacteristic
77C4⋊#92418—24no—
78#14519706—42no—
79C22⋊#92418—22no—
80C22⋊#92418D414yes—
81#14519689—24no—
82#14519689—24no—
83#14519689—42no—
84#14519687—42no—
85C4×#92418—24no—
86#14519684C2×C411yescharacteristic
87#14519684C2×C411yescharacteristic
88#14519684C2×C411yescharacteristic
89#14519684C2×C411yescharacteristic
90#14519684C2×C411yescharacteristic
91#14519684C2×C411yescharacteristic
92#14519684C2×C411yescharacteristic
93#14519684C2×C411yescharacteristic
94C23×C4×C61C2×C411yescharacteristic
95C23×C4×C61C2×C411yescharacteristic
96C23×C4×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C22×C8×C61C2312yes—
99C2×C42×C61—22no—
100C2×C42×C61—22no—
101C2×C42×C61C2×C411yescharacteristic
102C2×C42×C61C2×C411yescharacteristic
103C2×C42×C61C2311yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C22⋊C8×C61—22no—
106C22⋊C8×C61—22no—
107C24×C61C22×C411yescharacteristic
108C23×D61—22no—
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—41no—
114C23×D61—41no—
115C23×D61C22×C411yescharacteristic
116C23×D61C22×C411yescharacteristic
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—41no—
124C22×C4×C61M1614yescomplemented
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61#4112yes—
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—41no—
139C22×Dic61—41no—
140C22×Dic61—41no—
141C22×Dic61—41no—
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C22×C411yescharacteristic
147C22×Dic61C22×C411yescharacteristic
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—21no—
155C2×C4×D61—22no—
156C2×C4×D61—21no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—48no—
160C2×C4×D61—48no—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
163C2×C4×D61#4114yes—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176#11758198—22no—
177#11758198—22no—
178#11758198—22no—
179#11758198—22no—
180#11758198—22no—
181#11758198—22no—
182#11758198—22no—
183#11758198M1614yescomplemented
184C4×Dic61—22no—
185C4×Dic61—22no—
186C4×Dic61—41no—
187C4×Dic61—41no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418C2×D414yes—
194#11758189—48no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61C4×D412yes—
200C23×C61C2×M1612yes—
201C23×C61#13112yes—
202C23×C61C23×C411yescharacteristic
203C22×D61—28no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—44no—
217C22×D61—44no—
218C22×D61—44no—
219C22×D61—44no—
220C22×D61—44no—
221C22×D61—44no—
222C22×D61#11712yes—
223C22×D61C2×M1612yes—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—28no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—44no—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C2×M1612yes—
241C2×C4×C61C2×M1612yes—
242C2×C4×C61C2×M1612yes—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4112yes—
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61H(5,𝔽2)12yes—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—44no—
264C2×Dic61—44no—
265C2×Dic61—44no—
266C2×Dic61#11712yes—
267C2×Dic61C2×M1612yes—
268C4×D61—24no—
269C4×D61—24no—
270C4×D61—44no—
271C4×D61—44no—
272C4×D61—42no—
273C4×D61—42no—
274C8×C61—42no—
275C8×C61—42no—
276#92418—24no—
277#92418—42no—
278#22553—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61#43514yes—
286C22×C61C2×#11711yescharacteristic
287C22×C61C2×C4×D411yescharacteristic
288C22×C61#51811yescharacteristic
289C22×C61C22×M1611yescharacteristic
290C22×C61C22×M1611yescharacteristic
291C22×C61C2×#13111yescharacteristic
292C22×C61#56811yescharacteristic
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—416no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—41no—
313C4×C61#54612yes—
314C4×C61#55112yes—
315Dic61—22no—
316Dic61—22no—
317Dic61—41no—
318Dic61—41no—
319Dic61—41no—
320Dic61—41no—
321C2×#431—611no—
322C2×#407—611no—
323C2×C4×C22⋊C4—611no—
324#1924—612no—
325#1923—612no—
326#1851—612no—
327#1817D6111yescharacteristic, complemented
328#1797—611no—
329#1779—611no—
330#1753—612no—
331#1749—611no—
332C2×C61—24no—
333C2×C61#234211yescharacteristic
334C2×C61#292011yescharacteristic
335C2×C61#292411yescharacteristic
336C2×C61C2×#43511yescharacteristic
337C2×C61D4×M1611yescharacteristic
338C2×C61#297411yescharacteristic
339C2×C61#297511yescharacteristic
340D61—42no—
341D61—42no—
342C24×C4—611no—
343C22×M16—612no—
344C22×C22⋊C4—611no—
345C22×C22⋊C4—611no—
346C22×C42—611no—
347C22×C42C2×D6111yescharacteristic
348#431—1224no—
349C2×C4⋊C8—612no—
350C2×C4⋊C8—612no—
351#407—1224no—
352C2×C22⋊C8—611no—
353C2×C22⋊C8—611no—
354C2×C22⋊C8—612no—
355C2×C22⋊C8—612no—
356C2×C22⋊C8—612no—
357C2×C22⋊C8C2×D6111yescharacteristic
358C2×C22⋊C8C2×D6111yescharacteristic
359C2×#97—612no—
360C2×C4×C8—612no—
361C4×C22⋊C4—611no—
362C4×C22⋊C4—611no—
363C4×C22⋊C4—611no—
364C4×C22⋊C4—611no—
365C4×C22⋊C4—611no—
366C4×C22⋊C4—611no—
367C4×C22⋊C4—611no—
368C4×C22⋊C4—611no—
369C2×#95—611no—
370C2×#95—611no—
371#336—612no—
372#336—612no—
373#336C2×D6112yescomplemented
374#336C2×D6112yescomplemented
375C61#2255311yescharacteristic, complemented, Sylow
376C25—611no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—1224no—
385C23×C4—1222no—
386C23×C4—1222no—
387C23×C4—2442no—
388C23×C4C4×D6111yescharacteristic
389C23×C4C4×D6111yescharacteristic
390C23×C4C22×D6111yescharacteristic
391C2×M16—1224no—
392C2×M16—1224no—
393C2×M16—2442no—
394C22×C8—612no—
395C22×C8—612no—
396C22×C8C22×D6112yes—
397C22×C8C22×D6112yes—
398C2×C22⋊C4—611no—
399C2×C22⋊C4—611no—
400C2×C22⋊C4—611no—
401C2×C22⋊C4—611no—
402C2×C22⋊C4—611no—
403C2×C22⋊C4—611no—
404C2×C22⋊C4—611no—
405C2×C22⋊C4—611no—
406C2×C22⋊C4—1224no—
407C2×C22⋊C4—1224no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—2442no—
415C2×C42C4×D6111yescharacteristic
416C2×C42C4×D6111yescharacteristic
417C2×C42C22×D6111yescharacteristic
418C2×C42C22×D6111yescharacteristic
419C4⋊C8—1224no—
420C4⋊C8—2442no—
421C22⋊C8—22no—
422C22⋊C8—22no—
423C22⋊C8—614no—
424C22⋊C8—1224no—
425C22⋊C8—1224no—
426C22⋊C8—1222no—
427C22⋊C8—2442no—
428#97—2442no—
429C4×C8—1224no—
430#95—611no—
431#95—611no—
432#95—611no—
433#95—611no—
434#95—611no—
435#95—611no—
436#95—611no—
437#95—611no—
438C24—611no—
439C24—611no—
440C24—1222no—
441C24—1222no—
442C24—1222no—
443C24—1222no—
444C24—1222no—
445C24—2441no—
446C24—2441no—
447C24C2×C4×D6111yescharacteristic
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—41no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—614no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1221no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—2441no—
479C22×C4—2441no—
480C22×C4—2448no—
481C22×C4—2448no—
482C22×C4—2441no—
483C22×C4—2441no—
484C22×C4—2441no—
485C22×C4—2441no—
486C22×C4#1175818914yescomplemented
487C22×C4C2×C4×D6111yescharacteristic
488C22×C4C2×C4×D6111yescharacteristic
489C22×C4C2×C4×D6111yescharacteristic
490C22×C4C2×C4×D6111yescharacteristic
491C22×C4C2×C4×D6111yescharacteristic
492C22×C4D4×D6112yes—
493C22×C4#1175822412yes—
494C22×C4C23×D6111yescharacteristic
495M16—2448no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—614no—
503C2×C8—1222no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C22⋊C4—614no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—22no—
521C42—22no—
522C42—1222no—
523C42—1222no—
524C42—2441no—
525C42—2441no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—612no—
531C23—612no—
532C23—1228no—
533C23—1221no—
534C23—1221no—
535C23—1221no—
536C23—1221no—
537C23—1221no—
538C23—1221no—
539C23—1221no—
540C23—1221no—
541C23—1221no—
542C23—1221no—
543C23—1221no—
544C23—1221no—
545C23—2444no—
546C23—2444no—
547C23—2444no—
548C23—2444no—
549C23—2444no—
550C23—2444no—
551C23#1451973712yes—
552C23C2×#1175818912yes—
553C23#1451978512yes—
554C23C22×C4×D6111yescharacteristic
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—44no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—2444no—
591C2×C4—2444no—
592C2×C4—2444no—
593C2×C4—2444no—
594C2×C4—2444no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4#1451973712yes—
598C2×C4M16×D6112yes—
599C2×C4C2×#1175818912yes—
600C2×C4C2×#1175818912yes—
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4#41×D6112yes—
603C2×C4C2×#1175822411yescharacteristic
604C2×C4#1451982312yes—
605C8—42no—
606C8—42no—
607C8—1224no—
608C8—2442no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—1222no—
616C22—1222no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—1222no—
621C22—24416no—
622C22—2441no—
623C22—2441no—
624C22—2441no—
625C22—2441no—
626C22—2441no—
627C22—2441no—
628C22—2441no—
629C22—2441no—
630C22#1683983014yes—
631C22#117×D6111yescharacteristic
632C22C2×#1451973711yescharacteristic
633C22#1684023611yescharacteristic
634C22C2×M16×D6111yescharacteristic
635C22C22×#1175818911yescharacteristic
636C22C2×#1451978511yescharacteristic
637C22#1684052811yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—41no—
643C4—1222no—
644C4—1222no—
645C4—2441no—
646C4—2441no—
647C4—2441no—
648C4—2441no—
649C4#1684040412yes—
650C4#1684043512yes—
651C2—24no—
652C2—2442no—
653C2—2442no—
654C2#2144866111yescharacteristic
655C2#2145444711yescharacteristic
656C2#2145447811yescharacteristic
657C2C2×#1683983011yescharacteristic
658C2#2145455111yescharacteristic
659C2#2145485611yescharacteristic
660C2#2145485911yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC61×#1817C2
prime radicalC24×D61C2×C4
cosocleC22×C4×C61C24
hypercenter#1817D61
hypocenterC61#22553
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1817

Complements

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

NH ≅ G/Ntype
1. C2×#16839815C2semidirect
4. #21446433C2semidirect
5. #21446423C2semidirect
6. #21446003C2semidirect
7. #21445797C2semidirect
9. #21445599C2semidirect
10. #21445583C2semidirect
11. C61×#1817C2semidirect
18. C2×C4⋊#92418C22semidirect
19. C2×#14519706C22semidirect
26. C2×#14519687C22semidirect
27. C2×C4×#92418C22semidirect
38. #16839516C22semidirect
39. #16839516C22semidirect
43. C61×#336C22semidirect
44. C61×#336C22semidirect
124. C22×C4×C61M16semidirect
183. #11758198M16semidirect
327. #1817D61semidirect
373. #336C2×D61semidirect
374. #336C2×D61semidirect
375. C61#22553semidirect
486. C22×C4#11758189semidirect

Maximal subgroups

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

subgroupindex
C2×#168398152
C2×#168397242
C2×C4×#117581982
#214464332
#214464232
#214460032
#214457972
#214457282
#214455992
#214455832
C61×#18172
#2255361