Finite Group Databasebeta

← #29718391

Subgroups — #29718391

Order 15616.

986 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840091C211yesmaximal, characteristic
2C2×#103×D61C211yesmaximal, characteristic
3#21449460C211yesmaximal, characteristic
4#21448687C211yesmaximal, characteristic
5#21447197C211yesmaximal, characteristic, complemented
6#21447196C211yesmaximal, characteristic, complemented
7#21447064C211yesmaximal, characteristic, complemented
8#21446743C211yesmaximal, characteristic, complemented
9#21446740C211yesmaximal, characteristic, complemented
10#21446455C211yesmaximal, characteristic, complemented
11#21446454C211yesmaximal, characteristic, complemented
12#21446239C211yesmaximal, characteristic
13#21445998C211yesmaximal, characteristic
14#21445752C211yesmaximal, characteristic
15C61×#2043C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19C2×#14519753C2211yescharacteristic
20C2×#14519735—21no—
21C2×C4⋊C4×D61C2211yescharacteristic
22#16840091—24no—
23C2×#14519709C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35#103×D61—24no—
36C2×#14519689C2211yescharacteristic
37#16839682—21no—
38#16839682—21no—
39#16839669C2211yescharacteristic
40#16839668C2211yescharacteristic
41#16839661C2211yescharacteristic
42#16839649—21no—
43#16839643—21no—
44#16839633—21no—
45#16839633—21no—
46#16839624—21no—
47#16839619C2211yescharacteristic
48C4⋊C4×Dic61C2211yescharacteristic
49C4×C4⋊Dic61C2211yescharacteristic
50C4×#11758198C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4⋊Q8×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C2×Q8×D61—22no—
64C2×Q8×D61—22no—
65C2×Q8×D61—22no—
66C2×Q8×D61—22no—
67C2×Q8×D61—41no—
68C2×Q8×D61—41no—
69C22×C4×D61—21no—
70C22×C4×D61—21no—
71C22×C4×D61C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C2×C8×D61—22no—
75C2×C8×D61—41no—
76C2×C8×D61—41no—
77#14519775—22no—
78#14519775—41no—
79#14519753—22no—
80#14519753—22no—
81#14519735—44no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C4⋊C4×D61—22no—
91C4⋊C4×D61D414yes—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198D411yescharacteristic
95C2×#11758198D411yescharacteristic
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99#14519709—22no—
100#14519708—22no—
101C4⋊#92418—41no—
102#14519706—41no—
103#14519702—22no—
104#14519702—22no—
105#14519701—22no—
106#14519701—22no—
107#14519701—41no—
108#14519701—41no—
109#14519700—22no—
110#14519700—41no—
111#14519689—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684D411yescharacteristic
121#14519684D411yescharacteristic
122C22×Q8×C61C2311yescharacteristic
123C22×Q8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C4⋊Q8×C61—22no—
127C4⋊Q8×C61—41no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C42×C61C2311yescharacteristic
134#103×C61—22no—
135#103×C61—22no—
136#103×C61—22no—
137#103×C61—41no—
138#95×C61—21no—
139C23×D61C2×D411yescharacteristic
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61#4112yes—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166Q8×D61—28no—
167Q8×D61—44no—
168Q8×D61—44no—
169Q8×D61—44no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—24no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×C4×D61—42no—
179C2×C4×D61—42no—
180C2×C4×D61C2×D412yes—
181C2×C4×D61#4112yes—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122—42no—
188C2×Dic122—42no—
189C2×Dic122—42no—
190C2×Dic122#4112yes—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—42no—
201C4⋊C4×C61—42no—
202C4⋊C4×C61—42no—
203C4⋊C4×C61C2×D412yes—
204C42×C61—21no—
205C42×C61—21no—
206#11758198—22no—
207#11758198—42no—
208#11758198—42no—
209#11758198SD1614yes—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—42no—
216C4⋊Dic61C2×D412yes—
217#11758196—24no—
218#11758196—42no—
219#11758196—42no—
220#11758196—42no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61—42no—
224C4×Dic61—42no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C8×D61—44no—
232C23×C61C22×D411yescharacteristic
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61#12112yes—
238C22×D61C2×SD1612yes—
239C22×D61#13712yes—
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61—41no—
243Q8×C61—41no—
244Q8×C61—41no—
245Q8×C61—41no—
246Q8×C61—41no—
247Q8×C61—41no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61C2×SD1612yes—
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C2×C4×C61H(5,𝔽2)11yescharacteristic
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—22no—
277C2×Dic61—21no—
278C2×Dic61—22no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61#12112yes—
292C2×Dic61C2×SD1612yes—
293C4×D61—22no—
294C4×D61—22no—
295C4×D61—22no—
296C4×D61—22no—
297C4×D61—48no—
298C4×D61—48no—
299Dic122—22no—
300Dic122—22no—
301Dic122—48no—
302Dic122—41no—
303Dic122—41no—
304C8×C61—41no—
305C8×C61—41no—
306#92418—41no—
307#92418—41no—
308#24480—611nomaximal, Sylow
309C22×C61C2×#12111yescharacteristic
310C22×C61#54611yescharacteristic
311C22×C61#54611yescharacteristic
312C22×C61C22×SD1611yescharacteristic
313C22×C61C2×#13711yescharacteristic
314C22×C61#57811yescharacteristic
315C22×C61#57811yescharacteristic
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—22no—
319C2×D61—22no—
320C2×D61—22no—
321C2×D61—22no—
322C2×D61#46114yes—
323C4×C61—22no—
324C4×C61—42no—
325C4×C61—42no—
326C4×C61—41no—
327C4×C61—81no—
328C4×C61—81no—
329C4×C61#52511yescharacteristic
330C4×C61#53511yescharacteristic
331C4×C61#55111yescharacteristic
332C4×C61#55111yescharacteristic
333Dic61—22no—
334Dic61—22no—
335Dic61—42no—
336Dic61—42no—
337Dic61—41no—
338Dic61—81no—
339Dic61—81no—
340C2×#483—611no—
341C22×#103—611no—
342#2434—611no—
343#2344—611no—
344#2066—611no—
345#2066—611no—
346#2043—611no—
347#2043D6111yescharacteristic, complemented
348#1981—611no—
349#1981—611no—
350#1927—611no—
351#1927—611no—
352#1892—611no—
353#1850—611no—
354#1787—611no—
355C2×C61#241111yescharacteristic
356C2×C61C2×#46111yescharacteristic
357C2×C61#304411yescharacteristic
358C2×C61#329811yescharacteristic
359C2×C61#329811yescharacteristic
360C2×C61#331211yescharacteristic
361C2×C61#331211yescharacteristic
362D61—24no—
363C23×Q8—611no—
364C23×C8—611no—
365C2×C4⋊Q8—611no—
366C2×C4⋊Q8C2×D6111yescharacteristic
367C2×C22⋊Q8—611no—
368C2×#117—1221no—
369C22×C4⋊C4—611no—
370#483—1224no—
371C2×#106—611no—
372C2×#106—611no—
373C2×C4⋊C8—611no—
374C2×C4⋊C8—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#103—611no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103—611no—
383C2×#103—1224no—
384C2×#103C2×D6111yescharacteristic
385C2×#103C2×D6111yescharacteristic
386C2×#103C2×D6111yescharacteristic
387C2×#103C2×D6111yescharacteristic
388C2×C22⋊C8—611no—
389#395—1221no—
390#395—1221no—
391#391—611no—
392#391—611no—
393#389—611no—
394#389C2×D6111yescharacteristic
395#387—1221no—
396#387—1221no—
397#385—1221no—
398#385—1221no—
399#381—1221no—
400#380—611no—
401C4×C4⋊C4—611no—
402C4×C4⋊C4—611no—
403C4×C22⋊C4—611no—
404#340—611no—
405#340—611no—
406#336—611no—
407#336C2×D6111yescharacteristic
408C61#2448011yescharacteristic, complemented, Sylow
409C22×Q8—611no—
410C22×Q8—1221no—
411C22×Q8—1222no—
412C22×Q8—1222no—
413C22×Q8—1222no—
414C22×Q8—1222no—
415C22×Q8—2441no—
416C22×Q8—2441no—
417C22×Q8C22×D6111yescharacteristic
418C22×Q8C22×D6111yescharacteristic
419C23×C4—611no—
420C23×C4—1221no—
421C23×C4—1221no—
422C22×C8—611no—
423C22×C8—611no—
424C22×C8—1222no—
425C22×C8—2441no—
426C22×C8—2441no—
427C22×C8C22×D6111yescharacteristic
428C22×C8C22×D6111yescharacteristic
429C4⋊Q8—22no—
430C4⋊Q8—41no—
431C4⋊Q8—1222no—
432C4⋊Q8—2441no—
433C22⋊Q8—1222no—
434C22⋊Q8—1222no—
435#117—2444no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—614no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1222no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C22⋊C4—611no—
452C2×C22⋊C4—611no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C42—611no—
456C2×C42—1221no—
457C2×C42—1221no—
458C2×C42C22×D6111yescharacteristic
459#106—1222no—
460#106—1222no—
461C4⋊C8—2441no—
462C4⋊C8—2441no—
463#103—22no—
464#103—22no—
465#103—22no—
466#103—41no—
467#103—1222no—
468#103—1222no—
469#103—1222no—
470#103—1222no—
471#103—1222no—
472#103—2441no—
473#103—2441no—
474#103—2441no—
475C22⋊C8—1222no—
476#95—21no—
477#95—611no—
478#95—611no—
479#95—1221no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487C24—611no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—612no—
500C2×Q8—1221no—
501C2×Q8—1228no—
502C2×Q8—1221no—
503C2×Q8—1221no—
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—2442no—
507C2×Q8—2442no—
508C2×Q8—2442no—
509C2×Q8—2444no—
510C2×Q8—2444no—
511C2×Q8—2444no—
512C2×Q8#1175822412yes—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—612no—
520C22×C4—612no—
521C22×C4—611no—
522C22×C4—1224no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—2442no—
532C22×C4—2442no—
533C22×C4—2442no—
534C22×C4—2442no—
535C22×C4—2442no—
536C22×C4D4×D6111yescharacteristic
537C22×C4D4×D6111yescharacteristic
538C22×C4C23×D6111yescharacteristic
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—2444no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—612no—
559C4⋊C4—1224no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4—1221no—
565C4⋊C4—2442no—
566C4⋊C4—2442no—
567C4⋊C4—2442no—
568C4⋊C4—2442no—
569C4⋊C4D4×D6112yes—
570C22⋊C4—614no—
571C22⋊C4—1222no—
572C22⋊C4—2442no—
573C22⋊C4—2442no—
574C42—21no—
575C42—21no—
576C42—1221no—
577C42—1221no—
578C42—2442no—
579C42—2442no—
580C23—612no—
581C23—612no—
582C23—612no—
583C23—1221no—
584C23—1221no—
585C23—1221no—
586C23—1221no—
587C23C2×D4×D6111yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—41no—
591Q8—41no—
592Q8—41no—
593Q8—41no—
594Q8—41no—
595Q8—41no—
596Q8—1222no—
597Q8—1222no—
598Q8—2448no—
599Q8—2441no—
600Q8—2441no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—612no—
619C2×C4—612no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—1222no—
633C2×C4—2448no—
634C2×C4—2448no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4SD16×D6112yes—
646C2×C4C2×D4×D6111yescharacteristic
647C2×C4#41×D6111yescharacteristic
648C2×C4#41×D6111yescharacteristic
649C2×C4C2×#1175822411yescharacteristic
650C2×C4#1451982211yescharacteristic
651C2×C4#1451982311yescharacteristic
652C8—41no—
653C8—41no—
654C8—2441no—
655C8—2441no—
656C22—614no—
657C22—1222no—
658C22—1222no—
659C22—1222no—
660C22—1222no—
661C22—1222no—
662C22—1222no—
663C22#121×D6111yescharacteristic
664C22#1684039911yescharacteristic
665C22#1684040511yescharacteristic
666C22C2×SD16×D6111yescharacteristic
667C22#137×D6111yescharacteristic
668C22#1684057511yescharacteristic
669C22#1684057811yescharacteristic
670C4—22no—
671C4—42no—
672C4—41no—
673C4—42no—
674C4—81no—
675C4—81no—
676C4—1222no—
677C4—1222no—
678C4—4881no—
679C4—4881no—
680C4—2442no—
681C4—2441no—
682C4—2442no—
683C4#1684027211yescharacteristic
684C4#1684032811yescharacteristic
685C4#1684043511yescharacteristic
686C4#1684044011yescharacteristic
687C2—1224no—
688C2#2144919711yescharacteristic
689C2#461×D6111yescharacteristic
690C2#2145534411yescharacteristic
691C2#2145767711yescharacteristic
692C2#2145768311yescharacteristic
693C2#2145786311yescharacteristic
694C2#2145786911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2043C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2043D61
hypocenterC61#24480
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
5. #21447197C2semidirect
6. #21447196C2semidirect
7. #21447064C2semidirect
8. #21446743C2semidirect
9. #21446740C2semidirect
10. #21446455C2semidirect
11. #21446454C2semidirect
15. C61×#2043C2semidirect
347. #2043D61semidirect
408. C61#24480semidirect

Maximal subgroups

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

subgroupindex
C2×#168400912
C2×#103×D612
#214494602
#214486872
#214471972
#214471962
#214470642
#214467432
#214467402
#214464552
#214464542
#214462392
#214459982
#214457522
C61×#20432
#2448061