Finite Group Databasebeta

← #30615211

Subgroups — #30615211

Order 15616.

1370 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#431C211yesmaximal, characteristic
2#21462268C211yesmaximal, characteristic
3#21461723C212yesmaximal
4#21461688C214yesmaximal, complemented
5#21461658C212yesmaximal
6C4×#14519840C214yesmaximal, complemented
7C2×C4×Q8×D61C211yesmaximal, characteristic
8C2×#14519856—21no—
9#16840781—21no—
10#16840780—21no—
11#16840779—21no—
12#16840778—21no—
13D61⋊(C4⋊C8)C2212yes—
14C2×#14519840C2218yes—
15D61⋊(C22⋊C8)C2212yes—
16C4×D61⋊C8C2212yes—
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—
55C4×#92420—22no—
56C4×#92420—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—24no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—42no—
159C2×C4×D61—44no—
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22⋊C411yescharacteristic
163C2×C4×D61C22⋊C411yescharacteristic
164C2×C4×D61SD1614yes—
165C2×C4×D61Q1614yes—
166C2×C4×D61C22×C411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×C4×D61C2411yescharacteristic
174C2×Dic122—28no—
175C2×Dic122—21no—
176C2×Dic122—22no—
177C2×Dic122—22no—
178C2×Dic122—24no—
179C2×Dic122C22×C412yes—
180C22⋊#6923—44no—
181C2×#92420—24no—
182C4⋊#6923—28no—
183C4⋊#6923—44no—
184C4⋊#6923—44no—
185C4×#6923—44no—
186D61⋊C8—28no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61C22×C412yes—
191C42×C61—22no—
192C42×C61—22no—
193C42×C61C22⋊C411yescharacteristic
194C42×C61C22⋊C411yescharacteristic
195C42×C61C22×C411yescharacteristic
196C42×C61C22×C411yescharacteristic
197C4⋊Dic61—21no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61C22×C412yes—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22×C411yescharacteristic
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C23×C61C2×C22⋊C411yescharacteristic
226C22×D61—22no—
227C22×D61—22no—
228C22×D61C2×C22⋊C411yescharacteristic
229C22×D61C2×SD1612yes—
230C22×D61C2×Q1612yes—
231C22×D61C22×D411yescharacteristic
232C2×#6923—44no—
233C2×#6923—44no—
234C2×#6923—44no—
235Q8×C61—24no—
236Q8×C61—22no—
237Q8×C61#11714yescomplemented
238C2×C4×C61—24no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61#10314yescomplemented
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C23×C411yescharacteristic
251C2×Dic61—24no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—41no—
256C2×Dic61—42no—
257C2×Dic61#10314yescomplemented
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C22×D411yescharacteristic
264C4×D61—28no—
265C4×D61—28no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—44no—
274C4×D61—44no—
275C4×D61#10314yescomplemented
276C4×D61#10314yescomplemented
277C4×D61C2×SD1612yes—
278C4×D61C2×SD1612yes—
279C4×D61C2×Q1612yes—
280C4×D61C2×Q1612yes—
281C4×D61C2×#4114yes—
282Dic122—28no—
283Dic122—24no—
284Dic122—22no—
285Dic122—28no—
286Dic122#11714yescomplemented
287#92420—44no—
288#20734—611nomaximal, Sylow
289C22×C61—21no—
290C22×C61—21no—
291C22×C61C2×#10312yes—
292C22×C61C22×C22⋊C411yescharacteristic
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—24no—
296C2×D61—24no—
297C2×D61C2×#10312yes—
298C2×D61#52512yes—
299C2×D61C22×SD1611yescharacteristic
300C2×D61C22×Q1611yescharacteristic
301C2×D61#57512yes—
302#6923—84no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—24no—
306C4×C61—21no—
307C4×C61—42no—
308C4×C61C2×#10312yes—
309C4×C61C2×#10312yes—
310C4×C61C2×#11712yes—
311Dic61—21no—
312Dic61—21no—
313Dic61—21no—
314Dic61—24no—
315Dic61—42no—
316Dic61C2×#10312yes—
317Dic61C2×#10312yes—
318Dic61#52512yes—
319C22×C4×Q8—611no—
320C2×#431—611no—
321#2285—611no—
322#1824—612no—
323#1803—614no—
324#1787—612no—
325C4×#103—614no—
326C2×C61—22no—
327C2×C61#228011yescharacteristic
328C2×C61C22×#10311yescharacteristic
329C2×C61#289611yescharacteristic
330D61—22no—
331D61#310312yes—
332D61#310412yes—
333C23×Q8—611no—
334C2×C4×Q8—611no—
335C2×C4×Q8—612no—
336C2×C4×Q8—612no—
337C2×C4×Q8—612no—
338C2×C4×Q8—612no—
339C2×C4×Q8—611no—
340C2×C4×Q8—611no—
341C2×C4×Q8—1221no—
342C2×C4×Q8—1221no—
343C2×C4×Q8—1221no—
344C2×C4×Q8—1221no—
345C2×C4×Q8—1221no—
346C2×C4×Q8—1221no—
347C2×C4×Q8#692311yescharacteristic, complemented
348C2×#117—1221no—
349C22×C4⋊C4—611no—
350C22×C4⋊C4—1221no—
351C22×C42—611no—
352C22×C42—1221no—
353#431—1221no—
354#431—1221no—
355#431—1221no—
356#431—1221no—
357C2×C4⋊C8—612no—
358C2×#103—618no—
359C2×C22⋊C8—612no—
360C2×C4×C8—612no—
361#384—612no—
362#382—1221no—
363#380—612no—
364#380—1221no—
365C4×C4⋊C4—612no—
366C4×C4⋊C4—1221no—
367#340—618no—
368C61#2073411yescharacteristic, complemented, Sylow
369C22×Q8—612no—
370C22×Q8—612no—
371C22×Q8—611no—
372C22×Q8—1221no—
373C22×Q8—1221no—
374C22×Q8—1221no—
375C22×Q8—1228no—
376C22×Q8C2×#692311yescharacteristic
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—1221no—
381C23×C4—1221no—
382C22×C8—614no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—612no—
386C4×Q8—612no—
387C4×Q8—612no—
388C4×Q8—612no—
389C4×Q8—612no—
390C4×Q8—612no—
391C4×Q8—1222no—
392C4×Q8—1222no—
393C4×Q8—1222no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×Q8—1222no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8C2×#692312yes—
406C4×Q8C2×#692312yes—
407#117—2444no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—614no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1222no—
424C2×C4⋊C4—1222no—
425C2×C4⋊C4C2×#692311yescharacteristic
426C2×C22⋊C4—1222no—
427C2×C42—21no—
428C2×C42—611no—
429C2×C42—611no—
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—1222no—
440C2×C42—1222no—
441C2×C42—1222no—
442C2×C42—1222no—
443C2×C42—1221no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42—1222no—
447C2×C42C2×#692311yescharacteristic
448C4⋊C8—1222no—
449C4⋊C8—1222no—
450#103—12216no—
451C22⋊C8—1222no—
452C22⋊C8—1222no—
453C4×C8—1222no—
454C4×C8—1222no—
455#95—1222no—
456#95—1222no—
457C24—611no—
458C2×Q8—24no—
459C2×Q8—21no—
460C2×Q8—618no—
461C2×Q8—612no—
462C2×Q8—1222no—
463C2×Q8—1222no—
464C2×Q8—1224no—
465C2×Q8—1228no—
466C2×Q8—1224no—
467C2×Q8—1224no—
468C2×Q8—1228no—
469C2×Q8—1228no—
470C2×Q8—1224no—
471C2×Q8—1221no—
472C2×Q8C22×#692312yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—611no—
476C22×C4—614no—
477C22×C4—614no—
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—1222no—
494C22×C4—1222no—
495C22×C4—1228no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1224no—
501C22×C4—2442no—
502C22×C4—2444no—
503C22×C4C22⋊#692311yescharacteristic
504C22×C4C22⋊#692311yescharacteristic
505C22×C4C22×#692311yescharacteristic
506C2×C8—1224no—
507C2×C8—1228no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—612no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1228no—
519C4⋊C4—1221no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—2444no—
523C4⋊C4—2444no—
524C4⋊C4C22×#692312yes—
525C22⋊C4—2444no—
526C42—22no—
527C42—22no—
528C42—611no—
529C42—611no—
530C42—611no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—1222no—
541C42—1222no—
542C42—1222no—
543C42—1222no—
544C42—1222no—
545C42—1222no—
546C42—2444no—
547C42C22⋊#692311yescharacteristic
548C42C22⋊#692311yescharacteristic
549C42C22×#692311yescharacteristic
550C42C22×#692311yescharacteristic
551C23—611no—
552C23—612no—
553C23—612no—
554C23—611no—
555C23—1222no—
556C23—1222no—
557C23C2×C22⋊#692311yescharacteristic
558Q8—24no—
559Q8—22no—
560Q8—614no—
561Q8—1228no—
562Q8—1228no—
563Q8—1224no—
564Q8—1222no—
565Q8#1451985614yescomplemented
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—614no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—614no—
577C2×C4—614no—
578C2×C4—612no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—614no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1228no—
591C2×C4—1228no—
592C2×C4—1222no—
593C2×C4—1224no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2444no—
600C2×C4—2444no—
601C2×C4—2441no—
602C2×C4—2444no—
603C2×C4—2444no—
604C2×C4—2444no—
605C2×C4—2442no—
606C2×C4#1451984014yescomplemented
607C2×C4C2×C22⋊#692311yescharacteristic
608C2×C4C2×C22⋊#692311yescharacteristic
609C2×C4C2×C22⋊#692311yescharacteristic
610C2×C4C2×C22⋊#692311yescharacteristic
611C2×C4C2×C22⋊#692311yescharacteristic
612C2×C4C23×#692311yescharacteristic
613C8—2444no—
614C22—21no—
615C22—21no—
616C22—611no—
617C22—611no—
618C22—612no—
619C22—612no—
620C22—612no—
621C22—1224no—
622C22—1224no—
623C22—1221no—
624C22—1221no—
625C22C2×#1451984012yes—
626C22C22×C22⋊#692311yescharacteristic
627C4—21no—
628C4—21no—
629C4—24no—
630C4—21no—
631C4—42no—
632C4—612no—
633C4—612no—
634C4—612no—
635C4—1221no—
636C4—1221no—
637C4—1221no—
638C4—1224no—
639C4—4884no—
640C4—2442no—
641C4C2×#1451984012yes—
642C4C2×#1451984012yes—
643C4C2×#1451985612yes—
644C2—22no—
645C2—612no—
646C2—612no—
647C2—1222no—
648C2#2146226311yescharacteristic
649C2C22×#1451984011yescharacteristic
650C2#2146243911yescharacteristic
subgroupG/H
centerC2×C4C2×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#20734
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
4. #21461688C2semidirect
6. C4×#14519840C2semidirect
25. C2×Q8×Dic61C4semidirect
27. C2×C4×Dic122C4semidirect
29. C2×#14519742C4semidirect
34. C4×Q8×D61C4semidirect
35. C4×Q8×D61C4semidirect
42. C2×C4×Q8×C61C4semidirect
237. Q8×C61#117semidirect
244. C2×C4×C61#103semidirect
257. C2×Dic61#103semidirect
275. C4×D61#103semidirect
276. C4×D61#103semidirect
286. Dic122#117semidirect
347. C2×C4×Q8#6923semidirect
368. C61#20734semidirect
565. Q8#14519856semidirect
606. C2×C4#14519840semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4312
#214622682
#214617232
#214616882
#214616582
C4×#145198402
C2×C4×Q8×D612
#2073461