Finite Group Databasebeta

← #29709210

Subgroups — #29709210

Order 15616.

880 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758202C211yesmaximal, characteristic, complemented
2C2×C4×#11758200C211yesmaximal, characteristic
3#21448296C211yesmaximal, characteristic, complemented
4#21447301C211yesmaximal, characteristic, complemented
5#21447238C211yesmaximal, characteristic, complemented
6#21447207C211yesmaximal, characteristic
7#21447200C211yesmaximal, characteristic, complemented
8#21447141C211yesmaximal, characteristic, complemented
9#21447136C211yesmaximal, characteristic
10#21447055C211yesmaximal, characteristic
11#21446913C211yesmaximal, characteristic
12C4×C4⋊#92418C211yesmaximal, characteristic, complemented
13C4×#14519701C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic
15C4×#124×C61C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C4×#11758202—24no—
20C4×#11758200—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839646—21no—
40#16839646—21no—
41C4×#11758196—21no—
42C4×C4⋊Dic61C2211yescharacteristic
43#16839608—21no—
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×#124×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C61×#391C2211yescharacteristic
54C61×#390C2211yescharacteristic
55C61×#388—21no—
56C61×#388—21no—
57C61×#382C2211yescharacteristic
58C4×C22⋊C4×C61—21no—
59C4×C22⋊C4×C61—21no—
60C43×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758202—22no—
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758200—22no—
66C2×#11758200—22no—
67C2×#11758200—22no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C4×Dic122—22no—
73C4×Dic122D414yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82#14519710—22no—
83#14519708—22no—
84C4⋊#92418—22no—
85C4⋊#92418—22no—
86#14519701—22no—
87#14519701—22no—
88#14519701—22no—
89#14519700—22no—
90#14519700—22no—
91#14519698—22no—
92#14519698—22no—
93#14519698—22no—
94C4×#92418—22no—
95C4×#92418—22no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×Q8×C61C2311yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61—21no—
106#124×C61—22no—
107#124×C61—22no—
108C4×Q8×C61—22no—
109C4×Q8×C61—22no—
110C4×D4×C61—22no—
111C4×D4×C61—22no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61D411yescharacteristic
122C2×C42×C61D411yescharacteristic
123C2×C42×C61D411yescharacteristic
124C2×C42×C61D411yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C24×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61#4112yes—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61#4112yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—42no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122C2×D412yes—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—42no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—42no—
176C22⋊C4×C61—42no—
177C22⋊C4×C61—42no—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61—22no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—41no—
186C42×C61—42no—
187C42×C61C2×D412yes—
188C42×C61#4112yes—
189#11758202—28no—
190#11758200—28no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61C2×D412yes—
197#11758196—24no—
198#11758196—42no—
199#11758196—42no—
200C4×Dic61—24no—
201C4×Dic61—42no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418#4112yes—
213C2×#92418#4112yes—
214C23×C61—22no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61C22×D411yescharacteristic
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—41no—
225Q8×C61—41no—
226D4×C61—22no—
227D4×C61—22no—
228D4×C61—41no—
229D4×C61—41no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—84no—
253C2×C4×C61#12112yes—
254C2×C4×C61#12112yes—
255C2×C4×C61#13512yes—
256C2×C4×C61#13512yes—
257C2×C4×C61#13712yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×Dic61—22no—
265C2×Dic61—22no—
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—41no—
276Dic122—22no—
277Dic122—22no—
278Dic122—41no—
279Dic122—41no—
280#92418—22no—
281#92418—22no—
282#92418—41no—
283#92418—41no—
284#23681—611nomaximal, Sylow
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61C2×#12111yescharacteristic
293C22×C61#52511yescharacteristic
294C22×C61#53211yescharacteristic
295C22×C61C2×#13511yescharacteristic
296C22×C61C2×#13511yescharacteristic
297C22×C61C2×#13711yescharacteristic
298C22×C61#57511yescharacteristic
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—42no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61#46414yes—
308C4×C61#52511yescharacteristic
309C4×C61#52511yescharacteristic
310C4×C61#52911yescharacteristic
311C4×C61#52911yescharacteristic
312Dic61—42no—
313Dic61—42no—
314Dic61—81no—
315Dic61—81no—
316C2×C4×Q16—611no—
317C2×C4×SD16—611no—
318C4×#124D6111yescharacteristic
319#2295—611no—
320#2085—611no—
321#2074—611no—
322#2067—611no—
323#2066—611no—
324#2056—611no—
325#2055—611no—
326#2042—611no—
327#2015—611no—
328C4×C4⋊C8—611no—
329C4×#103—611no—
330C4×#102—611no—
331C2×C61—82no—
332C2×C61#239211yescharacteristic
333C2×C61C2×#46411yescharacteristic
334C2×C61#303511yescharacteristic
335C2×C61#310211yescharacteristic
336C2×C61#310711yescharacteristic
337C2×C61#316811yescharacteristic
338C2×C61#317211yescharacteristic
339C22×Q16—611no—
340C22×SD16—611no—
341C2×#124C2×D6111yescharacteristic
342C2×C4×Q8—611no—
343C2×C4×Q8C2×D6111yescharacteristic
344C2×C4×D4C2×D6111yescharacteristic
345C4×Q16—1224no—
346C4×SD16—1224no—
347C2×#107—611no—
348C2×#106—611no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×C4×C8—611no—
360C2×C4×C8—611no—
361#391—611no—
362#391—611no—
363#391C2×D6111yescharacteristic
364#390C2×D6111yescharacteristic
365#389—611no—
366#389—611no—
367#388—21no—
368#388—21no—
369#387—1221no—
370#387—1221no—
371#382C2×D6111yescharacteristic
372C4×C4⋊C4—611no—
373C4×C4⋊C4—1221no—
374C4×C22⋊C4—21no—
375C4×C22⋊C4—21no—
376#376—1221no—
377C43C2×D6111yescharacteristic
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#336—611no—
383#336—611no—
384C61#2368111yescharacteristic, complemented, Sylow
385C22×Q8—611no—
386C22×Q8C22×D6111yescharacteristic
387C22×D4C22×D6111yescharacteristic
388C23×C4—21no—
389C2×Q16—1222no—
390C2×Q16—1222no—
391C2×Q16—1222no—
392C2×SD16—1222no—
393C2×SD16—1222no—
394C2×SD16—1222no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—611no—
399#124—22no—
400#124—22no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—614no—
404C4×Q8—1222no—
405C4×D4—22no—
406C4×D4—22no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C42—21no—
422C2×C42—1221no—
423C2×C42—1221no—
424C2×C42C22⋊D6111yescharacteristic
425C2×C42C22⋊D6111yescharacteristic
426C2×C42C22⋊D6111yescharacteristic
427C2×C42C22⋊D6111yescharacteristic
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430C2×C42C22×D6111yescharacteristic
431#107—1222no—
432#106—1222no—
433C4⋊C8—1222no—
434C4⋊C8—1222no—
435#103—1222no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#102—1222no—
441#102—1222no—
442#102—1222no—
443C4×C8—1222no—
444C4×C8—1222no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456C24—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—612no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8#1175822212yes—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4#1175822212yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492Q16—1228no—
493SD16—1228no—
494C2×C8—612no—
495C2×C8—612no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—612no—
512C4⋊C4—1221no—
513C4⋊C4—1224no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—41no—
533C42—42no—
534C42—1224no—
535C42—2442no—
536C42D4×D6112yes—
537C42#1175822412yes—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C22×C22⋊D6111yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—1222no—
551Q8—1222no—
552Q8—2441no—
553Q8—2441no—
554D4—22no—
555D4—22no—
556D4—41no—
557D4—41no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
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—84no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4#1451974812yes—
594C2×C4#1451974812yes—
595C2×C4#1451979712yes—
596C2×C4#1451979712yes—
597C2×C4#1451980712yes—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4C2×#1175822411yescharacteristic
602C2×C4C2×#1175822211yescharacteristic
603C2×C4C2×#1175822211yescharacteristic
604C8—1222no—
605C8—1222no—
606C8—2441no—
607C8—2441no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22C2×#1451974811yescharacteristic
616C22#1684027311yescharacteristic
617C22#1684030811yescharacteristic
618C22C2×#1451979711yescharacteristic
619C22C2×#1451979711yescharacteristic
620C22C2×#1451980711yescharacteristic
621C22#1684056011yescharacteristic
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—4881no—
631C4—4881no—
632C4—2442no—
633C4—2442no—
634C4#1683998114yes—
635C4#1684027111yescharacteristic
636C4#1684027111yescharacteristic
637C4#1684029211yescharacteristic
638C4#1684029211yescharacteristic
639C2—82no—
640C2#2144905711yescharacteristic
641C2C2×#1683998111yescharacteristic
642C2#2145526811yescharacteristic
643C2#2145583711yescharacteristic
644C2#2145589011yescharacteristic
645C2#2145638311yescharacteristic
646C2#2145642611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#124×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC4×#124D61
hypocenterC61#23681
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. C2×C4×#11758202C2semidirect
3. #21448296C2semidirect
4. #21447301C2semidirect
5. #21447238C2semidirect
7. #21447200C2semidirect
8. #21447141C2semidirect
12. C4×C4⋊#92418C2semidirect
13. C4×#14519701C2semidirect
384. C61#23681semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582022
C2×C4×#117582002
#214482962
#214473012
#214472382
#214472072
#214472002
#214471412
#214471362
#214470552
#214469132
C4×C4⋊#924182
C4×#145197012
C4×#145196982
C4×#124×C612
#2368161