Finite Group Databasebeta

← #29843196

Subgroups — #29843196

Order 15616.

1998 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2#21448983C214yesmaximal
3#21448908C211yesmaximal, characteristic, complemented
4#21448908C211yesmaximal, characteristic, complemented
5#21448879C211yesmaximal, characteristic, complemented
6#21448783C212yesmaximal, complemented
7#21448737C212yesmaximal, complemented
8#21448708C212yesmaximal
9#21448703C212yesmaximal
10#21448559C212yesmaximal, complemented
11#21448556C212yesmaximal, complemented
12#21448504C212yesmaximal
13#21448495C212yesmaximal
14#21448468C214yesmaximal, complemented
15#21448223C211yesmaximal, characteristic
16#21448215C211yesmaximal, characteristic
17C61×#2365C211yesmaximal, characteristic, complemented
18#16840317C4116yescomplemented
19C2×#14519777C2211yescharacteristic
20C2×#14519768C2212yes—
21C2×#14519757C2212yes—
22C2×#14519753C2212yes—
23C2×Q8×Dic61C2212yes—
24C2×Q8×Dic61C2212yes—
25C2×#14519742C2212yes—
26C2×#14519735C2212yes—
27C2×#14519734C2212yes—
28C2×C42×D61C2211yescharacteristic
29#16839669C2212yes—
30#16839666C2214yes—
31#16839666C2214yes—
32#16839661C2212yes—
33#16839660C2212yes—
34#16839660C2214yes—
35#16839660C2212yes—
36#16839659C2214yes—
37#16839658C2212yes—
38#16839658C2214yes—
39#16839658C2212yes—
40#16839657C2214yes—
41#16839654C2214yes—
42#16839653C2214yes—
43#16839652C2214yes—
44#16839651C2214yes—
45#16839648C2214yes—
46#16839647C2214yes—
47#16839646C2214yes—
48#16839645C2214yes—
49#16839644C2214yes—
50#16839631C2211yescharacteristic
51#16839631C2211yescharacteristic
52#16839629C2214yes—
53#16839629C2212yes—
54#16839627C2214yes—
55#16839627C2211yescharacteristic
56#16839627C2211yescharacteristic
57#16839626C2212yes—
58#16839624C2212yes—
59#16839621C2212yes—
60#16839619C2211yescharacteristic
61#16839619C2211yescharacteristic
62#16839619C2211yescharacteristic
63#16839619C2211yescharacteristic
64C4⋊C4×Dic61C2212yes—
65C4⋊C4×Dic61C2212yes—
66C4⋊C4×Dic61C2212yes—
67C4×#11758196C2214yes—
68C4×#11758196C2212yes—
69C4×C4⋊Dic61C2211yescharacteristic
70C4×C4⋊Dic61C2211yescharacteristic
71C4×C4⋊Dic61C2212yes—
72C4×C4⋊Dic61C2212yes—
73C4×#11758198C2212yes—
74#16839608C2212yes—
75#16839607C2214yes—
76C2×C4⋊Q8×C61C2211yescharacteristic
77C2×C4×Q8×C61C2212yes—
78C61×#391C2212yes—
79C61×#389C2212yes—
80C61×#387C2214yes—
81C4×C4⋊C4×C61C2212yes—
82C4×C4⋊C4×C61C2211yescharacteristic
83C4×C4⋊C4×C61C2211yescharacteristic
84C22×Dic122C2312yes—
85C22×C4×D61C2311yescharacteristic
86C22×C4×D61C2311yescharacteristic
87C22×C4×D61C2311yescharacteristic
88#14519777C2×C418yes—
89#14519768C2×C4116yes—
90#14519757C2×C4116yes—
91#14519753C2×C4116yes—
92Q8×Dic61—44no—
93Q8×Dic61C2×C4116yes—
94#14519742C2×C4116yes—
95#14519735C2×C4116yes—
96#14519734—44no—
97C2×C4⋊Dic61C2311yescharacteristic
98C2×C4⋊Dic61C2311yescharacteristic
99C2×C4⋊Dic61C2311yescharacteristic
100C2×C4⋊Dic61C2311yescharacteristic
101C2×C4⋊Dic61C2314yes—
102C2×C4⋊Dic61C2312yes—
103C2×C4⋊Dic61C2312yes—
104C2×C4⋊Dic61C2314yes—
105C2×#11758196C2312yes—
106C2×#11758196C2312yes—
107C2×#11758196C2312yes—
108C2×#11758196C2312yes—
109C2×#11758196C2314yes—
110C2×#11758196C2314yes—
111C2×#11758196C2314yes—
112C2×#11758198C2312yes—
113C2×#11758198C2312yes—
114C2×#11758198C2312yes—
115C2×#11758198C2312yes—
116C2×#11758198C2314yes—
117C2×C4×Dic61C2312yes—
118C2×C4×Dic61C2312yes—
119C2×C4×Dic61C2314yes—
120C2×C4×Dic61C2314yes—
121C2×C4×Dic61C2314yes—
122C2×C4×Dic61C2311yescharacteristic
123C2×C4×Dic61C2311yescharacteristic
124C2×C4×Dic61C2311yescharacteristic
125C2×C4×Dic61C2314yes—
126C42×D61C2×C418yes—
127#14519684C2312yes—
128#14519684C2312yes—
129#14519684C2314yes—
130#14519684C2312yes—
131#14519684C2312yes—
132#14519684C2312yes—
133#14519684C2312yes—
134#14519684C2312yes—
135#14519684C2312yes—
136#14519684C2312yes—
137#14519684C2314yes—
138#14519684C2314yes—
139#14519684C2312yes—
140#14519684C2314yes—
141#14519684C2314yes—
142#14519684C2314yes—
143C22×Q8×C61C2312yes—
144C4⋊Q8×C61C2×C418yes—
145C4×Q8×C61—44no—
146C2×C4⋊C4×C61C2311yescharacteristic
147C2×C4⋊C4×C61C2311yescharacteristic
148C2×C4⋊C4×C61C2311yescharacteristic
149C2×C4⋊C4×C61C2311yescharacteristic
150C2×C4⋊C4×C61C2314yes—
151C2×C4⋊C4×C61C2312yes—
152C2×C4⋊C4×C61C2312yes—
153C2×C42×C61C2312yes—
154C2×C42×C61C2312yes—
155C2×C42×C61C2314yes—
156C2×C42×C61C2311yescharacteristic
157#95×C61C2312yes—
158#95×C61C2312yes—
159#95×C61C2314yes—
160#95×C61C2314yes—
161C23×D61C2411yescharacteristic
162C2×Q8×C61—24no—
163C2×Q8×C61—24no—
164C2×Q8×C61C22×C418yes—
165C22×C4×C61C2412yes—
166C22×C4×C61C2412yes—
167C22×C4×C61C2412yes—
168C22×C4×C61C2412yes—
169C22×C4×C61C2411yescharacteristic
170C22×C4×C61C2414yes—
171C22×C4×C61C2411yescharacteristic
172C22×C4×C61C2411yescharacteristic
173C22×Dic61C2412yes—
174C22×Dic61C2412yes—
175C22×Dic61C2412yes—
176C22×Dic61C2412yes—
177C22×Dic61C2411yescharacteristic
178C22×Dic61C2411yescharacteristic
179C22×Dic61C2411yescharacteristic
180C22×Dic61C2414yes—
181C2×C4×D61—22no—
182C2×C4×D61—22no—
183C2×C4×D61—22no—
184C2×C4×D61—22no—
185C2×C4×D61—41no—
186C2×C4×D61—41no—
187C2×C4×D61C22×C414yes—
188C2×C4×D61C22×C414yes—
189C2×C4×D61C22×C414yes—
190C2×Dic122—42no—
191C2×Dic122—42no—
192C2×Dic122C22×C418yes—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—44no—
198C4⋊C4×C61C22×C418yes—
199C4⋊C4×C61C22×C418yes—
200C42×C61—24no—
201C42×C61—24no—
202C42×C61—44no—
203C42×C61C22×C414yes—
204#11758198—42no—
205#11758198—42no—
206#11758198—42no—
207#11758198—42no—
208#11758198C22×C4116yes—
209C4⋊Dic61—22no—
210C4⋊Dic61—22no—
211C4⋊Dic61—22no—
212C4⋊Dic61—22no—
213C4⋊Dic61—44no—
214C4⋊Dic61C22×C418yes—
215C4⋊Dic61C22×C418yes—
216C4⋊Dic61C22×C4116yes—
217#11758196—44no—
218#11758196—44no—
219#11758196—42no—
220#11758196—42no—
221#11758196—42no—
222#11758196—42no—
223#11758196C22×C4116yes—
224C4×Dic61—24no—
225C4×Dic61—24no—
226C4×Dic61—44no—
227C4×Dic61—44no—
228C4×Dic61—44no—
229C4×Dic61C22×C414yes—
230C4×Dic61C22×C414yes—
231C4×Dic61C22×C414yes—
232C4×Dic61C22×C4116yes—
233C23×C61C2511yescharacteristic
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—21no—
240C22×D61C23×C412yes—
241Q8×C61—216no—
242C2×C4×C61—24no—
243C2×C4×C61—24no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61C23×C418yes—
259C2×C4×C61C23×C412yes—
260C2×C4×C61C23×C412yes—
261C2×C4×C61C23×C412yes—
262C2×C4×C61H(5,𝔽2)12yes—
263C2×C4×C612−1+412yes—
264C2×C4×C612−1+412yes—
265C2×C4×C612−1+412yes—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—24no—
271C2×Dic61—24no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—22no—
284C2×Dic61—22no—
285C2×Dic61—22no—
286C2×Dic61C23×C412yes—
287C2×Dic61C23×C412yes—
288C2×Dic61C23×C412yes—
289C2×Dic61C23×C418yes—
290C4×D61—28no—
291C4×D61—28no—
292C4×D61—44no—
293Dic122—48no—
294#34394—611nomaximal, Sylow
295C22×C61C24×C411yescharacteristic
296C22×C61C2×H(5,𝔽2)11yescharacteristic
297C22×C61C2×2−1+411yescharacteristic
298C22×C61C2×2−1+411yescharacteristic
299C22×C61C2×2−1+411yescharacteristic
300C22×C61#58511yescharacteristic
301C22×C61#58511yescharacteristic
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C4×C61—28no—
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—42no—
313C4×C61—42no—
314C4×C61—42no—
315C4×C61#51914yes—
316C4×C61#52014yes—
317Dic61—22no—
318Dic61—22no—
319Dic61—28no—
320Dic61—22no—
321Dic61—42no—
322Dic61—42no—
323Dic61—42no—
324Dic61—42no—
325C2×#533—611no—
326#2378—614no—
327#2369—611no—
328#2369—611no—
329#2365—611no—
330#2365D6111yescharacteristic, complemented
331#2356—612no—
332#2351—612no—
333#2347—612no—
334#2346—612no—
335#2330—612no—
336#2330—612no—
337#2323—612no—
338#2323—612no—
339#2320—614no—
340#2285—611no—
341#2283—611no—
342C2×C61C2×#51911yescharacteristic
343C2×C61C2×#52011yescharacteristic
344C2×C61#342811yescharacteristic
345C2×C61#354711yescharacteristic
346C2×C61#354711yescharacteristic
347C2×C61#355511yescharacteristic
348C2×C61#355511yescharacteristic
349D61—42no—
350#533—6116no—
351C2×C4⋊Q8—611no—
352C2×C4⋊Q8C2×D6111yescharacteristic
353C2×#125—612no—
354C2×C22⋊Q8—612no—
355C2×C22⋊Q8—612no—
356C2×C4×Q8—612no—
357C2×C4×Q8—612no—
358C2×C4×Q8—612no—
359C2×C4×Q8C2×D6112yes—
360C2×#117—612no—
361C2×#117—612no—
362C22×C42—611no—
363#391—612no—
364#391—614no—
365#391—614no—
366#391C2×D6112yes—
367#389—614no—
368#389—612no—
369#389—612no—
370#389—612no—
371#389—614no—
372#389—612no—
373#389—612no—
374#389—614no—
375#389—614no—
376#389C2×D6112yes—
377#388—614no—
378#388—614no—
379#388—614no—
380#388—614no—
381#387—614no—
382#387—614no—
383#387—614no—
384#387—614no—
385#387—614no—
386#387C2×D6114yes—
387#384—611no—
388#384—611no—
389#383—612no—
390#383—614no—
391#382—614no—
392#382—611no—
393#382—611no—
394#382—612no—
395#381—612no—
396#380—611no—
397#380—611no—
398#380—611no—
399#380—611no—
400#380—612no—
401C4×C4⋊C4—612no—
402C4×C4⋊C4—612no—
403C4×C4⋊C4—612no—
404C4×C4⋊C4—614no—
405C4×C4⋊C4—612no—
406C4×C4⋊C4—612no—
407C4×C4⋊C4—612no—
408C4×C4⋊C4—611no—
409C4×C4⋊C4—611no—
410C4×C4⋊C4C2×D6112yes—
411C4×C4⋊C4C2×D6111yescharacteristic
412C4×C4⋊C4C2×D6111yescharacteristic
413C4×C22⋊C4—612no—
414#376—612no—
415#376—614no—
416C61#3439411yescharacteristic, complemented, Sylow
417C22×Q8—612no—
418C22×Q8C22×D6112yes—
419C23×C4—611no—
420C23×C4—611no—
421C23×C4—611no—
422C4⋊Q8—618no—
423C4⋊Q8C4×D6118yes—
424#125—6116no—
425C22⋊Q8—6116no—
426C22⋊Q8—6116no—
427C4×Q8—44no—
428C4×Q8—6116no—
429C4×Q8—6116no—
430C4×Q8—2444no—
431#117—6116no—
432#117—2444no—
433C2×C4⋊C4—612no—
434C2×C4⋊C4—614no—
435C2×C4⋊C4—614no—
436C2×C4⋊C4—612no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—612no—
442C2×C4⋊C4—612no—
443C2×C4⋊C4—614no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—612no—
446C2×C4⋊C4—614no—
447C2×C4⋊C4—614no—
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C4⋊C4C22×D6111yescharacteristic
452C2×C4⋊C4C22×D6114yes—
453C2×C4⋊C4C22×D6112yes—
454C2×C4⋊C4C22×D6112yes—
455C2×C22⋊C4—614no—
456C2×C22⋊C4—612no—
457C2×C22⋊C4—612no—
458C2×C22⋊C4—612no—
459C2×C22⋊C4—612no—
460C2×C42—614no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—618no—
465C2×C42—612no—
466C2×C42—612no—
467C2×C42—614no—
468C2×C42—614no—
469C2×C42—614no—
470C2×C42C22×D6112yes—
471C2×C42C22×D6112yes—
472C2×C42C22×D6114yes—
473C2×C42C22×D6111yescharacteristic
474#95—612no—
475#95—612no—
476#95—612no—
477#95—612no—
478#95—612no—
479#95—612no—
480#95—614no—
481#95—612no—
482#95—612no—
483#95—612no—
484#95—612no—
485#95—614no—
486#95—614no—
487#95—614no—
488#95—614no—
489#95—614no—
490#95C22×D6114yes—
491#95C22×D6112yes—
492#95C22×D6112yes—
493#95C22×D6114yes—
494C24—611no—
495C2×Q8—24no—
496C2×Q8—24no—
497C2×Q8—618no—
498C2×Q8—2442no—
499C2×Q8—2442no—
500C2×Q8C2×C4×D6118yes—
501C22×C4—611no—
502C22×C4—614no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—614no—
506C22×C4—614no—
507C22×C4—614no—
508C22×C4—612no—
509C22×C4—612no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—2441no—
517C22×C4—2441no—
518C22×C4C23×D6112yes—
519C22×C4C23×D6112yes—
520C22×C4C23×D6112yes—
521C22×C4C23×D6112yes—
522C22×C4C23×D6111yescharacteristic
523C22×C4C23×D6114yes—
524C22×C4C23×D6111yescharacteristic
525C22×C4C23×D6111yescharacteristic
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—44no—
531C4⋊C4—6116no—
532C4⋊C4—618no—
533C4⋊C4—6116no—
534C4⋊C4—618no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C4⋊C4—1222no—
538C4⋊C4—1222no—
539C4⋊C4—2442no—
540C4⋊C4—2444no—
541C4⋊C4—2444no—
542C4⋊C4—2442no—
543C4⋊C4—2442no—
544C4⋊C4—2442no—
545C4⋊C4—2444no—
546C4⋊C4C2×C4×D6118yes—
547C4⋊C4C2×C4×D6118yes—
548C22⋊C4—6116no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C42—24no—
554C42—24no—
555C42—44no—
556C42—614no—
557C42—614no—
558C42—614no—
559C42—6116no—
560C42—1224no—
561C42—1224no—
562C42—2444no—
563C42—2444no—
564C42—2444no—
565C42C2×C4×D6114yes—
566C23—612no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23—1221no—
573C23C24×D6111yescharacteristic
574Q8—216no—
575Q8—2448no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—618no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1224no—
601C2×C4—1224no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1228no—
605C2×C4—1228no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—2444no—
619C2×C4C22×C4×D6112yes—
620C2×C4C22×C4×D6118yes—
621C2×C4C22×C4×D6112yes—
622C2×C4C22×C4×D6112yes—
623C2×C4#1451982212yes—
624C2×C4#1451982412yes—
625C2×C4#1451982412yes—
626C2×C4#1451982512yes—
627C22—1222no—
628C22—1222no—
629C22—1222no—
630C22—2441no—
631C22—2441no—
632C22—2441no—
633C22—2441no—
634C22C23×C4×D6111yescharacteristic
635C22C2×#1451982211yescharacteristic
636C22C2×#1451982411yescharacteristic
637C22C2×#1451982411yescharacteristic
638C22C2×#1451982511yescharacteristic
639C22#1684059911yescharacteristic
640C22#1684059911yescharacteristic
641C4—28no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—1222no—
648C4—1222no—
649C4—1222no—
650C4—1228no—
651C4—2442no—
652C4—2442no—
653C4—2442no—
654C4—2442no—
655C4#1684023814yes—
656C4#1684024114yes—
657C2—2442no—
658C2C2×#1684023811yescharacteristic
659C2C2×#1684024111yescharacteristic
660C2#2145902111yescharacteristic
661C2#2146019611yescharacteristic
662C2#2146020411yescharacteristic
663C2#2146029511yescharacteristic
664C2#2146030311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2365C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2365D61
hypocenterC61#34394
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2365

Complements

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

NH ≅ G/Ntype
3. #21448908C2semidirect
4. #21448908C2semidirect
5. #21448879C2semidirect
6. #21448783C2semidirect
7. #21448737C2semidirect
10. #21448559C2semidirect
11. #21448556C2semidirect
14. #21448468C2semidirect
17. C61×#2365C2semidirect
18. #16840317C4semidirect
330. #2365D61semidirect
416. C61#34394semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
#214489832
#214489082
#214488792
#214487832
#214487372
#214487082
#214487032
#214485592
#214485562
#214485042
#214484952
#214484682
#214482232
#214482152
C61×#23652
#3439461