Finite Group Databasebeta

← #29610856

Subgroups — #29610856

Order 15616.

1486 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21444087C211yesmaximal, characteristic, complemented
2#21444076C212yesmaximal, complemented
3#21444068C212yesmaximal, complemented
4#21444064C212yesmaximal, complemented
5#21444064C212yesmaximal, complemented
6#21444060C212yesmaximal
7#21444056C212yesmaximal, complemented
8#21444051C211yesmaximal, characteristic, complemented
9C2×C61×#394C211yesmaximal, characteristic
10C22×C4⋊Dic61C2212yes—
11C22×C22⋊Dic61C2212yes—
12C22×C22⋊Dic61C2212yes—
13C22×C22⋊Dic61C2212yes—
14C22×C22⋊Dic61C2212yes—
15C22×C4×Dic61C2211yescharacteristic
16C2×#14519684C2212yes—
17C2×#14519684C2212yes—
18C2×#14519684C2211yescharacteristic
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2212yes—
25C2×#14519684C2211yescharacteristic
26C23×D4×C61C2211yescharacteristic
27C22×C4⋊C4×C61C2211yescharacteristic
28C22×C22⋊C4×C61C2212yes—
29C22×C22⋊C4×C61C2212yes—
30C61×#394C418yescomplemented
31C2×#95×C61C2211yescharacteristic
32C23×Dic61C2312yes—
33C23×Dic61C2312yes—
34C23×Dic61C2312yes—
35C23×Dic61C2312yes—
36C2×C4⋊Dic61—42no—
37C2×C4⋊Dic61—42no—
38C2×C4⋊Dic61—42no—
39C2×C22⋊Dic61—24no—
40C2×C22⋊Dic61—24no—
41C2×C22⋊Dic61—42no—
42C2×C22⋊Dic61—42no—
43C2×C22⋊Dic61—42no—
44C2×C22⋊Dic61—42no—
45C2×C22⋊Dic61—42no—
46C2×C22⋊Dic61—42no—
47C2×C22⋊Dic61—42no—
48C2×C22⋊Dic61—42no—
49C2×C22⋊Dic61—42no—
50C2×C22⋊Dic61—42no—
51C2×C4×Dic61—42no—
52C2×C4×Dic61—41no—
53#14519684—24no—
54#14519684—42no—
55#14519684—42no—
56#14519684—42no—
57#14519684—42no—
58#14519684—42no—
59#14519684—42no—
60#14519684—41no—
61#14519684—41no—
62#14519684—41no—
63C25×C61C2312yes—
64C22×D4×C61—41no—
65C22×D4×C61—41no—
66C22×D4×C61—42no—
67C22×D4×C61—42no—
68C22×D4×C61C2×C414yes—
69C23×C4×C61C2311yescharacteristic
70C23×C4×C61C2312yes—
71C23×C4×C61C2311yescharacteristic
72C23×C4×C61C2311yescharacteristic
73C2×C4⋊C4×C61—41no—
74C2×C4⋊C4×C61—41no—
75C2×C4⋊C4×C61C2×C414yes—
76C2×C22⋊C4×C61—42no—
77C2×C22⋊C4×C61—42no—
78C2×C22⋊C4×C61—42no—
79C2×C22⋊C4×C61—42no—
80C2×C22⋊C4×C61C2×C418yes—
81C2×C22⋊C4×C61C2×C418yes—
82#95×C61C2×C414yes—
83C24×C61—22no—
84C24×C61—22no—
85C24×C61—22no—
86C24×C61—22no—
87C24×C61—22no—
88C24×C61—22no—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—22no—
94C24×C61—22no—
95C24×C61—22no—
96C24×C61—22no—
97C24×C61C22×C414yes—
98C24×C61C2411yescharacteristic
99C2×D4×C61—44no—
100C2×D4×C61—48no—
101C2×D4×C61—48no—
102C2×D4×C61—44no—
103C2×D4×C61—44no—
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—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61C22×C414yes—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C412yes—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
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#4114yes—
154C4⋊C4×C61—44no—
155C22⋊C4×C61—48no—
156C22⋊C4×C61—48no—
157C22⋊Dic61—48no—
158C22⋊Dic61—48no—
159C22⋊Dic61—48no—
160C22⋊Dic61—48no—
161C4⋊Dic61—48no—
162C4×Dic61—44no—
163C23×C61—24no—
164C23×C61—24no—
165C23×C61—24no—
166C23×C61—24no—
167C23×C61—24no—
168C23×C61—24no—
169C23×C61—24no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61C23×C411yescharacteristic
199C23×C61C2×#4112yes—
200C23×C61H(5,𝔽2)11yescharacteristic
201C23×C61H(5,𝔽2)12yes—
202C23×C61H(5,𝔽2)12yes—
203C23×C61H(5,𝔽2)11yescharacteristic
204C23×C61H(5,𝔽2)12yes—
205C23×C61H(5,𝔽2)12yes—
206C23×C612−1+412yes—
207D4×C61—416no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×Dic61—24no—
231C2×Dic61—24no—
232C2×Dic61—24no—
233C2×Dic61—42no—
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—
256C2×Dic61—42no—
257C2×Dic61—42no—
258#14641—611nomaximal, Sylow
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—44no—
262C22×C61—44no—
263C22×C61—44no—
264C22×C61—44no—
265C22×C61—44no—
266C22×C61—82no—
267C22×C61—82no—
268C22×C61—82no—
269C22×C61—82no—
270C22×C61—82no—
271C22×C61—82no—
272C22×C61—82no—
273C22×C61—82no—
274C22×C61C4×#4111yescharacteristic
275C22×C61#51811yescharacteristic
276C22×C61#51811yescharacteristic
277C22×C61#51812yes—
278C22×C61#52012yes—
279C22×C61#53812yes—
280C22×C61#53812yes—
281C22×C61#53912yes—
282C22×C61#54012yes—
283C22×C61#54012yes—
284C22×C61#54212yes—
285C22×C61#56012yes—
286C22×C61#56012yes—
287C22×C61#56012yes—
288C22×C61#56011yescharacteristic
289C22×C61H(3,𝔽4)11yescharacteristic
290C22×C61H(3,𝔽4)11yescharacteristic
291C22×C61#56212yes—
292C22×C61#56212yes—
293C22×C61#56212yes—
294C22×C61#56311yescharacteristic
295C4×C61—42no—
296C4×C61—42no—
297C4×C61—42no—
298C4×C61—44no—
299Dic61—44no—
300Dic61—82no—
301Dic61—82no—
302Dic61—82no—
303C2×#394D6111yescharacteristic
304#1445—611no—
305#1442—612no—
306#1440—612no—
307#1439—612no—
308#1439—612no—
309#1438—612no—
310#1437—612no—
311#1436—611no—
312C2×C61—84no—
313C2×C61#233311yescharacteristic
314C2×C61#233512yes—
315C2×C61#237511yescharacteristic
316C2×C61#237712yes—
317C2×C61#237911yescharacteristic
318C2×C61#282412yes—
319C2×C61#282812yes—
320C2×C61#282912yes—
321C2×C61#283712yes—
322C23×D4C2×D6111yescharacteristic
323C22×C4⋊C4—612no—
324C22×C4⋊C4C2×D6111yescharacteristic
325C22×C22⋊C4—612no—
326C22×C22⋊C4—612no—
327C22×C22⋊C4—612no—
328C22×C22⋊C4—612no—
329C22×C22⋊C4C2×D6112yes—
330C22×C22⋊C4C2×D6112yes—
331C22×C42—611no—
332#394Dic6118yescomplemented
333C2×#95—612no—
334C2×#95—612no—
335C2×#95—611no—
336C2×#95—612no—
337C2×#95—612no—
338C2×#95—612no—
339C2×#95—612no—
340C2×#95—611no—
341C2×#95—612no—
342C2×#95—611no—
343C2×#95C2×D6111yescharacteristic
344C61#1464111yescharacteristic, complemented, Sylow
345C25C22×D6112yes—
346C22×D4—42no—
347C22×D4—42no—
348C22×D4—41no—
349C22×D4—41no—
350C22×D4C2×Dic6114yes—
351C23×C4—612no—
352C23×C4—612no—
353C23×C4—612no—
354C23×C4—612no—
355C23×C4C22×D6112yes—
356C23×C4C22×D6111yescharacteristic
357C23×C4C22×D6111yescharacteristic
358C23×C4C22×D6111yescharacteristic
359C2×C4⋊C4—41no—
360C2×C4⋊C4—41no—
361C2×C4⋊C4—2442no—
362C2×C4⋊C4—2442no—
363C2×C4⋊C4—2442no—
364C2×C4⋊C4C2×Dic6114yes—
365C2×C22⋊C4—42no—
366C2×C22⋊C4—42no—
367C2×C22⋊C4—42no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4—1224no—
370C2×C22⋊C4—1224no—
371C2×C22⋊C4—2442no—
372C2×C22⋊C4—2442no—
373C2×C22⋊C4—2442no—
374C2×C22⋊C4—2442no—
375C2×C22⋊C4—2442no—
376C2×C22⋊C4—2442no—
377C2×C22⋊C4—2442no—
378C2×C22⋊C4—2442no—
379C2×C22⋊C4—2442no—
380C2×C22⋊C4—2442no—
381C2×C22⋊C4C2×Dic6118yes—
382C2×C22⋊C4C2×Dic6118yes—
383C2×C42—2442no—
384C2×C42—2441no—
385#95—1224no—
386#95—2442no—
387#95—2442no—
388#95—2442no—
389#95—2441no—
390#95—2441no—
391#95—2441no—
392#95—2442no—
393#95—2442no—
394#95—2442no—
395#95C2×Dic6114yes—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24C22×Dic6114yes—
411C24C23×D6111yescharacteristic
412C2×D4—44no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—44no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—614no—
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×C4—1222no—
462C22×C4—1222no—
463C22×C4C22×Dic6114yes—
464C22×C4C22×Dic6112yes—
465C22×C4C22×Dic6112yes—
466C22×C4C22×Dic6112yes—
467C4⋊C4—44no—
468C4⋊C4—2448no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—2448no—
472C22⋊C4—2448no—
473C22⋊C4—2448no—
474C22⋊C4—2448no—
475C42—2444no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23C23×Dic6111yescharacteristic
512C23#41×D6112yes—
513C23#1451982212yes—
514C23#1451982212yes—
515C23#1451982212yes—
516C23#1451982211yescharacteristic
517C23#1451982211yescharacteristic
518C23#1451982312yes—
519C23#1451982412yes—
520D4—416no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4—1224no—
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×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22#41×Dic6111yescharacteristic
587C22#1684023712yes—
588C22#1684023711yescharacteristic
589C22#1684023711yescharacteristic
590C22#1684024512yes—
591C22#1684034412yes—
592C22#1684034812yes—
593C22#1684035512yes—
594C22#1684036212yes—
595C22#1684036412yes—
596C22#1684038012yes—
597C22#1684050012yes—
598C22#1684050012yes—
599C22#1684050112yes—
600C22#1684050111yescharacteristic
601C22#1684050411yescharacteristic
602C22#1684050511yescharacteristic
603C22#1684050612yes—
604C22#1684050712yes—
605C22#1684050712yes—
606C22#1684051211yescharacteristic
607C4—42no—
608C4—42no—
609C4—44no—
610C4—42no—
611C4—4882no—
612C4—4882no—
613C4—4882no—
614C4—2444no—
615C2—84no—
616C2#2144858611yescharacteristic
617C2#2144861312yes—
618C2#2144896011yescharacteristic
619C2#2144897612yes—
620C2#2144899311yescharacteristic
621C2#2145389512yes—
622C2#2145393512yes—
623C2#2145395212yes—
624C2#2145402612yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#394C2
socle, cosocleC24×C61C24
prime radicalC23×D4×C61C22
hypercenterC2×#394D61
hypocenterC61#14641
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. #21444087C2semidirect
2. #21444076C2semidirect
3. #21444068C2semidirect
4. #21444064C2semidirect
5. #21444064C2semidirect
7. #21444056C2semidirect
8. #21444051C2semidirect
30. C61×#394C4semidirect
332. #394Dic61semidirect
344. C61#14641semidirect

Maximal subgroups

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

subgroupindex
#214440872
#214440762
#214440682
#214440642
#214440602
#214440562
#214440512
C2×C61×#3942
#1464161