Finite Group Databasebeta

← #29601341

Subgroups — #29601341

Order 15616.

1710 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839654C212yesmaximal
2C2×C4×#11758198C211yesmaximal, characteristic
3#21444055C212yesmaximal
4#21444053C212yesmaximal, complemented
5#21444049C211yesmaximal, characteristic, complemented
6#21444045C212yesmaximal, complemented
7#21444040C211yesmaximal, characteristic
8#21444040C211yesmaximal, characteristic
9#95×Dic61C211yesmaximal, characteristic, complemented
10#95×Dic61C211yesmaximal, characteristic, complemented
11C61×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×C4⋊Dic61C2212yes—
14C22×#11758198C2211yescharacteristic
15C22×#11758198C2212yes—
16C22×#11758198C2212yes—
17C22×#11758198C2211yescharacteristic
18C22×C4×Dic61C2212yes—
19C22×C4×Dic61C2211yescharacteristic
20C22×C4×Dic61C2212yes—
21#16839654C4116yescomplemented
22C4×#11758198C418yescomplemented
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2211yescharacteristic
31C2×#14519684C2212yes—
32C22×C42×C61C2211yescharacteristic
33C2×#95×C61C2211yescharacteristic
34C2×#95×C61C2211yescharacteristic
35C2×#95×C61C2212yes—
36C2×#95×C61C2212yes—
37C24×D61C2311yescharacteristic
38C23×Dic61C2312yes—
39C23×Dic61C2312yes—
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43C22×C4×D61—22no—
44C22×C4×D61—22no—
45C22×C4×D61—42no—
46C22×C4×D61—42no—
47C22×C4×D61C2×C414yes—
48C2×C4⋊Dic61—42no—
49C2×C4⋊Dic61—42no—
50C2×C4⋊Dic61C2×C418yescomplemented
51C2×#11758198—24no—
52C2×#11758198—42no—
53C2×#11758198—42no—
54C2×#11758198—42no—
55C2×#11758198—42no—
56C2×#11758198C2×C414yes—
57C2×#11758198C2×C418yes—
58C2×#11758198C2×C418yes—
59C2×#11758198C2×C414yes—
60C2×#11758198C2×C418yes—
61C2×C4×Dic61—24no—
62C2×C4×Dic61—42no—
63C2×C4×Dic61—42no—
64C2×C4×Dic61—42no—
65C2×C4×Dic61—42no—
66C2×C4×Dic61—42no—
67C2×C4×Dic61C2×C414yes—
68C2×C4×Dic61C2×C418yescomplemented
69#14519684—22no—
70#14519684—22no—
71#14519684—24no—
72#14519684—42no—
73#14519684—42no—
74#14519684—42no—
75#14519684C2×C414yes—
76#14519684C2×C414yes—
77#14519684C2×C418yescomplemented
78C23×C4×C61C2312yes—
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83C2×C42×C61—42no—
84C2×C42×C61C2×C414yes—
85#95×C61—22no—
86#95×C61—22no—
87#95×C61—42no—
88#95×C61C2×C418yescomplemented
89C24×C61C2411yescharacteristic
90C23×D61—21no—
91C23×D61—21no—
92C23×D61—22no—
93C23×D61—22no—
94C23×D61—22no—
95C23×D61—21no—
96C23×D61—21no—
97C23×D61—21no—
98C23×D61—21no—
99C23×D61C22×C412yes—
100C23×D61C22×C414yes—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
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—22no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61C22×C414yes—
123C22×C4×C61C22×C414yes—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C414yes—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C22×C412yes—
128C22×C4×C61#4112yes—
129C22×C4×C61#4112yes—
130C22×Dic61—22no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
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×Dic61C22×C412yes—
153C22×Dic61C22×C414yes—
154C22×Dic61C22×C414yes—
155C22×Dic61C22×C412yes—
156C22×Dic61C22×C414yes—
157C22×Dic61C22×C412yes—
158C22×Dic61#4114yes—
159C2×C4×D61—28no—
160C2×C4×D61—44no—
161C2×C4×D61—44no—
162C2×C4×D61—48no—
163C2×C4×D61—48no—
164C42×C61—44no—
165#11758198—28no—
166#11758198—48no—
167#11758198—48no—
168#11758198C42116yescomplemented
169C4⋊Dic61—48no—
170C4×Dic61—44no—
171C4×Dic61—48no—
172C4×Dic61—48no—
173C23×C61C23×C411yescharacteristic
174C23×C61C23×C412yes—
175C23×C61C2×#4111yescharacteristic
176C23×C61C2×#4111yescharacteristic
177C23×C61C2×#4112yes—
178C23×C61H(5,𝔽2)12yes—
179C23×C61H(5,𝔽2)12yes—
180C23×C61H(5,𝔽2)11yescharacteristic
181C23×C61H(5,𝔽2)11yescharacteristic
182C23×C612−1+411yescharacteristic
183C23×C612−1+411yescharacteristic
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—24no—
187C22×D61—24no—
188C22×D61—24no—
189C22×D61—24no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—24no—
195C22×D61—24no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—41no—
201C22×D61—42no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—42no—
206C22×D61C2×C4214yes—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—24no—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61C2×C4214yes—
235C2×C4×C61#11714yes—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61C2×C4214yes—
264C2×Dic61#11714yes—
265C4×D61—416no—
266#13788—611nomaximal, Sylow
267C22×C61C22×C4211yescharacteristic
268C22×C61C2×#11711yescharacteristic
269C22×C61C2×#11711yescharacteristic
270C22×C61C4×#4112yes—
271C22×C61C4×#4112yes—
272C22×C61#51812yes—
273C22×C61#51811yescharacteristic
274C22×C61#51811yescharacteristic
275C22×C61#51812yes—
276C22×C61#52012yes—
277C22×C61#52011yescharacteristic
278C22×C61#52011yescharacteristic
279C22×C61#52012yes—
280C22×C61#53812yes—
281C22×C61#53812yes—
282C22×C61#53812yes—
283C22×C61#53912yes—
284C22×C61#54012yes—
285C22×C61#54011yescharacteristic
286C22×C61#54111yescharacteristic
287C22×C61#54212yes—
288C22×C61#54211yescharacteristic
289C22×C61#54211yescharacteristic
290C2×D61—24no—
291C2×D61—24no—
292C2×D61—24no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—44no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—44no—
301C4×C61—24no—
302C4×C61—24no—
303C4×C61—42no—
304C4×C61—44no—
305C4×C61—44no—
306Dic61—24no—
307Dic61—24no—
308Dic61—42no—
309Dic61—44no—
310Dic61—44no—
311C2×#388—612no—
312C2×C4×C22⋊C4—611no—
313#1437—612no—
314#1436—612no—
315C4⋊#95—611no—
316#1434—612no—
317#1434D6111yescharacteristic, complemented
318C22⋊#95—611no—
319C22⋊#95—611no—
320C4×#95—611no—
321C4×#95—611no—
322C2×C61C4×#11711yescharacteristic
323C2×C61#227511yescharacteristic
324C2×C61#227511yescharacteristic
325C2×C61#231211yescharacteristic
326C2×C61#231411yescharacteristic
327C2×C61#231811yescharacteristic
328C2×C61#231911yescharacteristic
329C2×C61#233312yes—
330C2×C61#233412yes—
331C2×C61#233512yes—
332C2×C61#233712yes—
333D61—44no—
334C24×C4—611no—
335C22×C4⋊C4—612no—
336C22×C22⋊C4—611no—
337C22×C22⋊C4—611no—
338C22×C22⋊C4—612no—
339C22×C22⋊C4—612no—
340C22×C42—611no—
341C22×C42—612no—
342C22×C42—612no—
343C22×C42C2×D6111yescharacteristic
344#388—6116no—
345C4×C22⋊C4—618no—
346C2×#95—612no—
347C2×#95—611no—
348C2×#95—611no—
349C2×#95—611no—
350C2×#95—611no—
351C2×#95—612no—
352C2×#95—612no—
353C2×#95—612no—
354C2×#95—612no—
355C2×#95C2×D6111yescharacteristic
356C2×#95C2×D6111yescharacteristic
357C2×#95C2×D6112yes—
358C2×#95C2×D6112yes—
359C61#1378811yescharacteristic, complemented, Sylow
360C25—611no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—612no—
364C23×C4—612no—
365C23×C4—611no—
366C23×C4—614no—
367C23×C4—1222no—
368C23×C4—1222no—
369C23×C4—2442no—
370C23×C4—2442no—
371C23×C4C22×D6111yescharacteristic
372C23×C4C22×D6112yes—
373C23×C4C22×D6112yes—
374C23×C4C22×D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C2×C4⋊C4—618no—
377C2×C4⋊C4—2442no—
378C2×C4⋊C4—2442no—
379C2×C22⋊C4—614no—
380C2×C22⋊C4—618no—
381C2×C22⋊C4—614no—
382C2×C22⋊C4—618no—
383C2×C22⋊C4—618no—
384C2×C22⋊C4—1224no—
385C2×C22⋊C4—2442no—
386C2×C22⋊C4—2442no—
387C2×C22⋊C4—2442no—
388C2×C22⋊C4—2442no—
389C2×C42—42no—
390C2×C42—614no—
391C2×C42—618no—
392C2×C42—1224no—
393C2×C42—2442no—
394C2×C42—2442no—
395C2×C42—2442no—
396C2×C42—2442no—
397C2×C42—2442no—
398C2×C42C4×D6114yes—
399#95—22no—
400#95—22no—
401#95—42no—
402#95—614no—
403#95—614no—
404#95—618no—
405#95—1224no—
406#95—1222no—
407#95—1222no—
408#95—2442no—
409#95—2442no—
410#95—2442no—
411#95C4×D6118yescomplemented
412C24—614no—
413C24—612no—
414C24—1222no—
415C24—1221no—
416C24—1221no—
417C24—1222no—
418C24—1221no—
419C24—1221no—
420C24—1222no—
421C24—1221no—
422C24—1221no—
423C24C23×D6111yescharacteristic
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—612no—
446C22×C4—614no—
447C22×C4—612no—
448C22×C4—614no—
449C22×C4—612no—
450C22×C4—614no—
451C22×C4—614no—
452C22×C4—1222no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1221no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1228no—
475C22×C4—2444no—
476C22×C4—2444no—
477C22×C4—2448no—
478C22×C4—2448no—
479C22×C4C2×C4×D6114yes—
480C22×C4C2×C4×D6112yes—
481C22×C4C2×C4×D6114yes—
482C22×C4C2×C4×D6112yes—
483C22×C4C2×C4×D6114yes—
484C22×C4C2×C4×D6112yes—
485C22×C4#1175822412yes—
486C22×C4#1175822612yes—
487C4⋊C4—2448no—
488C22⋊C4—6116no—
489C22⋊C4—1228no—
490C22⋊C4—2448no—
491C22⋊C4—2448no—
492C42—44no—
493C42—2444no—
494C42—2448no—
495C42—2448no—
496C23—614no—
497C23—1222no—
498C23—1222no—
499C23—1224no—
500C23—1224no—
501C23—1224no—
502C23—1222no—
503C23—1224no—
504C23—1222no—
505C23—1224no—
506C23—1222no—
507C23—1222no—
508C23—1224no—
509C23—2442no—
510C23—2442no—
511C23—2442no—
512C23—2442no—
513C23—2442no—
514C23—2441no—
515C23—2441no—
516C23—2442no—
517C23—2441no—
518C23—2441no—
519C23C22×C4×D6111yescharacteristic
520C23C22×C4×D6112yes—
521C23#41×D6112yes—
522C23C2×#1175822411yescharacteristic
523C23C2×#1175822611yescharacteristic
524C23#1451982212yes—
525C23#1451982311yescharacteristic
526C23#1451982312yes—
527C23#1451982311yescharacteristic
528C23#1451982411yescharacteristic
529C23#1451982411yescharacteristic
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—614no—
558C2×C4—614no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1224no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1224no—
565C2×C4—1224no—
566C2×C4—1224no—
567C2×C4—1222no—
568C2×C4—1224no—
569C2×C4—1224no—
570C2×C4—1224no—
571C2×C4—1224no—
572C2×C4—1222no—
573C2×C4—1224no—
574C2×C4—1224no—
575C2×C4—1222no—
576C2×C4—24416no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4C42×D6114yes—
588C2×C4#1451973514yes—
589C22—1224no—
590C22—1224no—
591C22—1224no—
592C22—2442no—
593C22—2442no—
594C22—2444no—
595C22—2444no—
596C22—2444no—
597C22—2444no—
598C22—2442no—
599C22—2442no—
600C22C2×C42×D6111yescharacteristic
601C22#117×D6111yescharacteristic
602C22C2×#1451973511yescharacteristic
603C22C4×#1175822412yes—
604C22C4×#1175822612yes—
605C22#1684023412yes—
606C22#1684023511yescharacteristic
607C22#1684023611yescharacteristic
608C22#1684023612yes—
609C22#1684024112yes—
610C22#1684024211yescharacteristic
611C22#1684024211yescharacteristic
612C22#1684024212yes—
613C22#1684034912yes—
614C22#1684035112yes—
615C22#1684035212yes—
616C22#1684035312yes—
617C22#1684036412yes—
618C22#1684036811yescharacteristic
619C22#1684037511yescharacteristic
620C22#1684038012yes—
621C22#1684038111yescharacteristic
622C22#1684038111yescharacteristic
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—1224no—
629C4—1224no—
630C4—2444no—
631C4—2444no—
632C4—2442no—
633C2—2444no—
634C2C4×#1451973511yescharacteristic
635C2#2144817811yescharacteristic
636C2#2144818111yescharacteristic
637C2#2144841611yescharacteristic
638C2#2144842911yescharacteristic
639C2#2144845011yescharacteristic
640C2#2144845911yescharacteristic
641C2#2144858312yes—
642C2#2144859712yes—
643C2#2144860212yes—
644C2#2144862612yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC22×C61C22×C42
FittingC61×#1434C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1434D61
hypocenterC61#13788
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #21444053C2semidirect
5. #21444049C2semidirect
6. #21444045C2semidirect
9. #95×Dic61C2semidirect
10. #95×Dic61C2semidirect
11. C61×#1434C2semidirect
21. #16839654C4semidirect
22. C4×#11758198C4semidirect
50. C2×C4⋊Dic61C2×C4semidirect
68. C2×C4×Dic61C2×C4semidirect
77. #14519684C2×C4semidirect
88. #95×C61C2×C4semidirect
168. #11758198C42semidirect
317. #1434D61semidirect
359. C61#13788semidirect
411. #95C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396542
C2×C4×#117581982
#214440552
#214440532
#214440492
#214440452
#214440402
#95×Dic612
C61×#14342
#1378861