Finite Group Databasebeta

← #29840477

Subgroups — #29840477

Order 15616.

2422 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519776C211yesmaximal, characteristic
2#21448906C212yesmaximal, complemented
3#21448905C212yesmaximal, complemented
4#21448874C214yesmaximal, complemented
5#21448698C211yesmaximal, characteristic
6#21448696C211yesmaximal, characteristic
7#21448694C211yesmaximal, characteristic
8C22⋊C4×Dic122C212yesmaximal
9#21448651C212yesmaximal
10#21448564C214yesmaximal, complemented
11#21448436C212yesmaximal
12#21448390C212yesmaximal, complemented
13C4×#14519776C212yesmaximal, complemented
14#21448221C211yesmaximal, characteristic
15#21448174C212yesmaximal
16C4×#14519733C211yesmaximal, characteristic
17C61×#2345C211yesmaximal, characteristic
18C23×Dic122C2212yes—
19C2×#14519776C4116yescomplemented
20C2×#14519776C2212yes—
21C2×#14519776C2212yes—
22C2×#14519776C2211yescharacteristic
23C2×#14519776C2211yescharacteristic
24C2×#14519776C2211yescharacteristic
25C2×#14519776C2211yescharacteristic
26C2×C4×Dic122C2214yes—
27C2×#14519742C2214yes—
28C2×#14519733C2212yes—
29C2×#14519733C2211yescharacteristic
30C2×#14519733C2211yescharacteristic
31C22×C4⋊Dic61C2211yescharacteristic
32C22×#11758196C2212yes—
33C22×C4×Dic61C2211yescharacteristic
34#16839668C2214yes—
35#16839667C2214yes—
36#16839666C2214yes—
37#16839666C2214yes—
38#16839666C2214yes—
39#16839666C2214yes—
40#16839661C2212yes—
41#16839660C2212yes—
42#16839659C2214yes—
43#16839659C2214yes—
44#16839658C2212yes—
45#16839657C2214yes—
46#16839657C2214yes—
47#16839648C2214yes—
48#16839647C2214yes—
49#16839645C2214yes—
50#16839631C2212yes—
51#16839626C2212yes—
52#16839622C2211yescharacteristic
53#16839622C2212yes—
54#16839620C2212yes—
55#16839620C2212yes—
56#16839620C2212yes—
57C4⋊C4×Dic61C2212yes—
58C4⋊C4×Dic61C2212yes—
59C4⋊C4×Dic61C2212yes—
60C4×#11758196C2214yes—
61C4×#11758196C2214yes—
62C4×#11758196C2212yes—
63C4×C4⋊Dic61C2212yes—
64#16839613C2212yes—
65#16839613C2214yes—
66#16839613C2212yes—
67#16839613C2214yes—
68C22⋊C4×Dic61C2211yescharacteristic
69C22⋊C4×Dic61C2212yes—
70C22⋊C4×Dic61C2211yescharacteristic
71C22⋊C4×Dic61C2212yes—
72C4×C22⋊Dic61C2212yes—
73C4×C22⋊Dic61C2211yescharacteristic
74#16839607C2212yes—
75C42×Dic61C2212yes—
76C22×C4⋊C4×C61C2211yescharacteristic
77C61×#389C2212yes—
78C61×#387C2214yes—
79C61×#380C2211yescharacteristic
80C61×#380C2212yes—
81C4×C4⋊C4×C61C2212yes—
82C4×C22⋊C4×C61C2212yes—
83C4×C22⋊C4×C61C2211yescharacteristic
84C22×Dic122—48no—
85C22×Dic122C2×C4116yes—
86C22×Dic122C2312yes—
87C22×Dic122C2312yes—
88C22×Dic122C2312yes—
89C22×Dic122C2312yes—
90C23×Dic61C2311yescharacteristic
91C23×Dic61C2312yes—
92C23×Dic61C2311yescharacteristic
93#14519776C2×C4116yes—
94#14519776C2×C418yes—
95#14519776C2×C418yes—
96#14519776C2×C4116yes—
97#14519776C2×C418yes—
98#14519776C2×C418yes—
99C4×Dic122—48no—
100#14519742—48no—
101#14519733—28no—
102#14519733—24no—
103#14519733Q818yescomplemented
104C2×C4⋊Dic61C2×C418yes—
105C2×C4⋊Dic61C2312yes—
106C2×C4⋊Dic61C2314yes—
107C2×C4⋊Dic61C2311yescharacteristic
108C2×C4⋊Dic61C2311yescharacteristic
109C2×C4⋊Dic61C2312yes—
110C2×#11758196C2×C4116yes—
111C2×#11758196C2314yes—
112C2×#11758196C2312yes—
113C2×#11758196C2314yes—
114C2×#11758196C2314yes—
115C2×#11758196C2314yes—
116C2×#11758196C2312yes—
117C2×#11758196C2312yes—
118C2×#11758196C2312yes—
119C2×#11758196C2312yes—
120C2×#11758196C2312yes—
121C2×C22⋊Dic61C2312yes—
122C2×C22⋊Dic61C2311yescharacteristic
123C2×C22⋊Dic61C2311yescharacteristic
124C2×C4×Dic61C2×C418yes—
125C2×C4×Dic61C2312yes—
126C2×C4×Dic61C2314yes—
127C2×C4×Dic61C2312yes—
128C2×C4×Dic61C2312yes—
129C2×C4×Dic61C2312yes—
130C2×C4×Dic61C2314yes—
131C2×C4×Dic61C2314yes—
132C2×C4×Dic61C2314yes—
133C2×C4×Dic61C2311yescharacteristic
134C2×C4×Dic61C2311yescharacteristic
135C2×C4×Dic61C2311yescharacteristic
136C2×C4×Dic61C2311yescharacteristic
137C2×C4×Dic61C2312yes—
138#14519684C2312yes—
139#14519684C2312yes—
140#14519684C2312yes—
141#14519684C2314yes—
142#14519684C2314yes—
143#14519684C2314yes—
144#14519684C2312yes—
145#14519684C2314yes—
146#14519684C2314yes—
147#14519684C2314yes—
148#14519684C2314yes—
149C23×C4×C61C2311yescharacteristic
150C23×C4×C61C2312yes—
151C2×C4⋊C4×C61C2×C418yes—
152C2×C4⋊C4×C61C2312yes—
153C2×C4⋊C4×C61C2314yes—
154C2×C4⋊C4×C61C2311yescharacteristic
155C2×C4⋊C4×C61C2312yes—
156C2×C4⋊C4×C61C2311yescharacteristic
157C2×C22⋊C4×C61C2312yes—
158C2×C22⋊C4×C61C2311yescharacteristic
159C2×C22⋊C4×C61C2311yescharacteristic
160C2×C42×C61C2312yes—
161C2×C42×C61C2314yes—
162#95×C61C2312yes—
163#95×C61C2314yes—
164#95×C61C2314yes—
165#95×C61C2312yes—
166C24×C61C2411yescharacteristic
167C22×C4×C61—24no—
168C22×C4×C61—44no—
169C22×C4×C61C22×C414yes—
170C22×C4×C61C22×C418yes—
171C22×C4×C61C2412yes—
172C22×C4×C61C2414yes—
173C22×C4×C61C2412yes—
174C22×C4×C61C2412yes—
175C22×C4×C61C2411yescharacteristic
176C22×C4×C61C2412yes—
177C22×C4×C61C2411yescharacteristic
178C22×Dic61—24no—
179C22×Dic61—44no—
180C22×Dic61C22×C414yes—
181C22×Dic61C22×C418yes—
182C22×Dic61C22×C414yes—
183C22×Dic61C2×Q814yes—
184C22×Dic61#4114yes—
185C22×Dic61C2412yes—
186C22×Dic61C2414yes—
187C22×Dic61C2411yescharacteristic
188C22×Dic61C2412yes—
189C22×Dic61C2412yes—
190C22×Dic61C2412yes—
191C22×Dic61C2411yescharacteristic
192C22×Dic61C2411yescharacteristic
193C22×Dic61C2411yescharacteristic
194C2×Dic122—28no—
195C2×Dic122—28no—
196C2×Dic122—432no—
197C2×Dic122—44no—
198C2×Dic122—44no—
199C2×Dic122C22×C418yes—
200C2×Dic122C22×C418yes—
201C2×Dic122C22×C418yes—
202C2×Dic122C22×C418yes—
203C4⋊C4×C61—24no—
204C4⋊C4×C61—44no—
205C4⋊C4×C61C22×C418yes—
206C4⋊C4×C61C22×C414yes—
207C4⋊C4×C61C22×C414yes—
208C22⋊C4×C61—24no—
209C22⋊C4×C61—22no—
210C22⋊C4×C61C2×Q814yes—
211C42×C61—24no—
212C42×C61—44no—
213C22⋊Dic61—24no—
214C22⋊Dic61—22no—
215C22⋊Dic61C2×Q814yes—
216C4⋊Dic61—24no—
217C4⋊Dic61—44no—
218C4⋊Dic61C22×C418yes—
219C4⋊Dic61C22×C414yes—
220C4⋊Dic61C22×C414yes—
221#11758196—24no—
222#11758196—28no—
223#11758196—28no—
224#11758196—44no—
225#11758196—44no—
226#11758196C22×C418yes—
227#11758196C22×C418yes—
228#11758196C22×C418yes—
229#11758196C22×C418yes—
230#11758196C2×Q818yescomplemented
231C4×Dic61—24no—
232C4×Dic61—28no—
233C4×Dic61—24no—
234C4×Dic61—24no—
235C4×Dic61—24no—
236C4×Dic61—44no—
237C4×Dic61—44no—
238C4×Dic61—44no—
239C4×Dic61C22×C414yes—
240C4×Dic61C22×C414yes—
241C4×Dic61C22×C414yes—
242C4×Dic61C22×C414yes—
243C4×Dic61C2×Q818yescomplemented
244C23×C61—22no—
245C23×C61—22no—
246C23×C61C23×C412yes—
247C23×C61C22×Q812yes—
248C23×C61C2×#4112yes—
249C23×C61C2511yescharacteristic
250C2×C4×C61—22no—
251C2×C4×C61—24no—
252C2×C4×C61—24no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—28no—
256C2×C4×C61—22no—
257C2×C4×C61—24no—
258C2×C4×C61—22no—
259C2×C4×C61—24no—
260C2×C4×C61—22no—
261C2×C4×C61—24no—
262C2×C4×C61—48no—
263C2×C4×C61C23×C412yes—
264C2×C4×C61C23×C414yes—
265C2×C4×C61C23×C414yes—
266C2×C4×C61C23×C412yes—
267C2×C4×C61C22×Q814yes—
268C2×C4×C612−1+414yes—
269C2×Dic61—22no—
270C2×Dic61—24no—
271C2×Dic61—24no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—28no—
275C2×Dic61—22no—
276C2×Dic61—24no—
277C2×Dic61—22no—
278C2×Dic61—24no—
279C2×Dic61—22no—
280C2×Dic61—24no—
281C2×Dic61—48no—
282C2×Dic61C4×Q8116yescomplemented
283C2×Dic61C23×C412yes—
284C2×Dic61C23×C414yes—
285C2×Dic61C23×C414yes—
286C2×Dic61C23×C412yes—
287C2×Dic61C23×C412yes—
288C2×Dic61C23×C412yes—
289C2×Dic61C22×Q814yes—
290C2×Dic61C22×Q812yes—
291C2×Dic61C22×Q812yes—
292C2×Dic61C2×#4112yes—
293C2×Dic61C2×#4112yes—
294C2×Dic61H(5,𝔽2)14yes—
295Dic122—216no—
296Dic122—216no—
297Dic122—48no—
298Dic122—48no—
299#34216—611nomaximal, Sylow
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—24no—
303C22×C61—24no—
304C22×C61C2×C4×Q814yes—
305C22×C61C24×C411yescharacteristic
306C22×C61C23×Q811yescharacteristic
307C22×C61C22×#4111yescharacteristic
308C22×C61C2×H(5,𝔽2)12yes—
309C22×C61C2×2−1+412yes—
310C4×C61—24no—
311C4×C61—24no—
312C4×C61—22no—
313C4×C61—24no—
314C4×C61—42no—
315C4×C61—44no—
316C4×C61#51914yes—
317Dic61—24no—
318Dic61—24no—
319Dic61—22no—
320Dic61—24no—
321Dic61—42no—
322Dic61—44no—
323Dic61C2×C4×Q814yes—
324Dic61C2×C4×Q814yes—
325Dic61#51814yes—
326C22×C4⋊Q8—611no—
327#2369—612no—
328#2369—612no—
329#2365—614no—
330#2345—611no—
331#2345—611no—
332#2345—611no—
333#2345D6111yescharacteristic
334Q8×C22⋊C4—612no—
335Q8×C22⋊C4—612no—
336#2331—614no—
337#2316—612no—
338C4×C4⋊Q8—612no—
339C4×C4⋊Q8—612no—
340#2285—611no—
341#2274—612no—
342C4×#117—611no—
343C2×C61—24no—
344C2×C61C22×C4×Q811yescharacteristic
345C2×C61C2×#51811yescharacteristic
346C2×C61C2×#51911yescharacteristic
347C2×C61#350312yes—
348C2×C61#351112yes—
349C23×Q8—612no—
350C2×C4⋊Q8—6116no—
351C2×C4⋊Q8—611no—
352C2×C4⋊Q8—611no—
353C2×C4⋊Q8—612no—
354C2×C4⋊Q8—612no—
355C2×C4⋊Q8—611no—
356C2×C4⋊Q8—611no—
357C2×C4×Q8—614no—
358C2×C4×Q8—614no—
359C2×#117—611no—
360C2×#117—611no—
361C2×#117—612no—
362C22×C4⋊C4—612no—
363C22×C4⋊C4—611no—
364C22×C4⋊C4C2×D6111yescharacteristic
365C22×C42—611no—
366#391—614no—
367#391—614no—
368#391—614no—
369#391—614no—
370#391—614no—
371#391—614no—
372#389—614no—
373#389—612no—
374#389—612no—
375#389—614no—
376#389—612no—
377#389—614no—
378#389—614no—
379#389C2×D6112yes—
380#387—614no—
381#387—614no—
382#387—614no—
383#387C2×D6114yes—
384#384—612no—
385#382—612no—
386#380—612no—
387#380—611no—
388#380—612no—
389#380—612no—
390#380—612no—
391#380C2×D6112yes—
392#380C2×D6111yescharacteristic
393C4×C4⋊C4—612no—
394C4×C4⋊C4—612no—
395C4×C4⋊C4—612no—
396C4×C4⋊C4—614no—
397C4×C4⋊C4—612no—
398C4×C4⋊C4—614no—
399C4×C4⋊C4—612no—
400C4×C4⋊C4—614no—
401C4×C4⋊C4—612no—
402C4×C4⋊C4—612no—
403C4×C4⋊C4—614no—
404C4×C4⋊C4C2×D6112yes—
405C4×C22⋊C4—612no—
406C4×C22⋊C4—611no—
407C4×C22⋊C4—611no—
408C4×C22⋊C4—611no—
409C4×C22⋊C4—612no—
410C4×C22⋊C4—612no—
411C4×C22⋊C4C2×D6112yes—
412C4×C22⋊C4C2×D6111yescharacteristic
413#376—612no—
414C43—612no—
415C61#3421611yescharacteristic, complemented, Sylow
416C22×Q8—6116no—
417C22×Q8—612no—
418C22×Q8—612no—
419C22×Q8—612no—
420C22×Q8—612no—
421C22×Q8—2448no—
422C23×C4—611no—
423C23×C4—611no—
424C23×C4—612no—
425C23×C4C22×D6111yescharacteristic
426C23×C4C22×D6112yes—
427C4⋊Q8—618no—
428C4⋊Q8—618no—
429C4⋊Q8—6116no—
430C4⋊Q8—6116no—
431C4⋊Q8—618no—
432C4⋊Q8—618no—
433C4×Q8—2448no—
434C4×Q8—2448no—
435#117—618no—
436#117—1224no—
437#117—1228no—
438C2×C4⋊C4—6116no—
439C2×C4⋊C4—618no—
440C2×C4⋊C4—612no—
441C2×C4⋊C4—612no—
442C2×C4⋊C4—612no—
443C2×C4⋊C4—612no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—612no—
447C2×C4⋊C4—612no—
448C2×C4⋊C4—614no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—614no—
451C2×C4⋊C4—612no—
452C2×C4⋊C4—614no—
453C2×C4⋊C4—614no—
454C2×C4⋊C4—614no—
455C2×C4⋊C4C4×D6118yes—
456C2×C4⋊C4C22×D6112yes—
457C2×C4⋊C4C22×D6114yes—
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C4⋊C4C22×D6111yescharacteristic
460C2×C4⋊C4C22×D6112yes—
461C2×C22⋊C4—611no—
462C2×C22⋊C4—611no—
463C2×C22⋊C4—612no—
464C2×C22⋊C4C22×D6111yescharacteristic
465C2×C22⋊C4C22×D6112yes—
466C2×C22⋊C4C22×D6111yescharacteristic
467C2×C42—618no—
468C2×C42—612no—
469C2×C42—611no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—611no—
473C2×C42—612no—
474C2×C42—614no—
475C2×C42—612no—
476C2×C42—612no—
477C2×C42—612no—
478C2×C42—614no—
479C2×C42—614no—
480C2×C42—614no—
481C2×C42C22×D6112yes—
482C2×C42C22×D6114yes—
483#95—612no—
484#95—612no—
485#95—612no—
486#95—614no—
487#95—614no—
488#95—614no—
489#95—612no—
490#95—614no—
491#95—614no—
492#95—614no—
493#95—614no—
494#95C22×D6112yes—
495#95C22×D6114yes—
496#95C22×D6112yes—
497#95C22×D6114yes—
498C24C23×D6111yescharacteristic
499C2×Q8—618no—
500C2×Q8—618no—
501C2×Q8—618no—
502C2×Q8—618no—
503C2×Q8—1228no—
504C2×Q8—1228no—
505C2×Q8—2444no—
506C2×Q8—2444no—
507C2×Q8—24432no—
508C22×C4—24no—
509C22×C4—44no—
510C22×C4—614no—
511C22×C4—614no—
512C22×C4—614no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—614no—
517C22×C4—611no—
518C22×C4—612no—
519C22×C4—612no—
520C22×C4—612no—
521C22×C4—618no—
522C22×C4—612no—
523C22×C4—614no—
524C22×C4—1224no—
525C22×C4—2444no—
526C22×C4C2×C4×D6114yes—
527C22×C4C2×C4×D6118yes—
528C22×C4C23×D6114yes—
529C22×C4C23×D6112yes—
530C22×C4C23×D6111yescharacteristic
531C22×C4C23×D6112yes—
532C22×C4C23×D6112yes—
533C22×C4C23×D6112yes—
534C22×C4C23×D6111yescharacteristic
535C4⋊C4—24no—
536C4⋊C4—44no—
537C4⋊C4—618no—
538C4⋊C4—618no—
539C4⋊C4—618no—
540C4⋊C4—618no—
541C4⋊C4—618no—
542C4⋊C4—614no—
543C4⋊C4—614no—
544C4⋊C4—618no—
545C4⋊C4—1224no—
546C4⋊C4—1228no—
547C4⋊C4—1228no—
548C4⋊C4—1224no—
549C4⋊C4—2444no—
550C4⋊C4—2444no—
551C4⋊C4—2444no—
552C4⋊C4C2×C4×D6118yes—
553C4⋊C4C2×C4×D6114yes—
554C4⋊C4C2×C4×D6114yes—
555C22⋊C4—22no—
556C22⋊C4—24no—
557C22⋊C4—614no—
558C22⋊C4—1222no—
559C22⋊C4—1224no—
560C22⋊C4Q8×D6114yes—
561C42—24no—
562C42—44no—
563C42—614no—
564C42—614no—
565C42—614no—
566C42—614no—
567C42—618no—
568C42—1228no—
569C42—1224no—
570C42—1224no—
571C42—1224no—
572C42—1224no—
573C42—2444no—
574C42—2444no—
575C42—2444no—
576C23—22no—
577C23—22no—
578C23C22×C4×D6112yes—
579C23C2×Q8×D6112yes—
580C23#41×D6112yes—
581C23C24×D6111yescharacteristic
582Q8—12216no—
583Q8—12216no—
584Q8—2448no—
585Q8—2448no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—48no—
599C2×C4—6116no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—612no—
606C2×C4—612no—
607C2×C4—614no—
608C2×C4—612no—
609C2×C4—614no—
610C2×C4—614no—
611C2×C4—614no—
612C2×C4—1228no—
613C2×C4—1222no—
614C2×C4—1224no—
615C2×C4—1224no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1224no—
620C2×C4—1222no—
621C2×C4—1224no—
622C2×C4—1222no—
623C2×C4—1224no—
624C2×C4—2448no—
625C2×C4C22×C4×D6112yes—
626C2×C4C22×C4×D6114yes—
627C2×C4C22×C4×D6114yes—
628C2×C4C22×C4×D6112yes—
629C2×C4C2×Q8×D6114yes—
630C2×C4#1451982414yes—
631C22—24no—
632C22—22no—
633C22—22no—
634C22—24no—
635C22C4×Q8×D6114yes—
636C22C23×C4×D6111yescharacteristic
637C22C22×Q8×D6111yescharacteristic
638C22C2×#41×D6111yescharacteristic
639C22H(5,𝔽2)×D6112yes—
640C22C2×#1451982412yes—
641C4—24no—
642C4—24no—
643C4—22no—
644C4—24no—
645C4—44no—
646C4—42no—
647C4—614no—
648C4—614no—
649C4—614no—
650C4—1222no—
651C4—1224no—
652C4—1224no—
653C4—1224no—
654C4—2442no—
655C4—2444no—
656C4#1684023914yes—
657C2—24no—
658C2C2×C4×Q8×D6111yescharacteristic
659C2#518×D6111yescharacteristic
660C2C2×#1684023911yescharacteristic
661C2#2145963312yes—
662C2#2145972112yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2345C2
socle, cosocleC23×C61C25
prime radicalC24×C61C24
hypercenter#2345D61
hypocenterC61#34216
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2345

Complements

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

NH ≅ G/Ntype
2. #21448906C2semidirect
3. #21448905C2semidirect
4. #21448874C2semidirect
10. #21448564C2semidirect
12. #21448390C2semidirect
13. C4×#14519776C2semidirect
19. C2×#14519776C4semidirect
103. #14519733Q8semidirect
230. #11758196C2×Q8semidirect
243. C4×Dic61C2×Q8semidirect
282. C2×Dic61C4×Q8semidirect
415. C61#34216semidirect

Maximal subgroups

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

subgroupindex
C22×#145197762
#214489062
#214489052
#214488742
#214486982
#214486962
#214486942
C22⋊C4×Dic1222
#214486512
#214485642
#214484362
#214483902
C4×#145197762
#214482212
#214481742
C4×#145197332
C61×#23452
#3421661