Finite Group Databasebeta

← #29740192

Subgroups — #29740192

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×#16839971C211yesmaximal, characteristic
2C2×C4×Dic244C211yesmaximal, characteristic
3#21449596C211yesmaximal, characteristic
4#21448733C211yesmaximal, characteristic
5#21447355C211yesmaximal, characteristic
6#21447314C211yesmaximal, characteristic
7#21447231C211yesmaximal, characteristic
8#21447064C211yesmaximal, characteristic
9#21446376C211yesmaximal, characteristic
10#21446163C211yesmaximal, characteristic
11#21446161C211yesmaximal, characteristic
12#21446157C211yesmaximal, characteristic
13#21446074C211yesmaximal, characteristic
14#21446072C211yesmaximal, characteristic
15C61×#1933C211yesmaximal, characteristic
16C22×Dic244C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×#14519742C2211yescharacteristic
23#16839971—24no—
24C4×Dic244—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34#16839703—21no—
35#16839696—21no—
36#16839669C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839659—21no—
41#16839657—21no—
42#16839645—21no—
43#16839644—21no—
44#16839631—21no—
45#16839631C2211yescharacteristic
46C4×#11758196C2211yescharacteristic
47#16839613—21no—
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C2×Dic244—22no—
64C2×Dic244—22no—
65C2×Dic244D414yes—
66C2×#11758202—22no—
67C2×#11758202—22no—
68C2×#11758202—22no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519775—22no—
72#14519775—41no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75C4×Dic122—41no—
76C4×Dic122—41no—
77#14519742—22no—
78#14519742—22no—
79#14519742—22no—
80#14519742—41no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196D411yescharacteristic
94C2×#11758196D411yescharacteristic
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101#14519711—22no—
102#14519710—41no—
103#14519706—22no—
104#14519702—22no—
105#14519702—22no—
106#14519701—41no—
107#14519700—22no—
108#14519700—22no—
109#14519700—22no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684D411yescharacteristic
117#14519684D411yescharacteristic
118C22×Q8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126#103×C61—22no—
127#103×C61—22no—
128C4×C8×C61—22no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122C2×D412yes—
162C2×Dic122C2×D412yes—
163C2×Dic122#4112yes—
164C2×Dic122#4112yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D412yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—81no—
177C4⋊C4×C61#4112yes—
178C42×C61—21no—
179C42×C61—21no—
180#11758202—44no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—42no—
186C4⋊Dic61—81no—
187#11758196—24no—
188#11758196—41no—
189#11758196—42no—
190#11758196—42no—
191#11758196—42no—
192#11758196—42no—
193#11758196—42no—
194#11758196—42no—
195#11758196—81no—
196#11758196Q1614yescomplemented
197C4×Dic61—24no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—42no—
203C4×Dic61—42no—
204C4×Dic61—42no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211Dic244—28no—
212C23×C61C22×D411yescharacteristic
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
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—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61C2×Q1612yes—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C2×C4×C612−1+411yescharacteristic
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—22no—
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—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61C2×Q1612yes—
264C2×Dic61C2×Q1612yes—
265Dic122—22no—
266Dic122—22no—
267Dic122—22no—
268Dic122—22no—
269Dic122—22no—
270Dic122—41no—
271Dic122—41no—
272Dic122#12314yes—
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—22no—
286C4×C61—42no—
287C4×C61—41no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61C2×#12311yescharacteristic
292C4×C61#53911yescharacteristic
293C4×C61#54711yescharacteristic
294C4×C61#55311yescharacteristic
295Dic61—22no—
296Dic61—24no—
297Dic61—22no—
298Dic61—41no—
299Dic61—42no—
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303C2×#462—611no—
304C2×C4×Q16—611no—
305#2446—611no—
306Q8×C4⋊C4—611no—
307#2095—611no—
308#2087—611no—
309#2072—611no—
310#2043—611no—
311#1933D6111yescharacteristic
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×Q16—611no—
327C2×C4⋊Q8—611no—
328C2×C4×Q8—611no—
329C2×C4×Q8—611no—
330C2×C4×Q8—611no—
331C2×C4×Q8—611no—
332#462—1224no—
333C4×Q16—1224no—
334C2×#107—611no—
335C2×#107—611no—
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#391C2×D6111yescharacteristic
351#389—611no—
352#389—611no—
353#389—1221no—
354#389—1221no—
355#389C2×D6111yescharacteristic
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#340C2×D6111yescharacteristic
365#340C2×D6111yescharacteristic
366#336—611no—
367#336—611no—
368#336—611no—
369#336—611no—
370C61#2617111yescharacteristic, complemented, Sylow
371C22×Q8—611no—
372C22×Q8—611no—
373C22×Q8C22×D6111yescharacteristic
374C2×Q16—614no—
375C2×Q16—1222no—
376C2×Q16—1222no—
377C2×Q16—1222no—
378C2×Q16—1222no—
379C2×Q16—1222no—
380C22×C8—611no—
381C22×C8—611no—
382C22×C8C22×D6111yescharacteristic
383C22×C8C22×D6111yescharacteristic
384C4⋊Q8—1222no—
385C4⋊Q8—2441no—
386C4×Q8—1222no—
387C4×Q8—1222no—
388C4×Q8—1222no—
389C4×Q8—1222no—
390C4×Q8—1222no—
391C4×Q8—2441no—
392C4×Q8—2441no—
393C4×Q8—2441no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
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⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42—1221no—
417C2×C42—1221no—
418C2×C42C22×D6111yescharacteristic
419#107—1222no—
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—21no—
432#95—21no—
433#95—21no—
434#95—611no—
435#95—611no—
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—612no—
449C2×Q8—612no—
450C2×Q8—612no—
451C2×Q8—612no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
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—1228no—
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×C8D4×D6112yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—81no—
495C4⋊C4—614no—
496C4⋊C4—1224no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—4881no—
502C4⋊C4—4881no—
503C4⋊C4—2441no—
504C4⋊C4—2442no—
505C4⋊C4—2442no—
506C4⋊C4—2442no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C4⋊C4#1175822212yes—
512C42—21no—
513C42—21no—
514C42—1224no—
515C42—1221no—
516C42—1221no—
517C42—1221no—
518C42—1221no—
519C42—2442no—
520C42—2442no—
521C42—2442no—
522C23C2×D4×D6111yescharacteristic
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—614no—
528Q8—1222no—
529Q8—1222no—
530Q8—1222no—
531Q8—1222no—
532Q8—1222no—
533Q8—2441no—
534Q8—2441no—
535C2×C4—21no—
536C2×C4—21no—
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—41no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—1221no—
555C2×C4—1222no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1222no—
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×C4C2×D4×D6111yescharacteristic
578C2×C4#41×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C2×C4C2×#1175822211yescharacteristic
581C2×C4#1451982311yescharacteristic
582C2×C4#1451982411yescharacteristic
583C8—22no—
584C8—41no—
585C8—2441no—
586C8—2441no—
587C22#1684027111yescharacteristic
588C22#1684039611yescharacteristic
589C22#1684042011yescharacteristic
590C22C2×Q16×D6111yescharacteristic
591C22#1684055911yescharacteristic
592C22#1684056811yescharacteristic
593C22#1684057611yescharacteristic
594C4—22no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—1224no—
601C4—1222no—
602C4—1222no—
603C4—4881no—
604C4—4881no—
605C4—2442no—
606C4—2441no—
607C4—2441no—
608C4#123×D6111yescharacteristic
609C4#1684035311yescharacteristic
610C4#1684041711yescharacteristic
611C4#1684045611yescharacteristic
612C2#2144956111yescharacteristic
613C2#2145585611yescharacteristic
614C2#2145602911yescharacteristic
615C2#2145740111yescharacteristic
616C2#2145756811yescharacteristic
617C2#2145849411yescharacteristic
618C2#2145866711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1933C2
socle, prime radicalC23×C61C22×D4
hypercenter#1933D61
hypocenterC61#26171
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
196. #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×#168399712
C2×C4×Dic2442
#214495962
#214487332
#214473552
#214473142
#214472312
#214470642
#214463762
#214461632
#214461612
#214461572
#214460742
#214460722
C61×#19332
#2617161