Finite Group Databasebeta

← #30615102

Subgroups — #30615102

Order 15616.

1774 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#407C211yesmaximal, characteristic
2#21462263C211yesmaximal, characteristic
3#21461723C214yesmaximal, complemented
4#21461722C214yesmaximal, complemented
5#21461506C214yesmaximal, complemented
6C22×C4⋊C4×D61C211yesmaximal, characteristic
7C2×D61⋊M16C2212yes—
8C2×#14519856C2212yes—
9#16840734—21no—
10#16840733—21no—
11#16840733—21no—
12#16840732—21no—
13D61⋊(C22⋊C8)C2212yes—
14D61⋊(C22⋊C8)C2212yes—
15#16840705—21no—
16#16840704C2212yes—
17#16840704C2212yes—
18#16840703—21no—
19C4×C22⋊#6923—21no—
20C4×C22⋊#6923—21no—
21#16840635C22116yescomplemented
22C23×C4×D61—21no—
23C23×C4×D61C2211yescharacteristic
24C22×C4⋊Dic61C411yescharacteristic, complemented
25C22×#11758196—21no—
26C2×C4⋊C4×D61—21no—
27C2×C4⋊C4×D61—21no—
28C2×C4⋊C4×D61—21no—
29C2×C4⋊C4×D61—21no—
30C2×C4⋊C4×D61—21no—
31C2×C4⋊C4×D61—21no—
32C2×C4⋊C4×D61C412yescomplemented
33C2×C4⋊C4×D61C412yescomplemented
34C2×C4⋊C4×D61C412yescomplemented
35C2×C4⋊C4×D61C2212yes—
36C2×C4⋊C4×D61C2212yes—
37C2×C4⋊C4×D61C2212yes—
38C22×C4⋊C4×C61C411yescharacteristic, complemented
39C2×#11758213—22no—
40D61⋊M16—28no—
41C2×#11758217—22no—
42C2×D61⋊C8C2314yes—
43#14519856—28no—
44C2×C4⋊#6923C2314yes—
45C2×C22⋊#6923—22no—
46C2×C22⋊#6923—22no—
47C2×C22⋊#6923—22no—
48C2×C4×#6923—22no—
49#14519834—22no—
50#14519834—22no—
51C22⋊#92420—22no—
52C22⋊#92420—22no—
53(C2×#6923)⋊C4—22no—
54(C2×#6923)⋊C4—22no—
55C24×D61D411yescharacteristic
56C23×Dic61—21no—
57C23×Dic61D411yescharacteristic
58C22×C4×D61—28no—
59C22×C4×D61—22no—
60C22×C4×D61—22no—
61C22×C4×D61—22no—
62C22×C4×D61—22no—
63C22×C4×D61—22no—
64C22×C4×D61—22no—
65C22×C4×D61—48no—
66C22×C4×D61C2×C411yescharacteristic
67C22×C4×D61C2×C411yescharacteristic
68C22×C4×D61C2×C411yescharacteristic
69C22×C4×D61D411yescharacteristic
70C22×C4×D61D411yescharacteristic
71C22×C4×D61D411yescharacteristic
72C22×C4×D61D411yescharacteristic
73C22×C4×D61D411yescharacteristic
74C22×C4×D61D411yescharacteristic
75C22×C4×D61C2311yescharacteristic
76C22×C4×D61C2311yescharacteristic
77C22×C4×D61C2311yescharacteristic
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2×C412yes—
82C2×C4⋊Dic61C2×C412yes—
83C2×C4⋊Dic61C2×C412yes—
84C2×#11758196—22no—
85C2×#11758196—22no—
86C2×#11758196—22no—
87C2×#11758196—22no—
88C2×#11758196—22no—
89C2×#11758196—22no—
90C4⋊C4×D61—22no—
91C4⋊C4×D61—22no—
92C4⋊C4×D61—22no—
93C4⋊C4×D61—22no—
94C4⋊C4×D61—22no—
95C4⋊C4×D61—22no—
96C4⋊C4×D61—22no—
97C4⋊C4×D61—22no—
98C4⋊C4×D61—22no—
99C4⋊C4×D61—22no—
100C4⋊C4×D61—22no—
101C4⋊C4×D61—22no—
102C4⋊C4×D61C2×C414yes—
103C4⋊C4×D61C2×C414yes—
104C4⋊C4×D61C2×C414yes—
105C4⋊C4×D61C2314yes—
106C23×C4×C61—21no—
107C23×C4×C61C2×C411yescharacteristic
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2×C412yes—
112C2×C4⋊C4×C61C2×C412yes—
113C2×C4⋊C4×C61C2×C412yes—
114C24×C61C22⋊C411yescharacteristic
115C23×D61—24no—
116C23×D61—24no—
117C23×D61—24no—
118C23×D61C22⋊C411yescharacteristic
119C23×D61C22⋊C411yescharacteristic
120C23×D61C22⋊C411yescharacteristic
121C23×D61C2×D411yescharacteristic
122C23×D61C2×D411yescharacteristic
123C23×D61C2×D411yescharacteristic
124C22×#6923—24no—
125C22×C4×C61—24no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—44no—
130C22×C4×C61C22⋊C411yescharacteristic
131C22×C4×C61C22⋊C411yescharacteristic
132C22×C4×C61C22⋊C411yescharacteristic
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×Dic61—24no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—44no—
141C22×Dic61C22⋊C411yescharacteristic
142C22×Dic61C22⋊C411yescharacteristic
143C22×Dic61C22⋊C411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×C4×D61—24no—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—24no—
151C2×C4×D61—24no—
152C2×C4×D61—24no—
153C2×C4×D61—24no—
154C2×C4×D61—24no—
155C2×C4×D61—24no—
156C2×C4×D61—24no—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—48no—
163C2×C4×D61—48no—
164C2×C4×D61—48no—
165C2×C4×D61—48no—
166C2×C4×D61—48no—
167C2×C4×D61—48no—
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22⋊C411yescharacteristic
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22⋊C411yescharacteristic
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22×C411yescharacteristic
175C2×C4×D61C22×C411yescharacteristic
176C2×C4×D61C22×C411yescharacteristic
177C2×C4×D61C2×D411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179C2×C4×D61C2×D411yescharacteristic
180C2×C4×D61C2×D411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61#4118yes—
184C2×C4×D61C2411yescharacteristic
185C22⋊#6923—44no—
186C22⋊#6923—44no—
187C22⋊#6923—44no—
188#11758217—44no—
189C2×#92420—24no—
190C4⋊#6923—28no—
191C4×#6923—44no—
192#11758213—44no—
193D61⋊C8—28no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61C22×C414yes—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61C22×C414yes—
202#11758196—24no—
203#11758196—24no—
204#11758196—24no—
205#11758196—24no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61C2×C22⋊C411yescharacteristic
210C23×C61C2×C22⋊C411yescharacteristic
211C23×C61C2×C22⋊C411yescharacteristic
212C22×D61—232no—
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—22no—
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61C2×C22⋊C411yescharacteristic
229C22×D61C2×C22⋊C411yescharacteristic
230C22×D61C2×C22⋊C411yescharacteristic
231C22×D61#12314yes—
232C22×D61AGL(1,ℤ/8)14yes—
233C22×D61#13714yes—
234C22×D61C22×D411yescharacteristic
235C2×#6923—44no—
236C2×#6923—44no—
237C2×#6923—44no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—24no—
244C2×C4×C61—44no—
245C2×C4×C61—44no—
246C2×C4×C61—44no—
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61#11714yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—24no—
258C2×Dic61—44no—
259C2×Dic61—44no—
260C2×Dic61—44no—
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61#12314yes—
265C2×Dic61C22×D411yescharacteristic
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—48no—
279C4×D61—48no—
280C4×D61—48no—
281C4×D61—48no—
282C4×D61#11714yes—
283C4×D61#12314yes—
284C4×D61#12314yes—
285C4×D61C2×#4114yes—
286#92420—44no—
287#20700—611nomaximal, Sylow
288C22×C61—28no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61#38112yes—
296C22×C61#41812yes—
297C22×C61#41912yes—
298C22×C61C22×C22⋊C411yescharacteristic
299C2×D61—28no—
300C2×D61—28no—
301C2×D61—28no—
302C2×D61—21no—
303C2×D61—21no—
304C2×D61—21no—
305C2×D61—21no—
306C2×D61—21no—
307C2×D61—21no—
308C2×D61—28no—
309C2×D61—28no—
310C2×D61—28no—
311C2×D61#38112yes—
312C2×D61#41812yes—
313C2×D61#41912yes—
314C2×D61C2×#12312yes—
315C2×D61C2×AGL(1,ℤ/8)12yes—
316C2×D61C2×#13712yes—
317#6923—84no—
318C4×C61—21no—
319C4×C61—21no—
320C4×C61—44no—
321C4×C61#38112yes—
322C4×C61C2×#11712yes—
323Dic61—21no—
324Dic61—21no—
325Dic61—44no—
326Dic61#38112yes—
327Dic61C2×#12312yes—
328C23×C4⋊C4—611no—
329C2×#407—611no—
330#2280—611no—
331#1824—614no—
332#1823—614no—
333#1736—614no—
334C2×C61—22no—
335C2×C61—22no—
336C2×C61—22no—
337C2×C61C2×#38111yescharacteristic
338C2×C61C2×#41811yescharacteristic
339C2×C61C2×#41911yescharacteristic
340D61—22no—
341D61—22no—
342D61—22no—
343D61#309214yes—
344C24×C4—611no—
345C24×C4—1221no—
346C22×M16—612no—
347C2×#117—612no—
348C22×C4⋊C4—611no—
349C22×C4⋊C4—612no—
350C22×C4⋊C4—612no—
351C22×C4⋊C4—612no—
352C22×C4⋊C4—612no—
353C22×C4⋊C4—612no—
354C22×C4⋊C4—612no—
355C22×C4⋊C4—1221no—
356C22×C4⋊C4—1221no—
357C22×C4⋊C4—1221no—
358C22×C4⋊C4—1221no—
359C22×C4⋊C4—1221no—
360C22×C4⋊C4—1221no—
361C22×C4⋊C4—1221no—
362C22×C4⋊C4#692311yescharacteristic, complemented
363#407—1221no—
364#407—1221no—
365#407—1221no—
366#407—1221no—
367C2×C22⋊C8—612no—
368C2×C22⋊C8—612no—
369#381—1221no—
370#380—612no—
371#380—612no—
372#379—1221no—
373C4×C22⋊C4—1221no—
374C4×C22⋊C4—1221no—
375#340—6116no—
376C61#2070011yescharacteristic, complemented, Sylow
377C25—611no—
378C23×C4—21no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—611no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—1228no—
393C23×C4—1222no—
394C23×C4—1222no—
395C23×C4—1222no—
396C23×C4—1222no—
397C23×C4—1222no—
398C23×C4—1222no—
399C23×C4—1221no—
400C23×C4—2448no—
401C23×C4C2×#692311yescharacteristic
402C2×M16—1222no—
403C2×M16—1228no—
404C2×M16—1222no—
405C22×C8—614no—
406#117—1228no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—614no—
414C2×C4⋊C4—614no—
415C2×C4⋊C4—614no—
416C2×C4⋊C4—614no—
417C2×C4⋊C4—614no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1222no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4—1222no—
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1222no—
424C2×C4⋊C4—1222no—
425C2×C4⋊C4—1222no—
426C2×C4⋊C4—1222no—
427C2×C4⋊C4—1222no—
428C2×C4⋊C4—1222no—
429C2×C4⋊C4—1222no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1222no—
433C2×C4⋊C4—1222no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4C2×#692312yes—
440C2×C4⋊C4C2×#692312yes—
441C2×C4⋊C4C2×#692312yes—
442C2×C22⋊C4—1222no—
443C2×C22⋊C4—1222no—
444C2×C22⋊C4—1222no—
445C2×C42—1222no—
446C22⋊C8—1222no—
447C22⋊C8—1222no—
448C22⋊C8—1222no—
449C22⋊C8—1222no—
450#95—1222no—
451#95—1222no—
452C24—611no—
453C24—611no—
454C24—611no—
455C24—611no—
456C24—611no—
457C24—611no—
458C24—1224no—
459C24—1224no—
460C24—1224no—
461C24C22⋊#692311yescharacteristic
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—44no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—618no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—1224no—
491C22×C4—1224no—
492C22×C4—1224no—
493C22×C4—1224no—
494C22×C4—1224no—
495C22×C4—1224no—
496C22×C4—1224no—
497C22×C4—1224no—
498C22×C4—1224no—
499C22×C4—1224no—
500C22×C4—1224no—
501C22×C4—1224no—
502C22×C4—1224no—
503C22×C4—1224no—
504C22×C4—1224no—
505C22×C4—1224no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1224no—
510C22×C4—2448no—
511C22×C4—2448no—
512C22×C4—2448no—
513C22×C4—2448no—
514C22×C4—2448no—
515C22×C4—2448no—
516C22×C4—2444no—
517C22×C4C22⋊#692311yescharacteristic
518C22×C4C22⋊#692311yescharacteristic
519C22×C4C22⋊#692311yescharacteristic
520C22×C4C22×#692311yescharacteristic
521C22×C4C22×#692311yescharacteristic
522C22×C4C22×#692311yescharacteristic
523M16—2444no—
524M16—2444no—
525C2×C8—1224no—
526C2×C8—1228no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—614no—
531C4⋊C4—1224no—
532C4⋊C4—1224no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1228no—
536C4⋊C4—1222no—
537C4⋊C4—1222no—
538C4⋊C4—1222no—
539C4⋊C4C22×#692314yes—
540C22⋊C4—2444no—
541C22⋊C4—2444no—
542C22⋊C4—2444no—
543C42—2444no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—611no—
548C23—614no—
549C23—614no—
550C23—614no—
551C23—611no—
552C23—611no—
553C23—611no—
554C23—1222no—
555C23—1222no—
556C23—1222no—
557C23—1222no—
558C23—1222no—
559C23—1222no—
560C23—1222no—
561C23—1222no—
562C23—1222no—
563C23—1222no—
564C23—1222no—
565C23—1222no—
566C23—12232no—
567C23—1222no—
568C23—1222no—
569C23—1222no—
570C23C2×C22⋊#692311yescharacteristic
571C23C2×C22⋊#692311yescharacteristic
572C23C2×C22⋊#692311yescharacteristic
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—614no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—614no—
586C2×C4—614no—
587C2×C4—614no—
588C2×C4—614no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1224no—
609C2×C4—2448no—
610C2×C4—2448no—
611C2×C4—2448no—
612C2×C4—2448no—
613C2×C4—2444no—
614C2×C4—2444no—
615C2×C4—2444no—
616C2×C4—2444no—
617C2×C4—2444no—
618C2×C4—2444no—
619C2×C4C2×C22⋊#692311yescharacteristic
620C2×C4C2×C22⋊#692311yescharacteristic
621C2×C4C2×C22⋊#692311yescharacteristic
622C2×C4#1451985614yes—
623C2×C4C23×#692311yescharacteristic
624C8—2444no—
625C22—28no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—612no—
633C22—612no—
634C22—612no—
635C22—612no—
636C22—612no—
637C22—612no—
638C22—1228no—
639C22—1228no—
640C22—1228no—
641C22—1228no—
642C22—1228no—
643C22—1228no—
644C22—1221no—
645C22—1221no—
646C22—1221no—
647C22—1221no—
648C22—1221no—
649C22—1221no—
650C22#1684070512yes—
651C22#1684075412yes—
652C22#1684075512yes—
653C22C22×C22⋊#692311yescharacteristic
654C4—21no—
655C4—21no—
656C4—44no—
657C4—612no—
658C4—612no—
659C4—1221no—
660C4—1221no—
661C4—4884no—
662C4—2444no—
663C4#1684070512yes—
664C4C2×#1451985612yes—
665C2—22no—
666C2—22no—
667C2—22no—
668C2—614no—
669C2—1222no—
670C2—1222no—
671C2—1222no—
672C2C2×#1684070511yescharacteristic
673C2C2×#1684075411yescharacteristic
674C2C2×#1684075511yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×D61D4
cosocleC2×C4×D61C24
hypercenterC22×C4⋊C4#6923
hypocenterC61#20700
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21461723C2semidirect
4. #21461722C2semidirect
5. #21461506C2semidirect
21. #16840635C22semidirect
24. C22×C4⋊Dic61C4semidirect
32. C2×C4⋊C4×D61C4semidirect
33. C2×C4⋊C4×D61C4semidirect
34. C2×C4⋊C4×D61C4semidirect
38. C22×C4⋊C4×C61C4semidirect
362. C22×C4⋊C4#6923semidirect
376. C61#20700semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4072
#214622632
#214617232
#214617222
#214615062
C22×C4⋊C4×D612
#2070061