Finite Group Databasebeta

← D61⋊#1550 #30611744

Subgroups — D61⋊#1550 #30611744

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#423C211yesmaximal, characteristic
2C2×D61⋊(C4⋊C8)C211yesmaximal, characteristic
3#21461337C214yesmaximal, complemented
4#21461336C214yesmaximal, complemented
5D61⋊#333C212yesmaximal, complemented
6D61⋊#332C212yesmaximal, complemented
7#533×D61C211yesmaximal, characteristic
8C2×D61⋊M16—21no—
9C22×D61⋊C8—21no—
10#16840767C2212yes—
11#16840766—21no—
12#16840765—21no—
13#16840765C2212yes—
14C2×#14519844C2212yes—
15D61⋊(C4⋊C8)—21no—
16D61⋊(C4⋊C8)—21no—
17D61⋊(C4⋊C8)—21no—
18D61⋊(C4⋊C8)C2212yes—
19D61⋊(C4⋊C8)C2212yes—
20D61⋊(C4⋊C8)C2212yes—
21C2×C4⋊#92420—21no—
22#16840622C2218yescomplemented
23#16840621C2218yescomplemented
24#16840319C411yescharacteristic
25#16840318—21no—
26#16840317C411yescharacteristic
27#16840317C411yescharacteristic
28#16840316—21no—
29#16840316C2212yes—
30#16840315—21no—
31#16840314—21no—
32#16840314C2212yes—
33C4⋊Q8×D61C2211yescharacteristic
34C4⋊Q8×D61C2211yescharacteristic
35#125×D61—21no—
36C22⋊Q8×D61—21no—
37C22⋊Q8×D61C412yes—
38C4×Q8×D61—21no—
39C4×Q8×D61C412yes—
40#117×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic
43C2×#11758213—22no—
44D61⋊M16—44no—
45C2×#11758217—22no—
46C2×D61⋊C8—22no—
47C2×D61⋊C8—22no—
48C2×D61⋊C8—22no—
49C2×D61⋊C8—44no—
50C22×#92420—22no—
51#14519844—22no—
52#14519844—22no—
53#14519844C2314yes—
54#14519844C2314yes—
55C4⋊#92420—22no—
56C4⋊#92420—22no—
57C4⋊#92420—22no—
58C4⋊#92420—22no—
59C2×Q8×D61—21no—
60C2×Q8×D61C2×C412yes—
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64#14519777C2×C411yescharacteristic
65#14519777C2×C411yescharacteristic
66#14519776—21no—
67#14519776—21no—
68#14519776—21no—
69#14519776C2312yes—
70#14519775—21no—
71#14519775C2312yes—
72#14519769—21no—
73#14519768—21no—
74#14519768—21no—
75#14519767—22no—
76#14519767—22no—
77#14519757—21no—
78#14519757C2×C412yes—
79#14519756—21no—
80#14519756C2×C412yes—
81#14519755—22no—
82#14519755—22no—
83#14519754—22no—
84#14519754—22no—
85#14519753—21no—
86#14519753C2×C412yes—
87Q8×Dic61—21no—
88Q8×Dic61C2×C412yes—
89C4×Dic122—21no—
90C4×Dic122—22no—
91C4×Dic122—22no—
92C4×Dic122C2×C412yes—
93#14519742—21no—
94#14519742—22no—
95#14519742—22no—
96#14519742C2×C412yes—
97#14519736—21no—
98#14519735—21no—
99#14519735—21no—
100#14519734—22no—
101#14519733—22no—
102C4⋊C4×D61—22no—
103C4⋊C4×D61—22no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61—21no—
106C4⋊C4×D61—21no—
107C4⋊C4×D61C2×C412yes—
108C22⋊C4×D61—22no—
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C42×D61—22no—
113C42×D61C2×C411yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61C2311yescharacteristic
120C42×D61C2311yescharacteristic
121C4⋊Q8×C61C2×C411yescharacteristic
122C4⋊Q8×C61C2×C411yescharacteristic
123#125×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61C2×C412yes—
126C4×Q8×C61—21no—
127C4×Q8×C61C2×C412yes—
128#117×C61—21no—
129C2×C42×C61C2×C411yescharacteristic
130C23×D61C22⋊C411yescharacteristic
131C2×Q8×C61—21no—
132C2×Q8×C61C22×C412yes—
133C22×C4×C61C22⋊C411yescharacteristic
134C22×C4×C61C22⋊C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×Dic61C22⋊C411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139Q8×D61—44no—
140Q8×D61—42no—
141Q8×D61—42no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—42no—
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C22⋊C411yescharacteristic
152C2×C4×D61C22⋊C411yescharacteristic
153C2×C4×D61SD1614yes—
154C2×C4×D61Q1614yes—
155C2×C4×D61C22×C411yescharacteristic
156C2×C4×D61C22×C411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122C22×C412yes—
165#11758217—44no—
166C2×#92420—24no—
167C2×#92420—24no—
168C2×#92420—44no—
169#11758213—44no—
170D61⋊C8—44no—
171D61⋊C8—44no—
172D61⋊C8—44no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61C22×C412yes—
179C22⋊C4×C61—22no—
180C42×C61—22no—
181C42×C61C22⋊C411yescharacteristic
182C42×C61C22⋊C411yescharacteristic
183C42×C61C22×C411yescharacteristic
184C42×C61C22×C411yescharacteristic
185C22⋊Dic61—22no—
186#11758198—22no—
187#11758198—22no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61C22×C412yes—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61C22⋊C411yescharacteristic
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22×C411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2411yescharacteristic
217C23×C61C2×C22⋊C411yescharacteristic
218C22×D61—22no—
219C22×D61—22no—
220C22×D61#10314yes—
221C22×D61C2×C22⋊C411yescharacteristic
222C22×D61C2×C22⋊C411yescharacteristic
223Q8×C61—42no—
224Q8×C61—42no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—41no—
229C2×C4×C61#10314yes—
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C23×C411yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—41no—
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C2×C22⋊C411yescharacteristic
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C2×SD1612yes—
246C2×Dic61C2×Q1612yes—
247C2×Dic61C22×D411yescharacteristic
248C2×Dic61C22×D411yescharacteristic
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—21no—
254C4×D61—21no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—44no—
258C4×D61—44no—
259C4×D61#10314yes—
260C4×D61#10314yes—
261C4×D61C2×SD1612yes—
262C4×D61C2×SD1612yes—
263C4×D61C2×Q1612yes—
264C4×D61C2×Q1612yes—
265C4×D61AGL(1,ℤ/8)12yes—
266C4×D61#13712yes—
267Dic122—44no—
268Dic122—42no—
269Dic122—42no—
270Dic122—42no—
271Dic122—42no—
272#92420—44no—
273#92420—44no—
274C2×#1550—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61C2×#10312yes—
278C22×C61C22×C22⋊C411yescharacteristic
279C2×D61—21no—
280C2×D61—21no—
281C2×D61—28no—
282C2×D61#41312yes—
283C2×D61C2×#10312yes—
284C2×D61C2×#10312yes—
285C2×D61#41712yes—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61C2×#10312yes—
292C4×C61C2×#10312yes—
293C4×C61#41712yes—
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—42no—
298Dic61—42no—
299Dic61C2×#10312yes—
300Dic61C22×SD1611yescharacteristic
301Dic61C22×Q1611yescharacteristic
302Dic61C2×AGL(1,ℤ/8)11yescharacteristic
303Dic61C2×#13711yescharacteristic
304C2×#533—611no—
305C2×#423—611no—
306C22×C4⋊C8—611no—
307#1550—614no—
308#1550—614no—
309C2×#333—612no—
310C2×#332—612no—
311C2×C61—22no—
312C2×C61C2×#41311yescharacteristic
313C2×C61C22×#10311yescharacteristic
314C2×C61C2×#41711yescharacteristic
315D61—22no—
316D61#155014yescomplemented
317C22×M16—1221no—
318C23×C8—1221no—
319#533—611no—
320#533—611no—
321#533—612no—
322#533—612no—
323#533—611no—
324#533—1221no—
325#533—1221no—
326#533—1221no—
327#533—1221no—
328#533#692311yescharacteristic
329C2×C4⋊Q8—611no—
330C2×C4⋊Q8—611no—
331C2×#125—1221no—
332C2×C22⋊Q8—612no—
333C2×C22⋊Q8—1221no—
334C2×C4×Q8—612no—
335C2×C4×Q8—1221no—
336C2×#117—1221no—
337C22×C42—611no—
338#423—612no—
339#423—612no—
340#423—1221no—
341#423—1221no—
342C2×C4⋊C8—612no—
343C2×C4⋊C8—612no—
344C2×C4⋊C8—612no—
345C2×C4⋊C8—612no—
346C2×C4⋊C8—1221no—
347C2×C4⋊C8—1221no—
348C2×C4⋊C8—1221no—
349C2×C4⋊C8—1221no—
350#333—618no—
351#332—618no—
352C61C2×#155011yescharacteristic, complemented, Sylow
353C22×Q8—612no—
354C22×Q8—1221no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—611no—
358C2×M16—1222no—
359C2×M16—1222no—
360C2×M16—2444no—
361C22×C8—1222no—
362C22×C8—1222no—
363C22×C8—1222no—
364C22×C8—1222no—
365C22×C8—2444no—
366C4⋊Q8—612no—
367C4⋊Q8—612no—
368C4⋊Q8—611no—
369C4⋊Q8—611no—
370C4⋊Q8—1221no—
371C4⋊Q8—1221no—
372C4⋊Q8—1221no—
373C4⋊Q8—1221no—
374C4⋊Q8C2×#692311yescharacteristic
375C4⋊Q8C2×#692311yescharacteristic
376#125—21no—
377#125—1222no—
378#125—1222no—
379#125—1221no—
380#125—1221no—
381#125—1221no—
382C22⋊Q8—21no—
383C22⋊Q8—612no—
384C22⋊Q8—612no—
385C22⋊Q8—612no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388C22⋊Q8—1222no—
389C22⋊Q8—1222no—
390C22⋊Q8—1221no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8C2×#692312yes—
394C4×Q8—21no—
395C4×Q8—612no—
396C4×Q8—612no—
397C4×Q8—612no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8C2×#692312yes—
406#117—21no—
407#117—1222no—
408#117—1222no—
409#117—1221no—
410#117—1221no—
411#117—1221no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1221no—
418C2×C22⋊C4—1222no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—1222no—
431C2×C42C2×#692311yescharacteristic
432C4⋊C8—614no—
433C4⋊C8—614no—
434C4⋊C8—1222no—
435C4⋊C8—1222no—
436C4⋊C8—1222no—
437C4⋊C8—1222no—
438C4⋊C8—1222no—
439C4⋊C8—1222no—
440C24—611no—
441C2×Q8—21no—
442C2×Q8—612no—
443C2×Q8—1222no—
444C2×Q8—1222no—
445C2×Q8—1221no—
446C2×Q8—2442no—
447C2×Q8—2442no—
448C2×Q8—2444no—
449C2×Q8C22×#692312yes—
450C22×C4—611no—
451C22×C4—614no—
452C22×C4—614no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—2442no—
472C22×C4C22⋊#692311yescharacteristic
473C22×C4C22⋊#692311yescharacteristic
474C22×C4C22×#692311yescharacteristic
475M16—2444no—
476M16—2444no—
477C2×C8—1224no—
478C2×C8—1224no—
479C2×C8—2444no—
480C2×C8—2444no—
481C2×C8—2444no—
482C2×C8—2444no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—612no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4C22×#692312yes—
503C22⋊C4—22no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C42—22no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—1222no—
521C42—1222no—
522C42—1222no—
523C42C22⋊#692311yescharacteristic
524C42C22⋊#692311yescharacteristic
525C42C22×#692311yescharacteristic
526C42C22×#692311yescharacteristic
527C23—611no—
528C23—614no—
529C23—611no—
530C23—1222no—
531C23—1222no—
532C23C2×C22⋊#692311yescharacteristic
533Q8—42no—
534Q8—42no—
535Q8—2442no—
536Q8—2442no—
537Q8—2442no—
538Q8—2444no—
539Q8—2442no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—614no—
550C2×C4—614no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—611no—
558C2×C4—611no—
559C2×C4—611no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2444no—
573C2×C4—2444no—
574C2×C4—2441no—
575C2×C4#1451984014yes—
576C2×C4C2×C22⋊#692311yescharacteristic
577C2×C4C2×C22⋊#692311yescharacteristic
578C2×C4C2×C22⋊#692311yescharacteristic
579C2×C4C2×C22⋊#692311yescharacteristic
580C2×C4C2×C22⋊#692311yescharacteristic
581C2×C4C23×#692311yescharacteristic
582C8—2444no—
583C8—2444no—
584C22—21no—
585C22—21no—
586C22—612no—
587C22—612no—
588C22—612no—
589C22—612no—
590C22—1228no—
591C22—1221no—
592C22—1221no—
593C22C2×#1451984012yes—
594C22C22×C22⋊#692311yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—42no—
599C4—42no—
600C4—611no—
601C4—611no—
602C4—612no—
603C4—611no—
604C4—611no—
605C4—1221no—
606C4—1221no—
607C4—1221no—
608C4—2442no—
609C4—2442no—
610C4C2×#1451984012yes—
611C4C2×#1451984012yes—
612C4#1684075312yes—
613C2—22no—
614C2—614no—
615C2—1222no—
616C2D61⋊#41311yescharacteristic
617C2C22×#1451984011yescharacteristic
618C2C2×#1684075311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC4×Dic61C24
hypercenter#533#6923
hypocenterC61C2×#1550
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #21461337C2semidirect
4. #21461336C2semidirect
5. D61⋊#333C2semidirect
6. D61⋊#332C2semidirect
22. #16840622C22semidirect
23. #16840621C22semidirect
316. D61#1550semidirect
352. C61C2×#1550semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4232
C2×D61⋊(C4⋊C8)2
#214613372
#214613362
D61⋊#3332
D61⋊#3322
#533×D612
C2×#155061