Finite Group Databasebeta

← #29607945

Subgroups — #29607945

Order 15616.

1750 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839661C212yesmaximal, complemented
2C2×#16839655C212yesmaximal
3C2×#16839622C212yesmaximal
4#21444084C212yesmaximal, complemented
5#21444072C212yesmaximal, complemented
6#21444064C211yesmaximal, characteristic
7#21444064C211yesmaximal, characteristic
8#21444053C212yesmaximal, complemented
9C2×C61×#385C211yesmaximal, characteristic
10C22×C4⋊Dic61C2212yes—
11C22×C4⋊Dic61C2212yes—
12C22×C4⋊Dic61C2212yes—
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C22⋊Dic61C2212yes—
16C22×C22⋊Dic61C2212yes—
17C22×C4×Dic61C2212yes—
18#16839661—28no—
19#16839655—28no—
20#16839622—28no—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2212yes—
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2212yes—
29C24×C4×C61C2211yescharacteristic
30C22×C4⋊C4×C61C2212yes—
31C22×C22⋊C4×C61C2211yescharacteristic
32C22×C22⋊C4×C61C2211yescharacteristic
33C61×#385C418yescomplemented
34C2×#95×C61C2212yes—
35C23×Dic61C2312yes—
36C23×Dic61C2312yes—
37C23×Dic61C2312yes—
38C23×Dic61C2312yes—
39C2×C4⋊Dic61—24no—
40C2×C4⋊Dic61—24no—
41C2×C4⋊Dic61—24no—
42C2×C4⋊Dic61—24no—
43C2×C4⋊Dic61—24no—
44C2×C4⋊Dic61—24no—
45C2×C4⋊Dic61—24no—
46C2×C4⋊Dic61—41no—
47C2×C4⋊Dic61—42no—
48C2×C4⋊Dic61—42no—
49C2×C4⋊Dic61—42no—
50C2×C4⋊Dic61—42no—
51C2×C4⋊Dic61—41no—
52C2×C22⋊Dic61—24no—
53C2×C22⋊Dic61—24no—
54C2×C22⋊Dic61—24no—
55C2×C22⋊Dic61—24no—
56C2×C22⋊Dic61—42no—
57C2×C22⋊Dic61—42no—
58C2×C4×Dic61—24no—
59C2×C4×Dic61—24no—
60C2×C4×Dic61—42no—
61#14519684—24no—
62#14519684—24no—
63#14519684—24no—
64#14519684—24no—
65#14519684—42no—
66#14519684—41no—
67#14519684—41no—
68#14519684D418yes—
69C25×C61C2311yescharacteristic
70C23×C4×C61—24no—
71C23×C4×C61—24no—
72C23×C4×C61—41no—
73C23×C4×C61—41no—
74C23×C4×C61C2×C414yes—
75C23×C4×C61C2311yescharacteristic
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2312yes—
78C23×C4×C61C2312yes—
79C2×C4⋊C4×C61—24no—
80C2×C4⋊C4×C61—24no—
81C2×C4⋊C4×C61C2×C418yes—
82C2×C22⋊C4×C61—41no—
83C2×C22⋊C4×C61—41no—
84C2×C22⋊C4×C61C2×C414yes—
85C2×C22⋊C4×C61C2×C414yes—
86C2×C22⋊C4×C61D414yes—
87C2×C22⋊C4×C61Q814yes—
88#95×C61C2×C418yes—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—21no—
92C24×C61—21no—
93C24×C61—22no—
94C24×C61—21no—
95C24×C61—21no—
96C24×C61—22no—
97C24×C61C22×C412yes—
98C24×C61C2×D412yes—
99C24×C61C2×Q812yes—
100C24×C61C2411yescharacteristic
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—216no—
118C22×C4×C61—44no—
119C22×C4×C61—44no—
120C22×C4×C61—48no—
121C22×C4×C61—44no—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C22×C414yes—
125C22×C4×C61C22×C414yes—
126C22×C4×C61C2×D414yes—
127C22×C4×C61C2×D414yes—
128C22×C4×C61C2×Q814yes—
129C22×C4×C61#4114yes—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61C2×D414yes—
157C22×Dic61C2×D414yes—
158C4⋊C4×C61—216no—
159C22⋊C4×C61—44no—
160C22⋊C4×C61C4⋊C4116yescomplemented
161C22⋊Dic61—48no—
162C22⋊Dic61—48no—
163C4⋊Dic61—44no—
164C4⋊Dic61—48no—
165C4⋊Dic61—48no—
166C4⋊Dic61—48no—
167C4⋊Dic61—44no—
168C4×Dic61—48no—
169C23×C61—24no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—24no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—24no—
178C23×C61—24no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—41no—
186C23×C61—42no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—42no—
190C23×C61—41no—
191C23×C61C2×C4⋊C414yes—
192C23×C61C23×C411yescharacteristic
193C23×C61C22×D411yescharacteristic
194C23×C61C22×D412yes—
195C23×C61C22×Q811yescharacteristic
196C23×C61C2×#4112yes—
197C23×C61H(5,𝔽2)11yescharacteristic
198C23×C61H(5,𝔽2)11yescharacteristic
199C23×C61H(5,𝔽2)12yes—
200C23×C612−1+411yescharacteristic
201C23×C612−1+411yescharacteristic
202C23×C612−1+412yes—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—416no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61C2×C4⋊C418yes—
227C2×C4×C61C4×D418yes—
228C2×Dic61—24no—
229C2×Dic61—24no—
230C2×Dic61—24no—
231C2×Dic61—24no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256#14402—611nomaximal, Sylow
257C22×C61—24no—
258C22×C61—28no—
259C22×C61—44no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—44no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—44no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61C22×C4⋊C411yescharacteristic
269C22×C61C2×C4×D412yes—
270C22×C61#51811yescharacteristic
271C22×C61#51911yescharacteristic
272C22×C61#52012yes—
273C22×C61#53612yes—
274C22×C61#53712yes—
275C22×C61#53711yescharacteristic
276C22×C61#53711yescharacteristic
277C22×C61#54112yes—
278C22×C61#54212yes—
279C22×C61#54411yescharacteristic
280C22×C61#54411yescharacteristic
281C22×C61D4211yescharacteristic
282C22×C61D4212yes—
283C22×C61#54612yes—
284C22×C61#54712yes—
285C22×C61D4×Q812yes—
286C22×C61#55111yescharacteristic
287C22×C61#55412yes—
288C22×C61#56011yescharacteristic
289C22×C61#56212yes—
290C22×C61#56311yescharacteristic
291C4×C61—28no—
292C4×C61—28no—
293C4×C61—42no—
294C4×C61—42no—
295Dic61—44no—
296Dic61—44no—
297Dic61—82no—
298Dic61—82no—
299C2×#389—612no—
300C2×#388—612no—
301C2×#385D6111yescharacteristic
302C2×#380—612no—
303#1444—612no—
304#1441—612no—
305#1439—611no—
306#1439—611no—
307#1436—612no—
308C2×C61—44no—
309C2×C61#231611yescharacteristic
310C2×C61#232312yes—
311C2×C61D4×C4⋊C412yes—
312C2×C61#235811yescharacteristic
313C2×C61#237611yescharacteristic
314C2×C61#267212yes—
315C2×C61#267511yescharacteristic
316C2×C61#267711yescharacteristic
317C2×C61#270412yes—
318C2×C61#272211yescharacteristic
319C2×C61#274211yescharacteristic
320C24×C4C2×D6111yescharacteristic
321C22×C4⋊C4—612no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4—612no—
324C22×C4⋊C4—612no—
325C22×C4⋊C4—611no—
326C22×C4⋊C4C2×D6112yes—
327C22×C22⋊C4—612no—
328C22×C22⋊C4—612no—
329C22×C22⋊C4C2×D6111yescharacteristic
330C22×C22⋊C4C2×D6111yescharacteristic
331C22×C42—612no—
332#389—1228no—
333#388—1228no—
334#385Dic6118yescomplemented
335#380—1228no—
336C2×#95—612no—
337C2×#95—612no—
338C2×#95—612no—
339C2×#95—612no—
340C2×#95—612no—
341C2×#95—611no—
342C2×#95—612no—
343C2×#95—611no—
344C2×#95C2×D6112yes—
345C61#1440211yescharacteristic, complemented, Sylow
346C25C22×D6111yescharacteristic
347C23×C4—24no—
348C23×C4—24no—
349C23×C4—41no—
350C23×C4—41no—
351C23×C4—612no—
352C23×C4—612no—
353C23×C4—612no—
354C23×C4—612no—
355C23×C4C2×Dic6114yes—
356C23×C4C22×D6111yescharacteristic
357C23×C4C22×D6111yescharacteristic
358C23×C4C22×D6112yes—
359C23×C4C22×D6112yes—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—1224no—
363C2×C4⋊C4—1224no—
364C2×C4⋊C4—1224no—
365C2×C4⋊C4—1224no—
366C2×C4⋊C4—1224no—
367C2×C4⋊C4—1224no—
368C2×C4⋊C4—1224no—
369C2×C4⋊C4—2442no—
370C2×C4⋊C4—2442no—
371C2×C4⋊C4—2442no—
372C2×C4⋊C4—2441no—
373C2×C4⋊C4—2441no—
374C2×C4⋊C4—2442no—
375C2×C4⋊C4C2×Dic6118yes—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—41no—
378C2×C22⋊C4—1224no—
379C2×C22⋊C4—1224no—
380C2×C22⋊C4—1224no—
381C2×C22⋊C4—1224no—
382C2×C22⋊C4—2442no—
383C2×C22⋊C4—2442no—
384C2×C22⋊C4Dic12214yes—
385C2×C22⋊C4D24414yes—
386C2×C22⋊C4C2×Dic6114yes—
387C2×C22⋊C4C2×Dic6114yes—
388C2×C42—1224no—
389C2×C42—1224no—
390C2×C42—2442no—
391#95—618no—
392#95—1224no—
393#95—1224no—
394#95—1224no—
395#95—1224no—
396#95—2441no—
397#95—2441no—
398#95—2442no—
399#95C2×Dic6118yes—
400C24—22no—
401C24—22no—
402C24—21no—
403C24—21no—
404C24—22no—
405C24—22no—
406C24—21no—
407C24—21no—
408C24C2×Dic12212yes—
409C24C2×D24412yes—
410C24C22×Dic6112yes—
411C24C23×D6111yescharacteristic
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—216no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—614no—
434C22×C4—614no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4C2×Dic12214yes—
462C22×C4C2×D24414yes—
463C22×C4D4×D6114yes—
464C22×C4#1175822414yes—
465C22×C4C22×Dic6112yes—
466C22×C4C22×Dic6112yes—
467C22×C4C22×Dic6114yes—
468C22×C4C22×Dic6114yes—
469C4⋊C4—216no—
470C4⋊C4—2448no—
471C4⋊C4—2448no—
472C4⋊C4—2448no—
473C4⋊C4—2444no—
474C4⋊C4—2444no—
475C22⋊C4—44no—
476C22⋊C4—2448no—
477C22⋊C4—2448no—
478C22⋊C4C4⋊Dic61116yescomplemented
479C42—2448no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23C2×C4⋊Dic6114yes—
503C23C23×Dic6111yescharacteristic
504C23C2×D4×D6112yes—
505C23C22×D24411yescharacteristic
506C23C22×Dic12211yescharacteristic
507C23C2×#1175822412yes—
508C23#1451982211yescharacteristic
509C23#1451982211yescharacteristic
510C23#1451982312yes—
511C23#1451982412yes—
512C23#1451982511yescharacteristic
513C23#1451982511yescharacteristic
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—416no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1224no—
541C2×C4—1224no—
542C2×C4—1224no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4C2×C4⋊Dic6118yes—
566C2×C4D4×Dic6118yes—
567C22—28no—
568C22—24no—
569C22—42no—
570C22—44no—
571C22—44no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22C22×C4⋊Dic6111yescharacteristic
579C22C2×D4×Dic6112yes—
580C22#1684023711yescharacteristic
581C22#1684024011yescharacteristic
582C22#1684024512yes—
583C22#1684033512yes—
584C22#1684034212yes—
585C22#1684034311yescharacteristic
586C22#1684034311yescharacteristic
587C22#1684037212yes—
588C22#1684038512yes—
589C22#1684039311yescharacteristic
590C22#1684039311yescharacteristic
591C22#1684039411yescharacteristic
592C22D4×D24412yes—
593C22#1684039912yes—
594C22#1684041612yes—
595C22D4×Dic12212yes—
596C22#1684044311yescharacteristic
597C22#1684046912yes—
598C22#1684050211yescharacteristic
599C22#1684050912yes—
600C22#1684051511yescharacteristic
601C4—28no—
602C4—28no—
603C4—42no—
604C4—42no—
605C4—4882no—
606C4—4882no—
607C4—2444no—
608C4—2444no—
609C2—44no—
610C2#2144843711yescharacteristic
611C2#2144850312yes—
612C2D4×C4⋊Dic6112yes—
613C2#2144880311yescharacteristic
614C2#2144896711yescharacteristic
615C2#2145207312yes—
616C2#2145211011yescharacteristic
617C2#2145213411yescharacteristic
618C2#2145248112yes—
619C2#2145272511yescharacteristic
620C2#2145294811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#385C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#385D61
hypocenterC61#14402
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
1. C2×#16839661C2semidirect
4. #21444084C2semidirect
5. #21444072C2semidirect
8. #21444053C2semidirect
33. C61×#385C4semidirect
160. C22⋊C4×C61C4⋊C4semidirect
334. #385Dic61semidirect
345. C61#14402semidirect
478. C22⋊C4C4⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396612
C2×#168396552
C2×#168396222
#214440842
#214440722
#214440642
#214440532
C2×C61×#3852
#1440261