Finite Group Databasebeta

← #29687241

Subgroups — #29687241

Order 15616.

960 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic61C211yesmaximal, characteristic
2#117×Dic61C211yesmaximal, characteristic, complemented
3#21446448C211yesmaximal, characteristic, complemented
4#21446441C211yesmaximal, characteristic, complemented
5#21446399C211yesmaximal, characteristic, complemented
6#21446392C211yesmaximal, characteristic, complemented
7#21446142C211yesmaximal, characteristic
8#21446020C211yesmaximal, characteristic, complemented
9#21445914C211yesmaximal, characteristic, complemented
10#21445719C211yesmaximal, characteristic, complemented
11#21445718C211yesmaximal, characteristic, complemented
12#21445646C211yesmaximal, characteristic, complemented
13#21445646C211yesmaximal, characteristic, complemented
14#21445583C211yesmaximal, characteristic, complemented
15C2×C61×#482C211yesmaximal, characteristic
16C2×D4×Dic61—21no—
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C22×C4×Dic61C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27#16839665C2211yescharacteristic
28#16839661C2211yescharacteristic
29#16839655—21no—
30#16839642—21no—
31#16839622C2211yescharacteristic
32C4⋊C4×Dic61C2211yescharacteristic
33C4⋊C4×Dic61C2211yescharacteristic
34C4⋊C4×Dic61C2211yescharacteristic
35C4⋊C4×Dic61C2211yescharacteristic
36C4⋊C4×Dic61C2211yescharacteristic
37C22⋊C4×Dic61—21no—
38C22⋊C4×Dic61—21no—
39C22⋊C4×Dic61—21no—
40#16839608—21no—
41C42×Dic61—21no—
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#121×C61C2211yescharacteristic
53C2×#117×C61C2211yescharacteristic
54C61×#482C418yescomplemented
55C2×#107×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×C22⋊C8×C61C2211yescharacteristic
60C23×Dic61—21no—
61C23×Dic61D411yescharacteristic
62C23×Dic61D411yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65D4×Dic61—22no—
66D4×Dic61—22no—
67D4×Dic61—44no—
68#14519736—22no—
69#14519736—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519711—22no—
93#14519709—22no—
94#14519698—22no—
95#14519698—22no—
96C22⋊#92418—22no—
97#14519687—22no—
98#14519687—22no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106C22×D4×C61—21no—
107C22×D4×C61C2311yescharacteristic
108C23×C4×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111#121×C61—22no—
112#121×C61C2×C414yes—
113#117×C61—22no—
114#117×C61C2×C414yescomplemented
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121#107×C61C2×C414yescomplemented
122#106×C61C2×C414yescomplemented
123#102×C61C2×C414yescomplemented
124#102×C61C2×C414yescomplemented
125C22⋊C8×C61C2×C414yescomplemented
126C24×C61—21no—
127C24×C61C2×D411yescharacteristic
128C2×D4×C61—24no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—42no—
135C2×D4×C61—42no—
136C2×D4×C61C22×C412yes—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C22×C412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61#4112yes—
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—41no—
156C22×Dic61—41no—
157C22×Dic61—41no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61C22×C412yes—
170C2×C8×C61C22×C412yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22×C412yes—
175C4⋊C4×C61C22×C412yes—
176C4⋊C4×C61C22×C412yes—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—24no—
179C42×C61—24no—
180C22⋊Dic61—24no—
181C22⋊Dic61—42no—
182C22⋊Dic61—42no—
183C22⋊Dic61—42no—
184C22⋊Dic61—42no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—42no—
192C4⋊Dic61#4112yes—
193C4⋊Dic61#4112yes—
194C4×Dic61—24no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—42no—
200C4×Dic61—42no—
201C4×Dic61—42no—
202C4×Dic61—42no—
203C4×Dic61—42no—
204C4×Dic61#4112yes—
205C4×Dic61#4112yes—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61C4×D412yes—
226C23×C61C22×D411yescharacteristic
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—48no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61C4×D412yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—44no—
266C2×Dic61—44no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—84no—
278C2×Dic61#12312yes—
279C2×Dic61#12312yes—
280C2×Dic61AGL(1,ℤ/8)12yes—
281C8×C61—22no—
282C8×C61—22no—
283#92418—41no—
284#92418—41no—
285#21582—611nomaximal, Sylow
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61C2×C4×D411yescharacteristic
301C22×C61C2×#12311yescharacteristic
302C22×C61#52511yescharacteristic
303C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C61#57511yescharacteristic
305C22×C61#57511yescharacteristic
306C22×C61#57511yescharacteristic
307C4×C61—22no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61C4×#4111yescharacteristic
312C4×C61C4×#4111yescharacteristic
313C4×C61#52911yescharacteristic
314C4×C61#52911yescharacteristic
315Dic61—22no—
316Dic61—22no—
317Dic61—22no—
318Dic61—41no—
319Dic61—42no—
320Dic61—42no—
321Dic61—81no—
322Dic61—81no—
323C2×#482D6111yescharacteristic
324C4×#121—611no—
325C4×#117—611no—
326#1926—611no—
327#1925—611no—
328#1919—611no—
329#1918—611no—
330#1875—611no—
331#1854—611no—
332#1825—611no—
333#1776—611no—
334#1776—611no—
335#1762—611no—
336#1762—611no—
337#1749—611no—
338C2×C61—42no—
339C2×C61—82no—
340C2×C61C4×#12311yescharacteristic
341C2×C61#294311yescharacteristic
342C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
343C2×C61#309311yescharacteristic
344C2×C61#309411yescharacteristic
345C2×C61#311111yescharacteristic
346C2×C61#311511yescharacteristic
347C2×#121C2×D6111yescharacteristic
348C2×C4×D4—611no—
349C2×C4×D4—1221no—
350C2×#117—611no—
351C2×#117C2×D6111yescharacteristic
352C22×C42—611no—
353#482Dic6118yescomplemented
354C2×#107—611no—
355C2×#107C2×D6111yescharacteristic
356C2×#106—611no—
357C2×#106C2×D6111yescharacteristic
358C2×#102—611no—
359C2×#102—611no—
360C2×#102C2×D6111yescharacteristic
361C2×#102C2×D6111yescharacteristic
362C2×C22⋊C8—611no—
363C2×C22⋊C8C2×D6111yescharacteristic
364C2×#97—611no—
365C2×#97—611no—
366#390—611no—
367#389—611no—
368#388—1221no—
369#386—1221no—
370#380—611no—
371C4×C4⋊C4—611no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—611no—
374C4×C4⋊C4—611no—
375C4×C4⋊C4—611no—
376C4×C22⋊C4—1221no—
377C4×C22⋊C4—1221no—
378C4×C22⋊C4—1221no—
379#376—1221no—
380C43—1221no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#336—611no—
390#336—611no—
391C61#2158211yescharacteristic, complemented, Sylow
392C22×D4—21no—
393C22×D4C22×D6111yescharacteristic
394C23×C4—611no—
395C23×C4—611no—
396C23×C4—1221no—
397C23×C4C22×D6111yescharacteristic
398C22×C8—611no—
399C22×C8—611no—
400C22×C8C22×D6111yescharacteristic
401C22×C8C22×D6111yescharacteristic
402#121—22no—
403#121C2×Dic6114yes—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—2444no—
407#117—22no—
408#117—1222no—
409#117—1222no—
410#117C2×Dic6114yescomplemented
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
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—1222no—
432C2×C42—1222no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42—1221no—
439#107—1222no—
440#107C2×Dic6114yescomplemented
441#106—1222no—
442#106C2×Dic6114yescomplemented
443#102—1222no—
444#102—1222no—
445#102C2×Dic6114yescomplemented
446#102C2×Dic6114yescomplemented
447C22⋊C8—1222no—
448C22⋊C8C2×Dic6114yescomplemented
449#97—1222no—
450#97—1222no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458C24—21no—
459C24D4×D6111yescharacteristic
460C2×D4—24no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4C22×Dic6112yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—2442no—
488C22×C4—2441no—
489C22×C4—2441no—
490C22×C4—2441no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4D4×D6111yescharacteristic
494C22×C4#1175822412yes—
495C22×C4C22×Dic6112yes—
496C22×C4C23×D6111yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8C22×Dic6112yes—
508C2×C8C22×Dic6112yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—612no—
513C4⋊C4—612no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—2442no—
521C4⋊C4C22×Dic6112yes—
522C4⋊C4C22×Dic6112yes—
523C4⋊C4C22×Dic6112yes—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—1224no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C42—24no—
532C42—612no—
533C42—612no—
534C42—1222no—
535C42—1222no—
536C42—1224no—
537C42—1221no—
538C42—1221no—
539C42—2442no—
540C42—2442no—
541C42—2442no—
542C42—2442no—
543C42—2442no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23D4×Dic6112yes—
558C23C2×D4×D6111yescharacteristic
559D4—22no—
560D4—22no—
561D4—48no—
562D4—41no—
563D4—41no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—4884no—
594C2×C4—2444no—
595C2×C4—2444no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4D4×Dic6112yes—
607C2×C4C23×Dic6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4#41×D6111yescharacteristic
611C2×C4#41×D6111yescharacteristic
612C2×C4C2×#1175822411yescharacteristic
613C8—22no—
614C8—22no—
615C8—2441no—
616C8—2441no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×D4×Dic6111yescharacteristic
632C22#123×D6111yescharacteristic
633C22#1684027011yescharacteristic
634C22AGL(1,ℤ/8)×D6111yescharacteristic
635C22#1684055811yescharacteristic
636C22#1684055911yescharacteristic
637C22#1684055911yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—1222no—
643C4—1222no—
644C4—1222no—
645C4—4881no—
646C4—4881no—
647C4—2442no—
648C4—2442no—
649C4—2441no—
650C4#41×Dic6111yescharacteristic
651C4#41×Dic6111yescharacteristic
652C4#1684028911yescharacteristic
653C4#1684028911yescharacteristic
654C2—42no—
655C2—82no—
656C2#123×Dic6111yescharacteristic
657C2#2145462011yescharacteristic
658C2AGL(1,ℤ/8)×Dic6111yescharacteristic
659C2#2145574711yescharacteristic
660C2#2145575811yescharacteristic
661C2#2145592311yescharacteristic
662C2#2145596511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#482C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#482D61
hypocenterC61#21582
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#482

Complements

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

NH ≅ G/Ntype
2. #117×Dic61C2semidirect
3. #21446448C2semidirect
4. #21446441C2semidirect
5. #21446399C2semidirect
6. #21446392C2semidirect
8. #21446020C2semidirect
9. #21445914C2semidirect
10. #21445719C2semidirect
11. #21445718C2semidirect
12. #21445646C2semidirect
13. #21445646C2semidirect
14. #21445583C2semidirect
54. C61×#482C4semidirect
114. #117×C61C2×C4semidirect
121. #107×C61C2×C4semidirect
122. #106×C61C2×C4semidirect
123. #102×C61C2×C4semidirect
124. #102×C61C2×C4semidirect
125. C22⋊C8×C61C2×C4semidirect
353. #482Dic61semidirect
391. C61#21582semidirect
410. #117C2×Dic61semidirect
440. #107C2×Dic61semidirect
442. #106C2×Dic61semidirect
445. #102C2×Dic61semidirect
446. #102C2×Dic61semidirect
448. C22⋊C8C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#121×Dic612
#117×Dic612
#214464482
#214464412
#214463992
#214463922
#214461422
#214460202
#214459142
#214457192
#214457182
#214456462
#214455832
C2×C61×#4822
#2158261