Finite Group Databasebeta

← #29688489

Subgroups — #29688489

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840144C211yesmaximal, characteristic
2C2×#16839872C211yesmaximal, characteristic
3C2×#16839867C211yesmaximal, characteristic, complemented
4C2×#16839862C211yesmaximal, characteristic, complemented
5C2×#16839860C211yesmaximal, characteristic, complemented
6C4×#14519765C211yesmaximal, characteristic, complemented
7#21446703C211yesmaximal, characteristic, complemented
8#21446699C211yesmaximal, characteristic, complemented
9#21446694C211yesmaximal, characteristic
10#21446689C211yesmaximal, characteristic, complemented
11#21446631C211yesmaximal, characteristic, complemented
12#21446628C211yesmaximal, characteristic, complemented
13#21445981C211yesmaximal, characteristic, complemented
14C4×#14519688C211yesmaximal, characteristic, complemented
15C4×#124×C61C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C22×#11758199C2211yescharacteristic
20C2×#14519765C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23#16840144C418yescomplemented
24#16839872—24no—
25#16839867—24no—
26#16839862—24no—
27#16839860—24no—
28C2×#14519710C2211yescharacteristic
29C2×#14519710C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31C2×#14519708C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519701C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519700C2211yescharacteristic
36C2×#14519698C2211yescharacteristic
37C2×#14519698C2211yescharacteristic
38C2×#14519697C2211yescharacteristic
39C2×#14519697C2211yescharacteristic
40C2×#14519688C2211yescharacteristic
41C2×#14519688C2211yescharacteristic
42C2×#14519688C2211yescharacteristic
43C2×#14519688C2211yescharacteristic
44C2×C4×#92418C2211yescharacteristic
45C2×C4×#92418C2211yescharacteristic
46#16839668C2211yescharacteristic
47#16839664C2211yescharacteristic
48#16839653—22no—
49#16839628C2211yescharacteristic
50C4×#11758198—22no—
51C2×#124×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C61×#391C2211yescharacteristic
55C61×#390C2211yescharacteristic
56C61×#388—22no—
57C61×#382C2211yescharacteristic
58C4×C22⋊C4×C61—22no—
59C43×C61C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202C2×C414yescomplemented
66C2×#11758200—22no—
67C2×#11758200—22no—
68C2×#11758200C2×C414yescomplemented
69C2×#11758201—22no—
70C2×#11758201—22no—
71C2×#11758201C2×C414yescomplemented
72C2×#11758199—22no—
73C2×#11758199—22no—
74C2×#11758199C2×C414yes—
75C22×#92418C2311yescharacteristic
76C22×#92418C2311yescharacteristic
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79#14519765—22no—
80#14519765C2×C414yescomplemented
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×D244—22no—
84C4×D244—22no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758198—22no—
89C2×#11758198—21no—
90C2×#11758198—22no—
91C2×C4×Dic61—21no—
92#14519710—22no—
93#14519710—22no—
94#14519708—22no—
95#14519708—22no—
96#14519701—22no—
97#14519701—22no—
98#14519700—22no—
99#14519700—22no—
100#14519698—22no—
101#14519698—22no—
102#14519697—22no—
103#14519697—22no—
104#14519688—22no—
105#14519688—22no—
106#14519688D414yes—
107#14519688D414yes—
108C4×#92418—22no—
109C4×#92418C2×C414yescomplemented
110#14519684—22no—
111#14519684—22no—
112C22×Q8×C61C2311yescharacteristic
113C22×D4×C61C2311yescharacteristic
114C23×C4×C61—21no—
115#124×C61—22no—
116#124×C61C2×C414yescomplemented
117C4×Q8×C61—22no—
118C4×Q8×C61—22no—
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—22no—
125C2×C22⋊C4×C61—22no—
126C2×C22⋊C4×C61—21no—
127C2×C42×C61—21no—
128C2×C42×C61D412yes—
129C2×C42×C61D412yes—
130C2×C42×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132C2×C42×C61C2311yescharacteristic
133#95×C61—22no—
134#95×C61—22no—
135C24×C61—21no—
136C23×D61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C412yes—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61C22×C412yes—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—44no—
153C22×C4×C61—42no—
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244C22×C412yes—
168C2×C4×D61—44no—
169C2×C4×D61—42no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122C22×C412yes—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—42no—
181C22⋊C4×C61—28no—
182C22⋊C4×C61—44no—
183C22⋊C4×C61—42no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—42no—
187C42×C61—42no—
188C42×C61—42no—
189C42×C61C22×C412yes—
190C42×C61C2×D412yes—
191C42×C61C2×D412yes—
192C42×C61#4112yes—
193#11758202—28no—
194#11758201—28no—
195#11758200—28no—
196#11758199—28no—
197#11758198—28no—
198#11758198—44no—
199#11758198—42no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204#11758196—42no—
205C4×Dic61—42no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418C22×C412yes—
211C2×#92418C22×C412yes—
212C2×#92418C2×D412yes—
213C2×#92418C2×D412yes—
214C2×#92418C2×D412yes—
215C2×#92418C2×D412yes—
216C2×#92418#4112yes—
217C2×#92418#4112yes—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—42no—
222C23×C61—42no—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—42no—
228C22×D61—42no—
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—41no—
232Q8×C61—41no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—41no—
236D4×C61—41no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—84no—
252C2×C4×C61C4×D414yes—
253C2×C4×C61#12712yes—
254C2×C4×C61#12712yes—
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268D244—22no—
269D244—22no—
270D244—41no—
271D244—41no—
272C4×D61—84no—
273Dic122—22no—
274Dic122—22no—
275Dic122—41no—
276Dic122—41no—
277#92418—22no—
278#92418—22no—
279#92418C4×D414yes—
280#92418C4×D414yes—
281#21693—611nomaximal, Sylow
282C22×C61—42no—
283C22×C61—44no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—82no—
287C22×C61C2×C4×D411yescharacteristic
288C22×C61C2×#12711yescharacteristic
289C22×C61#53211yescharacteristic
290C22×C61#57512yes—
291C22×C61#57512yes—
292C2×D61—42no—
293C2×D61—44no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—82no—
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—24no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61—81no—
303C4×C61—81no—
304C4×C61C2×C4×D411yescharacteristic
305C4×C61C2×C4×D411yescharacteristic
306C4×C61#53211yescharacteristic
307C4×C61#53211yescharacteristic
308Dic61—42no—
309Dic61—81no—
310Dic61—81no—
311C2×#495—611no—
312C2×#442—611no—
313C2×#441—611no—
314C2×#440—611no—
315C2×#440—611no—
316C4×#124—611no—
317C4×#124D6111yescharacteristic, complemented
318#1973—611no—
319#1973—611no—
320#1972—611no—
321#1971—611no—
322#1958—611no—
323#1958—611no—
324#1846—611no—
325C4×#97—611no—
326C2×C61—82no—
327C2×C61C4×#12711yescharacteristic
328C2×C61#294312yes—
329C2×C61#315311yescharacteristic
330C2×C61#315311yescharacteristic
331C2×C61#318312yes—
332D61—82no—
333C22×Q16—611no—
334C22×SD16—611no—
335C22×SD16—611no—
336C22×D8—611no—
337C2×#124—611no—
338C2×#124C2×D6111yescharacteristic
339C2×C4×Q8—611no—
340C2×C4×Q8C2×D6111yescharacteristic
341C2×C4×D4—611no—
342C2×C4×D4C2×D6111yescharacteristic
343#495—618no—
344#442—1224no—
345#441—1224no—
346#440—1224no—
347#440—1224no—
348C2×#107—611no—
349C2×#107—611no—
350C2×#106—611no—
351C2×#106—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#97—611no—
361C2×#97—611no—
362C2×#97—611no—
363C2×#97—611no—
364C2×C4×C8—611no—
365C2×C4×C8—611no—
366#391—611no—
367#391C2×D6111yescharacteristic
368#390—611no—
369#390C2×D6111yescharacteristic
370#388—22no—
371#388—1222no—
372#382—611no—
373#382C2×D6111yescharacteristic
374C4×C22⋊C4—22no—
375C4×C22⋊C4—1222no—
376C43C2×D6111yescharacteristic
377C61#2169311yescharacteristic, complemented, Sylow
378C22×Q8—611no—
379C22×Q8C22×D6111yescharacteristic
380C22×D4—611no—
381C22×D4C22×D6111yescharacteristic
382C23×C4—21no—
383C23×C4—1221no—
384C2×Q16—614no—
385C2×Q16—1222no—
386C2×Q16—1222no—
387C2×SD16—614no—
388C2×SD16—614no—
389C2×SD16—1222no—
390C2×SD16—1222no—
391C2×SD16—1222no—
392C2×SD16—1222no—
393C2×D8—614no—
394C2×D8—1222no—
395C2×D8—1222no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400#124—22no—
401#124—614no—
402#124—1222no—
403#124C4×D6114yescomplemented
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—1222no—
407C4×Q8—1222no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—1222no—
411C4×D4—1222no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4—22no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—1222no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1222no—
424C2×C42—21no—
425C2×C42—1221no—
426C2×C42C22⋊D6112yes—
427C2×C42C22⋊D6112yes—
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430C2×C42C22×D6111yescharacteristic
431#107—1222no—
432#107—1222no—
433#106—1222no—
434#106—1222no—
435#103—1222no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#102—1222no—
440#102—1222no—
441#102—1222no—
442#102—1222no—
443#97—614no—
444#97—614no—
445#97—1222no—
446#97—1222no—
447C4×C8—614no—
448C4×C8—1222no—
449#95—22no—
450#95—22no—
451#95—1222no—
452#95—1222no—
453C24—21no—
454C24—1221no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—612no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8C2×C4×D6112yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—612no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4C2×C4×D6112yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—2444no—
488C22×C4—2442no—
489C22×C4C2×C22⋊D6112yes—
490C22×C4C2×C22⋊D6112yes—
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C23×D6111yescharacteristic
494Q16—1228no—
495SD16—1228no—
496SD16—1228no—
497D8—1228no—
498C2×C8—612no—
499C2×C8—612no—
500C2×C8—612no—
501C2×C8—612no—
502C2×C8—612no—
503C2×C8—612no—
504C2×C8—612no—
505C2×C8—612no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—2442no—
520C22⋊C4—28no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—1228no—
524C22⋊C4—2444no—
525C22⋊C4—2442no—
526C42—21no—
527C42—21no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—2442no—
532C42C2×C4×D6112yes—
533C42#1175822212yes—
534C42D4×D6112yes—
535C42D4×D6112yes—
536C23—22no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—1222no—
542C23—2441no—
543C23—2442no—
544C23—2442no—
545C23—2441no—
546C23C22×C22⋊D6111yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—1222no—
552Q8—1222no—
553Q8—2441no—
554Q8—2441no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—1222no—
560D4—1222no—
561D4—2441no—
562D4—2441no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—84no—
578C2×C4—1222no—
579C2×C4—4884no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4C4×C22⋊D6114yes—
587C2×C4#1451977312yes—
588C2×C4#1451977312yes—
589C2×C4C22×C4×D6111yescharacteristic
590C2×C4C2×D4×D6111yescharacteristic
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4#41×D6111yescharacteristic
593C2×C4#41×D6111yescharacteristic
594C2×C4C2×#1175822211yescharacteristic
595C8—614no—
596C8—614no—
597C8—1222no—
598C8—1222no—
599C22—44no—
600C22—42no—
601C22—81no—
602C22—82no—
603C22—81no—
604C22—4881no—
605C22—4882no—
606C22—4881no—
607C22—2444no—
608C22—2442no—
609C22C2×C4×C22⋊D6111yescharacteristic
610C22C2×#1451977311yescharacteristic
611C22#1684030811yescharacteristic
612C22#1684056012yes—
613C22#1684056012yes—
614C4—24no—
615C4—22no—
616C4—22no—
617C4—42no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—4881no—
622C4—4881no—
623C4—2442no—
624C4C4×D4×D6111yescharacteristic
625C4C4×D4×D6111yescharacteristic
626C4#1684030911yescharacteristic
627C4#1684030911yescharacteristic
628C2—82no—
629C2—4882no—
630C2C4×#1451977311yescharacteristic
631C2#2145461712yes—
632C2#2145628311yescharacteristic
633C2#2145628711yescharacteristic
634C2#2145654212yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#124×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
cosocleC22×C4×C61C24
hypercenterC4×#124D61
hypocenterC61#21693
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
3. C2×#16839867C2semidirect
4. C2×#16839862C2semidirect
5. C2×#16839860C2semidirect
6. C4×#14519765C2semidirect
7. #21446703C2semidirect
8. #21446699C2semidirect
10. #21446689C2semidirect
11. #21446631C2semidirect
12. #21446628C2semidirect
13. #21445981C2semidirect
14. C4×#14519688C2semidirect
15. C4×#124×C61C2semidirect
23. #16840144C4semidirect
65. C2×#11758202C2×C4semidirect
68. C2×#11758200C2×C4semidirect
71. C2×#11758201C2×C4semidirect
80. #14519765C2×C4semidirect
109. C4×#92418C2×C4semidirect
116. #124×C61C2×C4semidirect
317. C4×#124D61semidirect
377. C61#21693semidirect
403. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401442
C2×#168398722
C2×#168398672
C2×#168398622
C2×#168398602
C4×#145197652
#214467032
#214466992
#214466942
#214466892
#214466312
#214466282
#214459812
C4×#145196882
C4×#124×C612
#2169361