Finite Group Databasebeta

← #29699369

Subgroups — #29699369

Order 15616.

866 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839861C211yesmaximal, characteristic
2#21448772C211yesmaximal, characteristic, complemented
3#21448728C211yesmaximal, characteristic
4#21446510C211yesmaximal, characteristic, complemented
5#21446494C211yesmaximal, characteristic, complemented
6#21446392C211yesmaximal, characteristic, complemented
7#21446154C211yesmaximal, characteristic
8#21446152C211yesmaximal, characteristic
9#21445951C211yesmaximal, characteristic, complemented
10#21445833C211yesmaximal, characteristic, complemented
11#21445818C211yesmaximal, characteristic
12#21445690C211yesmaximal, characteristic, complemented
13#21445643C211yesmaximal, characteristic
14C4×#14519697C211yesmaximal, characteristic
15C61×#1921C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×#14519742C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21#16839861C418yescomplemented
22C2×#14519706C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×C4×#92418C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839667C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839646—21no—
36#16839644—21no—
37#16839636—21no—
38#16839633—21no—
39#16839631C2211yescharacteristic
40#16839627C2211yescharacteristic
41#16839626—21no—
42#16839619—21no—
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758196—21no—
45C4×#11758196C2211yescharacteristic
46#95×D61—21no—
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C4×C4⋊C4×C61C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C2×#11758190—22no—
66C2×#11758190—22no—
67C2×#11758190C2×C414yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519742—22no—
71#14519742C2×C414yescomplemented
72C4×D244—22no—
73C4×D244—41no—
74#14519739—22no—
75#14519739—22no—
76#14519739—41no—
77#14519739C2×C414yes—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196D411yescharacteristic
85C2×#11758196D411yescharacteristic
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519706—22no—
94#14519702—41no—
95#14519700—41no—
96#14519700C2×C414yescomplemented
97#14519699—22no—
98#14519697—22no—
99#14519697C2×C414yes—
100#14519687—41no—
101#14519687C2×C414yescomplemented
102C4×#92418—41no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684D411yescharacteristic
112#14519684D411yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#107×C61C2×C414yescomplemented
123#95×C61—21no—
124#95×C61—21no—
125C23×D61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C22×C412yes—
144C2×D244#4112yes—
145C2×D244#4112yes—
146C2×C4×D61—24no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61—42no—
151C2×C4×D61—81no—
152C2×C4×D61—81no—
153C2×C4×D61—81no—
154C2×C4×D61—81no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122C22×C412yes—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C22×C412yes—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—81no—
174C4⋊C4×C61C22×C412yes—
175C4⋊C4×C61C22×C412yes—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—81no—
179#11758198—24no—
180#11758198—42no—
181#11758198—42no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—81no—
187#11758196—22no—
188#11758196—24no—
189#11758196—41no—
190#11758196—42no—
191#11758196—81no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—24no—
195C4×Dic61—81no—
196C4×Dic61C22×C412yes—
197C4×Dic61#4112yes—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C22×C412yes—
204#11758190—28no—
205C23×C61C22×D411yescharacteristic
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
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#13712yes—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61C4×D412yes—
257C2×Dic61C4×D412yes—
258D244—22no—
259D244—22no—
260D244—22no—
261D244#11714yes—
262C4×D61—84no—
263C4×D61—84no—
264C4×D61—84no—
265Dic122—22no—
266Dic122—22no—
267Dic122—41no—
268Dic122—41no—
269C8×C61—22no—
270C8×C61—41no—
271#92418—22no—
272#92418—41no—
273#22776—611nomaximal, Sylow
274C22×C61C2×C4×D411yescharacteristic
275C22×C61#54611yescharacteristic
276C22×C61#54811yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57511yescharacteristic
279C22×C61#57611yescharacteristic
280C22×C61#57711yescharacteristic
281C2×D61—24no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—42no—
288C4×C61—22no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61—41no—
292C4×C61—81no—
293C4×C61—81no—
294C4×C61C2×#11711yescharacteristic
295C4×C61#52011yescharacteristic
296C4×C61#55211yescharacteristic
297C4×C61#55311yescharacteristic
298Dic61—22no—
299Dic61—22no—
300Dic61—42no—
301Dic61—41no—
302Dic61—41no—
303Dic61—81no—
304Dic61—81no—
305C2×#440—611no—
306#2355—611no—
307#2349—611no—
308#1935—611no—
309#1933—611no—
310#1921D6111yescharacteristic, complemented
311#1918—611no—
312#1877—611no—
313#1877—611no—
314#1836—611no—
315#1805—611no—
316#1802—611no—
317#1771—611no—
318#1762—611no—
319C4×#102—611no—
320C2×C61#235211yescharacteristic
321C2×C61C4×#13711yescharacteristic
322C2×C61#294611yescharacteristic
323C2×C61#331111yescharacteristic
324C2×C61#331411yescharacteristic
325C2×C61#337011yescharacteristic
326C2×C61#337211yescharacteristic
327D61—44no—
328C22×SD16—611no—
329C2×C4×Q8—611no—
330C2×C4×D4—611no—
331C2×C4×D4—611no—
332C2×C4×D4—611no—
333#440—618no—
334C2×#107C2×D6111yescharacteristic
335C2×C4⋊C8—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#102—611no—
340C2×#102—611no—
341C2×#102—611no—
342C2×#97—611no—
343C2×#97—611no—
344C2×C4×C8—611no—
345#391—611no—
346#391—611no—
347#389—611no—
348#389C2×D6111yescharacteristic
349#387—1221no—
350#387—1221no—
351#385—1221no—
352#385—1221no—
353#384—611no—
354#382—611no—
355#382—1221no—
356#380—1221no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—1221no—
360C4×C4⋊C4C2×D6111yescharacteristic
361C2×#95—1221no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340C2×D6111yescharacteristic
368#340C2×D6111yescharacteristic
369#340C2×D6111yescharacteristic
370#336—611no—
371#336C2×D6111yescharacteristic
372C61#2277611yescharacteristic, complemented, Sylow
373C22×Q8—611no—
374C22×D4—611no—
375C23×C4—1221no—
376C23×C4—1221no—
377C23×C4—1221no—
378C2×SD16—614no—
379C2×SD16—1222no—
380C2×SD16—1222no—
381C22×C8—611no—
382C22×C8—611no—
383C22×C8C22×D6111yescharacteristic
384C22×C8C22×D6111yescharacteristic
385C4×Q8—614no—
386C4×Q8—1222no—
387C4×D4—614no—
388C4×D4—1222no—
389C4×D4—1222no—
390C4×D4—1222no—
391C4×D4—2441no—
392C4×D4—2441no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C42—21no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415#107C4×D6114yescomplemented
416C4⋊C8—1222no—
417#103—614no—
418#103—2441no—
419#103—2441no—
420#102—614no—
421#102—1222no—
422#102—1222no—
423#97—614no—
424#97—2441no—
425C4×C8—2441no—
426#95—21no—
427#95—21no—
428#95—611no—
429#95—611no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438C24—1221no—
439C2×Q8—612no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—1221no—
445C2×D4—612no—
446C2×D4—612no—
447C2×D4—612no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1224no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—4881no—
464C22×C4—4881no—
465C22×C4—4881no—
466C22×C4—4881no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4C23×D6111yescharacteristic
474SD16—1228no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—612no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8C2×C4×D6112yes—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—81no—
494C4⋊C4—1224no—
495C4⋊C4—1222no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—4881no—
501C4⋊C4—4881no—
502C4⋊C4—2441no—
503C4⋊C4—2442no—
504C4⋊C4C2×C4×D6112yes—
505C4⋊C4C2×C4×D6112yes—
506C22⋊C4—1224no—
507C22⋊C4—2442no—
508C22⋊C4—2442no—
509C42—21no—
510C42—21no—
511C42—81no—
512C42—612no—
513C42—612no—
514C42—1224no—
515C42—1221no—
516C42—1221no—
517C42—4881no—
518C23—1222no—
519C23—1222no—
520C23—1222no—
521C23—2441no—
522C23—2441no—
523C23—2441no—
524C23—2441no—
525C23C2×D4×D6111yescharacteristic
526Q8—1222no—
527Q8—1222no—
528Q8—2441no—
529Q8—2441no—
530D4—614no—
531D4—1222no—
532D4—1222no—
533D4—1222no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1222no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—4884no—
565C2×C4—4884no—
566C2×C4—4884no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4#1451980912yes—
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4#41×D6111yescharacteristic
578C2×C4#41×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C2×C4#1451982311yescharacteristic
581C2×C4#1451982511yescharacteristic
582C8—22no—
583C8—41no—
584C8—1222no—
585C8—2441no—
586C22—1224no—
587C22—2442no—
588C22—2442no—
589C22—2442no—
590C22—2442no—
591C22—2442no—
592C22—2442no—
593C22C4×D4×D6111yescharacteristic
594C22#1684040411yescharacteristic
595C22#1684042211yescharacteristic
596C22C2×#1451980911yescharacteristic
597C22#1684055911yescharacteristic
598C22#1684056611yescharacteristic
599C22#1684057211yescharacteristic
600C4—22no—
601C4—42no—
602C4—42no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—1222no—
607C4—1222no—
608C4—4881no—
609C4—4881no—
610C4—2442no—
611C4—2441no—
612C4—2441no—
613C4#117×D6111yescharacteristic
614C4#1684024411yescharacteristic
615C4#1684044411yescharacteristic
616C4#1684046311yescharacteristic
617C2—2444no—
618C2#2144874311yescharacteristic
619C2C4×#1451980911yescharacteristic
620C2#2145464811yescharacteristic
621C2#2145785311yescharacteristic
622C2#2145789811yescharacteristic
623C2#2145857011yescharacteristic
624C2#2145859911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1921C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1921D61
hypocenterC61#22776
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
2. #21448772C2semidirect
4. #21446510C2semidirect
5. #21446494C2semidirect
6. #21446392C2semidirect
9. #21445951C2semidirect
10. #21445833C2semidirect
12. #21445690C2semidirect
15. C61×#1921C2semidirect
21. #16839861C4semidirect
71. #14519742C2×C4semidirect
96. #14519700C2×C4semidirect
101. #14519687C2×C4semidirect
122. #107×C61C2×C4semidirect
310. #1921D61semidirect
372. C61#22776semidirect
415. #107C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398612
#214487722
#214487282
#214465102
#214464942
#214463922
#214461542
#214461522
#214459512
#214458332
#214458182
#214456902
#214456432
C4×#145196972
C61×#19212
#2277661