Finite Group Databasebeta

← #30610402

Subgroups — #30610402

Order 15616.

1314 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#431C211yesmaximal, characteristic, complemented
2D61⋊#431C211yesmaximal, characteristic, complemented
3#21461121C212yesmaximal, complemented
4#21461120C212yesmaximal, complemented
5#21461117C214yesmaximal, complemented
6#21461113C214yesmaximal, complemented
7C2×C4×#11758226C211yesmaximal, characteristic
8#16840781—21no—
9#16840781C2212yescomplemented
10#16840780—21no—
11#16840780C2212yescomplemented
12#16840779—21no—
13#16840779—21no—
14#16840778C2212yes—
15#16840778C2212yes—
16D61⋊(C4⋊C8)—21no—
17D61⋊(C4⋊C8)C2212yescomplemented
18D61⋊(C22⋊C8)—21no—
19D61⋊(C22⋊C8)—21no—
20C4×D61⋊C8—21no—
21C4×D61⋊C8C2212yescomplemented
22Q8⋊#92420C2218yescomplemented
23#16840610C2218yescomplemented
24C22×#11758226C411yescharacteristic
25C4×#11758226—21no—
26C4×#11758226—21no—
27C4×#11758226—21no—
28C4×#11758226—21no—
29C4×#11758226C412yes—
30C4×#11758226C412yes—
31C4×#11758226C2212yes—
32C4×#11758226C2212yes—
33C2×Q8×Dic61C2211yescharacteristic
34C2×C4×D244—21no—
35C2×C4×D244C411yescharacteristic
36C2×#14519739—21no—
37C2×#14519739C2211yescharacteristic
38C2×#14519735—21no—
39C2×#14519735C411yescharacteristic
40C2×C42×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C2×C4×Q8×C61C411yescharacteristic
43C2×D61⋊C8—22no—
44C2×D61⋊C8—22no—
45#14519844—22no—
46#14519844—22no—
47C4⋊#92420—22no—
48C4⋊#92420C2314yes—
49#14519834—22no—
50#14519834—22no—
51C22⋊#92420—22no—
52C22⋊#92420—22no—
53C4×#92420—22no—
54C4×#92420—22no—
55C4×#92420—22no—
56C4×#92420C2314yes—
57C2×#11758226—21no—
58C2×#11758226—21no—
59C2×#11758226—24no—
60C2×#11758226C818yescomplemented
61C2×#11758226C2×C412yes—
62C2×#11758226C2×C412yes—
63C22×D244—21no—
64C22×D244C2×C411yescharacteristic
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61—21no—
68C22×C4×D61C2×C411yescharacteristic
69C22×C4×D61D411yescharacteristic
70C22×C4×D61D411yescharacteristic
71Q8×Dic61—21no—
72Q8×Dic61—21no—
73Q8×Dic61C2×C412yes—
74Q8×Dic61C2312yes—
75C4×D244—22no—
76C4×D244—22no—
77C4×D244—22no—
78C4×D244—22no—
79C4×D244—21no—
80C4×D244—21no—
81C4×D244C2×C412yes—
82C4×D244C2×C412yes—
83#14519739—22no—
84#14519739—22no—
85#14519739—22no—
86#14519739—22no—
87#14519739—21no—
88#14519739—21no—
89#14519739C2×C412yes—
90#14519739C2312yes—
91#14519735—21no—
92#14519735—21no—
93#14519735—22no—
94#14519735—22no—
95#14519735—22no—
96#14519735—22no—
97#14519735C2×C412yes—
98#14519735C2×C412yes—
99C2×C4⋊Dic61—21no—
100C2×C4⋊Dic61C2×C411yescharacteristic
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109C42×D61—22no—
110C42×D61—22no—
111C42×D61—22no—
112C42×D61—22no—
113C42×D61C2×C411yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61C2311yescharacteristic
120C42×D61C2311yescharacteristic
121C22×Q8×C61C2×C411yescharacteristic
122C4×Q8×C61—21no—
123C4×Q8×C61—21no—
124C4×Q8×C61C2×C412yes—
125C4×Q8×C61C2×C412yes—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2×C411yescharacteristic
128C2×C42×C61—21no—
129C2×C42×C61C2×C411yescharacteristic
130C23×D61—21no—
131C23×D61C22⋊C411yescharacteristic
132C2×Q8×C61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61C2×C814yescomplemented
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×Dic61C22⋊C411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145#11758226—28no—
146#11758226—28no—
147#11758226—24no—
148#11758226—24no—
149#11758226C2×C818yescomplemented
150#11758226C2×C818yescomplemented
151C2×D244—28no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—21no—
155C2×D244—22no—
156C2×D244C2×C814yescomplemented
157C2×D244C22×C412yes—
158C2×C4×D61—28no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—44no—
168C2×C4×D61—44no—
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×D61C2×C814yes—
176C2×C4×D61M1614yes—
177C2×C4×D61C22×C411yescharacteristic
178C2×C4×D61C22×C411yescharacteristic
179C2×C4×D61C2×D411yescharacteristic
180C2×C4×D61C2×D411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×#92420—24no—
184C2×#92420—24no—
185D61⋊C8—44no—
186D61⋊C8—44no—
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
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206C4⋊Dic61C22×C412yes—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22×C411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2411yescharacteristic
223C23×C61C2×C22⋊C411yescharacteristic
224C22×D61—22no—
225C22×D61—24no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—42no—
229C22×D61—42no—
230C22×D61C22⋊C814yes—
231C22×D61C2×C22⋊C411yescharacteristic
232C22×D61C2×C22⋊C411yescharacteristic
233Q8×C61—22no—
234Q8×C61—24no—
235Q8×C61C22×C814yes—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C22×C812yes—
248C2×C4×C61C2×M1612yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—41no—
253C2×Dic61—42no—
254C2×Dic61C22⋊C814yes—
255C2×Dic61C2×C22⋊C411yescharacteristic
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C22×D411yescharacteristic
260C2×Dic61C22×D411yescharacteristic
261D244—28no—
262D244—28no—
263D244—22no—
264D244—24no—
265D244C22×C814yes—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—28no—
271C4×D61—28no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—44no—
277C4×D61—44no—
278C4×D61—44no—
279C4×D61—44no—
280C4×D61C22⋊C814yes—
281C4×D61C22⋊C814yes—
282C4×D61C22×C812yes—
283C4×D61C22×C812yes—
284C4×D61C2×M1612yes—
285C4×D61C2×M1612yes—
286#92420—44no—
287#92420—44no—
288#17437—611nomaximal, Sylow
289C22×C61—21no—
290C22×C61—21no—
291C22×C61C2×C22⋊C812yes—
292C22×C61C22×C22⋊C411yescharacteristic
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—24no—
298C2×D61—44no—
299C2×D61—48no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61C2×C22⋊C812yes—
303C2×D61C2×C22⋊C812yes—
304C4×C61—24no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—42no—
309C4×C61C2×C22⋊C812yes—
310C4×C61#42112yes—
311C4×C61C23×C811yescharacteristic
312C4×C61C22×M1611yescharacteristic
313Dic61—21no—
314Dic61—21no—
315Dic61—21no—
316Dic61—42no—
317Dic61C2×C22⋊C812yes—
318Dic61C2×C22⋊C812yes—
319Dic61#57512yes—
320C2×C4×#41—611no—
321C2×#431—611no—
322C2×#431—611no—
323#1479—612no—
324#1478—612no—
325#1477—614no—
326#1476—614no—
327C2×C61—22no—
328C2×C61C22×C22⋊C811yescharacteristic
329C2×C61#289611yescharacteristic
330C2×C61C2×#42111yescharacteristic
331D61—22no—
332D61—22no—
333D61—44no—
334C22×#41—611no—
335C4×#41—612no—
336C4×#41—612no—
337C4×#41—612no—
338C4×#41—612no—
339C4×#41—1221no—
340C4×#41—1221no—
341C4×#41—1221no—
342C4×#41—1221no—
343C2×C4×Q8—611no—
344C2×C4×Q8#692311yescharacteristic
345C2×C4×D4—611no—
346C2×C4×D4—611no—
347C2×C4×D4—1221no—
348C2×C4×D4—1221no—
349C2×#117—611no—
350C2×#117—1221no—
351C22×C42—611no—
352C22×C42—1221no—
353#431—612no—
354#431—612no—
355#431—612no—
356#431—612no—
357#431—1221no—
358#431—1221no—
359#431—1221no—
360#431—1221no—
361C2×C4⋊C8—612no—
362C2×C4⋊C8—1221no—
363C2×C22⋊C8—1221no—
364C2×C22⋊C8—1221no—
365C2×C4×C8—612no—
366C2×C4×C8—1221no—
367Q8⋊C8—618no—
368D4⋊C8—618no—
369C61#1743711yescharacteristic, complemented, Sylow
370C2×#41—618no—
371C2×#41—612no—
372C2×#41—612no—
373C2×#41—1221no—
374C2×#41—1221no—
375C2×#41—1224no—
376C22×Q8C2×#692311yescharacteristic
377C22×D4—611no—
378C22×D4—1221no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—1221no—
383C23×C4—1221no—
384C23×C4—1221no—
385C22×C8—1222no—
386C22×C8—1222no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8—612no—
390C4×Q8—612no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8C2×#692312yes—
394C4×Q8C2×#692312yes—
395C4×D4—612no—
396C4×D4—612no—
397C4×D4—612no—
398C4×D4—612no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—1222no—
402C4×D4—1222no—
403C4×D4—1221no—
404C4×D4—1221no—
405C4×D4—1222no—
406C4×D4—1222no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1221no—
410C4×D4—1221no—
411#117—612no—
412#117—612no—
413#117—1222no—
414#117—1222no—
415#117—1222no—
416#117—1222no—
417#117—1221no—
418#117—1221no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4C2×#692311yescharacteristic
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C42—21no—
427C2×C42—611no—
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—1222no—
439C2×C42—1222no—
440C2×C42—1222no—
441C2×C42—1222no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42C2×#692311yescharacteristic
445C4⋊C8—614no—
446C4⋊C8—1222no—
447C4⋊C8—1222no—
448C4⋊C8—1222no—
449C22⋊C8—1222no—
450C22⋊C8—1222no—
451C22⋊C8—1222no—
452C22⋊C8—1222no—
453C4×C8—614no—
454C4×C8—1222no—
455C4×C8—1222no—
456C4×C8—1222no—
457C24—611no—
458C24—1221no—
459#41—618no—
460#41—618no—
461#41—1228no—
462#41—1228no—
463#41—1224no—
464#41—1224no—
465C2×Q8—21no—
466C2×Q8—22no—
467C2×Q8D61⋊C814yescomplemented
468C2×Q8C22×#692312yes—
469C2×D4—614no—
470C2×D4—612no—
471C2×D4—1222no—
472C2×D4—1222no—
473C2×D4—1228no—
474C2×D4—1221no—
475C2×D4—1222no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—614no—
482C22×C4—614no—
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—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1228no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1221no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—2442no—
504C22×C4—2444no—
505C22×C4—2442no—
506C22×C4—2444no—
507C22×C4C22⋊#692311yescharacteristic
508C22×C4C22⋊#692311yescharacteristic
509C22×C4C22×#692311yescharacteristic
510C2×C8—1224no—
511C2×C8—1224no—
512C2×C8—2444no—
513C2×C8—2444no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—612no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1221no—
521C4⋊C4C22×#692312yes—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C42—22no—
529C42—22no—
530C42—611no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—1222no—
543C42—1222no—
544C42—1222no—
545C42—1222no—
546C42C22⋊#692311yescharacteristic
547C42C22⋊#692311yescharacteristic
548C42C22×#692311yescharacteristic
549C42C22×#692311yescharacteristic
550C23—611no—
551C23—611no—
552C23—614no—
553C23—1224no—
554C23—1222no—
555C23—2441no—
556C23—2441no—
557C23—2442no—
558C23—2442no—
559C23C2×C22⋊#692311yescharacteristic
560Q8—24no—
561Q8—22no—
562Q8C2×D61⋊C814yes—
563D4—614no—
564D4—1228no—
565D4—1228no—
566D4—1224no—
567D4—1222no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—614no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—614no—
579C2×C4—614no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—612no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—611no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1228no—
590C2×C4—1228no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2444no—
600C2×C4—2444no—
601C2×C4—2444no—
602C2×C4—2441no—
603C2×C4—2442no—
604C2×C4—2444no—
605C2×C4C2×C22⋊#692311yescharacteristic
606C2×C4C2×C22⋊#692311yescharacteristic
607C2×C4C2×C22⋊#692311yescharacteristic
608C2×C4C2×C22⋊#692311yescharacteristic
609C2×C4C2×C22⋊#692311yescharacteristic
610C2×C4C2×D61⋊C812yes—
611C2×C4D61⋊M1612yes—
612C2×C4C23×#692311yescharacteristic
613C8—2444no—
614C8—2444no—
615C22—21no—
616C22—21no—
617C22—612no—
618C22—612no—
619C22—1221no—
620C22—1221no—
621C22—1221no—
622C22—1221no—
623C22—1224no—
624C22—2444no—
625C22—2448no—
626C22—2442no—
627C22—2442no—
628C22D61⋊(C22⋊C8)12yes—
629C22C22×C22⋊#692311yescharacteristic
630C4—24no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—42no—
635C4—612no—
636C4—612no—
637C4—612no—
638C4—1221no—
639C4—1221no—
640C4—1221no—
641C4—2442no—
642C4D61⋊(C22⋊C8)12yes—
643C4#1684076112yes—
644C4C22×D61⋊C811yescharacteristic
645C4C2×D61⋊M1611yescharacteristic
646C2—22no—
647C2—1222no—
648C2—1222no—
649C2—2444no—
650C2C2×D61⋊(C22⋊C8)11yescharacteristic
651C2#2146243911yescharacteristic
652C2C2×#1684076111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC4×C61C23×C8
FrattiniC2×C4C23×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758226C4
cosocleC4×Dic61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#17437
LCSG ▷ 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
1. D61⋊#431C2semidirect
2. D61⋊#431C2semidirect
3. #21461121C2semidirect
4. #21461120C2semidirect
5. #21461117C2semidirect
6. #21461113C2semidirect
9. #16840781C22semidirect
11. #16840780C22semidirect
17. D61⋊(C4⋊C8)C22semidirect
21. C4×D61⋊C8C22semidirect
22. Q8⋊#92420C22semidirect
23. #16840610C22semidirect
60. C2×#11758226C8semidirect
134. C2×Q8×C61C2×C8semidirect
149. #11758226C2×C8semidirect
150. #11758226C2×C8semidirect
156. C2×D244C2×C8semidirect
369. C61#17437semidirect
467. C2×Q8D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4312
#214611212
#214611202
#214611172
#214611132
C2×C4×#117582262
#1743761