Finite Group Databasebeta

← #30620277

Subgroups — #30620277

Order 15616.

1424 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840793C212yesmaximal, complemented
2#21461993C212yesmaximal, complemented
3#21461796C212yesmaximal, complemented
4#21461792C212yesmaximal, complemented
5#21461710C211yesmaximal, characteristic, complemented
6#21461709C211yesmaximal, characteristic, complemented
7#21461675C211yesmaximal, characteristic, complemented
8#21461545C211yesmaximal, characteristic, complemented
9#21461482C211yesmaximal, characteristic, complemented
10C2×#16840628C211yesmaximal, characteristic, complemented
11C2×#41×Dic61C211yesmaximal, characteristic
12C22×#11758217C2212yes—
13C23×#92420C2212yes—
14#16840793—28no—
15C2×#14519844C2212yes—
16C2×C22⋊#92420C2212yes—
17C2×C22⋊#92420C2212yes—
18C2×C22⋊#92420C2212yes—
19C2×C22⋊#92420C2212yes—
20C2×#14519832C2212yes—
21#16840628C2212yes—
22#16840628C2212yes—
23#16840628C2212yes—
24#16840628C2212yes—
25#16840628C2211yescharacteristic
26#16840628C2211yescharacteristic
27#16840628C2211yescharacteristic
28#16840628C2211yescharacteristic
29#41×Dic61—28no—
30C2×Q8×Dic61C411yescharacteristic
31C2×Q8×Dic61C2211yescharacteristic
32C2×D4×Dic61C411yescharacteristic
33C2×D4×Dic61C411yescharacteristic
34C2×D4×Dic61C411yescharacteristic
35C2×D4×Dic61C2211yescharacteristic
36C2×D4×Dic61C2211yescharacteristic
37C2×D4×Dic61C2211yescharacteristic
38C2×#14519736C411yescharacteristic
39C2×#14519736C411yescharacteristic
40C2×#14519736C411yescharacteristic
41C22×C4×Dic61C2211yescharacteristic
42C22×C4×Dic61C2211yescharacteristic
43C22×C4×Dic61C2211yescharacteristic
44C22×#41×C61C411yescharacteristic
45C2×#11758217—24no—
46C2×#11758217—24no—
47C2×#11758217—42no—
48C22×#92420—24no—
49C22×#92420—24no—
50C22×#92420—42no—
51C22×#92420C2312yes—
52C22×#92420C2312yes—
53C22×#92420C2312yes—
54C22×#92420C2312yes—
55#14519844—24no—
56C22⋊#92420—24no—
57C22⋊#92420—24no—
58C22⋊#92420—24no—
59C22⋊#92420—42no—
60#14519832—24no—
61C23×Dic61C2×C411yescharacteristic
62C23×Dic61C2×C411yescharacteristic
63C23×Dic61C2×C411yescharacteristic
64C23×Dic61C2311yescharacteristic
65C23×Dic61C2311yescharacteristic
66C23×Dic61C2311yescharacteristic
67Q8×Dic61—24no—
68Q8×Dic61—24no—
69D4×Dic61—24no—
70D4×Dic61—24no—
71D4×Dic61—24no—
72D4×Dic61—24no—
73D4×Dic61—24no—
74D4×Dic61D418yes—
75#14519736—24no—
76#14519736—24no—
77#14519736—24no—
78C2×C4⋊Dic61C2×C411yescharacteristic
79C2×C4⋊Dic61C2×C411yescharacteristic
80C2×C4⋊Dic61C2×C411yescharacteristic
81C2×C4⋊Dic61C2×C411yescharacteristic
82C2×C4⋊Dic61C2×C411yescharacteristic
83C2×C4⋊Dic61C2×C411yescharacteristic
84C2×C22⋊Dic61C2×C411yescharacteristic
85C2×C22⋊Dic61C2×C411yescharacteristic
86C2×C22⋊Dic61C2×C411yescharacteristic
87C2×C22⋊Dic61C2×C411yescharacteristic
88C2×C22⋊Dic61C2×C411yescharacteristic
89C2×C22⋊Dic61C2×C411yescharacteristic
90C2×C4×Dic61—24no—
91C2×C4×Dic61—24no—
92C2×C4×Dic61—24no—
93C2×C4×Dic61C2×C411yescharacteristic
94C2×C4×Dic61C2×C411yescharacteristic
95C2×C4×Dic61C2×C411yescharacteristic
96C2×C4×Dic61C2×C411yescharacteristic
97C2×C4×Dic61C2×C411yescharacteristic
98C2×C4×Dic61C2×C411yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103C2×#41×C61—24no—
104C2×#41×C61—24no—
105C2×#41×C61—24no—
106C22×Q8×C61C2×C411yescharacteristic
107C22×D4×C61C2×C411yescharacteristic
108C22×D4×C61C2×C411yescharacteristic
109C22×D4×C61C2×C411yescharacteristic
110C23×C4×C61C2×C411yescharacteristic
111C23×C4×C61C2×C411yescharacteristic
112C23×C4×C61C2×C411yescharacteristic
113C24×C61C22×C411yescharacteristic
114C24×C61C22×C411yescharacteristic
115C24×C61C22×C411yescharacteristic
116#41×C61—232no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61C22⋊C414yes—
128C2×D4×C61M1614yescomplemented
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61M1614yescomplemented
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×C4×C61C22×C411yescharacteristic
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
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×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—41no—
159C22×Dic61—41no—
160C22×Dic61—41no—
161C22×Dic61C22×C411yescharacteristic
162C22×Dic61C22×C411yescharacteristic
163C22×Dic61C22×C411yescharacteristic
164C22×Dic61C22×C411yescharacteristic
165C22×Dic61C22×C411yescharacteristic
166C22×Dic61C22×C411yescharacteristic
167C22×Dic61C22×C411yescharacteristic
168C22×Dic61C2×D414yes—
169C22×Dic61C2×D414yes—
170C22×Dic61C2411yescharacteristic
171#11758217—48no—
172C2×#92420—22no—
173C2×#92420—22no—
174C2×#92420—22no—
175C2×#92420—22no—
176C2×#92420—22no—
177C2×#92420—22no—
178C2×#92420—22no—
179C2×#92420—22no—
180C2×#92420—22no—
181C2×#92420—22no—
182C2×#92420—22no—
183C2×#92420—48no—
184C2×#92420#4114yescomplemented
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61C22⋊C414yes—
190C22⋊Dic61C22⋊C414yes—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61C22⋊C414yes—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61C2×D414yes—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61C2×C22⋊C412yes—
224C23×C61C2×C22⋊C412yes—
225C23×C61C2×M1612yes—
226C23×C61C2×M1612yes—
227C23×C61C2×M1612yes—
228C23×C61C23×C411yescharacteristic
229Q8×C61—28no—
230D4×C61—28no—
231D4×C61—28no—
232D4×C61—28no—
233C2×C4×C61—28no—
234C2×C4×C61—28no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—28no—
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C2×M1612yes—
246C2×C4×C61C2×M1612yes—
247C2×C4×C61C2×M1612yes—
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—24no—
257C2×Dic61—24no—
258C2×Dic61—24no—
259C2×Dic61—24no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—44no—
270C2×Dic61—44no—
271C2×Dic61—44no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61C2×C22⋊C412yes—
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C2×C22⋊C412yes—
277C2×Dic61#13112yes—
278C2×Dic61C22×D412yes—
279C2×Dic61C2×#4112yes—
280C2×Dic61H(5,𝔽2)11yescharacteristic
281C2×Dic612−1+411yescharacteristic
282#92420—24no—
283#92420—42no—
284#92420—42no—
285#92420—42no—
286#22471—611nomaximal, Sylow
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61#40714yes—
308C22×C61C22×C22⋊C411yescharacteristic
309C22×C61C4×#4111yescharacteristic
310C22×C61#52011yescharacteristic
311C22×C61C22×M1611yescharacteristic
312C22×C61C22×M1611yescharacteristic
313C22×C61C2×#13111yescharacteristic
314C22×C61#56811yescharacteristic
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—22no—
318C4×C61—22no—
319Dic61—22no—
320Dic61—22no—
321Dic61—22no—
322Dic61—22no—
323Dic61—41no—
324Dic61—41no—
325Dic61—41no—
326Dic61#54612yes—
327Dic61#54712yes—
328C2×C4×#41—611no—
329C2×#435—612no—
330#1985—612no—
331#1870—612no—
332#1869—612no—
333#1817—611no—
334#1817—611no—
335#1797—611no—
336#1749—611no—
337#1728—611no—
338C2×#336—611no—
339C2×C61—24no—
340C2×C61—42no—
341C2×C61—42no—
342C2×C61#233911yescharacteristic
343C2×C61C2×#40711yescharacteristic
344C2×C61#288011yescharacteristic
345C2×C61#296511yescharacteristic
346C2×C61#297011yescharacteristic
347C2×C61#298911yescharacteristic
348C2×C61#299011yescharacteristic
349C22×#41#692311yescharacteristic
350C22×M16—612no—
351C23×C8—612no—
352C4×#41—1228no—
353C2×C4×Q8—611no—
354C2×C4×Q8—611no—
355C2×C4×D4—611no—
356C2×C4×D4—611no—
357C2×C4×D4—611no—
358C2×C4×D4—611no—
359C2×C4×D4—611no—
360C2×C4×D4—611no—
361C2×#117—611no—
362C2×#117—611no—
363C2×#117—611no—
364C22×C42—611no—
365C22×C42—611no—
366C22×C42—611no—
367#435—1228no—
368C2×C4⋊C8—612no—
369C2×C22⋊C8—612no—
370C2×C22⋊C8—612no—
371C2×C22⋊C8—612no—
372C2×C22⋊C8—612no—
373C2×#97—612no—
374#336—611no—
375#336—611no—
376#336—611no—
377#336—612no—
378#336—612no—
379#336—612no—
380#336—612no—
381#336—611no—
382C61#2247111yescharacteristic, complemented, Sylow
383C2×#41—24no—
384C2×#41—24no—
385C2×#41—24no—
386C22×Q8C2×#692311yescharacteristic
387C22×D4C2×#692311yescharacteristic
388C22×D4C2×#692311yescharacteristic
389C22×D4C2×#692311yescharacteristic
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—611no—
395C23×C4—611no—
396C23×C4C2×#692311yescharacteristic
397C23×C4C2×#692311yescharacteristic
398C23×C4C2×#692311yescharacteristic
399C2×M16—1224no—
400C2×M16—1224no—
401C2×M16—2442no—
402C22×C8—612no—
403C22×C8—612no—
404C22×C8—612no—
405C22×C8—612no—
406C22×C8—1224no—
407C22×C8—1224no—
408C22×C8—2442no—
409C4×Q8—1224no—
410C4×Q8—1224no—
411C4×D4—618no—
412C4×D4—1224no—
413C4×D4—1224no—
414C4×D4—1224no—
415C4×D4—1224no—
416C4×D4—1224no—
417#117—1224no—
418#117—1224no—
419#117—1224no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C22⋊C4—611no—
427C2×C22⋊C4—611no—
428C2×C22⋊C4—611no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4—611no—
431C2×C22⋊C4—611no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—1224no—
443C2×C42—1224no—
444C2×C42—1224no—
445C4⋊C8—1224no—
446C22⋊C8—1224no—
447C22⋊C8—1224no—
448C22⋊C8—1224no—
449C22⋊C8—2442no—
450#97—1224no—
451C24C22×#692311yescharacteristic
452C24C22×#692311yescharacteristic
453C24C22×#692311yescharacteristic
454#41—232no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4#1175821714yescomplemented
466C2×D4C22⋊#692314yes—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—611no—
476C22×C4—614no—
477C22×C4—614no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—1222no—
486C22×C4—1221no—
487C22×C4—1222no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1222no—
492C22×C4—1221no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1221no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—2441no—
502C22×C4—2441no—
503C22×C4—2441no—
504C22×C4#1175821714yescomplemented
505C22×C4C22×#692311yescharacteristic
506C22×C4C22×#692311yescharacteristic
507C22×C4C22×#692311yescharacteristic
508C22×C4C22×#692311yescharacteristic
509M16—2448no—
510C2×C8—614no—
511C2×C8—1222no—
512C2×C8—1222no—
513C2×C8—1222no—
514C2×C8—1222no—
515C2×C8—1222no—
516C2×C8—1222no—
517C2×C8—1222no—
518C2×C8—1222no—
519C2×C8—1222no—
520C2×C8—1222no—
521C2×C8—1222no—
522C2×C8—2448no—
523C4⋊C4—614no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C22⋊C4—614no—
530C22⋊C4—614no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C42—614no—
536C42—1222no—
537C42—1222no—
538C42—1222no—
539C42—1222no—
540C42—1222no—
541C42—1222no—
542C42—1222no—
543C42—1222no—
544C42—1222no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23C2×C22⋊#692312yes—
562C23C2×C22⋊#692312yes—
563C23C2×#1175821712yes—
564C23C2×#1175821712yes—
565C23C2×#1175821712yes—
566C23C23×#692311yescharacteristic
567Q8—28no—
568D4—28no—
569D4—28no—
570D4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—612no—
583C2×C4—612no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2444no—
614C2×C4—2444no—
615C2×C4—2444no—
616C2×C4C2×C22⋊#692312yes—
617C2×C4C2×#1175821712yes—
618C2×C4C2×#1175821712yes—
619C2×C4C2×#1175821712yes—
620C8—1224no—
621C8—2442no—
622C8—2442no—
623C8—2442no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22#1684073214yes—
645C22C22×C22⋊#692311yescharacteristic
646C22#41×#692311yescharacteristic
647C22#1684083111yescharacteristic
648C22C22×#1175821711yescharacteristic
649C22C22×#1175821711yescharacteristic
650C22D61⋊#13111yescharacteristic
651C22#1684084111yescharacteristic
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—612no—
657C4—612no—
658C4—1222no—
659C4—1222no—
660C4—1222no—
661C4—1222no—
662C4—2441no—
663C4—2441no—
664C4—2441no—
665C2—24no—
666C2—42no—
667C2—42no—
668C2#2146232211yescharacteristic
669C2C2×#1684073211yescharacteristic
670C2#2146240911yescharacteristic
671C2#2146262811yescharacteristic
672C2#2146265611yescharacteristic
673C2#2146273811yescharacteristic
674C2#2146274511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
Fitting, prime radicalC22×#41×C61C4
cosocleC22×Dic61C24
hypercenterC22×#41#6923
hypocenterC61#22471
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16840793C2semidirect
2. #21461993C2semidirect
3. #21461796C2semidirect
4. #21461792C2semidirect
5. #21461710C2semidirect
6. #21461709C2semidirect
7. #21461675C2semidirect
8. #21461545C2semidirect
9. #21461482C2semidirect
10. C2×#16840628C2semidirect
128. C2×D4×C61M16semidirect
137. C22×C4×C61M16semidirect
184. C2×#92420#41semidirect
382. C61#22471semidirect
465. C2×D4#11758217semidirect
504. C22×C4#11758217semidirect

Maximal subgroups

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

subgroupindex
C2×#168407932
#214619932
#214617962
#214617922
#214617102
#214617092
#214616752
#214615452
#214614822
C2×#168406282
C2×#41×Dic612
#2247161