Finite Group Databasebeta

← #29601291

Subgroups — #29601291

Order 15616.

2102 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839633C212yesmaximal
2C2×C4×#11758198C211yesmaximal, characteristic
3C2×#16839609C211yesmaximal, characteristic
4C2×#95×D61C211yesmaximal, characteristic
5#21444054C212yesmaximal, complemented
6#21444051C212yesmaximal, complemented
7#21444047C211yesmaximal, characteristic, complemented
8#21444046C211yesmaximal, characteristic, complemented
9#21444045C211yesmaximal, characteristic, complemented
10#21444040C212yesmaximal
11C61×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×C4×D61C2212yes—
13C23×C4×D61C2211yescharacteristic
14C22×#11758196C2212yes—
15C22×#11758198C2212yes—
16C22×#11758198C2211yescharacteristic
17C22×#11758198C2211yescharacteristic
18C22×C4×Dic61C2212yes—
19C22×C4×Dic61C2211yescharacteristic
20#16839633C4116yescomplemented
21C4×#11758198C418yescomplemented
22#16839609—24no—
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2211yescharacteristic
32C2×#14519684C2211yescharacteristic
33#95×D61—24no—
34C22×C4⋊C4×C61C2212yes—
35C22×C42×C61C2211yescharacteristic
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C24×D61C2311yescharacteristic
40C23×Dic61C2312yes—
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2312yes—
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2311yescharacteristic
45C22×C4×D61—24no—
46C22×C4×D61—24no—
47C22×C4×D61—42no—
48C22×C4×D61—42no—
49C22×C4×D61—42no—
50C22×C4×D61—42no—
51C22×C4×D61—42no—
52C22×C4×D61—42no—
53C22×C4×D61C2×C418yes—
54C22×C4×D61C2×C414yes—
55C22×C4×D61D414yes—
56C22×C4×D61Q814yes—
57C2×#11758196—42no—
58C2×#11758196—42no—
59C2×#11758196C2×C418yescomplemented
60C2×#11758198—24no—
61C2×#11758198—24no—
62C2×#11758198—24no—
63C2×#11758198C2×C414yes—
64C2×#11758198C2×C418yes—
65C2×#11758198C2×C414yes—
66C2×#11758198C2×C418yes—
67C2×C4×Dic61—24no—
68C2×C4×Dic61—42no—
69C2×C4×Dic61—42no—
70C2×C4×Dic61—42no—
71C2×C4×Dic61C2×C414yescomplemented
72#14519684—22no—
73#14519684—22no—
74#14519684—24no—
75#14519684—42no—
76#14519684—42no—
77#14519684—42no—
78#14519684C2×C414yescomplemented
79#14519684C2×C414yescomplemented
80#14519684C2×C418yescomplemented
81#14519684C2×C418yescomplemented
82C23×C4×C61C2312yes—
83C23×C4×C61C2311yescharacteristic
84C23×C4×C61C2312yes—
85C23×C4×C61C2311yescharacteristic
86C23×C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61—42no—
88C2×C4⋊C4×C61—42no—
89C2×C4⋊C4×C61C2×C418yescomplemented
90C2×C42×C61—42no—
91C2×C42×C61C2×C414yescomplemented
92#95×C61—22no—
93#95×C61—22no—
94#95×C61—42no—
95C24×C61C2411yescharacteristic
96C23×D61—22no—
97C23×D61—22no—
98C23×D61—21no—
99C23×D61—21no—
100C23×D61—21no—
101C23×D61—21no—
102C23×D61C22×C412yes—
103C23×D61C22×C414yes—
104C23×D61C2×D412yes—
105C23×D61C2×Q812yes—
106C23×D61#4114yes—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
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×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61C22×C414yes—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C22×C414yes—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C22×C414yes—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×Q812yes—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C22×Dic61—21no—
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—21no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—21no—
157C22×Dic61C22×C414yes—
158C22×Dic61C22×C414yes—
159C22×Dic61C22×C412yes—
160C22×Dic61C22×C412yes—
161C22×Dic61C22×C414yes—
162C22×Dic61C22×C412yes—
163C22×Dic61C2×D412yes—
164C22×Dic61C2×Q812yes—
165C2×C4×D61—216no—
166C2×C4×D61—44no—
167C2×C4×D61—44no—
168C2×C4×D61—48no—
169C2×C4×D61—48no—
170C2×C4×D61—48no—
171C2×C4×D61—48no—
172C2×C4×D61—48no—
173C2×C4×D61—48no—
174C2×C4×D61—48no—
175C2×C4×D61—48no—
176C2×C4×D61C4⋊C4116yescomplemented
177C4⋊C4×C61—48no—
178C42×C61—44no—
179#11758198—28no—
180#11758198—216no—
181#11758198C42116yescomplemented
182#11758196—48no—
183C4×Dic61—44no—
184C4×Dic61—48no—
185C23×C61C23×C411yescharacteristic
186C23×C61C23×C412yes—
187C23×C61C22×D411yescharacteristic
188C23×C61C22×Q811yescharacteristic
189C23×C61C2×#4112yes—
190C23×C61H(5,𝔽2)12yes—
191C23×C61H(5,𝔽2)12yes—
192C23×C61H(5,𝔽2)11yescharacteristic
193C23×C61H(5,𝔽2)11yescharacteristic
194C23×C612−1+411yescharacteristic
195C23×C612−1+411yescharacteristic
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—24no—
199C22×D61—24no—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—24no—
203C22×D61—24no—
204C22×D61—24no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—24no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61—24no—
211C22×D61—24no—
212C22×D61—22no—
213C22×D61—22no—
214C22×D61C2×C4214yes—
215C22×D61C2×C4⋊C414yes—
216C22×D61#11714yes—
217C22×D61C4×D418yes—
218C22×D61C4×Q818yes—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61C2×C4214yes—
247C2×C4×C61C2×C4⋊C414yes—
248C2×Dic61—22no—
249C2×Dic61—24no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—22no—
257C2×Dic61—24no—
258C2×Dic61—24no—
259C2×Dic61—22no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—22no—
263C2×Dic61—24no—
264C2×Dic61—24no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61C2×C4214yes—
276C2×Dic61C2×C4⋊C414yes—
277C4×D61—416no—
278C4×D61—432no—
279#13782—611nomaximal, Sylow
280C22×C61C22×C4211yescharacteristic
281C22×C61C22×C4⋊C411yescharacteristic
282C22×C61C2×#11711yescharacteristic
283C22×C61C2×C4×D412yes—
284C22×C61C2×C4×Q812yes—
285C22×C61#51812yes—
286C22×C61#51811yescharacteristic
287C22×C61#51812yes—
288C22×C61#51811yescharacteristic
289C22×C61#52011yescharacteristic
290C22×C61#52011yescharacteristic
291C22×C61#52012yes—
292C22×C61#52012yes—
293C22×C61#53412yes—
294C22×C61#53511yescharacteristic
295C22×C61#53611yescharacteristic
296C22×C61#53812yes—
297C22×C61#53812yes—
298C22×C61#53912yes—
299C22×C61#53912yes—
300C22×C61#54312yes—
301C22×C61#54411yescharacteristic
302C22×C61#54411yescharacteristic
303C2×D61—24no—
304C2×D61—24no—
305C2×D61—24no—
306C2×D61—24no—
307C2×D61—28no—
308C2×D61—28no—
309C2×D61—28no—
310C2×D61—28no—
311C2×D61—24no—
312C2×D61—24no—
313C2×D61C4×C4⋊C418yes—
314C4×C61—24no—
315C4×C61—24no—
316C4×C61—42no—
317C4×C61—44no—
318C4×C61—44no—
319Dic61—24no—
320Dic61—24no—
321Dic61—42no—
322Dic61—44no—
323Dic61—44no—
324C2×#385—612no—
325C2×C4×C22⋊C4—611no—
326C2×C4×C22⋊C4—611no—
327C22×#95—611no—
328#1436—612no—
329#1436—612no—
330C4⋊#95—611no—
331C4⋊#95D6111yescharacteristic, complemented
332#1434—611no—
333#1434—611no—
334C22⋊#95—612no—
335C2×C61C2×C4×C4⋊C411yescharacteristic
336C2×C61#227511yescharacteristic
337C2×C61#227511yescharacteristic
338C2×C61#231511yescharacteristic
339C2×C61#231711yescharacteristic
340C2×C61#231811yescharacteristic
341C2×C61#231811yescharacteristic
342C2×C61#232112yes—
343C2×C61#232212yes—
344C2×C61#232912yes—
345C2×C61#232912yes—
346D61—28no—
347C24×C4—611no—
348C24×C4—612no—
349C22×C4⋊C4—612no—
350C22×C4⋊C4C2×D6112yes—
351C22×C22⋊C4—611no—
352C22×C22⋊C4—611no—
353C22×C22⋊C4—612no—
354C22×C42—611no—
355C22×C42—612no—
356C22×C42C2×D6111yescharacteristic
357#385—6116no—
358C4×C22⋊C4—618no—
359C4×C22⋊C4—1224no—
360C2×#95—612no—
361C2×#95—612no—
362C2×#95—611no—
363C2×#95—611no—
364C2×#95—611no—
365C2×#95—611no—
366C2×#95—612no—
367C2×#95—612no—
368C2×#95—612no—
369C2×#95—612no—
370C2×#95—1224no—
371C2×#95C2×D6111yescharacteristic
372C2×#95C2×D6111yescharacteristic
373C2×#95C2×D6112yes—
374C61#1378211yescharacteristic, complemented, Sylow
375C25—611no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—614no—
379C23×C4—614no—
380C23×C4—614no—
381C23×C4—612no—
382C23×C4—611no—
383C23×C4—612no—
384C23×C4—618no—
385C23×C4—1224no—
386C23×C4—1224no—
387C23×C4—2442no—
388C23×C4—2442no—
389C23×C4—2442no—
390C23×C4—2442no—
391C23×C4—2442no—
392C23×C4—2442no—
393C23×C4C22×D6112yes—
394C23×C4C22×D6111yescharacteristic
395C23×C4C22×D6112yes—
396C23×C4C22×D6111yescharacteristic
397C23×C4C22×D6111yescharacteristic
398C2×C4⋊C4—42no—
399C2×C4⋊C4—42no—
400C2×C4⋊C4—618no—
401C2×C4⋊C4—2442no—
402C2×C4⋊C4—2442no—
403C2×C4⋊C4C4×D6118yescomplemented
404C2×C22⋊C4—614no—
405C2×C22⋊C4—618no—
406C2×C22⋊C4—614no—
407C2×C22⋊C4—618no—
408C2×C22⋊C4—1224no—
409C2×C22⋊C4—1224no—
410C2×C22⋊C4—1224no—
411C2×C42—42no—
412C2×C42—614no—
413C2×C42—1224no—
414C2×C42—2442no—
415C2×C42—2442no—
416C2×C42—2442no—
417C2×C42C4×D6114yescomplemented
418#95—22no—
419#95—22no—
420#95—42no—
421#95—614no—
422#95—614no—
423#95—618no—
424#95—618no—
425#95—1224no—
426#95—1222no—
427#95—1222no—
428#95—2442no—
429#95—2442no—
430#95—2442no—
431C24—612no—
432C24—614no—
433C24—612no—
434C24—612no—
435C24—614no—
436C24—1221no—
437C24—1222no—
438C24—1221no—
439C24—1222no—
440C24—1221no—
441C24—1221no—
442C24C23×D6111yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—614no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—6116no—
470C22×C4—614no—
471C22×C4—612no—
472C22×C4—614no—
473C22×C4—12216no—
474C22×C4—1222no—
475C22×C4—1221no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1221no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1221no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—2444no—
496C22×C4—2444no—
497C22×C4—2448no—
498C22×C4—2448no—
499C22×C4—2448no—
500C22×C4—2448no—
501C22×C4—2448no—
502C22×C4—2448no—
503C22×C4—2448no—
504C22×C4—2448no—
505C22×C4C2×C4×D6112yes—
506C22×C4C2×C4×D6114yes—
507C22×C4C2×C4×D6114yes—
508C22×C4C2×C4×D6112yes—
509C22×C4C2×C4×D6114yes—
510C22×C4C2×C4×D6112yes—
511C22×C4D4×D6112yes—
512C22×C4Q8×D6112yes—
513C4⋊C4—48no—
514C4⋊C4—2448no—
515C22⋊C4—6116no—
516C22⋊C4—1228no—
517C22⋊C4—12216no—
518C42—44no—
519C42—2444no—
520C42—2448no—
521C23—614no—
522C23—614no—
523C23—614no—
524C23—618no—
525C23—618no—
526C23—1224no—
527C23—1222no—
528C23—1224no—
529C23—1222no—
530C23—1222no—
531C23—1222no—
532C23—1224no—
533C23—1224no—
534C23—1224no—
535C23—1222no—
536C23—1222no—
537C23—1224no—
538C23—1224no—
539C23—1224no—
540C23—1224no—
541C23—1224no—
542C23—1222no—
543C23—1222no—
544C23C22×C4×D6111yescharacteristic
545C23C22×C4×D6112yes—
546C23C2×D4×D6111yescharacteristic
547C23C2×Q8×D6111yescharacteristic
548C23#41×D6112yes—
549C23#1451982211yescharacteristic
550C23#1451982212yes—
551C23#1451982211yescharacteristic
552C23#1451982312yes—
553C23#1451982511yescharacteristic
554C23#1451982511yescharacteristic
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—614no—
583C2×C4—614no—
584C2×C4—1224no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1222no—
593C2×C4—1224no—
594C2×C4—1224no—
595C2×C4—1224no—
596C2×C4—1224no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1224no—
600C2×C4—1224no—
601C2×C4—24416no—
602C2×C4—24432no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4C42×D6114yes—
614C2×C4C4⋊C4×D6114yes—
615C22—618no—
616C22—1224no—
617C22—1224no—
618C22—1228no—
619C22—1228no—
620C22—1228no—
621C22—1228no—
622C22—1224no—
623C22—1224no—
624C22—1224no—
625C22—1224no—
626C22C2×C42×D6111yescharacteristic
627C22C2×C4⋊C4×D6111yescharacteristic
628C22#117×D6111yescharacteristic
629C22C4×D4×D6112yes—
630C22C4×Q8×D6112yes—
631C22#1684023412yes—
632C22#1684023511yescharacteristic
633C22#1684023512yes—
634C22#1684023611yescharacteristic
635C22#1684024311yescharacteristic
636C22#1684024311yescharacteristic
637C22#1684024312yes—
638C22#1684024412yes—
639C22#1684032012yes—
640C22#1684032511yescharacteristic
641C22#1684033711yescharacteristic
642C22#1684034412yes—
643C22#1684034612yes—
644C22#1684035812yes—
645C22#1684036012yes—
646C22#1684038712yes—
647C22#1684038911yescharacteristic
648C22#1684038911yescharacteristic
649C4—24no—
650C4—24no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—1224no—
655C4—1224no—
656C4—2442no—
657C4—2444no—
658C4—2444no—
659C2—1228no—
660C2C4×C4⋊C4×D6111yescharacteristic
661C2#2144817711yescharacteristic
662C2#2144817911yescharacteristic
663C2#2144843211yescharacteristic
664C2#2144844011yescharacteristic
665C2#2144845411yescharacteristic
666C2#2144845511yescharacteristic
667C2#2144847812yes—
668C2#2144848312yes—
669C2#2144855112yes—
670C2#2144855312yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC22×C61C22×C42
FittingC61×C4⋊#95C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenterC4⋊#95D61
hypocenterC61#13782
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
5. #21444054C2semidirect
6. #21444051C2semidirect
7. #21444047C2semidirect
8. #21444046C2semidirect
9. #21444045C2semidirect
11. C61×C4⋊#95C2semidirect
20. #16839633C4semidirect
21. C4×#11758198C4semidirect
59. C2×#11758196C2×C4semidirect
71. C2×C4×Dic61C2×C4semidirect
78. #14519684C2×C4semidirect
79. #14519684C2×C4semidirect
80. #14519684C2×C4semidirect
81. #14519684C2×C4semidirect
89. C2×C4⋊C4×C61C2×C4semidirect
91. C2×C42×C61C2×C4semidirect
176. C2×C4×D61C4⋊C4semidirect
181. #11758198C42semidirect
331. C4⋊#95D61semidirect
374. C61#13782semidirect
403. C2×C4⋊C4C4×D61semidirect
417. C2×C42C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396332
C2×C4×#117581982
C2×#168396092
C2×#95×D612
#214440542
#214440512
#214440472
#214440462
#214440452
#214440402
C61×C4⋊#952
#1378261