Finite Group Databasebeta

← #29681234

Subgroups — #29681234

Order 15616.

1986 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840273C211yesmaximal, characteristic
2C2×#16839756C212yesmaximal, complemented
3C22×C22⋊#92418C211yesmaximal, characteristic, complemented
4#21446141C212yesmaximal, complemented
5#21446136C212yesmaximal, complemented
6#21446131C212yesmaximal, complemented
7#21446126C212yesmaximal, complemented
8#21445535C212yesmaximal, complemented
9C22×#117×C61C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×#92418C2212yes—
12#16840273—24no—
13#16840273C418yescomplemented
14C2×#14519760—21no—
15C2×#14519756C2211yescharacteristic
16C2×#14519756C2211yescharacteristic
17C2×#14519751C2211yescharacteristic
18C2×#14519751C2211yescharacteristic
19C2×#14519746—21no—
20C2×C4×C22⋊D61—21no—
21C2×C4×C22⋊D61—21no—
22C2×#14519736C2211yescharacteristic
23#16839756—28no—
24C2×#14519700C2214yescomplemented
25C2×#14519697C2214yescomplemented
26C2×C22⋊#92418—24no—
27C2×C22⋊#92418C2212yes—
28C2×C22⋊#92418C2212yes—
29#16839524C2214yescomplemented
30#16839524C2214yescomplemented
31C24×C4×C61C2211yescharacteristic
32C2×#117×C61—21no—
33C2×#117×C61—21no—
34C2×#117×C61—28no—
35C2×#117×C61C2212yes—
36C2×#117×C61C2212yes—
37C22×C4⋊C4×C61C2212yes—
38C22×C22⋊C4×C61—21no—
39C22×C42×C61—21no—
40C2×#11758222—22no—
41C2×#11758222—24no—
42C2×#11758222—24no—
43C2×#11758222C2×C414yescomplemented
44C22×Dic122C2311yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—21no—
47C22×C22⋊D61—21no—
48C22×C22⋊D61—21no—
49C22×C4×D61—21no—
50C22×#92418—24no—
51C22×#92418—24no—
52C22×#92418—24no—
53C22×#92418C2314yes—
54#14519760—28no—
55#14519756—22no—
56#14519756—22no—
57#14519756C2×C414yes—
58#14519756C2×C414yescomplemented
59#14519751—22no—
60#14519751—22no—
61#14519751C2×C414yes—
62#14519751C2×C414yescomplemented
63#14519746—28no—
64C4×C22⋊D61—28no—
65C4×C22⋊D61—28no—
66#14519736—22no—
67#14519736C2×C414yescomplemented
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×C4×Dic61—21no—
78#14519700—28no—
79#14519697—28no—
80C22⋊#92418—24no—
81C22⋊#92418—24no—
82C25×C61D411yescharacteristic
83C23×C4×C61—24no—
84C23×C4×C61—24no—
85C23×C4×C61—22no—
86C23×C4×C61C2×C414yes—
87C23×C4×C61D414yes—
88C23×C4×C61D411yescharacteristic
89C23×C4×C61D411yescharacteristic
90C23×C4×C61D411yescharacteristic
91C23×C4×C61C2311yescharacteristic
92C23×C4×C61C2311yescharacteristic
93C23×C4×C61C2311yescharacteristic
94#117×C61—28no—
95#117×C61—28no—
96#117×C61—24no—
97#117×C61—28no—
98#117×C61D418yescomplemented
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61—28no—
101C2×C4⋊C4×C61C2314yes—
102C2×C22⋊C4×C61—28no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—22no—
105C2×C42×C61—28no—
106C2×C42×C61—22no—
107C2×C42×C61—22no—
108C24×C61—22no—
109C24×C61—22no—
110C24×C61—24no—
111C24×C61—42no—
112C24×C61C22⋊C414yes—
113C24×C61C2×D411yescharacteristic
114C24×C61C2×D411yescharacteristic
115C24×C61C2×D411yescharacteristic
116C23×D61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—216no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—216no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—28no—
130C22×C4×C61—24no—
131C22×C4×C61—44no—
132C22×C4×C61—44no—
133C22×C4×C61C22⋊C414yes—
134C22×C4×C61C22⋊C414yes—
135C22×C4×C61C22⋊C414yes—
136C22×C4×C61C22×C412yes—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D414yes—
146C22×C4×C61#4114yes—
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—24no—
149C2×C22⋊D61—24no—
150C2×C22⋊D61—24no—
151C2×C22⋊D61—44no—
152C2×C22⋊D61—42no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—44no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158#11758222—216no—
159#11758222—48no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—21no—
163C2×D244C22×C412yes—
164C2×C4×D61—24no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×Dic122—22no—
168C2×Dic122—22no—
169C2×Dic122—21no—
170C2×Dic122C22×C412yes—
171C4⋊C4×C61—28no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61C2×D418yes—
174C22⋊C4×C61—28no—
175C22⋊C4×C61—28no—
176C42×C61—28no—
177C42×C61—28no—
178C22⋊Dic61—24no—
179C22⋊Dic61—24no—
180C22⋊Dic61—24no—
181#11758198—24no—
182#11758198—24no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61C22×C412yes—
186C4⋊Dic61C22×C412yes—
187#11758196—24no—
188#11758196—24no—
189C4×Dic61—24no—
190C2×#92418—24no—
191C2×#92418—24no—
192C2×#92418—24no—
193C2×#92418—48no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—28no—
204C23×C61—28no—
205C23×C61—44no—
206C23×C61—48no—
207C23×C61—48no—
208C23×C61C2×C22⋊C412yes—
209C23×C61C2×C22⋊C412yes—
210C23×C61C2×C22⋊C412yes—
211C23×C61#12012yes—
212C23×C61#12312yes—
213C23×C61#13514yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—42no—
219C22×D61—42no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—28no—
225C2×C4×C61—24no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—28no—
230C2×C4×C61—28no—
231C2×C4×C61—28no—
232C2×C4×C61—416no—
233C2×C4×C61—44no—
234C2×C4×C61—44no—
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C4×D418yes—
239C2×C4×C61#12012yes—
240C2×C4×C61#12118yes—
241C2×C4×C61#12312yes—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D412yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4112yes—
246C22⋊D61—48no—
247C22⋊D61—48no—
248C22⋊D61—48no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256D244—24no—
257D244—42no—
258C4×D61—48no—
259Dic122—24no—
260Dic122—42no—
261#92418—44no—
262#20908—611nomaximal, Sylow
263C22×C61—24no—
264C22×C61—24no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—24no—
269C22×C61—24no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—416no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61#38614yes—
278C22×C61#41614yescomplemented
279C22×C61C22×C22⋊C411yescharacteristic
280C22×C61C2×#12011yescharacteristic
281C22×C61C2×#12311yescharacteristic
282C22×C61C2×#13512yes—
283C22×C61#57512yes—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—82no—
288C2×D61—82no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—48no—
292C4×C61#38614yes—
293C4×C61C2×C4×D412yes—
294C4×C61C2×#12112yes—
295Dic61—42no—
296Dic61—42no—
297Dic61—42no—
298C2×#525—611no—
299C22×#117D6111yescharacteristic, complemented
300C2×#416—612no—
301C22×C22⋊C8—611no—
302#1875—612no—
303#1874—612no—
304#1873—612no—
305#1872—612no—
306#1735—612no—
307C2×C61—24no—
308C2×C61—24no—
309C2×C61—42no—
310C2×C61C2×#38611yescharacteristic
311C2×C61C2×#41611yescharacteristic
312C2×C61#289611yescharacteristic
313C2×C61#300212yes—
314C2×C61#309112yes—
315D61—82no—
316C22×#41—611no—
317C24×C4C2×D6111yescharacteristic
318C23×C8—612no—
319#525—618no—
320#525—1224no—
321C2×#123—1221no—
322C2×C22⋊Q8—611no—
323C2×C22⋊Q8—611no—
324C2×#121—611no—
325C2×#121—611no—
326C2×#120—1221no—
327C2×C4×D4—1221no—
328C2×C4×D4—1221no—
329C2×#117—21no—
330C2×#117—21no—
331C2×#117—28no—
332C2×#117—611no—
333C2×#117C2×D6112yes—
334C2×#117C2×D6112yes—
335C22×C4⋊C4C2×D6112yes—
336C22×C22⋊C4—21no—
337C22×C42—21no—
338#416—1228no—
339C2×#103—614no—
340C2×#102—614no—
341C2×C22⋊C8—612no—
342C2×C22⋊C8—612no—
343C2×C22⋊C8—1224no—
344#340—614no—
345#340—614no—
346C61#2090811yescharacteristic, complemented, Sylow
347C25C22⋊D6111yescharacteristic
348C2×#41—614no—
349C2×#41—1222no—
350C2×#41—1224no—
351C2×#41—1224no—
352C22×Q8—611no—
353C22×D4—611no—
354C22×D4—1221no—
355C22×D4—1221no—
356C22×D4—1221no—
357C23×C4—24no—
358C23×C4—24no—
359C23×C4—22no—
360C23×C4—1221no—
361C23×C4C4×D6114yes—
362C23×C4D24414yes—
363C23×C4C22⋊D6111yescharacteristic
364C23×C4C22⋊D6111yescharacteristic
365C23×C4C22⋊D6111yescharacteristic
366C23×C4C22×D6111yescharacteristic
367C23×C4C22×D6111yescharacteristic
368C23×C4C22×D6111yescharacteristic
369C22×C8—614no—
370C22×C8—1224no—
371C22×C8—1224no—
372C22×C8—1224no—
373#123—1228no—
374C22⋊Q8—614no—
375C22⋊Q8—614no—
376C22⋊Q8—1222no—
377C22⋊Q8—1222no—
378#121—614no—
379#121—614no—
380#121—1222no—
381#121—1222no—
382#120—1228no—
383C4×D4—1228no—
384C4×D4—1228no—
385#117—28no—
386#117—28no—
387#117—24no—
388#117—28no—
389#117—614no—
390#117—1222no—
391#117C22⋊D6118yescomplemented
392C2×C4⋊C4—22no—
393C2×C4⋊C4—28no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4C22×D6114yes—
399C2×C22⋊C4—28no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C42—28no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—1221no—
411#103—1228no—
412#102—1228no—
413C22⋊C8—1224no—
414C22⋊C8—1224no—
415C24—22no—
416C24—22no—
417C24—24no—
418C24—42no—
419C24—1221no—
420C24#1175819814yes—
421C24C2×C22⋊D6111yescharacteristic
422C24C2×C22⋊D6111yescharacteristic
423C24C2×C22⋊D6111yescharacteristic
424#41—12216no—
425#41—2448no—
426C2×Q8—612no—
427C2×Q8—1221no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×D4—612no—
431C2×D4—1224no—
432C2×D4—1224no—
433C2×D4—1224no—
434C2×D4—1222no—
435C2×D4—1221no—
436C2×D4—1222no—
437C2×D4—2444no—
438C2×D4—2442no—
439C2×D4—2442no—
440C2×D4—2444no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—216no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—216no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1224no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4#1175819814yes—
464C22×C4#1175819814yes—
465C22×C4#1175819814yes—
466C22×C4C2×C4×D6112yes—
467C22×C4C2×C4×D6112yes—
468C22×C4C2×C4×D6112yes—
469C22×C4C2×D24412yes—
470C22×C4C2×D24412yes—
471C22×C4C2×D24412yes—
472C22×C4#1175822214yes—
473C22×C4C2×C22⋊D6111yescharacteristic
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C2×C22⋊D6114yes—
476C22×C4C2×C22⋊D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478C2×C8—1224no—
479C2×C8—1224no—
480C2×C8—1224no—
481C2×C8—2448no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—612no—
485C4⋊C4—612no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1224no—
489C4⋊C4—1224no—
490C4⋊C4C2×C22⋊D6118yes—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C22⋊C4—1224no—
496C22⋊C4—1224no—
497C22⋊C4—1224no—
498C42—28no—
499C42—28no—
500C42—1224no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—21no—
507C23—21no—
508C23—28no—
509C23—22no—
510C23—22no—
511C23—28no—
512C23—44no—
513C23—48no—
514C23—48no—
515C23—1222no—
516C23—2441no—
517C23—2441no—
518C23—2442no—
519C23—2442no—
520C23C2×#1175819812yes—
521C23C2×#1175819812yes—
522C23C2×#1175819812yes—
523C23#1451974612yes—
524C23#1451976012yes—
525C23#1451979714yes—
526C23C22×C22⋊D6111yescharacteristic
527Q8—1224no—
528Q8—2442no—
529D4—1224no—
530D4—2448no—
531D4—2448no—
532D4—2448no—
533D4—2442no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—416no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2448no—
557C2×C4C2×#1175819812yes—
558C2×C4C2×#1175819812yes—
559C2×C4C2×#1175819812yes—
560C2×C4C4×C22⋊D6118yes—
561C2×C4#1451974612yes—
562C2×C4#1451975118yes—
563C2×C4#1451976012yes—
564C2×C4C22×C4×D6111yescharacteristic
565C2×C4C22×C22⋊D6112yes—
566C2×C4C22×D24411yescharacteristic
567C2×C4C2×#1175822212yes—
568C8—2444no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—416no—
581C22—42no—
582C22—42no—
583C22—4882no—
584C22—4882no—
585C22—2442no—
586C22—2442no—
587C22—2442no—
588C22#1683964114yes—
589C22#1683975614yescomplemented
590C22C22×#1175819811yescharacteristic
591C22C2×#1451974611yescharacteristic
592C22C2×#1451976011yescharacteristic
593C22C2×#1451979712yes—
594C22#1684056012yes—
595C4—22no—
596C4—22no—
597C4—48no—
598C4—2442no—
599C4—2442no—
600C4—2442no—
601C4#1683964114yes—
602C4C2×C4×C22⋊D6112yes—
603C4C2×#1451975112yes—
604C2—24no—
605C2—24no—
606C2—42no—
607C2—4882no—
608C2C2×#1683964111yescharacteristic
609C2C2×#1683975611yescharacteristic
610C2#2145433611yescharacteristic
611C2#2145503712yes—
612C2#2145573212yes—
subgroupG/H
centerC22×C4C2×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#117×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840273C2
cosocleC22×C4×C61C24
hypercenterC22×#117D61
hypocenterC61#20908
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#16839756C2semidirect
3. C22×C22⋊#92418C2semidirect
4. #21446141C2semidirect
5. #21446136C2semidirect
6. #21446131C2semidirect
7. #21446126C2semidirect
8. #21445535C2semidirect
9. C22×#117×C61C2semidirect
13. #16840273C4semidirect
24. C2×#14519700C22semidirect
25. C2×#14519697C22semidirect
29. #16839524C22semidirect
30. #16839524C22semidirect
43. C2×#11758222C2×C4semidirect
58. #14519756C2×C4semidirect
62. #14519751C2×C4semidirect
67. #14519736C2×C4semidirect
98. #117×C61D4semidirect
278. C22×C61#416semidirect
299. C22×#117D61semidirect
346. C61#20908semidirect
391. #117C22⋊D61semidirect
589. C22#16839756semidirect

Maximal subgroups

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

subgroupindex
C2×#168402732
C2×#168397562
C22×C22⋊#924182
#214461412
#214461362
#214461312
#214461262
#214455352
C22×#117×C612
#2090861