Finite Group Databasebeta

← #29709583

Subgroups — #29709583

Order 15616.

874 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×#16839869C211yesmaximal, characteristic, complemented
2C2×#16839818C211yesmaximal, characteristic
3#21448357C211yesmaximal, characteristic
4#21448357C211yesmaximal, characteristic
5#21447317C211yesmaximal, characteristic, complemented
6#21447254C211yesmaximal, characteristic
7#21447214C211yesmaximal, characteristic, complemented
8#21447200C211yesmaximal, characteristic, complemented
9#21447020C211yesmaximal, characteristic
10#21447013C211yesmaximal, characteristic
11#21446826C211yesmaximal, characteristic
12#21445715C211yesmaximal, characteristic, complemented
13#21445651C211yesmaximal, characteristic, complemented
14#21445651C211yesmaximal, characteristic, complemented
15C61×#2056C211yesmaximal, characteristic, complemented
16C2×#14519753C2211yescharacteristic
17C2×#14519753C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×C42×D61C2211yescharacteristic
21#16839869—24no—
22#16839818—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519689C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839649—21no—
38#16839649—21no—
39#16839636—21no—
40#16839636—21no—
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45#16839613—21no—
46#16839613—21no—
47#16839609—21no—
48#16839609—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#391C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519753—22no—
67#14519753—22no—
68#14519753—22no—
69#14519753—22no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—22no—
73Q8×Dic61—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C42×D61—22no—
96C42×D61—22no—
97#14519710—22no—
98#14519709—22no—
99#14519708—22no—
100C4⋊#92418—22no—
101#14519701—22no—
102#14519701—22no—
103#14519701—22no—
104#14519701—22no—
105#14519689—22no—
106#14519689—22no—
107#14519687—22no—
108#14519687D414yes—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115C22×Q8×C61C2311yescharacteristic
116C22×Q8×C61C2311yescharacteristic
117C2×Q16×C61—22no—
118C2×Q16×C61—22no—
119C2×Q16×C61—22no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C4⋊C8×C61—22no—
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×D61C2×D411yescharacteristic
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61#4112yes—
144C2×Q8×C61#4112yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—41no—
170C2×C4×D61—41no—
171Q16×C61—28no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D412yes—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184#11758198—24no—
185#11758198—24no—
186#11758198—42no—
187#11758198—42no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—42no—
195C4⋊Dic61—42no—
196C4⋊Dic61#4112yes—
197C4⋊Dic61#4112yes—
198#11758196—24no—
199#11758196—24no—
200#11758196—42no—
201#11758196—42no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C4×Dic61—42no—
208C4×Dic61C2×D412yes—
209C4×Dic61#4112yes—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D412yes—
216C23×C61C22×D411yescharacteristic
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61#13712yes—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—22no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61—41no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
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×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61#13712yes—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—41no—
266C2×Dic61—41no—
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#12112yes—
276C2×Dic61#12112yes—
277C4×D61—44no—
278C4×D61—44no—
279C4×D61—41no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—41no—
283C8×C61—22no—
284C8×C61—41no—
285#92418—22no—
286#92418—41no—
287#23709—611nomaximal, Sylow
288C22×C61C2×#12111yescharacteristic
289C22×C61#52511yescharacteristic
290C22×C61#52511yescharacteristic
291C22×C61C2×#13711yescharacteristic
292C22×C61C2×#13711yescharacteristic
293C22×C61#57511yescharacteristic
294C22×C61#57511yescharacteristic
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—41no—
302C4×C61—22no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61#52511yescharacteristic
309C4×C61#52911yescharacteristic
310C4×C61#52911yescharacteristic
311C4×C61#53211yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—42no—
315Dic61—42no—
316Dic61—41no—
317Dic61—81no—
318Dic61—81no—
319C2×#441—611no—
320C2×#432—611no—
321C4×C22⋊Q8—611no—
322C4×C22⋊Q8—611no—
323#2088—611no—
324#2077—611no—
325#2068—611no—
326#2066—611no—
327#2056D6111yescharacteristic, complemented
328#2036—611no—
329#2035—611no—
330#2001—611no—
331#1776—611no—
332#1763—611no—
333#1763—611no—
334C2×C61#239211yescharacteristic
335C2×C61#305611yescharacteristic
336C2×C61#305611yescharacteristic
337C2×C61#311111yescharacteristic
338C2×C61#311311yescharacteristic
339C2×C61#311611yescharacteristic
340C2×C61#311711yescharacteristic
341D61—42no—
342C22×Q16C2×D6111yescharacteristic
343C2×C22⋊Q8—611no—
344C2×C22⋊Q8—611no—
345C2×C4×Q8—611no—
346C2×C4×Q8—611no—
347C22×C42—611no—
348#441—1224no—
349#432—1224no—
350C2×#107—611no—
351C2×#106—611no—
352C2×#106—611no—
353C2×C4⋊C8—611no—
354C2×C4⋊C8C2×D6111yescharacteristic
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103C2×D6111yescharacteristic
360C2×#103C2×D6111yescharacteristic
361C2×C22⋊C8—611no—
362C2×C22⋊C8—611no—
363C2×#97—611no—
364C2×#97—611no—
365#391C2×D6111yescharacteristic
366#391C2×D6111yescharacteristic
367#389—611no—
368#389—611no—
369#387—1221no—
370#387—1221no—
371#385—1221no—
372#385—1221no—
373#380—611no—
374#380—611no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4—611no—
377C4×C4⋊C4—1221no—
378C4×C4⋊C4—1221no—
379C4×C22⋊C4—1221no—
380C4×C22⋊C4—1221no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340C2×D6111yescharacteristic
387C61#2370911yescharacteristic, complemented, Sylow
388C22×Q8C22×D6111yescharacteristic
389C22×Q8C22×D6111yescharacteristic
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—611no—
393C2×Q16—22no—
394C2×Q16—22no—
395C2×Q16—22no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8C22×D6111yescharacteristic
399C22×C8C22×D6111yescharacteristic
400C22⋊Q8—1222no—
401C22⋊Q8—1222no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—1222no—
407C4×Q8—1222no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—1222no—
428C2×C42—1222no—
429C2×C42—1221no—
430C2×C42—1221no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42C22×D6111yescharacteristic
434#107—1222no—
435#106—1222no—
436#106—1222no—
437C4⋊C8—22no—
438C4⋊C8—1222no—
439#103—22no—
440#103—22no—
441#103—1222no—
442#103—1222no—
443#103—1222no—
444#103—1222no—
445C22⋊C8—1222no—
446C22⋊C8—1222no—
447#97—614no—
448#97—1222no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459C24—611no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8#1175822412yes—
471C2×Q8#1175822412yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—2441no—
494C22×C4—2441no—
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498Q16—28no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—612no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8D4×D6112yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—612no—
516C4⋊C4—612no—
517C4⋊C4—1224no—
518C4⋊C4—1224no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4—2442no—
529C22⋊C4—1224no—
530C22⋊C4—1224no—
531C22⋊C4—2442no—
532C22⋊C4—2442no—
533C42—21no—
534C42—21no—
535C42—612no—
536C42—612no—
537C42—1222no—
538C42—1222no—
539C42—2442no—
540C42—2442no—
541C42—2442no—
542C42—2442no—
543C23—612no—
544C23—1221no—
545C23—1221no—
546C23—1221no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23C2×D4×D6111yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8—41no—
556Q8—41no—
557Q8—41no—
558Q8—41no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4#1451980912yes—
604C2×C4C2×D4×D6111yescharacteristic
605C2×C4#41×D6111yescharacteristic
606C2×C4#41×D6111yescharacteristic
607C2×C4#41×D6111yescharacteristic
608C2×C4C2×#1175822411yescharacteristic
609C2×C4C2×#1175822411yescharacteristic
610C8—22no—
611C8—41no—
612C8—1222no—
613C8—2441no—
614C22—1222no—
615C22—1222no—
616C22—1222no—
617C22—2441no—
618C22—2441no—
619C22—2441no—
620C22—2441no—
621C22#121×D6111yescharacteristic
622C22#1684027211yescharacteristic
623C22#1684027211yescharacteristic
624C22#137×D6111yescharacteristic
625C22C2×#1451980911yescharacteristic
626C22#1684055911yescharacteristic
627C22#1684056111yescharacteristic
628C4—22no—
629C4—41no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—1222no—
635C4—1222no—
636C4—4881no—
637C4—4881no—
638C4—2442no—
639C4—2441no—
640C4—2442no—
641C4#1684026811yescharacteristic
642C4#1684028911yescharacteristic
643C4#1684028911yescharacteristic
644C4#1684031111yescharacteristic
645C2—2442no—
646C2#2144905811yescharacteristic
647C2#2145544611yescharacteristic
648C2#2145544811yescharacteristic
649C2#2145592711yescharacteristic
650C2#2145594811yescharacteristic
651C2#2145598011yescharacteristic
652C2#2145598911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2056C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2056D61
hypocenterC61#23709
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
1. C2×#16839869C2semidirect
5. #21447317C2semidirect
7. #21447214C2semidirect
8. #21447200C2semidirect
12. #21445715C2semidirect
13. #21445651C2semidirect
14. #21445651C2semidirect
15. C61×#2056C2semidirect
327. #2056D61semidirect
387. C61#23709semidirect

Maximal subgroups

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

subgroupindex
C2×#168398692
C2×#168398182
#214483572
#214473172
#214472542
#214472142
#214472002
#214470202
#214470132
#214468262
#214457152
#214456512
C61×#20562
#2370961