Finite Group Databasebeta

← #30616703

Subgroups — #30616703

Order 15616.

1238 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#432C211yesmaximal, characteristic
2C2×C4×C4⋊#6923C211yesmaximal, characteristic
3#21461998C212yesmaximal, complemented
4#21461985C212yesmaximal, complemented
5#21461804C214yesmaximal
6#21461591C214yesmaximal, complemented
7#533×D61C211yesmaximal, characteristic
8C22×C4⋊#6923C2212yes—
9C22×C4×#6923—21no—
10#16840785—21no—
11#16840784—21no—
12#16840783—21no—
13#16840782—21no—
14D61⋊(C4⋊C8)C2212yes—
15C2×#14519840C2218yes—
16D61⋊(C22⋊C8)C2212yes—
17#16840721C2212yes—
18C4×C4⋊#6923—21no—
19C4×C4⋊#6923—21no—
20C4×C4⋊#6923C2212yes—
21C4×C4⋊#6923C2212yes—
22C2×(C2×#6923)⋊C4—21no—
23#16840635C2218yes—
24#16840319C411yescharacteristic, complemented
25#16840318—21no—
26#16840317C411yescharacteristic, complemented
27#16840317C411yescharacteristic, complemented
28#16840316—21no—
29#16840316C412yescomplemented
30#16840315—21no—
31#16840314—21no—
32#16840314C412yescomplemented
33C4⋊Q8×D61C2211yescharacteristic
34C4⋊Q8×D61C2211yescharacteristic
35#125×D61—21no—
36C22⋊Q8×D61—21no—
37C22⋊Q8×D61C2212yes—
38C4×Q8×D61—21no—
39C4×Q8×D61C2212yes—
40#117×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic, complemented
43C23×#6923—22no—
44C2×D61⋊C8C2314yes—
45C2×C4⋊#6923—22no—
46C2×C4⋊#6923—28no—
47C2×C4⋊#6923C2314yes—
48C2×C4×#6923—22no—
49C2×C4×#6923—22no—
50C2×C4×#6923—44no—
51#14519844—22no—
52C4⋊#92420—22no—
53#14519840—216no—
54#14519834—22no—
55C22⋊#92420—22no—
56#14519832—22no—
57#14519832—22no—
58(C2×#6923)⋊C4—22no—
59(C2×#6923)⋊C4—22no—
60C2×Q8×D61—21no—
61C2×Q8×D61C2312yes—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65#14519777C2×C411yescharacteristic, complemented
66#14519777C2×C411yescharacteristic, complemented
67#14519776—21no—
68#14519776—21no—
69#14519776—21no—
70#14519776C2×C412yescomplemented
71#14519775—21no—
72#14519775C2×C412yescomplemented
73#14519769—21no—
74#14519768—21no—
75#14519768—21no—
76#14519767—22no—
77#14519767—22no—
78#14519757—21no—
79#14519757C2×C412yes—
80#14519756—21no—
81#14519756C2×C412yes—
82#14519755—22no—
83#14519755—22no—
84#14519754—22no—
85#14519754—22no—
86#14519753—21no—
87#14519753C2×C412yes—
88Q8×Dic61—21no—
89Q8×Dic61C2×C412yescomplemented
90C4×Dic122—21no—
91C4×Dic122—22no—
92C4×Dic122—22no—
93C4×Dic122C2×C412yescomplemented
94#14519742—21no—
95#14519742—22no—
96#14519742—22no—
97#14519742C2×C412yescomplemented
98#14519736—21no—
99#14519735—21no—
100#14519735—21no—
101#14519734—22no—
102#14519733—22no—
103C4⋊C4×D61—21no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61—21no—
106C4⋊C4×D61—22no—
107C4⋊C4×D61—22no—
108C4⋊C4×D61C2312yes—
109C22⋊C4×D61—22no—
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C42×D61—22no—
114C42×D61C2×C411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61C2311yescharacteristic
121C42×D61C2311yescharacteristic
122C4⋊Q8×C61C2×C411yescharacteristic, complemented
123C4⋊Q8×C61C2×C411yescharacteristic, complemented
124#125×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61C2×C412yes—
127C4×Q8×C61—21no—
128C4×Q8×C61C2×C412yescomplemented
129#117×C61—21no—
130C2×C42×C61C2×C411yescharacteristic
131C23×D61C2×D411yescharacteristic
132C22×#6923—28no—
133C22×#6923—24no—
134C22×#6923—44no—
135C22×#6923—44no—
136C2×Q8×C61—21no—
137C2×Q8×C61C22×C412yes—
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×Dic61C22⋊C411yescharacteristic
142C22×Dic61C22⋊C411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144Q8×D61—28no—
145Q8×D61—42no—
146Q8×D61—42no—
147C2×C4×D61—24no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—24no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61C22⋊C411yescharacteristic
154C2×C4×D61C22⋊C411yescharacteristic
155C2×C4×D61C22⋊C411yescharacteristic
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22×C411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2411yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—22no—
167C2×Dic122—22no—
168C2×Dic122C22×C412yes—
169C2×#92420—24no—
170C4⋊#6923—28no—
171C4⋊#6923—24no—
172C4⋊#6923#4118yes—
173C4×#6923—44no—
174C4×#6923—44no—
175D61⋊C8—28no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61C22×C412yes—
182C22⋊C4×C61—22no—
183C42×C61—22no—
184C42×C61C22⋊C411yescharacteristic
185C42×C61C22⋊C411yescharacteristic
186C42×C61C22×C411yescharacteristic
187C42×C61C22×C411yescharacteristic
188C22⋊Dic61—22no—
189#11758198—22no—
190#11758198—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61C22×C412yes—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
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
220C23×C61C2×C22⋊C411yescharacteristic
221C22×D61—22no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61C2×C22⋊C411yescharacteristic
225C22×D61#13714yes—
226C22×D61C22×D411yescharacteristic
227C2×#6923—28no—
228C2×#6923—416no—
229C2×#6923—44no—
230C2×#6923—44no—
231Q8×C61—24no—
232Q8×C61—42no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C23×C411yescharacteristic
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C22×D411yescharacteristic
255C4×D61—22no—
256C4×D61—21no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—21no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—44no—
267C4×D61—42no—
268C4×D61—42no—
269C4×D61AGL(1,ℤ/8)12yes—
270C4×D61#13712yes—
271C4×D61C2×#4114yes—
272Dic122—24no—
273Dic122—42no—
274Dic122—44no—
275Dic122—44no—
276#92420—44no—
277#21394—611nomaximal, Sylow
278C22×C61—21no—
279C22×C61—21no—
280C22×C61#41912yes—
281C22×C61C22×C22⋊C411yescharacteristic
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—24no—
285C2×D61—24no—
286C2×D61#41912yes—
287C2×D61#52512yes—
288C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
289C2×D61C2×#13712yes—
290C2×D61C2×#13711yescharacteristic
291#6923—48no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—42no—
297C4×C61—42no—
298C4×C61#41712yes—
299C4×C61C4×#4112yes—
300Dic61—21no—
301Dic61—21no—
302Dic61—21no—
303Dic61—21no—
304Dic61—42no—
305Dic61—42no—
306Dic61#41712yes—
307Dic61#53212yes—
308C2×#533—611no—
309C2×#432—611no—
310C2×C4×C4⋊C4—611no—
311#1988—612no—
312#1981—612no—
313#1872—614no—
314#1763—614no—
315C2×C61—22no—
316C2×C61#229711yescharacteristic
317C2×C61C2×#41911yescharacteristic
318C2×C61C2×#41711yescharacteristic
319D61—22no—
320D61#311012yes—
321D61#311312yes—
322#533—611no—
323#533—611no—
324#533—612no—
325#533—612no—
326#533—611no—
327#533—1221no—
328#533—1221no—
329#533—1221no—
330#533—1221no—
331#533#692311yescharacteristic, complemented
332C2×C4⋊Q8—611no—
333C2×C4⋊Q8—611no—
334C2×#125—1221no—
335C2×C22⋊Q8—612no—
336C2×C22⋊Q8—1221no—
337C2×C4×Q8—612no—
338C2×C4×Q8—1221no—
339C2×#117—1221no—
340C22×C4⋊C4—612no—
341C22×C42—611no—
342C22×C42—1221no—
343#432—1221no—
344#432—1221no—
345#432—1221no—
346#432—1221no—
347C2×C4⋊C8—612no—
348C2×#103—618no—
349C2×C22⋊C8—612no—
350C2×#97—612no—
351C4×C4⋊C4—612no—
352C4×C4⋊C4—612no—
353C4×C4⋊C4—1221no—
354C4×C4⋊C4—1221no—
355C2×#95—1221no—
356#340—618no—
357C61#2139411yescharacteristic, complemented, Sylow
358C22×Q8—612no—
359C22×Q8—1221no—
360C23×C4—611no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—1222no—
364C22×C8—614no—
365C4⋊Q8—612no—
366C4⋊Q8—612no—
367C4⋊Q8—611no—
368C4⋊Q8—611no—
369C4⋊Q8—1221no—
370C4⋊Q8—1221no—
371C4⋊Q8—1221no—
372C4⋊Q8—1221no—
373C4⋊Q8C2×#692311yescharacteristic, complemented
374C4⋊Q8C2×#692311yescharacteristic, complemented
375#125—21no—
376#125—1222no—
377#125—1222no—
378#125—1221no—
379#125—1221no—
380#125—1221no—
381C22⋊Q8—21no—
382C22⋊Q8—612no—
383C22⋊Q8—612no—
384C22⋊Q8—612no—
385C22⋊Q8—1222no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388C22⋊Q8—1222no—
389C22⋊Q8—1221no—
390C22⋊Q8—1221no—
391C22⋊Q8—1221no—
392C22⋊Q8C2×#692312yes—
393C4×Q8—21no—
394C4×Q8—612no—
395C4×Q8—612no—
396C4×Q8—612no—
397C4×Q8—1222no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×Q8—1221no—
401C4×Q8—1222no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8C2×#692312yescomplemented
405#117—21no—
406#117—1222no—
407#117—1222no—
408#117—1221no—
409#117—1221no—
410#117—1221no—
411C2×C4⋊C4—614no—
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×C4⋊C4—1228no—
419C2×C4⋊C4—1222no—
420C2×C22⋊C4—1222no—
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—611no—
431C2×C42—611no—
432C2×C42—1222no—
433C2×C42—1222no—
434C2×C42—1222no—
435C2×C42—2444no—
436C2×C42C2×#692311yescharacteristic
437C4⋊C8—1222no—
438C4⋊C8—1222no—
439#103—12216no—
440C22⋊C8—1222no—
441C22⋊C8—1222no—
442#97—1222no—
443#97—1222no—
444#95—1222no—
445#95—1222no—
446C24—611no—
447C2×Q8—21no—
448C2×Q8—612no—
449C2×Q8—1222no—
450C2×Q8—1222no—
451C2×Q8—1228no—
452C2×Q8—1221no—
453C2×Q8—2442no—
454C2×Q8—2442no—
455C2×Q8C22×#692312yes—
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—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—1222no—
472C22×C4—1224no—
473C22×C4—1222no—
474C22×C4—1224no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1224no—
478C22×C4—1228no—
479C22×C4—2444no—
480C22×C4—2444no—
481C22×C4C22⋊#692311yescharacteristic
482C22×C4C22⋊#692311yescharacteristic
483C22×C4C22×#692311yescharacteristic
484C2×C8—1224no—
485C2×C8—1228no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—618no—
492C4⋊C4—612no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1224no—
502C4⋊C4—1228no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4C22×#692312yes—
509C22⋊C4—22no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C42—22no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—611no—
526C42—1222no—
527C42—1222no—
528C42—1222no—
529C42—2444no—
530C42—2444no—
531C42C22⋊#692311yescharacteristic
532C42C22⋊#692311yescharacteristic
533C42C22×#692311yescharacteristic
534C42C22×#692311yescharacteristic
535C23—611no—
536C23—614no—
537C23—611no—
538C23—1222no—
539C23—1221no—
540C23—1221no—
541C23C2×C22⋊#692311yescharacteristic
542Q8—24no—
543Q8—42no—
544Q8—1224no—
545Q8—2444no—
546Q8—2444no—
547Q8—2442no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—611no—
554C2×C4—611no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—614no—
558C2×C4—611no—
559C2×C4—611no—
560C2×C4—611no—
561C2×C4—611no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1221no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1228no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—2444no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2444no—
583C2×C4—2444no—
584C2×C4—24416no—
585C2×C4C2×C22⋊#692311yescharacteristic
586C2×C4C2×C22⋊#692311yescharacteristic
587C2×C4C2×C22⋊#692311yescharacteristic
588C2×C4C2×C22⋊#692311yescharacteristic
589C2×C4C2×C22⋊#692311yescharacteristic
590C2×C4C23×#692311yescharacteristic
591C8—2444no—
592C22—21no—
593C22—21no—
594C22—612no—
595C22—612no—
596C22—611no—
597C22—611no—
598C22—612no—
599C22—1224no—
600C22—1224no—
601C22—1221no—
602C22—1221no—
603C22#1684075512yes—
604C22C22×C22⋊#692311yescharacteristic
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—42no—
610C4—42no—
611C4—612no—
612C4—612no—
613C4—1221no—
614C4—1221no—
615C4—1221no—
616C4—1221no—
617C4—2448no—
618C4—2442no—
619C4—2442no—
620C4#1684075312yes—
621C4#41×#692312yes—
622C2—22no—
623C2—612no—
624C2—612no—
625C2—1222no—
626C2#2146228011yescharacteristic
627C2C2×#1684075511yescharacteristic
628C2C2×#1684075311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#533#6923
hypocenterC61#21394
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #21461998C2semidirect
4. #21461985C2semidirect
6. #21461591C2semidirect
24. #16840319C4semidirect
26. #16840317C4semidirect
27. #16840317C4semidirect
29. #16840316C4semidirect
32. #16840314C4semidirect
42. C61×#533C4semidirect
65. #14519777C2×C4semidirect
66. #14519777C2×C4semidirect
70. #14519776C2×C4semidirect
72. #14519775C2×C4semidirect
89. Q8×Dic61C2×C4semidirect
93. C4×Dic122C2×C4semidirect
97. #14519742C2×C4semidirect
122. C4⋊Q8×C61C2×C4semidirect
123. C4⋊Q8×C61C2×C4semidirect
128. C4×Q8×C61C2×C4semidirect
331. #533#6923semidirect
357. C61#21394semidirect
373. C4⋊Q8C2×#6923semidirect
374. C4⋊Q8C2×#6923semidirect
404. C4×Q8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4322
C2×C4×C4⋊#69232
#214619982
#214619852
#214618042
#214615912
#533×D612
#2139461