Finite Group Databasebeta

← #29689020

Subgroups — #29689020

Order 15616.

1570 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D61C211yesmaximal, characteristic
2C22×#14519689C211yesmaximal, characteristic
3#380×D61C211yesmaximal, characteristic
4#21446243C211yesmaximal, characteristic, complemented
5#21446242C211yesmaximal, characteristic, complemented
6#21446240C211yesmaximal, characteristic, complemented
7#21446125C212yesmaximal, complemented
8#21446123C212yesmaximal, complemented
9#21445908C212yesmaximal, complemented
10#21445752C212yesmaximal, complemented
11C61×#1892C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15C2×#14519757C2211yescharacteristic
16C2×#14519756C2211yescharacteristic
17C2×#14519755—21no—
18C2×#14519754—21no—
19C2×#14519753C2211yescharacteristic
20C22⋊Q8×D61—24no—
21C22⋊Q8×D61C418yescomplemented
22C2×C4⋊C4×D61—21no—
23C2×C4⋊C4×D61C2211yescharacteristic
24C2×C4⋊C4×D61C2212yes—
25C2×C22⋊C4×D61—21no—
26C2×C22⋊C4×D61—21no—
27C2×#14519702C2212yescomplemented
28C2×#14519701C2212yescomplemented
29C2×#14519700C2212yescomplemented
30C2×#14519689—24no—
31C2×#14519689C2212yes—
32C2×#14519689C2212yes—
33#16839622C2211yescharacteristic
34#16839621—21no—
35#16839620—21no—
36#16839619C2212yes—
37#95×D61—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C23×C8×C61C2211yescharacteristic
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yescomplemented
44C61×#380C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C2×Q8×D61—22no—
50C2×Q8×D61—42no—
51C2×Q8×D61—42no—
52C2×Q8×D61C2×C414yescomplemented
53C23×Dic61D411yescharacteristic
54C22×C4×D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61—22no—
57C22×C4×D61—42no—
58C22×C4×D61—42no—
59C22×C4×D61C2×C414yes—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D414yes—
63C22×C4×D61C2311yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C22×#92418—22no—
66C22×#92418—42no—
67C22×#92418C2312yes—
68#14519757—22no—
69#14519757C2×C414yes—
70#14519756—22no—
71#14519756C2×C414yescomplemented
72#14519755—28no—
73#14519754—28no—
74#14519753—22no—
75#14519753C2×C414yes—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—22no—
83C4⋊C4×D61—28no—
84C4⋊C4×D61—22no—
85C4⋊C4×D61—42no—
86C4⋊C4×D61—42no—
87C4⋊C4×D61C2×C414yescomplemented
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—22no—
93C22⋊C4×D61—28no—
94C22⋊C4×D61—44no—
95#14519702—42no—
96#14519701—42no—
97#14519700—42no—
98#14519689—24no—
99#14519689—24no—
100#14519684—21no—
101#14519684—22no—
102C22×Q8×C61C2311yescharacteristic
103C23×C4×C61C2311yescharacteristic
104C22×C8×C61—22no—
105C22×C8×C61—42no—
106C22×C8×C61C2312yes—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61C2×C414yescomplemented
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2312yes—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114#103×C61—42no—
115#95×C61—21no—
116C24×C61C2×D411yescharacteristic
117C23×D61—42no—
118C23×D61—42no—
119C23×D61C22⋊C414yes—
120C23×D61SD1614yes—
121C23×D61Q1614yes—
122C23×D61C2×D411yescharacteristic
123C23×D61C2×D411yescharacteristic
124C2×Q8×C61—21no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61C22×C412yes—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—22no—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—22no—
140C22×Dic61—42no—
141C22×Dic61C22⋊C414yes—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144Q8×D61—48no—
145Q8×D61—44no—
146Q8×D61—44no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—24no—
152C2×C4×D61—24no—
153C2×C4×D61—48no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—44no—
159C2×C4×D61—44no—
160C2×C4×D61—44no—
161C2×C4×D61—44no—
162C2×C4×D61—44no—
163C2×C4×D61C22⋊C414yes—
164C2×C4×D61C22⋊C414yes—
165C2×C4×D61C22×C412yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C2×D412yes—
168C2×C4×D61C2×D412yes—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—21no—
172C2×Dic122—24no—
173C2×Dic122—44no—
174C2×Dic122C22×C412yes—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—44no—
179C4⋊C4×C61—24no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61C22×C412yes—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—42no—
186C22⋊Dic61—24no—
187C22⋊Dic61—42no—
188#11758198—24no—
189#11758198—24no—
190#11758198—44no—
191C4⋊Dic61—21no—
192C4⋊Dic61—24no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22×C412yes—
196#11758196—24no—
197#11758196—24no—
198#11758196—24no—
199#11758196—44no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—44no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61C2×C22⋊C412yes—
207C23×C61C2×SD1612yes—
208C23×C61C2×Q1612yes—
209C23×C61C22×D411yescharacteristic
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—44no—
215C22×D61—44no—
216C22×D61—48no—
217C22×D61—48no—
218C22×D61#103116yescomplemented
219C22×D61C2×C22⋊C412yes—
220C22×D61C2×C22⋊C412yes—
221C22×D61C2×SD1612yes—
222C22×D61C2×SD1612yes—
223C22×D61C2×Q1612yes—
224C22×D61C2×Q1612yes—
225Q8×C61—24no—
226Q8×C61—41no—
227Q8×C61—41no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)12yes—
243C2×C4×C61H(5,𝔽2)12yes—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—44no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256C4×D61—24no—
257C4×D61—24no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—44no—
263C4×D61—44no—
264C4×D61—48no—
265C4×D61—48no—
266C4×D61—88no—
267Dic122—24no—
268Dic122—48no—
269Dic122—41no—
270Dic122—41no—
271C8×C61—42no—
272#92418—42no—
273#21759—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—24no—
277C22×C61—24no—
278C22×C61C2×#10314yes—
279C22×C61C22×C22⋊C411yescharacteristic
280C22×C61#53412yes—
281C22×C61C22×SD1611yescharacteristic
282C22×C61C22×Q1611yescharacteristic
283C22×C61#57512yes—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—24no—
289C2×D61—24no—
290C2×D61—24no—
291C2×D61—24no—
292C2×D61—416no—
293C2×D61C2×#10314yes—
294C2×D61C2×#10314yes—
295C4×C61—22no—
296C4×C61—42no—
297C4×C61—42no—
298C4×C61—82no—
299C4×C61#51812yes—
300C4×C61#53411yescharacteristic
301C4×C61#53411yescharacteristic
302Dic61—22no—
303Dic61—22no—
304Dic61—42no—
305Dic61—42no—
306Dic61—82no—
307C22×C22⋊Q8—611no—
308C22×C22⋊C8—611no—
309C2×#380—611no—
310#1892—611no—
311#1892—611no—
312#1892—611no—
313#1892D6111yescharacteristic, complemented
314#1872—612no—
315#1872—612no—
316#1824—612no—
317#1787—612no—
318C2×C61—24no—
319C2×C61#230811yescharacteristic
320C2×C61C22×#10311yescharacteristic
321C2×C61#289611yescharacteristic
322C2×C61#318712yes—
323C2×C61#318912yes—
324D61—24no—
325D61—24no—
326C23×Q8—611no—
327C24×C4—611no—
328C23×C8—611no—
329C23×C8C2×D6111yescharacteristic
330C2×C22⋊Q8—611no—
331C2×C22⋊Q8—611no—
332C2×C22⋊Q8—618no—
333C2×C22⋊Q8—611no—
334C2×C22⋊Q8—1221no—
335C2×C22⋊Q8—1221no—
336C2×C22⋊Q8—1224no—
337C2×C22⋊Q8C2×D6111yescharacteristic
338C22×C4⋊C4—611no—
339C22×C4⋊C4—612no—
340C22×C4⋊C4—1221no—
341C22×C22⋊C4—1221no—
342C22×C22⋊C4—1221no—
343C2×#103—612no—
344C2×#103—612no—
345C2×#103—612no—
346C2×#103C2×D6112yescomplemented
347C2×C22⋊C8—612no—
348C2×C22⋊C8—612no—
349C2×C22⋊C8—1224no—
350#380—611no—
351#380—612no—
352#380—1221no—
353#380—1221no—
354#380C2×D6111yescharacteristic
355C2×#95—1221no—
356#340—612no—
357#340—612no—
358#340—612no—
359#340C2×D6112yescomplemented
360C61#2175911yescharacteristic, complemented, Sylow
361C25—611no—
362C22×Q8—614no—
363C22×Q8—611no—
364C22×Q8—1221no—
365C22×Q8—1222no—
366C22×Q8—2442no—
367C22×Q8—2442no—
368C22×Q8C22×D6111yescharacteristic
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—611no—
372C23×C4—614no—
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—614no—
376C23×C4—1221no—
377C23×C4—1221no—
378C23×C4—1222no—
379C23×C4—2442no—
380C23×C4—2442no—
381C23×C4C22×D6111yescharacteristic
382C22×C8—22no—
383C22×C8—42no—
384C22×C8—612no—
385C22×C8—1222no—
386C22×C8—2442no—
387C22×C8C22×D6112yes—
388C22⋊Q8—22no—
389C22⋊Q8—614no—
390C22⋊Q8—614no—
391C22⋊Q8—614no—
392C22⋊Q8—1228no—
393C22⋊Q8—1228no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396C22⋊Q8—1222no—
397C22⋊Q8C4×D6114yescomplemented
398C2×C4⋊C4—21no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—614no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—1222no—
403C2×C4⋊C4—1228no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—2442no—
410C2×C4⋊C4—2442no—
411C2×C4⋊C4C22×D6112yes—
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—1228no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1222no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—2444no—
422#103—42no—
423#103—2442no—
424#103—2442no—
425#103—2442no—
426C22⋊C8—1224no—
427C22⋊C8—1224no—
428#95—21no—
429#95—1222no—
430#95—1221no—
431C24—611no—
432C24—611no—
433C24—614no—
434C24—614no—
435C24—614no—
436C24—2442no—
437C24—2442no—
438C24D4×D6111yescharacteristic
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—612no—
443C2×Q8—1224no—
444C2×Q8—1221no—
445C2×Q8—1222no—
446C2×Q8—1222no—
447C2×Q8—2444no—
448C2×Q8—2448no—
449C2×Q8—2444no—
450C2×Q8—2444no—
451C2×Q8C2×C4×D6112yes—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—614no—
460C22×C4—612no—
461C22×C4—612no—
462C22×C4—612no—
463C22×C4—612no—
464C22×C4—614no—
465C22×C4—614no—
466C22×C4—1224no—
467C22×C4—1224no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—2448no—
476C22×C4—2444no—
477C22×C4—2444no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4—2444no—
481C22×C4—2444no—
482C22×C4—2444no—
483C22×C4—2444no—
484C22×C4—2444no—
485C22×C4—2442no—
486C22×C4C2×C4×D6112yes—
487C22×C4D4×D6112yes—
488C22×C4D4×D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—44no—
494C2×C8—1222no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8—2444no—
498C4⋊C4—21no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—612no—
503C4⋊C4—1224no—
504C4⋊C4—1224no—
505C4⋊C4—1224no—
506C4⋊C4—1221no—
507C4⋊C4—1224no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—2444no—
511C4⋊C4C2×C4×D6112yes—
512C22⋊C4—24no—
513C22⋊C4—42no—
514C22⋊C4—1224no—
515C22⋊C4—1224no—
516C22⋊C4—1224no—
517C22⋊C4—2444no—
518C22⋊C4—2442no—
519C23—22no—
520C23—22no—
521C23—612no—
522C23—612no—
523C23—612no—
524C23—612no—
525C23—612no—
526C23—612no—
527C23—6116no—
528C23—1222no—
529C23—1222no—
530C23—1222no—
531C23—1222no—
532C23—2444no—
533C23—2448no—
534C23—2448no—
535C23—2444no—
536C23C22⋊C4×D6112yes—
537C23SD16×D6112yes—
538C23Q16×D6112yes—
539C23C2×D4×D6111yescharacteristic
540Q8—24no—
541Q8—41no—
542Q8—41no—
543Q8—1224no—
544Q8—2448no—
545Q8—2441no—
546Q8—2441no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—612no—
559C2×C4—612no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—4888no—
571C2×C4—2444no—
572C2×C4—2444no—
573C2×C4—2448no—
574C2×C4—2448no—
575C2×C4—2444no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4C22⋊C4×D6112yes—
582C2×C4C22×C4×D6111yescharacteristic
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4#1451982212yes—
585C2×C4#1451982312yes—
586C8—42no—
587C8—2442no—
588C22—24no—
589C22—22no—
590C22—24no—
591C22—22no—
592C22—614no—
593C22—614no—
594C22—1224no—
595C22—1224no—
596C22—1222no—
597C22—1222no—
598C22—1224no—
599C22—1224no—
600C22—1222no—
601C22—1222no—
602C22—24416no—
603C22#103×D6114yes—
604C22C2×C22⋊C4×D6111yescharacteristic
605C22#1684032012yes—
606C22C2×SD16×D6111yescharacteristic
607C22C2×Q16×D6111yescharacteristic
608C22#1684055912yes—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—1222no—
614C4—1222no—
615C4—4882no—
616C4—2442no—
617C4—2442no—
618C4#1684023412yes—
619C4#1684032211yescharacteristic
620C4#1684032311yescharacteristic
621C2—24no—
622C2—1224no—
623C2—1224no—
624C2#2144839311yescharacteristic
625C2C2×#103×D6111yescharacteristic
626C2#2145433411yescharacteristic
627C2#2145657112yes—
628C2#2145659012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1892C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1892D61
hypocenterC61#21759
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
4. #21446243C2semidirect
5. #21446242C2semidirect
6. #21446240C2semidirect
7. #21446125C2semidirect
8. #21446123C2semidirect
9. #21445908C2semidirect
10. #21445752C2semidirect
11. C61×#1892C2semidirect
21. C22⋊Q8×D61C4semidirect
27. C2×#14519702C22semidirect
28. C2×#14519701C22semidirect
29. C2×#14519700C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
43. C2×#103×C61C22semidirect
45. C61×#340C22semidirect
52. C2×Q8×D61C2×C4semidirect
71. #14519756C2×C4semidirect
87. C4⋊C4×D61C2×C4semidirect
108. C22⋊Q8×C61C2×C4semidirect
218. C22×D61#103semidirect
313. #1892D61semidirect
346. C2×#103C2×D61semidirect
359. #340C2×D61semidirect
360. C61#21759semidirect
397. C22⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D612
C22×#145196892
#380×D612
#214462432
#214462422
#214462402
#214461252
#214461232
#214459082
#214457522
C61×#18922
#2175961