Finite Group Databasebeta

← #29602366

Subgroups — #29602366

Order 15616.

1426 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839661C211yesmaximal, characteristic
2C2×#16839627C212yesmaximal
3C2×C4⋊C4×Dic61C211yesmaximal, characteristic
4#21444078C212yesmaximal
5#21444075C212yesmaximal
6#21444049C211yesmaximal, characteristic
7#21444048C211yesmaximal, characteristic
8#21444048C211yesmaximal, characteristic
9#21444045C212yesmaximal
10C4×#14519684C211yesmaximal, characteristic
11C2×C61×#389C211yesmaximal, characteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×C4⋊Dic61C2212yes—
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C4×Dic61C2211yescharacteristic
16C22×C4×Dic61C2212yes—
17C22×C4×Dic61C2211yescharacteristic
18#16839661—24no—
19#16839627—28no—
20C4⋊C4×Dic61C418yescomplemented
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2211yescharacteristic
33C22×C4⋊C4×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2212yes—
35C22×C4⋊C4×C61C2211yescharacteristic
36C22×C42×C61C2211yescharacteristic
37C61×#389—24no—
38C2×#95×C61C2212yes—
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2311yescharacteristic
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—22no—
47C2×C4⋊Dic61—24no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—24no—
52C2×C4⋊Dic61—42no—
53C2×C4⋊Dic61C2×C414yes—
54C2×C4×Dic61—22no—
55C2×C4×Dic61—22no—
56C2×C4×Dic61—24no—
57C2×C4×Dic61—24no—
58C2×C4×Dic61—22no—
59C2×C4×Dic61—22no—
60C2×C4×Dic61—22no—
61C2×C4×Dic61C2×C414yes—
62C2×C4×Dic61C2×C418yes—
63#14519684—24no—
64#14519684—24no—
65#14519684—22no—
66#14519684—24no—
67#14519684—24no—
68#14519684—24no—
69#14519684—22no—
70#14519684—24no—
71#14519684—22no—
72#14519684—42no—
73#14519684C2×C418yes—
74#14519684D414yes—
75C23×C4×C61C2312yes—
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2312yes—
79C23×C4×C61C2311yescharacteristic
80C2×C4⋊C4×C61—24no—
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61—24no—
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61—42no—
86C2×C4⋊C4×C61C2×C414yes—
87C2×C4⋊C4×C61D414yes—
88C2×C4⋊C4×C61Q814yes—
89C2×C42×C61—22no—
90C2×C42×C61—22no—
91C2×C42×C61—22no—
92#95×C61—24no—
93C24×C61C2411yescharacteristic
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61C22×C414yes—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D414yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×Q814yes—
123C22×C4×C61C2×Q812yes—
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×C4×C61#4112yes—
127C22×C4×C61#4112yes—
128C22×C4×C61#4112yes—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61C22×C412yes—
165C22×Dic61C22×C414yes—
166C22×Dic61C22×C412yes—
167C22×Dic61C2×D412yes—
168C22×Dic61C2×D412yes—
169C22×Dic61#4112yes—
170C22×Dic61#4112yes—
171C4⋊C4×C61—48no—
172C4⋊C4×C61—44no—
173C4⋊C4×C61C4⋊C4116yes—
174C42×C61—44no—
175C4⋊Dic61—28no—
176C4⋊Dic61—48no—
177C4⋊Dic61—44no—
178C4×Dic61—28no—
179C4×Dic61—216no—
180C4×Dic61—44no—
181C23×C61C23×C411yescharacteristic
182C23×C61C22×D411yescharacteristic
183C23×C61C22×D411yescharacteristic
184C23×C61C22×Q811yescharacteristic
185C23×C61C2×#4111yescharacteristic
186C23×C61C2×#4111yescharacteristic
187C23×C61C2×#4111yescharacteristic
188C23×C61C2×#4111yescharacteristic
189C23×C61C2×#4111yescharacteristic
190C23×C61C2×#4111yescharacteristic
191C23×C61C2×#4111yescharacteristic
192C23×C61H(5,𝔽2)11yescharacteristic
193C23×C612−1+411yescharacteristic
194C23×C612−1+411yescharacteristic
195C23×C612−1+411yescharacteristic
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61C2×C4⋊C418yes—
224C2×C4×C61C2×C4⋊C414yes—
225C2×C4×C61#11714yes—
226C2×C4×C61#11714yes—
227C2×C4×C61#12414yes—
228C2×C4×C61#12414yes—
229C2×C4×C61#12514yes—
230C2×C4×C61#12514yes—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—22no—
241C2×Dic61—24no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61C4×D414yes—
273#13918—611nomaximal, Sylow
274C22×C61C22×C4⋊C411yescharacteristic
275C22×C61C2×#11711yescharacteristic
276C22×C61C2×#11711yescharacteristic
277C22×C61C2×C4×D411yescharacteristic
278C22×C61C4×#4111yescharacteristic
279C22×C61#51911yescharacteristic
280C22×C61#52011yescharacteristic
281C22×C61C2×#12411yescharacteristic
282C22×C61C2×#12411yescharacteristic
283C22×C61C2×#12511yescharacteristic
284C22×C61C2×#12511yescharacteristic
285C22×C61#52911yescharacteristic
286C22×C61#52911yescharacteristic
287C22×C61#53211yescharacteristic
288C22×C61#53211yescharacteristic
289C22×C61#54111yescharacteristic
290C22×C61#54111yescharacteristic
291C22×C61#54211yescharacteristic
292C22×C61#54211yescharacteristic
293C22×C61D4211yescharacteristic
294C22×C61#54711yescharacteristic
295C22×C61#54711yescharacteristic
296C22×C61#54711yescharacteristic
297C22×C61#54811yescharacteristic
298C22×C61#54811yescharacteristic
299C22×C61D4×Q811yescharacteristic
300C22×C61#55211yescharacteristic
301C22×C61#55211yescharacteristic
302C22×C61#55211yescharacteristic
303C22×C61#55211yescharacteristic
304C22×C61#55411yescharacteristic
305C22×C61#55511yescharacteristic
306C22×C61#55511yescharacteristic
307C22×C61#55711yescharacteristic
308C22×C61#55711yescharacteristic
309C4×C61—28no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—82no—
313C4×C61#38218yescomplemented
314Dic61—24no—
315Dic61—42no—
316Dic61—42no—
317Dic61—42no—
318Dic61—44no—
319Dic61—82no—
320C2×#389—611no—
321C2×#389D6111yescharacteristic
322C2×#382—612no—
323C2×C4×C4⋊C4—611no—
324#1442—612no—
325#1442—612no—
326C4⋊#95—611no—
327C4⋊#95—611no—
328C4⋊#95—611no—
329#1434—612no—
330C4×#95—611no—
331C2×C61C2×#38211yescharacteristic
332C2×C61#229311yescharacteristic
333C2×C61#232011yescharacteristic
334C2×C61#234411yescharacteristic
335C2×C61#234611yescharacteristic
336C2×C61D4×C4⋊C411yescharacteristic
337C2×C61#235111yescharacteristic
338C2×C61#248911yescharacteristic
339C2×C61#248911yescharacteristic
340C2×C61#249111yescharacteristic
341C2×C61#249111yescharacteristic
342C2×C61#250611yescharacteristic
343C2×C61#250611yescharacteristic
344C2×C61#250811yescharacteristic
345C2×C61#250811yescharacteristic
346C22×C4⋊C4—611no—
347C22×C4⋊C4—611no—
348C22×C4⋊C4—612no—
349C22×C4⋊C4C2×D6111yescharacteristic
350C22×C4⋊C4C2×D6112yes—
351C22×C4⋊C4C2×D6111yescharacteristic
352C22×C42—611no—
353C22×C42—612no—
354C22×C42—611no—
355C22×C42C2×D6111yescharacteristic
356#389—24no—
357#389—1224no—
358#382—1228no—
359C4×C4⋊C4—618no—
360C2×#95—611no—
361C2×#95—612no—
362C2×#95—611no—
363C2×#95—611no—
364C2×#95—611no—
365C2×#95—612no—
366C2×#95—612no—
367C2×#95—612no—
368C2×#95—612no—
369C2×#95—612no—
370C2×#95—612no—
371C2×#95—612no—
372C2×#95C2×D6112yes—
373C61#1391811yescharacteristic, complemented, Sylow
374C23×C4—611no—
375C23×C4—611no—
376C23×C4—612no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—612no—
380C23×C4C22×D6112yes—
381C23×C4C22×D6111yescharacteristic
382C23×C4C22×D6111yescharacteristic
383C23×C4C22×D6112yes—
384C23×C4C22×D6111yescharacteristic
385C2×C4⋊C4—24no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—42no—
391C2×C4⋊C4—614no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1224no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1224no—
398C2×C4⋊C4—1222no—
399C2×C4⋊C4—2442no—
400C2×C4⋊C4Dic12214yes—
401C2×C4⋊C4C4×D6114yes—
402C2×C4⋊C4C22⋊D6114yes—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—618no—
407C2×C42—614no—
408C2×C42—1224no—
409C2×C42—1224no—
410C2×C42—1222no—
411C2×C42—1222no—
412C2×C42—1222no—
413C2×C42—1222no—
414C2×C42—1222no—
415#95—24no—
416#95—618no—
417#95—614no—
418#95—1224no—
419#95—1224no—
420#95—1222no—
421#95—1222no—
422#95—1224no—
423#95—1224no—
424#95—1222no—
425#95—1224no—
426#95—1224no—
427#95—2442no—
428C24C23×D6111yescharacteristic
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—614no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1222no—
468C22×C4—1221no—
469C22×C4—1222no—
470C22×C4—1221no—
471C22×C4—1222no—
472C22×C4—1221no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1222no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4C2×Dic12214yes—
495C22×C4C2×Dic12212yes—
496C22×C4C2×C4×D6114yes—
497C22×C4C2×C4×D6112yes—
498C22×C4D4×D6112yes—
499C22×C4#1175822412yes—
500C22×C4#1175822412yes—
501C22×C4#1175822612yes—
502C22×C4#1175822612yes—
503C22×C4#1175822612yes—
504C22×C4C2×C22⋊D6114yes—
505C22×C4C2×C22⋊D6112yes—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—1228no—
509C4⋊C4—2448no—
510C4⋊C4—2444no—
511C4⋊C4#11758196116yes—
512C42—44no—
513C42—12216no—
514C42—1228no—
515C42—2444no—
516C23C22×C4×D6111yescharacteristic
517C23C2×D4×D6111yescharacteristic
518C23C22×C22⋊D6111yescharacteristic
519C23C22×Dic12211yescharacteristic
520C23#41×D6111yescharacteristic
521C23#41×D6111yescharacteristic
522C23C2×#1175822411yescharacteristic
523C23C2×#1175822411yescharacteristic
524C23C2×#1175822611yescharacteristic
525C23C2×#1175822611yescharacteristic
526C23C2×#1175822611yescharacteristic
527C23#1451982211yescharacteristic
528C23#1451982411yescharacteristic
529C23#1451982411yescharacteristic
530C23#1451982511yescharacteristic
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—614no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—1224no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4C2×#1175819618yes—
601C2×C4C2×#1175819614yes—
602C2×C4#1451973514yes—
603C2×C4#1451973514yes—
604C2×C4#1451976414yes—
605C2×C4#1451976614yes—
606C2×C4#1451976814yes—
607C2×C4#1451976814yes—
608C22C22×#1175819611yescharacteristic
609C22C2×#1451973511yescharacteristic
610C22C2×#1451973511yescharacteristic
611C22C4×D4×D6111yescharacteristic
612C22C4×#1175822611yescharacteristic
613C22#1684023811yescharacteristic
614C22#1684024111yescharacteristic
615C22C2×#1451976411yescharacteristic
616C22C2×#1451976611yescharacteristic
617C22C2×#1451976811yescharacteristic
618C22C2×#1451976811yescharacteristic
619C22#1684029711yescharacteristic
620C22#1684029811yescharacteristic
621C22#1684030711yescharacteristic
622C22#1684031311yescharacteristic
623C22#1684037211yescharacteristic
624C22#1684037511yescharacteristic
625C22#1684037911yescharacteristic
626C22#1684038611yescharacteristic
627C22D4×C22⋊D6111yescharacteristic
628C22#1684041111yescharacteristic
629C22#1684041211yescharacteristic
630C22#1684041511yescharacteristic
631C22#1684041911yescharacteristic
632C22#1684042311yescharacteristic
633C22D4×Dic12211yescharacteristic
634C22#1684044611yescharacteristic
635C22#1684045011yescharacteristic
636C22#1684045011yescharacteristic
637C22#1684045411yescharacteristic
638C22#1684047111yescharacteristic
639C22#1684047611yescharacteristic
640C22#1684047811yescharacteristic
641C22#1684048811yescharacteristic
642C22#1684049111yescharacteristic
643C4—28no—
644C4—42no—
645C4—42no—
646C4—82no—
647C4—1224no—
648C4—4882no—
649C4—2444no—
650C4—2442no—
651C4—2442no—
652C4—2442no—
653C4#1683962718yescomplemented
654C2C2×#1683962711yescharacteristic
655C2#2144827711yescharacteristic
656C2#2144846611yescharacteristic
657C2#2144868711yescharacteristic
658C2#2144870311yescharacteristic
659C2D4×#1175819611yescharacteristic
660C2#2144873711yescharacteristic
661C2#2145016211yescharacteristic
662C2#2145016311yescharacteristic
663C2#2145018411yescharacteristic
664C2#2145018511yescharacteristic
665C2#2145033711yescharacteristic
666C2#2145033811yescharacteristic
667C2#2145035911yescharacteristic
668C2#2145036011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#389C2
socle, prime radical, cosocleC24×C61C24
hypercenterC2×#389D61
hypocenterC61#13918
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
20. C4⋊C4×Dic61C4semidirect
313. C4×C61#382semidirect
373. C61#13918semidirect
653. C4#16839627semidirect

Maximal subgroups

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

subgroupindex
C2×#168396612
C2×#168396272
C2×C4⋊C4×Dic612
#214440782
#214440752
#214440492
#214440482
#214440452
C4×#145196842
C2×C61×#3892
#1391861