Finite Group Databasebeta

← #29689731

Subgroups — #29689731

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840307C211yesmaximal, characteristic
2C2×#16839816C211yesmaximal, characteristic, complemented
3C2×#16839627C211yesmaximal, characteristic, complemented
4#21446790C212yesmaximal, complemented
5#21446641C212yesmaximal, complemented
6#21446137C211yesmaximal, characteristic, complemented
7#21446136C211yesmaximal, characteristic, complemented
8#21446134C211yesmaximal, characteristic, complemented
9#21445828C212yesmaximal, complemented
10#21445827C212yesmaximal, complemented
11C61×#1874C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13#16840307—24no—
14#16840307C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519773C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519751C2211yescharacteristic
19C2×#14519749—21no—
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519737—21no—
23C22×C4⋊Dic61C2211yescharacteristic
24C22×C4×Dic61C2211yescharacteristic
25#16839816—24no—
26C2×#14519706C2212yes—
27C2×#14519699C2212yes—
28C2×#14519698C2212yes—
29C2×#14519697C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519687C2212yes—
32#16839627—21no—
33#16839627—21no—
34#16839627C2212yes—
35#16839627C2212yes—
36C2×#14519684—21no—
37C2×#14519684—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#121×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758224—28no—
47C2×#11758224—44no—
48C2×#11758224—44no—
49C22×Dic122—21no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C23×Dic61—21no—
56C23×Dic61D411yescharacteristic
57C23×Dic61D411yescharacteristic
58C22×C4×D61—21no—
59C22×#92418C2312yes—
60#14519776—22no—
61#14519776C2×C414yescomplemented
62#14519773—22no—
63#14519773C2×C414yescomplemented
64#14519763—28no—
65#14519751—22no—
66#14519751C2×C414yes—
67#14519749—28no—
68D4×Dic61—22no—
69D4×Dic61C2×C414yescomplemented
70#14519739—22no—
71#14519739C2×C414yescomplemented
72#14519737—28no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—42no—
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61C2×C414yes—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D414yes—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519706—24no—
88#14519699—42no—
89#14519698—42no—
90#14519697—42no—
91C22⋊#92418—22no—
92#14519687—24no—
93#14519684—22no—
94#14519684—22no—
95#14519684—22no—
96#14519684—22no—
97C22×D4×C61—21no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C22×C8×C61C2312yes—
102#121×C61—22no—
103#121×C61C2×C414yes—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—42no—
106C2×C4⋊C4×C61C2312yes—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109#102×C61—42no—
110C22⋊C8×C61—22no—
111C24×C61—21no—
112C24×C61C2×D411yescharacteristic
113C23×D61—21no—
114C2×D4×C61—24no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61C22×C412yes—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—24no—
133C2×C22⋊D61—24no—
134C2×C22⋊D61—24no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61—42no—
138C2×C22⋊D61—42no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—42no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—24no—
145C22×Dic61—22no—
146C22×Dic61—41no—
147C22×Dic61—41no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61C22⋊C414yes—
154C22×Dic61C22⋊C414yes—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159#11758224—432no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—21no—
163C2×D244C22×C412yes—
164C2×C4×D61—24no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×Dic122—24no—
168C2×Dic122—42no—
169C2×Dic122—42no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61C22×C412yes—
178C22⋊C4×C61—24no—
179C22⋊Dic61—24no—
180#11758198—24no—
181#11758198—24no—
182C4⋊Dic61—21no—
183C4⋊Dic61—22no—
184C4⋊Dic61—22no—
185C4⋊Dic61—42no—
186C4⋊Dic61C22×C412yes—
187#11758196—24no—
188C4×Dic61C22⋊C414yes—
189C4×Dic61C22⋊C414yes—
190C4×Dic61C22×C412yes—
191C4×Dic61C22×C412yes—
192C4×Dic61C2×D412yes—
193C4×Dic61C2×D412yes—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—22no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61C2×C22⋊C412yes—
209C23×C61AGL(1,ℤ/8)12yes—
210C23×C61AGL(1,ℤ/8)12yes—
211C23×C61C22×D411yescharacteristic
212C22×D61—22no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219D4×C61—24no—
220D4×C61—48no—
221D4×C61—42no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—84no—
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C23×C411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61H(5,𝔽2)12yes—
237C2×C4×C612−1+412yes—
238C22⋊D61—48no—
239C22⋊D61—48no—
240C22⋊D61—48no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—28no—
250C2×Dic61—22no—
251C2×Dic61—44no—
252C2×Dic61—48no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—84no—
259C2×Dic61C2×C22⋊C412yes—
260C2×Dic61C2×C22⋊C412yes—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263D244—24no—
264D244—42no—
265C4×D61—48no—
266Dic122—48no—
267C8×C61—42no—
268#92418—42no—
269#21865—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61#41814yes—
284C22×C61C22×C22⋊C411yescharacteristic
285C22×C61#53511yescharacteristic
286C22×C61#53611yescharacteristic
287C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C61#57511yescharacteristic
290C22×C61#57511yescharacteristic
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C4×C61—22no—
299C4×C61—42no—
300C4×C61—82no—
301C4×C61#52012yes—
302C4×C61#53712yes—
303Dic61—22no—
304Dic61—22no—
305Dic61—22no—
306Dic61—22no—
307Dic61—42no—
308Dic61—82no—
309C2×#532—611no—
310C2×#432—611no—
311C2×#382—611no—
312#1993—612no—
313#1961—612no—
314#1874—611no—
315#1874—611no—
316#1874—611no—
317#1874D6111yescharacteristic, complemented
318#1804—612no—
319#1804—612no—
320C2×C61—24no—
321C2×C61—82no—
322C2×C61#231111yescharacteristic
323C2×C61C2×#41811yescharacteristic
324C2×C61#289611yescharacteristic
325C2×C61#321311yescharacteristic
326C2×C61#321311yescharacteristic
327C2×C61#323411yescharacteristic
328C2×C61#323411yescharacteristic
329D61—82no—
330C22×#41—1221no—
331#532—618no—
332#532—1224no—
333C2×C4⋊Q8—611no—
334C2×#127—611no—
335C2×#124—1221no—
336C2×#121—611no—
337C2×#121—1221no—
338C2×#121C2×D6111yescharacteristic
339C2×C4×D4—611no—
340C2×C4×D4—611no—
341C2×C4×D4—1221no—
342C22×C4⋊C4—611no—
343C22×C4⋊C4C2×D6111yescharacteristic
344C22×C42—611no—
345#432—1224no—
346C2×C4⋊C8—612no—
347C2×#102—612no—
348C2×#102—612no—
349C2×#102—612no—
350C2×#102C2×D6112yes—
351C2×C22⋊C8—611no—
352C2×C22⋊C8C2×D6111yescharacteristic
353C2×#97—612no—
354#382—612no—
355#382—612no—
356#382—1221no—
357#382—1221no—
358C2×#95—1221no—
359C2×#95—1221no—
360#340—612no—
361#340—612no—
362#340—612no—
363#340C2×D6112yes—
364C61#2186511yescharacteristic, complemented, Sylow
365C2×#41—1228no—
366C2×#41—2444no—
367C2×#41—2444no—
368C22×Q8—1221no—
369C22×D4—21no—
370C22×D4—611no—
371C22×D4—1221no—
372C22×D4—1221no—
373C22×D4—1221no—
374C22×D4C22×D6111yescharacteristic
375C23×C4—21no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—1221no—
379C23×C4—1221no—
380C23×C4—1221no—
381C23×C4C22×D6111yescharacteristic
382C22×C8—612no—
383C22×C8C22×D6112yes—
384C4⋊Q8—614no—
385C4⋊Q8—1222no—
386#127—614no—
387#127—1222no—
388#124—1228no—
389#121—22no—
390#121—614no—
391#121—1228no—
392#121—1222no—
393#121C4×D6114yes—
394C4×D4—614no—
395C4×D4—614no—
396C4×D4—1228no—
397C4×D4—1222no—
398C4×D4—1222no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—42no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—612no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—2442no—
406C2×C4⋊C4C22×D6112yes—
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C42—614no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—614no—
416C2×C42—611no—
417C2×C42—611no—
418C4⋊C8—1224no—
419#102—42no—
420#102—2442no—
421#102—2442no—
422#102—2442no—
423C22⋊C8—22no—
424C22⋊C8—1222no—
425#97—1224no—
426#95—1222no—
427#95—1222no—
428#95—1222no—
429#95—1222no—
430C24—21no—
431C24—1221no—
432C24D4×D6111yescharacteristic
433#41—24432no—
434C2×Q8—1224no—
435C2×Q8—2442no—
436C2×Q8—2442no—
437C2×D4—24no—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—612no—
444C2×D4—1224no—
445C2×D4—1224no—
446C2×D4—1224no—
447C2×D4—1221no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—2442no—
451C2×D4—2442no—
452C2×D4—2442no—
453C2×D4—2442no—
454C2×D4—2442no—
455C2×D4—2442no—
456C2×D4C2×C4×D6112yes—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—614no—
469C22×C4—614no—
470C22×C4—1224no—
471C22×C4—1224no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—2442no—
477C22×C4—2442no—
478C22×C4—2441no—
479C22×C4—2441no—
480C22×C4—2442no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4C2×C4×D6112yes—
486C22×C4D4×D6112yes—
487C22×C4D4×D6111yescharacteristic
488C22×C4C23×D6111yescharacteristic
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—1222no—
493C2×C8—1222no—
494C2×C8—1222no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—42no—
499C4⋊C4—612no—
500C4⋊C4—1224no—
501C4⋊C4—1221no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—2442no—
505C4⋊C4C2×C4×D6112yes—
506C22⋊C4—24no—
507C22⋊C4—1224no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C42—612no—
511C42—612no—
512C42—612no—
513C42—612no—
514C42—614no—
515C42—614no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—1222no—
528C23—2441no—
529C23—2441no—
530C23—2441no—
531C23—2441no—
532C23—2441no—
533C23—2441no—
534C23C22⋊C4×D6112yes—
535C23#1451980112yes—
536C23#1451980412yes—
537C23C2×D4×D6111yescharacteristic
538Q8—2448no—
539D4—24no—
540D4—48no—
541D4—42no—
542D4—1224no—
543D4—2448no—
544D4—2448no—
545D4—2448no—
546D4—2442no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—84no—
558C2×C4—612no—
559C2×C4—612no—
560C2×C4—612no—
561C2×C4—612no—
562C2×C4—1228no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—4884no—
573C2×C4—2448no—
574C2×C4—2444no—
575C2×C4—2448no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4C22⋊C4×D6112yes—
582C2×C4C22×C4×D6111yescharacteristic
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4#1451982212yes—
585C2×C4#1451982412yes—
586C8—42no—
587C8—2442no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—4881no—
602C22—4881no—
603C22—4881no—
604C22—4881no—
605C22—2442no—
606C22—2442no—
607C22—2442no—
608C22#1683976514yes—
609C22C2×C22⋊C4×D6111yescharacteristic
610C22#1684032711yescharacteristic
611C22#1684033411yescharacteristic
612C22C2×#1451980111yescharacteristic
613C22C2×#1451980411yescharacteristic
614C22#1684055811yescharacteristic
615C22#1684056111yescharacteristic
616C4—22no—
617C4—42no—
618C4—82no—
619C4—1222no—
620C4—1222no—
621C4—1222no—
622C4—1222no—
623C4—4882no—
624C4—2442no—
625C4#1684024112yes—
626C4#1684033912yes—
627C2—24no—
628C2—82no—
629C2—4882no—
630C2#2144840611yescharacteristic
631C2C2×#1683976511yescharacteristic
632C2#2145433511yescharacteristic
633C2#2145680311yescharacteristic
634C2#2145680511yescharacteristic
635C2#2145700611yescharacteristic
636C2#2145700811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1874C2
socleC23×C61C22×D4
prime radicalC2×#16840307C2
cosocleC22×C4×C61C24
hypercenter#1874D61
hypocenterC61#21865
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C2×#16839816C2semidirect
3. C2×#16839627C2semidirect
4. #21446790C2semidirect
5. #21446641C2semidirect
6. #21446137C2semidirect
7. #21446136C2semidirect
8. #21446134C2semidirect
9. #21445828C2semidirect
10. #21445827C2semidirect
11. C61×#1874C2semidirect
14. #16840307C4semidirect
61. #14519776C2×C4semidirect
63. #14519773C2×C4semidirect
69. D4×Dic61C2×C4semidirect
71. #14519739C2×C4semidirect
317. #1874D61semidirect
364. C61#21865semidirect

Maximal subgroups

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

subgroupindex
C2×#168403072
C2×#168398162
C2×#168396272
#214467902
#214466412
#214461372
#214461362
#214461342
#214458282
#214458272
C61×#18742
#2186561