Finite Group Databasebeta

← #29729862

Subgroups — #29729862

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840111C211yesmaximal, characteristic, complemented
2C2×#16839966C211yesmaximal, characteristic
3#21450991C211yesmaximal, characteristic
4#21448876C211yesmaximal, characteristic, complemented
5#21447241C211yesmaximal, characteristic, complemented
6#21447138C211yesmaximal, characteristic
7#21447134C211yesmaximal, characteristic
8#21446765C211yesmaximal, characteristic, complemented
9#21446536C211yesmaximal, characteristic, complemented
10#21446173C211yesmaximal, characteristic
11#21446172C211yesmaximal, characteristic
12#21445832C211yesmaximal, characteristic, complemented
13#21445831C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic
15C61×#2074C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519776C2211yescharacteristic
19C2×#14519775C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23#16840111—24no—
24#16839966—24no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839675—21no—
36#16839675—21no—
37#16839667C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839663C2211yescharacteristic
40#16839659—21no—
41#16839649—21no—
42#16839644—21no—
43#16839627C2211yescharacteristic
44#16839621—21no—
45#16839621—21no—
46C4⋊C4×Dic61C2211yescharacteristic
47C4×#11758196C2211yescharacteristic
48#16839613—21no—
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#382C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×#11758190—22no—
66C2×#11758190—22no—
67C2×#11758190—41no—
68C2×#11758201—22no—
69C2×#11758201—22no—
70C2×#11758201—41no—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519776—22no—
74#14519776—22no—
75#14519775—22no—
76#14519775D414yes—
77Q8×Dic61—22no—
78Q8×Dic61—41no—
79C4×Dic122—22no—
80C4×Dic122—41no—
81#14519739—22no—
82#14519739—22no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196D411yescharacteristic
91C2×#11758196D411yescharacteristic
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C4⋊#92418—41no—
103#14519706—22no—
104#14519706—41no—
105#14519702—22no—
106#14519701—22no—
107#14519700—22no—
108#14519700—22no—
109#14519699—22no—
110#14519697—22no—
111C8×Dic61—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684D411yescharacteristic
118#14519684D411yescharacteristic
119C22×Q8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126#103×C61—22no—
127#103×C61—22no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C23×D61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61C2×D412yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244#4112yes—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—42no—
168C2×Dic122—81no—
169C2×Dic122—81no—
170C2×Dic122C2×D412yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185#11758201—44no—
186#11758198—42no—
187#11758198—81no—
188#11758198—81no—
189#11758198—81no—
190#11758198—81no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—42no—
195C4⋊Dic61#4112yes—
196#11758196—24no—
197#11758196—24no—
198#11758196—22no—
199#11758196—22no—
200#11758196—42no—
201#11758196—42no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C4×Dic61C2×D412yes—
208C4×Dic61#4112yes—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215#11758190—44no—
216C23×C61C22×D411yescharacteristic
217C22×D61—22no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—41no—
227Q8×C61—41no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61#12112yes—
270C2×Dic61#12112yes—
271C2×Dic61#13512yes—
272C2×Dic61#13712yes—
273D244—22no—
274D244—22no—
275D244—41no—
276D244—41no—
277C4×D61—84no—
278Dic122—22no—
279Dic122—22no—
280Dic122—41no—
281Dic122—41no—
282Dic122—84no—
283C8×C61—41no—
284C8×C61—41no—
285#92418—41no—
286#92418—41no—
287#25319—611nomaximal, Sylow
288C22×C61C2×#12111yescharacteristic
289C22×C61#54611yescharacteristic
290C22×C61#54811yescharacteristic
291C22×C61C2×#13511yescharacteristic
292C22×C61C2×#13711yescharacteristic
293C22×C61#57711yescharacteristic
294C22×C61#57711yescharacteristic
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—81no—
302C4×C61—41no—
303C4×C61—42no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61#52911yescharacteristic
309C4×C61#54211yescharacteristic
310C4×C61#54611yescharacteristic
311C4×C61#54711yescharacteristic
312Dic61—22no—
313Dic61—41no—
314Dic61—42no—
315Dic61—41no—
316Dic61—42no—
317Dic61—81no—
318Dic61#46414yes—
319C2×#487—611no—
320C2×#461—611no—
321#2563—611no—
322#2365—611no—
323#2074—611no—
324#2074D6111yescharacteristic, complemented
325#2055—611no—
326#2055—611no—
327#1987—611no—
328#1939—611no—
329#1880—611no—
330#1880—611no—
331#1805—611no—
332#1805—611no—
333C4×#102—611no—
334C2×C61#242711yescharacteristic
335C2×C61C2×#46411yescharacteristic
336C2×C61#304311yescharacteristic
337C2×C61#330311yescharacteristic
338C2×C61#332911yescharacteristic
339C2×C61#336011yescharacteristic
340C2×C61#337211yescharacteristic
341D61—82no—
342C22×SD16—611no—
343C22×SD16—611no—
344C2×C4⋊Q8—611no—
345C2×C4⋊Q8—611no—
346C2×C4×Q8—611no—
347C2×C4×Q8—611no—
348C2×C4×D4—611no—
349#487—1224no—
350#461—1224no—
351C2×C4⋊C8—611no—
352C2×C4⋊C8—611no—
353C2×C4⋊C8—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103C2×D6111yescharacteristic
359C2×#103C2×D6111yescharacteristic
360C2×#102—611no—
361C2×#102—611no—
362C2×C4×C8—611no—
363#393—1221no—
364#393—1221no—
365#391—611no—
366#391C2×D6111yescharacteristic
367#390—611no—
368#390—611no—
369#389—1221no—
370#387—1221no—
371#387—1221no—
372#382—611no—
373#382C2×D6111yescharacteristic
374#380—1221no—
375#380—1221no—
376C4×C4⋊C4—611no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—1221no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340C2×D6111yescharacteristic
384#340C2×D6111yescharacteristic
385#336—611no—
386#336C2×D6111yescharacteristic
387C61#2531911yescharacteristic, complemented, Sylow
388C22×Q8—611no—
389C22×Q8—1221no—
390C22×Q8C22×D6111yescharacteristic
391C22×D4—611no—
392C23×C4—1221no—
393C2×SD16—1222no—
394C2×SD16—1222no—
395C2×SD16—1222no—
396C2×SD16—1222no—
397C2×SD16—2441no—
398C2×SD16—2441no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8C22×D6111yescharacteristic
402C22×C8C22×D6111yescharacteristic
403C4⋊Q8—614no—
404C4⋊Q8—1222no—
405C4⋊Q8—1222no—
406C4⋊Q8—1222no—
407C4×Q8—1222no—
408C4×Q8—1222no—
409C4×Q8—2441no—
410C4×Q8—2441no—
411C4×D4—1222no—
412C4×D4—1222no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—1221no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42C22×D6111yescharacteristic
436C4⋊C8—1222no—
437C4⋊C8—2441no—
438C4⋊C8—2441no—
439#103—22no—
440#103—22no—
441#103—1222no—
442#103—1222no—
443#103—1222no—
444#103—1222no—
445#102—1222no—
446#102—1222no—
447C4×C8—1222no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—611no—
453#95—611no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459C24—1221no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—612no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—4881no—
472C2×Q8—4881no—
473C2×Q8—2442no—
474C2×Q8D4×D6112yes—
475C2×D4—612no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498SD16—2444no—
499SD16—2444no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—612no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1224no—
522C4⋊C4—1224no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4—2442no—
529C22⋊C4—4881no—
530C22⋊C4—4881no—
531C22⋊C4—4881no—
532C22⋊C4—4881no—
533C22⋊C4—2442no—
534C42—21no—
535C42—21no—
536C42—612no—
537C42—612no—
538C42—1221no—
539C42—1221no—
540C42—2442no—
541C42—2442no—
542C42—2442no—
543C23—1222no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23—2441no—
549C23—2441no—
550C23C2×D4×D6111yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—41no—
554Q8—41no—
555Q8—1222no—
556Q8—1222no—
557Q8—4884no—
558Q8—2441no—
559Q8—2441no—
560D4—1222no—
561D4—1222no—
562D4—2441no—
563D4—2441no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—612no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1222no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—4884no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4C2×D4×D6111yescharacteristic
605C2×C4#41×D6111yescharacteristic
606C2×C4#41×D6111yescharacteristic
607C2×C4#41×D6111yescharacteristic
608C2×C4#1451982311yescharacteristic
609C2×C4#1451982411yescharacteristic
610C8—41no—
611C8—41no—
612C8—2441no—
613C8—2441no—
614C22—4881no—
615C22—4881no—
616C22—4881no—
617C22—4881no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22#121×D6111yescharacteristic
622C22#1684040411yescharacteristic
623C22#1684042011yescharacteristic
624C22#135×D6111yescharacteristic
625C22#137×D6111yescharacteristic
626C22#1684056911yescharacteristic
627C22#1684057111yescharacteristic
628C4—41no—
629C4—42no—
630C4—41no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—614no—
635C4—1222no—
636C4—4881no—
637C4—2441no—
638C4—2442no—
639C4—2442no—
640C4—2441no—
641C4#1684029511yescharacteristic
642C4#1684037811yescharacteristic
643C4#1684040411yescharacteristic
644C4#1684041011yescharacteristic
645C2—4882no—
646C2#2144939111yescharacteristic
647C2#464×D6111yescharacteristic
648C2#2145533811yescharacteristic
649C2#2145774511yescharacteristic
650C2#2145811111yescharacteristic
651C2#2145846511yescharacteristic
652C2#2145859711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2074C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2074D61
hypocenterC61#25319
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2074

Complements

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

NH ≅ G/Ntype
1. C2×#16840111C2semidirect
4. #21448876C2semidirect
5. #21447241C2semidirect
8. #21446765C2semidirect
9. #21446536C2semidirect
12. #21445832C2semidirect
13. #21445831C2semidirect
15. C61×#2074C2semidirect
324. #2074D61semidirect
387. C61#25319semidirect

Maximal subgroups

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

subgroupindex
C2×#168401112
C2×#168399662
#214509912
#214488762
#214472412
#214471382
#214471342
#214467652
#214465362
#214461732
#214461722
#214458322
#214458312
#214456332
C61×#20742
#2531961