Finite Group Databasebeta

← #29699352

Subgroups — #29699352

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839862C211yesmaximal, characteristic
2#21448792C211yesmaximal, characteristic
3#21448731C211yesmaximal, characteristic, complemented
4#21446513C211yesmaximal, characteristic
5#21446499C211yesmaximal, characteristic
6#21446409C211yesmaximal, characteristic, complemented
7#21446388C211yesmaximal, characteristic, complemented
8#21446171C211yesmaximal, characteristic
9#21446171C211yesmaximal, characteristic
10#21445954C211yesmaximal, characteristic, complemented
11#21445835C211yesmaximal, characteristic, complemented
12#21445696C211yesmaximal, characteristic, complemented
13#21445648C211yesmaximal, characteristic, complemented
14#21445638C211yesmaximal, characteristic, complemented
15C61×#1802C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20#16839862C418yescomplemented
21C2×#14519710C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519688C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839665C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839657—21no—
34#16839657—21no—
35#16839655—21no—
36#16839655—21no—
37#16839631—21no—
38#16839628C2211yescharacteristic
39#16839625—21no—
40C4⋊C4×Dic61—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C22⋊Dic61—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#384C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C2×#11758200—22no—
62C2×#11758200—22no—
63C2×#11758200C2×C414yes—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C4×Dic122—22no—
67C4×Dic122C2×C414yes—
68#14519742—22no—
69#14519742—22no—
70#14519742—41no—
71#14519742—41no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519710C2×C414yes—
94#14519702—22no—
95#14519702—22no—
96#14519700C2×C414yes—
97#14519698C2×C414yes—
98#14519688C2×C414yes—
99#14519687—41no—
100C8×Dic61—41no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C23×C4×C61—21no—
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C4×D4×C61—22no—
114C4×D4×C61C2×C414yes—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C42×C61C2311yescharacteristic
122C4⋊C8×C61—22no—
123#102×C61—41no—
124#102×C61—41no—
125C24×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61C22×C412yes—
132C22×C4×C61—21no—
133C22×C4×C61—24no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—81no—
137C22×C4×C61—81no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122C22×C412yes—
153C2×Dic122#4112yes—
154C2×Dic122#4112yes—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—42no—
167C4⋊C4×C61—81no—
168C4⋊C4×C61C22×C412yes—
169C22⋊C4×C61—24no—
170C42×C61C22×C412yes—
171C42×C61#4112yes—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174C22⋊Dic61—81no—
175C22⋊Dic61—81no—
176#11758200—28no—
177C4⋊Dic61—21no—
178C4⋊Dic61—22no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—41no—
184C4⋊Dic61C22×C412yes—
185#11758196—24no—
186#11758196—42no—
187#11758196—42no—
188#11758196—81no—
189#11758196—81no—
190C4×Dic61—24no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—42no—
196C4×Dic61—42no—
197C4×Dic61—81no—
198C4×Dic61—81no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C22×C412yes—
204C2×#92418C22×C412yes—
205C23×C61—22no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61C22×D411yescharacteristic
213D4×C61—22no—
214D4×C61—22no—
215D4×C61—41no—
216D4×C61—41no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—84no—
231C2×C4×C61C4×D412yes—
232C2×C4×C61C4×D412yes—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×Dic61—22no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61#13712yes—
263Dic122—22no—
264Dic122—22no—
265Dic122—22no—
266Dic122#11714yes—
267C8×C61—41no—
268C8×C61—41no—
269#92418—22no—
270#92418—22no—
271#22775—611nomaximal, Sylow
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61C2×C4×D411yescharacteristic
280C22×C61#54611yescharacteristic
281C22×C61#54811yescharacteristic
282C22×C61C2×#13711yescharacteristic
283C22×C61#57511yescharacteristic
284C22×C61#57611yescharacteristic
285C22×C61#57711yescharacteristic
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—42no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61C2×#11711yescharacteristic
292C4×C61#52011yescharacteristic
293C4×C61#55211yescharacteristic
294C4×C61#55311yescharacteristic
295Dic61—24no—
296Dic61—22no—
297Dic61—42no—
298Dic61—41no—
299Dic61—41no—
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303C2×#440—611no—
304#2357—611no—
305Q8×C4⋊C4—611no—
306#1936—611no—
307#1934—611no—
308#1921—611no—
309#1918—611no—
310#1880—611no—
311#1880—611no—
312#1836—611no—
313#1805—611no—
314#1802D6111yescharacteristic
315#1771—611no—
316#1763—611no—
317C4×#103—611no—
318C2×C61—82no—
319C2×C61#235211yescharacteristic
320C2×C61C4×#13711yescharacteristic
321C2×C61#294611yescharacteristic
322C2×C61#330911yescharacteristic
323C2×C61#332011yescharacteristic
324C2×C61#337111yescharacteristic
325C2×C61#337511yescharacteristic
326C22×SD16—611no—
327C2×C4×Q8—611no—
328C2×C4×Q8—611no—
329C2×C4×Q8—611no—
330C2×C4×D4C2×D6111yescharacteristic
331#440—618no—
332C2×#107—611no—
333C2×C4⋊C8C2×D6111yescharacteristic
334C2×#103—611no—
335C2×#103—611no—
336C2×#103—611no—
337C2×#102—611no—
338C2×#102C2×D6111yescharacteristic
339C2×#102C2×D6111yescharacteristic
340C2×#97—611no—
341C2×#97—611no—
342C2×C4×C8—611no—
343#390—611no—
344#390—611no—
345#389—611no—
346#389—611no—
347#389—1221no—
348#389—1221no—
349#388—1221no—
350#388—1221no—
351#384—1221no—
352#384C2×D6111yescharacteristic
353#382—611no—
354#381—1221no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—1221no—
359C4×C22⋊C4—1221no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340C2×D6111yescharacteristic
367#340C2×D6111yescharacteristic
368#336—611no—
369#336—611no—
370C61#2277511yescharacteristic, complemented, Sylow
371C22×Q8—611no—
372C22×D4C22×D6111yescharacteristic
373C23×C4—21no—
374C2×SD16—614no—
375C2×SD16—1222no—
376C2×SD16—1222no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8C22×D6111yescharacteristic
380C22×C8C22×D6111yescharacteristic
381C4×Q8—614no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×Q8—2441no—
386C4×Q8—2441no—
387C4×D4—22no—
388C4×D4C4×D6114yes—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—1221no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42C22×D6111yescharacteristic
417#107—614no—
418C4⋊C8—22no—
419#103—614no—
420#103—1222no—
421#103—1222no—
422#102—41no—
423#102—41no—
424#102—614no—
425#97—614no—
426#97—2441no—
427C4×C8—2441no—
428#95—611no—
429#95—611no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436C24—21no—
437C2×Q8—612no—
438C2×Q8—612no—
439C2×Q8—612no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D6112yes—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—1228no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—612no—
474C2×C8—612no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—612no—
487C4⋊C4—1222no—
488C4⋊C4—1221no—
489C4⋊C4—1224no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—4881no—
495C4⋊C4—4881no—
496C4⋊C4—2441no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C4⋊C4C2×C4×D6112yes—
500C22⋊C4—24no—
501C22⋊C4—4881no—
502C22⋊C4—4881no—
503C22⋊C4—2442no—
504C22⋊C4—2442no—
505C42—1221no—
506C42—1221no—
507C42—1221no—
508C42—1221no—
509C42—1224no—
510C42—4881no—
511C42—4881no—
512C42—2442no—
513C42—2442no—
514C42C2×C4×D6112yes—
515C42#1175822412yes—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C2×D4×D6111yescharacteristic
524Q8—614no—
525Q8—1222no—
526Q8—1222no—
527Q8—1222no—
528D4—22no—
529D4—22no—
530D4—41no—
531D4—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—612no—
547C2×C4—1222no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4#1451973712yes—
571C2×C4#1451973712yes—
572C2×C4C22×C4×D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4#1451982211yescharacteristic
577C2×C4#1451982511yescharacteristic
578C8—41no—
579C8—41no—
580C8—1222no—
581C8—1222no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22C2×#1451973711yescharacteristic
590C22#1684040911yescharacteristic
591C22#1684042211yescharacteristic
592C22#137×D6111yescharacteristic
593C22#1684055811yescharacteristic
594C22#1684056311yescharacteristic
595C22#1684056811yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—1224no—
602C4—1222no—
603C4—4881no—
604C4—4881no—
605C4—2442no—
606C4—2441no—
607C4—2441no—
608C4—2441no—
609C4#117×D6111yescharacteristic
610C4#1684024311yescharacteristic
611C4#1684044611yescharacteristic
612C4#1684045911yescharacteristic
613C2—82no—
614C2#2144874111yescharacteristic
615C2#2145464311yescharacteristic
616C2#2145464511yescharacteristic
617C2#2145783211yescharacteristic
618C2#2145799711yescharacteristic
619C2#2145859411yescharacteristic
620C2#2145863611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1802D61
hypocenterC61#22775
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
3. #21448731C2semidirect
6. #21446409C2semidirect
7. #21446388C2semidirect
10. #21445954C2semidirect
11. #21445835C2semidirect
12. #21445696C2semidirect
13. #21445648C2semidirect
14. #21445638C2semidirect
20. #16839862C4semidirect
370. C61#22775semidirect

Maximal subgroups

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

subgroupindex
C2×#168398622
#214487922
#214487312
#214465132
#214464992
#214464092
#214463882
#214461712
#214459542
#214458352
#214456962
#214456482
#214456382
C61×#18022
#2277561