Finite Group Databasebeta

← #29679844

Subgroups — #29679844

Order 15616.

1470 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D61C211yesmaximal, characteristic
2C2×#16839725C211yesmaximal, characteristic
3#21448198C211yesmaximal, characteristic
4#21445917C218yesmaximal, complemented
5#21445530C211yesmaximal, characteristic, complemented
6#21445529C211yesmaximal, characteristic, complemented
7#21445528C211yesmaximal, characteristic, complemented
8C61×#1733C211yesmaximal, characteristic, complemented
9C23×C4×D61C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C2×#14519736C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13C2×#14519735C412yescomplemented
14C2×#14519734—21no—
15C2×#14519733—21no—
16#117×D61—22no—
17#117×D61—22no—
18#117×D61C414yescomplemented
19#117×D61C2214yes—
20C2×C4⋊C4×D61C412yescomplemented
21C2×C22⋊C4×D61—21no—
22C2×C42×D61—21no—
23#16839725—21no—
24#16839725—21no—
25#16839725C2214yes—
26C2×#14519689—22no—
27#16839623—21no—
28#16839619—22no—
29#16839617—21no—
30C4×#11758198C2214yes—
31#16839523C2214yescomplemented
32#16839522C2214yescomplemented
33#16839521C2214yescomplemented
34C22×M16×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#339C2214yescomplemented
38C24×D61D411yescharacteristic
39C23×Dic61D411yescharacteristic
40C22×C4×D61—22no—
41C22×C4×D61—22no—
42C22×C4×D61—22no—
43C22×C4×D61—22no—
44C22×C4×D61—22no—
45C22×C4×D61C2×C411yescharacteristic
46C22×C4×D61C2×C412yes—
47C22×C4×D61C2×C411yescharacteristic
48C22×C4×D61D412yes—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D412yes—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61C2312yes—
53C2×#11758194—21no—
54C2×#11758194—22no—
55C2×#11758194—22no—
56C2×#11758194C2312yes—
57C22×#92418—21no—
58#14519736—21no—
59#14519736—21no—
60#14519736—22no—
61#14519736—22no—
62#14519736C2×C412yes—
63#14519736C2312yes—
64#14519735—24no—
65#14519735C2×C418yes—
66#14519734—24no—
67#14519734—24no—
68#14519733—24no—
69#14519733—24no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2×C412yes—
72C2×#11758196—22no—
73C4⋊C4×D61—22no—
74C4⋊C4×D61—22no—
75C4⋊C4×D61C2×C414yes—
76C4⋊C4×D61C2×C414yes—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—22no—
80C2×#11758198—24no—
81C22⋊C4×D61—24no—
82C22⋊C4×D61—24no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61C2312yes—
86C42×D61—24no—
87C42×D61—24no—
88#14519689—24no—
89#14519689—24no—
90#14519684—24no—
91C23×C4×C61C2311yescharacteristic
92C2×M16×C61—21no—
93C2×M16×C61—22no—
94C2×M16×C61—22no—
95C2×M16×C61C2312yes—
96C22×C8×C61—21no—
97#117×C61—21no—
98#117×C61—21no—
99#117×C61—22no—
100#117×C61—22no—
101#117×C61C2×C412yes—
102#117×C61C2312yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2×C412yes—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2312yes—
109C24×C61C2×D411yescharacteristic
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—44no—
113C23×D61C22⋊C411yescharacteristic
114C23×D61C22⋊C412yes—
115C23×D61C22⋊C411yescharacteristic
116C23×D61C2×D412yes—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×C4×C61#4112yes—
127C22×C4×C61#4112yes—
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61C22⋊C411yescharacteristic
134C22×Dic61C22⋊C411yescharacteristic
135C22×Dic61C22⋊C412yes—
136C22×Dic61C2×D412yes—
137C2×C4×D61—22no—
138C2×C4×D61—24no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—24no—
142C2×C4×D61—22no—
143C2×C4×D61—24no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—24no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—24no—
150C2×C4×D61—24no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—24no—
156C2×C4×D61—24no—
157C2×C4×D61—48no—
158C2×C4×D61C22⋊C412yes—
159C2×C4×D61C22⋊C412yes—
160C2×C4×D61C22⋊C412yes—
161C2×C4×D61C22⋊C412yes—
162C2×C4×D61C22×C412yes—
163C2×C4×D61C22×C412yes—
164C2×C4×D61C2×D412yes—
165C2×C4×D61C2×D412yes—
166M16×C61—22no—
167M16×C61—22no—
168M16×C61—42no—
169C2×C8×C61—22no—
170C2×C8×C61—42no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—42no—
173C4⋊C4×C61C22×C414yes—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—42no—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—42no—
184#11758198—28no—
185#11758198—48no—
186C4⋊Dic61—22no—
187C4⋊Dic61—42no—
188C4⋊Dic61C22×C414yes—
189#11758196—24no—
190#11758196—24no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193C4×Dic61—28no—
194C4×Dic61—22no—
195C4×Dic61—22no—
196#11758194—22no—
197#11758194—22no—
198#11758194—42no—
199C2×#92418—22no—
200C2×#92418—42no—
201C23×C61—42no—
202C23×C61C2×C22⋊C411yescharacteristic
203C23×C61C2×C22⋊C411yescharacteristic
204C23×C61#12312yes—
205C23×C61#12312yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—24no—
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—24no—
212C22×D61—22no—
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—48no—
219C22×D61—48no—
220C22×D61C2×C22⋊C412yes—
221C22×D61C2×C22⋊C412yes—
222C2×C4×C61—24no—
223C2×C4×C61—44no—
224C2×C4×C61—44no—
225C2×C4×C61—42no—
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61#11712yes—
229C2×C4×C61#11712yes—
230C2×C4×C61#11712yes—
231C2×C4×C61#11712yes—
232C2×C4×C61#12312yes—
233C2×C4×C61#12312yes—
234C2×C4×C61#12312yes—
235C2×C4×C61#12312yes—
236C2×C4×C61#12312yes—
237C2×C4×C61#12312yes—
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—44no—
251C2×Dic61—44no—
252C2×Dic61—42no—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C412yes—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—24no—
264C4×D61—24no—
265C4×D61—24no—
266C4×D61—24no—
267C4×D61—48no—
268C4×D61—48no—
269C8×C61—42no—
270#92418—42no—
271#20705—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61#38112yes—
278C22×C61#38112yes—
279C22×C61C22×C22⋊C411yescharacteristic
280C22×C61C2×#12311yescharacteristic
281C22×C61C2×#12311yescharacteristic
282C2×D61—28no—
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—44no—
289C2×D61—44no—
290C2×D61—48no—
291C2×D61—44no—
292C2×D61#42114yes—
293C4×C61—44no—
294C4×C61—82no—
295C4×C61#38112yes—
296C4×C61#38112yes—
297C4×C61C2×#11711yescharacteristic
298C4×C61C2×#11711yescharacteristic
299C4×C61C2×#12311yescharacteristic
300C4×C61C2×#12311yescharacteristic
301Dic61—22no—
302Dic61—22no—
303Dic61—44no—
304Dic61—82no—
305C22×#117—611no—
306C2×#407—611no—
307#2280—611no—
308#1826—618no—
309#1733—611no—
310#1733—611no—
311#1733—611no—
312#1733D6111yescharacteristic, complemented
313C2×C61—21no—
314C2×C61—21no—
315C2×C61—22no—
316C2×C61—22no—
317C2×C61C2×#38111yescharacteristic
318C2×C61C2×#42112yes—
319D61—24no—
320D61—42no—
321C24×C4—611no—
322C22×M16—611no—
323C22×M16C2×D6111yescharacteristic
324C2×#117—611no—
325C2×#117—611no—
326C2×#117—614no—
327C2×#117—614no—
328C2×#117—612no—
329C2×#117—1221no—
330C2×#117—1221no—
331C2×#117—1222no—
332C2×#117—1222no—
333C2×#117C2×D6111yescharacteristic
334C2×#117C2×D6111yescharacteristic
335C22×C4⋊C4—612no—
336C22×C22⋊C4—1221no—
337C22×C42—1221no—
338#407—614no—
339#407—1221no—
340#407—1221no—
341C2×C22⋊C8—1222no—
342#381—1221no—
343#380—1222no—
344#379—1221no—
345C4×C22⋊C4—614no—
346#339—614no—
347#339—614no—
348#339—614no—
349#339C2×D6114yescomplemented
350C61#2070511yescharacteristic, complemented, Sylow
351C25—611no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—612no—
356C23×C4—612no—
357C23×C4—612no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—612no—
361C23×C4—1222no—
362C23×C4—1222no—
363C23×C4—1222no—
364C23×C4—1222no—
365C23×C4—1222no—
366C23×C4C22×D6111yescharacteristic
367C2×M16—21no—
368C2×M16—22no—
369C2×M16—22no—
370C2×M16—612no—
371C2×M16—1221no—
372C2×M16—1222no—
373C2×M16—1222no—
374C2×M16C22×D6112yes—
375C22×C8—21no—
376C22×C8—1221no—
377#117—21no—
378#117—21no—
379#117—22no—
380#117—22no—
381#117—612no—
382#117—612no—
383#117—618no—
384#117—1224no—
385#117—1224no—
386#117—1224no—
387#117—1224no—
388#117—1224no—
389#117—1221no—
390#117—1221no—
391#117—1222no—
392#117—1222no—
393#117C4×D6112yes—
394#117C22×D6112yes—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—614no—
398C2×C4⋊C4—614no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C4×D6112yes—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1224no—
407C2×C22⋊C4—1224no—
408C2×C22⋊C4—1222no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1224no—
411C2×C22⋊C4—1221no—
412C2×C42—21no—
413C2×C42—612no—
414C2×C42—1224no—
415C2×C42—1224no—
416C2×C42—1222no—
417C2×C42—1221no—
418C2×C42C22×D6112yes—
419C22⋊C8—1224no—
420C22⋊C8—1224no—
421#95—1224no—
422C24—611no—
423C24—612no—
424C24—612no—
425C24—611no—
426C24—1222no—
427C24—1222no—
428C24—2444no—
429C24D4×D6111yescharacteristic
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—612no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1224no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1224no—
458C22×C4—1224no—
459C22×C4—1224no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1224no—
463C22×C4—1224no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—2448no—
469C22×C4C2×C4×D6111yescharacteristic
470C22×C4C2×C4×D6111yescharacteristic
471C22×C4#1175822212yes—
472C22×C4#1175822212yes—
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6111yescharacteristic
475C22×C4D4×D6111yescharacteristic
476C22×C4#1175822412yes—
477C22×C4#1175822612yes—
478C22×C4C23×D6111yescharacteristic
479M16—22no—
480M16—22no—
481M16—42no—
482M16—1222no—
483M16—1222no—
484M16—2442no—
485C2×C8—22no—
486C2×C8—42no—
487C2×C8—1222no—
488C2×C8—2442no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—614no—
492C4⋊C4—1224no—
493C4⋊C4—1224no—
494C4⋊C4—1222no—
495C4⋊C4—2442no—
496C4⋊C4C2×C4×D6114yes—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—42no—
500C22⋊C4—1228no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C22⋊C4—2448no—
504C22⋊C4—2442no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—1228no—
510C42—1222no—
511C42—1222no—
512C42—1222no—
513C42—1222no—
514C23—42no—
515C23—612no—
516C23—612no—
517C23—1224no—
518C23—1224no—
519C23—1224no—
520C23—1222no—
521C23—1222no—
522C23—1222no—
523C23—1224no—
524C23—1222no—
525C23—2442no—
526C23—2442no—
527C23—2442no—
528C23—2448no—
529C23—2448no—
530C23C22⋊C4×D6111yescharacteristic
531C23C22⋊C4×D6111yescharacteristic
532C23#1451975912yes—
533C23#1451976212yes—
534C23C2×D4×D6111yescharacteristic
535C2×C4—24no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1224no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1224no—
555C2×C4—1224no—
556C2×C4—1224no—
557C2×C4—1224no—
558C2×C4—1224no—
559C2×C4—2448no—
560C2×C4—2448no—
561C2×C4—2444no—
562C2×C4—2442no—
563C2×C4—2444no—
564C2×C4C22⋊C4×D6111yescharacteristic
565C2×C4C22⋊C4×D6111yescharacteristic
566C2×C4#1451973412yes—
567C2×C4#1451973412yes—
568C2×C4#1451973512yes—
569C2×C4#1451973512yes—
570C2×C4#1451975912yes—
571C2×C4#1451975912yes—
572C2×C4#1451975912yes—
573C2×C4#1451976212yes—
574C2×C4#1451976212yes—
575C2×C4#1451976212yes—
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×#1175822411yescharacteristic
579C2×C4C2×#1175822211yescharacteristic
580C2×C4C2×#1175822211yescharacteristic
581C2×C4C2×#1175822611yescharacteristic
582C8—42no—
583C8—2442no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—42no—
588C22—42no—
589C22—614no—
590C22—1228no—
591C22—1222no—
592C22—1222no—
593C22—1222no—
594C22—1222no—
595C22—1222no—
596C22—2444no—
597C22—2444no—
598C22—2448no—
599C22—2444no—
600C22#1683962412yes—
601C22#1683962412yes—
602C22C2×C22⋊C4×D6111yescharacteristic
603C22C2×#1451975911yescharacteristic
604C22C2×#1451976211yescharacteristic
605C4—44no—
606C4—82no—
607C4—1222no—
608C4—1222no—
609C4—4882no—
610C4—2444no—
611C4#1683962412yes—
612C4#1683962412yes—
613C4C2×#1451973411yescharacteristic
614C4C2×#1451973511yescharacteristic
615C4C2×#1451975911yescharacteristic
616C4C2×#1451976211yescharacteristic
617C2—21no—
618C2—21no—
619C2—22no—
620C2—22no—
621C2—1224no—
622C2—2442no—
623C2C2×#1683962411yescharacteristic
624C2#421×D6112yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1733C2
socleC22×C61C22×C22⋊C4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1733D61
hypocenterC61#20705
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
4. #21445917C2semidirect
5. #21445530C2semidirect
6. #21445529C2semidirect
7. #21445528C2semidirect
8. C61×#1733C2semidirect
13. C2×#14519735C4semidirect
18. #117×D61C4semidirect
20. C2×C4⋊C4×D61C4semidirect
31. #16839523C22semidirect
32. #16839522C22semidirect
33. #16839521C22semidirect
37. C61×#339C22semidirect
312. #1733D61semidirect
349. #339C2×D61semidirect
350. C61#20705semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D612
C2×#168397252
#214481982
#214459172
#214455302
#214455292
#214455282
C61×#17332
#2070561