Finite Group Databasebeta

← #30615217

Subgroups — #30615217

Order 15616.

1350 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#432C211yesmaximal, characteristic
2#21462268C211yesmaximal, characteristic
3#21461722C212yesmaximal
4#21461688C214yesmaximal, complemented
5#21461658C212yesmaximal
6#21461591C214yesmaximal, complemented
7C2×C4×Q8×D61C211yesmaximal, characteristic
8C2×#14519856—21no—
9#16840785—21no—
10#16840784—21no—
11#16840783—21no—
12#16840782—21no—
13D61⋊(C4⋊C8)C2212yes—
14C2×#14519840C2218yes—
15D61⋊(C22⋊C8)C2212yes—
16#16840721C2212yes—
17#16840708C2212yes—
18#16840706—21no—
19#16840704—21no—
20#16840704C2212yes—
21C4×C4⋊#6923—21no—
22C4×C4⋊#6923C2212yes—
23#16840635C2218yes—
24C22×Q8×D61C2211yescharacteristic
25C2×Q8×Dic61C411yescharacteristic, complemented
26C2×C4×Dic122—21no—
27C2×C4×Dic122C411yescharacteristic, complemented
28C2×#14519742—21no—
29C2×#14519742C411yescharacteristic, complemented
30C4×Q8×D61—21no—
31C4×Q8×D61—21no—
32C4×Q8×D61—21no—
33C4×Q8×D61—21no—
34C4×Q8×D61C412yescomplemented
35C4×Q8×D61C412yescomplemented
36C4×Q8×D61C2212yes—
37C4×Q8×D61C2212yes—
38C2×C4⋊C4×D61—21no—
39C2×C4⋊C4×D61C2211yescharacteristic
40C2×C42×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C2×C4×Q8×C61C411yescharacteristic, complemented
43C2×D61⋊C8C2314yes—
44#14519856—44no—
45C2×C4⋊#6923—22no—
46C2×C4⋊#6923—22no—
47C2×C4⋊#6923C2314yes—
48C2×C22⋊#6923—22no—
49C2×C4×#6923—22no—
50#14519844—22no—
51C4⋊#92420—22no—
52#14519840—216no—
53#14519834—22no—
54C22⋊#92420—22no—
55#14519832—22no—
56#14519832—22no—
57(C2×#6923)⋊C4—22no—
58(C2×#6923)⋊C4—22no—
59C22×Dic122—21no—
60C22×Dic122C2×C411yescharacteristic
61C2×Q8×D61—28no—
62C2×Q8×D61—21no—
63C2×Q8×D61—21no—
64C2×Q8×D61C2×C412yes—
65C2×Q8×D61C2312yes—
66C22×C4×D61—21no—
67C22×C4×D61—21no—
68C22×C4×D61D411yescharacteristic
69C22×C4×D61D411yescharacteristic
70C22×C4×D61C2311yescharacteristic
71Q8×Dic61—21no—
72Q8×Dic61—21no—
73Q8×Dic61C2×C412yes—
74Q8×Dic61C2×C412yes—
75C4×Dic122—21no—
76C4×Dic122—21no—
77C4×Dic122—22no—
78C4×Dic122—22no—
79C4×Dic122—22no—
80C4×Dic122—22no—
81C4×Dic122C2×C412yes—
82C4×Dic122C2×C412yes—
83#14519742—21no—
84#14519742—21no—
85#14519742—22no—
86#14519742—22no—
87#14519742—22no—
88#14519742—22no—
89#14519742C2×C412yes—
90#14519742C2×C412yes—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2×C411yescharacteristic
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C4⋊C4×D61—22no—
97C4⋊C4×D61—22no—
98C4⋊C4×D61—22no—
99C4⋊C4×D61—22no—
100C4⋊C4×D61—21no—
101C4⋊C4×D61—21no—
102C4⋊C4×D61C2×C412yes—
103C4⋊C4×D61C2312yes—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C42×D61—22no—
111C42×D61—22no—
112C42×D61—22no—
113C42×D61—22no—
114C42×D61C2×C411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61C2311yescharacteristic
121C42×D61C2311yescharacteristic
122C22×Q8×C61C2×C411yescharacteristic
123C4×Q8×C61—21no—
124C4×Q8×C61—21no—
125C4×Q8×C61C2×C412yes—
126C4×Q8×C61C2×C412yes—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2×C411yescharacteristic
129C2×C42×C61—21no—
130C2×C42×C61C2×C411yescharacteristic
131C23×D61C2×D411yescharacteristic
132C22×#6923—24no—
133C2×Q8×C61—24no—
134C2×Q8×C61—21no—
135C2×Q8×C61C22×C412yes—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61C22⋊C411yescharacteristic
144C22×Dic61C22⋊C411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146Q8×D61—28no—
147Q8×D61—28no—
148Q8×D61—24no—
149Q8×D61—24no—
150Q8×D61—24no—
151Q8×D61#4118yes—
152C2×C4×D61—28no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61C22⋊C411yescharacteristic
164C2×C4×D61C22⋊C411yescharacteristic
165C2×C4×D61C22⋊C411yescharacteristic
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C22×C411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2411yescharacteristic
175C2×Dic122—28no—
176C2×Dic122—21no—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C2×Dic122—24no—
180C2×Dic122C22×C412yes—
181C22⋊#6923—44no—
182C2×#92420—24no—
183C4⋊#6923—28no—
184C4⋊#6923—44no—
185C4⋊#6923—44no—
186C4×#6923—44no—
187D61⋊C8—28no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61C22×C412yes—
192C42×C61—22no—
193C42×C61—22no—
194C42×C61C22⋊C411yescharacteristic
195C42×C61C22⋊C411yescharacteristic
196C42×C61C22×C411yescharacteristic
197C42×C61C22×C411yescharacteristic
198C4⋊Dic61—21no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61C22×C412yes—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C22×C411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C23×C61C2×C22⋊C411yescharacteristic
227C22×D61—22no—
228C22×D61—22no—
229C22×D61C2×C22⋊C411yescharacteristic
230C22×D61AGL(1,ℤ/8)12yes—
231C22×D61#13712yes—
232C22×D61C22×D411yescharacteristic
233C2×#6923—44no—
234C2×#6923—44no—
235C2×#6923—44no—
236Q8×C61—24no—
237Q8×C61—22no—
238Q8×C61#11714yescomplemented
239C2×C4×C61—24no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—42no—
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C23×C411yescharacteristic
252C2×Dic61—24no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—42no—
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C22×D411yescharacteristic
265C4×D61—28no—
266C4×D61—28no—
267C4×D61—22no—
268C4×D61—21no—
269C4×D61—21no—
270C4×D61—22no—
271C4×D61—21no—
272C4×D61—21no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—42no—
280C4×D61—42no—
281C4×D61—44no—
282C4×D61#13714yes—
283C4×D61C2×#4114yes—
284Dic122—28no—
285Dic122—24no—
286Dic122—22no—
287Dic122—28no—
288Dic122#11714yescomplemented
289#92420—44no—
290#20740—611nomaximal, Sylow
291C22×C61—21no—
292C22×C61—21no—
293C22×C61#41712yes—
294C22×C61C22×C22⋊C411yescharacteristic
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—24no—
298C2×D61—24no—
299C2×D61#41712yes—
300C2×D61#52512yes—
301C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
302C2×D61C2×#13712yes—
303C2×D61C2×#13711yescharacteristic
304#6923—84no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—24no—
309C4×C61—21no—
310C4×C61—42no—
311C4×C61#41912yes—
312C4×C61C2×#11712yes—
313Dic61—21no—
314Dic61—21no—
315Dic61—21no—
316Dic61—21no—
317Dic61—24no—
318Dic61—42no—
319Dic61#41912yes—
320Dic61#52512yes—
321C22×C4×Q8—611no—
322C2×#432—611no—
323#2285—611no—
324#1823—612no—
325#1803—614no—
326#1787—612no—
327#1763—614no—
328C2×C61—22no—
329C2×C61#228011yescharacteristic
330C2×C61C2×#41911yescharacteristic
331C2×C61C2×#41711yescharacteristic
332D61—22no—
333D61#311212yes—
334D61#311312yes—
335C23×Q8—611no—
336C2×C4×Q8—611no—
337C2×C4×Q8—612no—
338C2×C4×Q8—612no—
339C2×C4×Q8—612no—
340C2×C4×Q8—612no—
341C2×C4×Q8—611no—
342C2×C4×Q8—611no—
343C2×C4×Q8—1221no—
344C2×C4×Q8—1221no—
345C2×C4×Q8—1221no—
346C2×C4×Q8—1221no—
347C2×C4×Q8—1221no—
348C2×C4×Q8—1221no—
349C2×C4×Q8#692311yescharacteristic, complemented
350C2×#117—1221no—
351C22×C4⋊C4—611no—
352C22×C4⋊C4—1221no—
353C22×C42—611no—
354C22×C42—1221no—
355#432—1221no—
356#432—1221no—
357#432—1221no—
358#432—1221no—
359C2×C4⋊C8—612no—
360C2×#103—618no—
361C2×C22⋊C8—612no—
362C2×#97—612no—
363#384—612no—
364#382—1221no—
365#380—612no—
366#380—1221no—
367C4×C4⋊C4—612no—
368C4×C4⋊C4—1221no—
369#340—618no—
370C61#2074011yescharacteristic, complemented, Sylow
371C22×Q8—612no—
372C22×Q8—612no—
373C22×Q8—611no—
374C22×Q8—1221no—
375C22×Q8—1221no—
376C22×Q8—1221no—
377C22×Q8—1228no—
378C22×Q8C2×#692311yescharacteristic
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—1221no—
383C23×C4—1221no—
384C22×C8—614no—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×Q8—612no—
388C4×Q8—612no—
389C4×Q8—612no—
390C4×Q8—612no—
391C4×Q8—612no—
392C4×Q8—612no—
393C4×Q8—1222no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×Q8—1221no—
407C4×Q8C2×#692312yes—
408C4×Q8C2×#692312yes—
409#117—2444no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—614no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1222no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1222no—
426C2×C4⋊C4—1222no—
427C2×C4⋊C4C2×#692311yescharacteristic
428C2×C22⋊C4—1222no—
429C2×C42—21no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—1222no—
442C2×C42—1222no—
443C2×C42—1222no—
444C2×C42—1222no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42—1222no—
449C2×C42C2×#692311yescharacteristic
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452#103—12216no—
453C22⋊C8—1222no—
454C22⋊C8—1222no—
455#97—1222no—
456#97—1222no—
457#95—1222no—
458#95—1222no—
459C24—611no—
460C2×Q8—24no—
461C2×Q8—21no—
462C2×Q8—618no—
463C2×Q8—612no—
464C2×Q8—1222no—
465C2×Q8—1222no—
466C2×Q8—1224no—
467C2×Q8—1228no—
468C2×Q8—1224no—
469C2×Q8—1224no—
470C2×Q8—1228no—
471C2×Q8—1228no—
472C2×Q8—1224no—
473C2×Q8—1221no—
474C2×Q8C22×#692312yes—
475C22×C4—21no—
476C22×C4—21no—
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—611no—
491C22×C4—611no—
492C22×C4—1224no—
493C22×C4—1224no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1228no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1224no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4C22⋊#692311yescharacteristic
507C22×C4C22⋊#692311yescharacteristic
508C22×C4C22×#692311yescharacteristic
509C2×C8—1224no—
510C2×C8—1228no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—612no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1228no—
522C4⋊C4—1221no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—2444no—
526C4⋊C4—2444no—
527C4⋊C4C22×#692312yes—
528C22⋊C4—2444no—
529C42—22no—
530C42—22no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—1222no—
544C42—1222no—
545C42—1222no—
546C42—1222no—
547C42—1222no—
548C42—1222no—
549C42—2444no—
550C42C22⋊#692311yescharacteristic
551C42C22⋊#692311yescharacteristic
552C42C22×#692311yescharacteristic
553C42C22×#692311yescharacteristic
554C23—611no—
555C23—612no—
556C23—612no—
557C23—611no—
558C23—1222no—
559C23—1222no—
560C23C2×C22⋊#692311yescharacteristic
561Q8—24no—
562Q8—22no—
563Q8—614no—
564Q8—1228no—
565Q8—1228no—
566Q8—1224no—
567Q8—1222no—
568Q8#1451985614yescomplemented
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—42no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—614no—
579C2×C4—614no—
580C2×C4—611no—
581C2×C4—611no—
582C2×C4—611no—
583C2×C4—611no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1222no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1228no—
595C2×C4—1228no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1224no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2444no—
607C2×C4—2444no—
608C2×C4—2444no—
609C2×C4—2444no—
610C2×C4—2442no—
611C2×C4C2×C22⋊#692311yescharacteristic
612C2×C4C2×C22⋊#692311yescharacteristic
613C2×C4C2×C22⋊#692311yescharacteristic
614C2×C4C2×C22⋊#692311yescharacteristic
615C2×C4C2×C22⋊#692311yescharacteristic
616C2×C4C23×#692311yescharacteristic
617C8—2444no—
618C22—21no—
619C22—21no—
620C22—612no—
621C22—611no—
622C22—611no—
623C22—612no—
624C22—612no—
625C22—1224no—
626C22—1224no—
627C22—1221no—
628C22—1221no—
629C22#1684075312yes—
630C22C22×C22⋊#692311yescharacteristic
631C4—21no—
632C4—21no—
633C4—21no—
634C4—24no—
635C4—21no—
636C4—42no—
637C4—612no—
638C4—612no—
639C4—1221no—
640C4—1221no—
641C4—1221no—
642C4—1221no—
643C4—1224no—
644C4—4884no—
645C4—2442no—
646C4#1684075512yes—
647C4C2×#1451985612yes—
648C2—22no—
649C2—612no—
650C2—612no—
651C2—1222no—
652C2#2146226311yescharacteristic
653C2C2×#1684075511yescharacteristic
654C2C2×#1684075311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#20740
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
4. #21461688C2semidirect
6. #21461591C2semidirect
25. C2×Q8×Dic61C4semidirect
27. C2×C4×Dic122C4semidirect
29. C2×#14519742C4semidirect
34. C4×Q8×D61C4semidirect
35. C4×Q8×D61C4semidirect
42. C2×C4×Q8×C61C4semidirect
238. Q8×C61#117semidirect
288. Dic122#117semidirect
349. C2×C4×Q8#6923semidirect
370. C61#20740semidirect
568. Q8#14519856semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4322
#214622682
#214617222
#214616882
#214616582
#214615912
C2×C4×Q8×D612
#2074061