Finite Group Databasebeta

← #29737087

Subgroups — #29737087

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840045C211yesmaximal, characteristic, complemented
2C2×#16839758C211yesmaximal, characteristic
3#21449511C211yesmaximal, characteristic, complemented
4#21448660C211yesmaximal, characteristic
5#21447338C211yesmaximal, characteristic, complemented
6#21447191C211yesmaximal, characteristic, complemented
7#21447189C211yesmaximal, characteristic, complemented
8#21446953C211yesmaximal, characteristic, complemented
9#21446085C211yesmaximal, characteristic, complemented
10#21446075C211yesmaximal, characteristic, complemented
11#21446020C211yesmaximal, characteristic, complemented
12#21445996C211yesmaximal, characteristic, complemented
13#21445842C211yesmaximal, characteristic, complemented
14#21445832C211yesmaximal, characteristic, complemented
15C61×#1788C211yesmaximal, characteristic
16C23×#92418C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16840045—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28#16839758—24no—
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33#16839685—21no—
34#16839674—21no—
35#16839666—21no—
36#16839665—21no—
37#16839665—21no—
38#16839665C2211yescharacteristic
39#16839665C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839645—21no—
43#16839627C2211yescharacteristic
44#16839622—21no—
45#16839622C2211yescharacteristic
46C4×C22⋊Dic61—21no—
47C4×C22⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×#41×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×C22⋊C8×C61C2211yescharacteristic
60C61×#380C2211yescharacteristic
61C22×Dic122—21no—
62C22×#92418—22no—
63C22×#92418—22no—
64C22×#92418—41no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519776—22no—
68#14519776—41no—
69Q8×Dic61—22no—
70Q8×Dic61—22no—
71Q8×Dic61—41no—
72Q8×Dic61—41no—
73#14519736—22no—
74#14519736—41no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61D411yescharacteristic
91C2×C22⋊Dic61D411yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97#14519711—41no—
98#14519709—22no—
99#14519708—22no—
100#14519708—41no—
101#14519706—22no—
102#14519706—41no—
103#14519701—22no—
104#14519701D414yes—
105#14519698—22no—
106#14519698—22no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114C2×#41×C61—22no—
115C2×#41×C61—22no—
116C2×#41×C61—22no—
117C2×#41×C61—41no—
118C2×#41×C61—41no—
119C2×#41×C61Q814yescomplemented
120C22×Q8×C61C2311yescharacteristic
121C22×D4×C61—21no—
122C22×D4×C61C2311yescharacteristic
123C23×C4×C61—21no—
124C23×C4×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C22⋊C4×C61—21no—
130#103×C61—22no—
131#103×C61—22no—
132#102×C61—41no—
133#102×C61—41no—
134C22⋊C8×C61—22no—
135#95×C61—21no—
136C24×C61—21no—
137C24×C61C2×D411yescharacteristic
138#41×C61—28no—
139#41×C61—44no—
140#41×C61—44no—
141#41×C61—44no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61C2×D412yes—
146C2×Q8×C61C2×Q812yes—
147C2×Q8×C61#4112yes—
148C2×D4×C61—24no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—42no—
155C2×D4×C61—42no—
156C2×D4×C61C2×Q812yes—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—24no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—42no—
166C22×C4×C61—42no—
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×Q812yes—
169C22×C4×C61#4112yes—
170C22×C4×C61C2411yescharacteristic
171C22×Dic61—21no—
172C22×Dic61—21no—
173C22×Dic61—21no—
174C22×Dic61—21no—
175C22×Dic61C2×D411yescharacteristic
176C22×Dic61C2×D411yescharacteristic
177C22×Dic61C2×D411yescharacteristic
178C22×Dic61C2×D411yescharacteristic
179C2×Dic122—42no—
180C2×Dic122—42no—
181C2×Dic122—81no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C22⋊C4×C61—81no—
193C22⋊Dic61—22no—
194C22⋊Dic61—41no—
195C22⋊Dic61—42no—
196C22⋊Dic61—42no—
197C22⋊Dic61—42no—
198C22⋊Dic61—81no—
199C22⋊Dic61—81no—
200C22⋊Dic61—81no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—42no—
209C4⋊Dic61—42no—
210C4⋊Dic61—42no—
211C4⋊Dic61C2×D412yes—
212#11758196—42no—
213#11758196—81no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—42no—
219C4×Dic61—42no—
220C4×Dic61—42no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—44no—
227C2×#92418C2×D412yes—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—22no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61—41no—
240C23×C61—41no—
241C23×C61C22⋊Q812yes—
242C23×C61C22×D411yescharacteristic
243Q8×C61—22no—
244Q8×C61—22no—
245Q8×C61—22no—
246Q8×C61C22⋊Q814yes—
247D4×C61—22no—
248D4×C61—22no—
249D4×C61—48no—
250D4×C61—41no—
251D4×C61—41no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—21no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—48no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61C22⋊Q812yes—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×Q811yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61H(5,𝔽2)11yescharacteristic
274C2×C4×C61H(5,𝔽2)11yescharacteristic
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—22no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—21no—
286C2×Dic61—21no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61—41no—
297C2×Dic61—41no—
298C2×Dic61#13512yes—
299Dic122—84no—
300C8×C61—41no—
301C8×C61—41no—
302#92418—22no—
303#92418—41no—
304#25951—611nomaximal, Sylow
305C22×C61—22no—
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61—42no—
313C22×C61—42no—
314C22×C61—42no—
315C22×C61—81no—
316C22×C61—81no—
317C22×C61—81no—
318C22×C61—81no—
319C22×C61C2×C22⋊Q811yescharacteristic
320C22×C61D4211yescharacteristic
321C22×C61#54611yescharacteristic
322C22×C61C2×#13511yescharacteristic
323C22×C61#57511yescharacteristic
324C22×C61#57711yescharacteristic
325C22×C61#57711yescharacteristic
326C4×C61—22no—
327C4×C61—24no—
328C4×C61—81no—
329C4×C61—81no—
330C4×C61C2×C22⋊Q811yescharacteristic
331C4×C61#54311yescharacteristic
332C4×C61#54611yescharacteristic
333C4×C61#55111yescharacteristic
334Dic61—22no—
335Dic61—41no—
336Dic61—42no—
337Dic61—41no—
338Dic61—41no—
339Dic61—81no—
340Dic61—81no—
341Dic61—81no—
342C2×#475—611no—
343C2×#416—611no—
344#2439—611no—
345#2341—611no—
346#2092—611no—
347#2065—611no—
348#2065—611no—
349#2023—611no—
350#1865—611no—
351#1863—611no—
352#1854—611no—
353#1850—611no—
354#1807—611no—
355#1805—611no—
356#1788D6111yescharacteristic
357C2×C61—42no—
358C2×C61—82no—
359C2×C61#243711yescharacteristic
360C2×C61#307611yescharacteristic
361C2×C61#308411yescharacteristic
362C2×C61#327111yescharacteristic
363C2×C61#328211yescharacteristic
364C2×C61#330411yescharacteristic
365C2×C61#332111yescharacteristic
366C22×#41C2×D6111yescharacteristic
367C23×C8—611no—
368C2×C4⋊Q8—611no—
369C2×C4×Q8—611no—
370C2×C4×Q8—611no—
371C2×#117—611no—
372#475—1224no—
373C2×#107—611no—
374C2×#106—611no—
375C2×#106—611no—
376C2×#106—611no—
377C2×C4⋊C8—611no—
378C2×C4⋊C8—611no—
379#416—1224no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103C2×D6111yescharacteristic
383C2×#103C2×D6111yescharacteristic
384C2×#102—611no—
385C2×#102—611no—
386C2×#102C2×D6111yescharacteristic
387C2×#102C2×D6111yescharacteristic
388C2×C22⋊C8C2×D6111yescharacteristic
389#396—1221no—
390#393—1221no—
391#391—1221no—
392#390—611no—
393#390—611no—
394#390—1221no—
395#390—1221no—
396#389—611no—
397#389—611no—
398#387—1221no—
399#382—611no—
400#380—611no—
401#380—1221no—
402#380C2×D6111yescharacteristic
403C4×C22⋊C4—611no—
404C4×C22⋊C4—1221no—
405#340—611no—
406#340—611no—
407#340—611no—
408#340—611no—
409#336—611no—
410#336—611no—
411C61#2595111yescharacteristic, complemented, Sylow
412C2×#41—22no—
413C2×#41—22no—
414C2×#41—22no—
415C2×#41—41no—
416C2×#41—41no—
417C2×#41Dic12214yescomplemented
418C22×Q8—1221no—
419C22×Q8C22×D6111yescharacteristic
420C22×D4—21no—
421C22×D4C22×D6111yescharacteristic
422C23×C4—21no—
423C23×C4C22×D6111yescharacteristic
424C22×C8—611no—
425C22×C8—611no—
426C22×C8—1222no—
427C22×C8—1222no—
428C22×C8—2441no—
429C22×C8C22×D6111yescharacteristic
430C22×C8C22×D6111yescharacteristic
431C4⋊Q8—1222no—
432C4⋊Q8—2441no—
433C4×Q8—1222no—
434C4×Q8—1222no—
435C4×Q8—2441no—
436C4×Q8—2441no—
437#117—1222no—
438#117—2441no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C22⋊C4—21no—
451C2×C22⋊C4—611no—
452C2×C22⋊C4—611no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—1221no—
462C2×C42—1221no—
463C2×C42—1221no—
464#107—2441no—
465#106—1222no—
466#106—1222no—
467#106—2441no—
468C4⋊C8—1222no—
469C4⋊C8—2441no—
470#103—22no—
471#103—22no—
472#103—614no—
473#103—1222no—
474#102—41no—
475#102—41no—
476#102—1222no—
477#102—1222no—
478C22⋊C8—22no—
479#95—21no—
480#95—611no—
481#95—611no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487C24—21no—
488C24D4×D6111yescharacteristic
489#41—28no—
490#41—44no—
491#41—44no—
492#41—44no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—4881no—
497C2×Q8—2442no—
498C2×Q8—2442no—
499C2×Q8C2×Dic12212yes—
500C2×Q8D4×D6112yes—
501C2×Q8#1175822412yes—
502C2×D4—24no—
503C2×D4—21no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—42no—
509C2×D4—42no—
510C2×D4C2×Dic12212yes—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—24no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4C2×Dic12212yes—
530C22×C4D4×D6111yescharacteristic
531C22×C4#1175822412yes—
532C22×C4C23×D6111yescharacteristic
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—612no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—2444no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—612no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—4881no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C4⋊C4—2442no—
562C4⋊C4—2442no—
563C22⋊C4—81no—
564C22⋊C4—1222no—
565C22⋊C4—4881no—
566C22⋊C4—4881no—
567C22⋊C4—4881no—
568C22⋊C4—2442no—
569C22⋊C4—2441no—
570C22⋊C4—2442no—
571C22⋊C4—2442no—
572C42—1221no—
573C42—1221no—
574C42—1221no—
575C42—1221no—
576C42—2442no—
577C42—2442no—
578C42—2442no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—22no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23#1451975412yes—
593C23C2×D4×D6111yescharacteristic
594Q8—22no—
595Q8—22no—
596Q8—22no—
597Q8—4884no—
598Q8#1451975414yes—
599D4—22no—
600D4—22no—
601D4—48no—
602D4—41no—
603D4—41no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—48no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—612no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1222no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—2441no—
634C2×C4—2441no—
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#1451975412yes—
645C2×C4C2×D4×D6111yescharacteristic
646C2×C4C22×Dic12211yescharacteristic
647C2×C4C2×#1175822411yescharacteristic
648C2×C4C2×#1175822411yescharacteristic
649C2×C4#1451982211yescharacteristic
650C2×C4#1451982211yescharacteristic
651C8—41no—
652C8—41no—
653C8—1222no—
654C8—2441no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22C2×#1451975411yescharacteristic
670C22#1684039411yescharacteristic
671C22#1684039911yescharacteristic
672C22#135×D6111yescharacteristic
673C22#1684055911yescharacteristic
674C22#1684057111yescharacteristic
675C22#1684057211yescharacteristic
676C4—22no—
677C4—24no—
678C4—81no—
679C4—81no—
680C4—1222no—
681C4—4881no—
682C4—4881no—
683C4—4881no—
684C4—2441no—
685C4—2441no—
686C4—2441no—
687C4—2442no—
688C4C2×#1451975411yescharacteristic
689C4#1684038811yescharacteristic
690C4#1684039911yescharacteristic
691C4#1684044311yescharacteristic
692C2—42no—
693C2—82no—
694C2#2144948511yescharacteristic
695C2#2145560311yescharacteristic
696C2#2145568511yescharacteristic
697C2#2145740811yescharacteristic
698C2#2145752911yescharacteristic
699C2#2145775111yescharacteristic
700C2#2145800911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1788C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenter#1788D61
hypocenterC61#25951
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
1. C2×#16840045C2semidirect
3. #21449511C2semidirect
5. #21447338C2semidirect
6. #21447191C2semidirect
7. #21447189C2semidirect
8. #21446953C2semidirect
9. #21446085C2semidirect
10. #21446075C2semidirect
11. #21446020C2semidirect
12. #21445996C2semidirect
13. #21445842C2semidirect
14. #21445832C2semidirect
119. C2×#41×C61Q8semidirect
411. C61#25951semidirect
417. C2×#41Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168400452
C2×#168397582
#214495112
#214486602
#214473382
#214471912
#214471892
#214469532
#214460852
#214460752
#214460202
#214459962
#214458422
#214458322
C61×#17882
#2595161