Finite Group Databasebeta

← #29740179

Subgroups — #29740179

Order 15616.

800 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839970C211yesmaximal, characteristic
2#21449589C211yesmaximal, characteristic
3Q8×#11758196C211yesmaximal, characteristic
4#21447354C211yesmaximal, characteristic
5#21447313C211yesmaximal, characteristic
6#21447230C211yesmaximal, characteristic
7#21447063C211yesmaximal, characteristic
8#21446495C211yesmaximal, characteristic
9#21446376C211yesmaximal, characteristic
10#21446162C211yesmaximal, characteristic
11#21446160C211yesmaximal, characteristic
12#21446158C211yesmaximal, characteristic
13#21446073C211yesmaximal, characteristic
14#21446071C211yesmaximal, characteristic
15C2×C4×Q16×C61C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20#16839970—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29#16839700—21no—
30#16839693—21no—
31#16839669C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839667C2211yescharacteristic
34#16839660—21no—
35#16839660C2211yescharacteristic
36#16839658—21no—
37#16839658C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839646—21no—
40#16839646—21no—
41#16839631—21no—
42#16839631C2211yescharacteristic
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×Q16×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C4×Q16×C61—24no—
57C2×#107×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758202—22no—
63C2×#11758202—22no—
64C2×#11758202—22no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519775—22no—
68#14519775—41no—
69Q8×Dic61—22no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—41no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196D411yescharacteristic
86C2×#11758196D411yescharacteristic
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519710—41no—
92#14519706—22no—
93#14519702—22no—
94#14519702—22no—
95#14519701—22no—
96#14519701—22no—
97#14519701—22no—
98#14519700—41no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×Q8×C61C2311yescharacteristic
111C22×Q8×C61C2311yescharacteristic
112C2×Q16×C61—22no—
113C2×Q16×C61—22no—
114C2×Q16×C61D414yes—
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C4×Q8×C61—22no—
118C4×Q8×C61—22no—
119C4×Q8×C61—41no—
120C4×Q8×C61—41no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#107×C61—22no—
129#103×C61—22no—
130#103×C61—22no—
131C4×C8×C61—22no—
132C2×Q8×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—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61C2×D412yes—
141C2×Q8×C61C2×D412yes—
142C2×Q8×C61#4112yes—
143C2×Q8×C61#4112yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165Q16×C61—28no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D412yes—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61—42no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—42no—
181C42×C61—42no—
182#11758202—44no—
183C4⋊Dic61—24no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—42no—
190C4⋊Dic61#4112yes—
191#11758196—41no—
192#11758196—42no—
193#11758196—42no—
194#11758196—42no—
195#11758196—42no—
196#11758196—81no—
197#11758196—81no—
198#11758196—81no—
199#11758196Q1614yescomplemented
200C4×Dic61—24no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—42no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C23×C61C22×D411yescharacteristic
213Q8×C61—22no—
214Q8×C61—22no—
215Q8×C61—22no—
216Q8×C61—22no—
217Q8×C61—22no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61#12314yes—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61C2×Q1612yes—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×C4×C612−1+411yescharacteristic
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61C2×Q1612yes—
268C2×Dic61C2×Q1612yes—
269Dic122—41no—
270Dic122—41no—
271Dic122—41no—
272Dic122—41no—
273C8×C61—22no—
274C8×C61—41no—
275#92418—41no—
276#92418—41no—
277#26171—611nomaximal, Sylow
278C22×C61#52511yescharacteristic
279C22×C61D4211yescharacteristic
280C22×C61#54811yescharacteristic
281C22×C61C22×Q1611yescharacteristic
282C22×C61#57511yescharacteristic
283C22×C61#57711yescharacteristic
284C22×C61#57811yescharacteristic
285C4×C61—24no—
286C4×C61—22no—
287C4×C61—42no—
288C4×C61—41no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61C2×#12311yescharacteristic
292C4×C61#53911yescharacteristic
293C4×C61#54711yescharacteristic
294C4×C61#55311yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—41no—
298Dic61—42no—
299Dic61—41no—
300Dic61—81no—
301Dic61—81no—
302Dic61—81no—
303C2×#462—611no—
304C2×C4×Q16D6111yescharacteristic
305#2446—611no—
306Q8×C4⋊C4—611no—
307#2095—611no—
308#2087—611no—
309#2072—611no—
310#2043—611no—
311#1933—611no—
312#1916—611no—
313#1878—611no—
314#1878—611no—
315#1878—611no—
316#1863—611no—
317#1863—611no—
318C2×C61#244311yescharacteristic
319C2×C61#310411yescharacteristic
320C2×C61#312011yescharacteristic
321C2×C61#327111yescharacteristic
322C2×C61D4×Q1611yescharacteristic
323C2×C61#336311yescharacteristic
324C2×C61#337711yescharacteristic
325C22×Q16—611no—
326C22×Q16C2×D6111yescharacteristic
327C2×C4⋊Q8—611no—
328C2×C4×Q8—611no—
329C2×C4×Q8—611no—
330C2×C4×Q8C2×D6111yescharacteristic
331C2×C4×Q8C2×D6111yescharacteristic
332#462—1224no—
333C4×Q16—24no—
334C2×#107—611no—
335C2×#107C2×D6111yescharacteristic
336C2×C4⋊C8—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103C2×D6111yescharacteristic
344C2×#103C2×D6111yescharacteristic
345C2×C4×C8C2×D6111yescharacteristic
346#400—1221no—
347#398—1221no—
348#391—611no—
349#391—611no—
350#391—611no—
351#389—611no—
352#389—611no—
353#389—611no—
354#389—1221no—
355#389—1221no—
356#387—1221no—
357#387—1221no—
358#384—611no—
359#384—1221no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—1221no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#336—611no—
367#336—611no—
368#336—611no—
369#336—611no—
370C61#2617111yescharacteristic, complemented, Sylow
371C22×Q8—611no—
372C22×Q8C22×D6111yescharacteristic
373C22×Q8C22×D6111yescharacteristic
374C2×Q16—22no—
375C2×Q16—22no—
376C2×Q16—1222no—
377C2×Q16—1222no—
378C2×Q16—1222no—
379C2×Q16C22⋊D6114yes—
380C22×C8—611no—
381C22×C8—611no—
382C22×C8C22×D6111yescharacteristic
383C22×C8C22×D6111yescharacteristic
384C4⋊Q8—1222no—
385C4⋊Q8—2441no—
386C4×Q8—22no—
387C4×Q8—22no—
388C4×Q8—41no—
389C4×Q8—41no—
390C4×Q8—1222no—
391C4×Q8—1222no—
392C4×Q8—1222no—
393C4×Q8—2441no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—1221no—
417C2×C42—1221no—
418C2×C42C22×D6111yescharacteristic
419#107—22no—
420#107—2441no—
421C4⋊C8—1222no—
422#103—22no—
423#103—22no—
424#103—1222no—
425#103—1222no—
426#103—1222no—
427#103—1222no—
428#103—1222no—
429#103—2441no—
430C4×C8—22no—
431#95—611no—
432#95—611no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8#1175822212yes—
457C2×Q8#1175822212yes—
458C2×Q8C2×C22⋊D6112yes—
459C2×Q8C2×C22⋊D6112yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4D4×D6111yescharacteristic
473C22×C4D4×D6111yescharacteristic
474C22×C4C23×D6111yescharacteristic
475Q16—28no—
476Q16—2444no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8C2×C22⋊D6112yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—614no—
496C4⋊C4—612no—
497C4⋊C4—1224no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—4881no—
504C4⋊C4—4881no—
505C4⋊C4—4881no—
506C4⋊C4—2441no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C4⋊C4—2442no—
512C42—21no—
513C42—21no—
514C42—42no—
515C42—42no—
516C42—1224no—
517C42—1221no—
518C42—1221no—
519C42—1221no—
520C42—1221no—
521C42—2442no—
522C23C2×D4×D6111yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—22no—
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—2441no—
531Q8—2441no—
532Q8—2441no—
533Q8—2441no—
534Q8#1451976014yes—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1222no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—2441no—
566C2×C4—2441no—
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—2441no—
576C2×C4Q16×D6112yes—
577C2×C4C22×C22⋊D6111yescharacteristic
578C2×C4#41×D6111yescharacteristic
579C2×C4C2×#1175822211yescharacteristic
580C2×C4C2×#1175822211yescharacteristic
581C2×C4#1451982211yescharacteristic
582C2×C4#1451982411yescharacteristic
583C8—22no—
584C8—41no—
585C8—2441no—
586C8—2441no—
587C22#1684027111yescharacteristic
588C22D4×C22⋊D6111yescharacteristic
589C22#1684042111yescharacteristic
590C22C2×Q16×D6111yescharacteristic
591C22#1684055911yescharacteristic
592C22#1684056911yescharacteristic
593C22#1684057511yescharacteristic
594C4—24no—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—1222no—
601C4—1222no—
602C4—4881no—
603C4—4881no—
604C4—4881no—
605C4—2441no—
606C4—2442no—
607C4—2441no—
608C4C2×#1451976011yescharacteristic
609C4#1684035411yescharacteristic
610C4#1684041111yescharacteristic
611C4#1684046111yescharacteristic
612C2#2144954811yescharacteristic
613C2#2145585611yescharacteristic
614C2#2145603911yescharacteristic
615C2#2145740411yescharacteristic
616C2Q16×C22⋊D6111yescharacteristic
617C2#2145849511yescharacteristic
618C2#2145865711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×Q16×C61C2
socle, prime radicalC23×C61C22×D4
hypercenterC2×C4×Q16D61
hypocenterC61#26171
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
199. #11758196Q16semidirect
370. C61#26171semidirect

Maximal subgroups

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

subgroupindex
C2×#168399702
#214495892
Q8×#117581962
#214473542
#214473132
#214472302
#214470632
#214464952
#214463762
#214461622
#214461602
#214461582
#214460732
#214460712
C2×C4×Q16×C612
#2617161