Finite Group Databasebeta

← #29714332

Subgroups — #29714332

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840319C211yesmaximal, characteristic, complemented
2C2×#16840053C211yesmaximal, characteristic, complemented
3C2×#16839968C211yesmaximal, characteristic, complemented
4C4×#14519751C211yesmaximal, characteristic
5#21447016C212yesmaximal, complemented
6#21446824C212yesmaximal
7#21446766C211yesmaximal, characteristic, complemented
8#21446741C211yesmaximal, characteristic, complemented
9#21446173C212yesmaximal, complemented
10#21446090C212yesmaximal, complemented
11C61×#1797C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13#16840319—24no—
14#16840319—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519769—21no—
18C2×#14519756—21no—
19C2×#14519756C2212yes—
20C2×#14519751C2211yescharacteristic
21C2×C4×Dic122—21no—
22C2×C4×Dic122C2212yes—
23C2×C4×D244C2211yescharacteristic
24C2×C4×C22⋊D61—21no—
25C2×#14519736—21no—
26#16840053—24no—
27#16839968—24no—
28C2×#14519709C2212yes—
29C2×#14519702C2212yes—
30C2×#14519702C2212yes—
31C2×#14519702C2212yes—
32C2×#14519699C2212yes—
33#16839664C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839653—21no—
36#16839641—21no—
37#16839622C2211yescharacteristic
38C4×C4⋊Dic61C2211yescharacteristic
39C4×#11758198—21no—
40#16839525C2212yes—
41#16839525C2212yes—
42C22×C42×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic
45C2×C22⋊C8×C61C2212yes—
46C61×#336C2212yes—
47C22×Dic122—21no—
48C22×Dic122C2312yes—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C22×C4×D61—21no—
52C2×#11758190—24no—
53C2×#11758190—24no—
54C2×#11758190—42no—
55#14519777—22no—
56#14519777—22no—
57#14519777D414yescomplemented
58#14519777D414yescomplemented
59#14519769—28no—
60#14519756—28no—
61#14519756—24no—
62#14519756—24no—
63#14519751—22no—
64#14519751—22no—
65C4×Dic122—28no—
66C4×Dic122—24no—
67C4×Dic122—24no—
68C4×D244—22no—
69C4×D244—22no—
70C4×C22⋊D61—44no—
71#14519736—28no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—22no—
80C2×#11758196—22no—
81C2×#11758196—21no—
82C2×C22⋊Dic61—22no—
83C2×C22⋊Dic61—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—22no—
89C2×C4×Dic61—21no—
90#14519709—24no—
91#14519702—24no—
92#14519702—24no—
93#14519702—42no—
94#14519699—24no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98C23×C4×C61D411yescharacteristic
99C23×C4×C61D411yescharacteristic
100C23×C4×C61C2311yescharacteristic
101C22×C8×C61C2312yes—
102C22×C8×C61C2312yes—
103C2×C42×C61—22no—
104C2×C42×C61—22no—
105C2×C42×C61D411yescharacteristic
106C2×C42×C61D411yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109C4⋊C8×C61—22no—
110C4⋊C8×C61—22no—
111C22⋊C8×C61—42no—
112C24×C61C2×D411yescharacteristic
113C23×D61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—41no—
124C22×C4×C61—41no—
125C22×C4×C61SD1614yescomplemented
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—24no—
133C2×C22⋊D61—42no—
134C2×C22⋊D61—42no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244#4112yes—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—24no—
154C2×Dic122—22no—
155C2×Dic122—42no—
156C2×Dic122—42no—
157C2×Dic122C2×D414yes—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61—21no—
167C42×C61—21no—
168C42×C61C2×D412yes—
169C42×C61C2×D412yes—
170C22⋊Dic61—28no—
171C22⋊Dic61—42no—
172#11758198—24no—
173#11758198—42no—
174#11758198—42no—
175#11758198—42no—
176C4⋊Dic61—24no—
177C4⋊Dic61—24no—
178C4⋊Dic61—24no—
179C4⋊Dic61—22no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61C2×D414yes—
186C4⋊Dic61#4112yes—
187#11758196—28no—
188#11758196—28no—
189#11758196—42no—
190C4×Dic61—28no—
191C4×Dic61—42no—
192#11758190—48no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61C2×SD1612yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—22no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—28no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—44no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61#12012yes—
225C2×C4×C61#12012yes—
226C2×C4×C61C2×SD1612yes—
227C2×C4×C61C2×SD1612yes—
228C2×C4×C61#13712yes—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C22⋊D61—48no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—22no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—41no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251D244—22no—
252D244—22no—
253D244—41no—
254D244—41no—
255C4×D61—84no—
256Dic122—24no—
257Dic122—24no—
258Dic122—48no—
259Dic122—42no—
260Dic122—42no—
261C8×C61—42no—
262C8×C61—42no—
263#24151—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61C2×#12011yescharacteristic
272C22×C61#52511yescharacteristic
273C22×C61#53711yescharacteristic
274C22×C61C22×SD1611yescharacteristic
275C22×C61C2×#13711yescharacteristic
276C22×C61#57511yescharacteristic
277C22×C61#57511yescharacteristic
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61#44814yes—
290C4×C61#54611yescharacteristic
291C4×C61#54711yescharacteristic
292C4×C61#54811yescharacteristic
293C4×C61#55011yescharacteristic
294Dic61—42no—
295Dic61—44no—
296Dic61—44no—
297Dic61—81no—
298Dic61—81no—
299C2×#533—611no—
300C2×#476—611no—
301C2×#461—611no—
302C4×#121—611no—
303#2035—612no—
304#2000—612no—
305#1987—611no—
306#1981—611no—
307#1880—612no—
308#1865—612no—
309#1797D6111yescharacteristic, complemented
310C2×C61—42no—
311C2×C61#240711yescharacteristic
312C2×C61C2×#44811yescharacteristic
313C2×C61#300511yescharacteristic
314C2×C61#310111yescharacteristic
315C2×C61#311611yescharacteristic
316C2×C61#324611yescharacteristic
317C2×C61#325211yescharacteristic
318D61—82no—
319C22×SD16—612no—
320#533—1224no—
321#533—1224no—
322C2×C4⋊Q8—611no—
323C2×C4⋊Q8—611no—
324C2×#125—1221no—
325C2×C22⋊Q8—612no—
326C2×C22⋊Q8—1221no—
327C2×#121—611no—
328C2×C4×Q8—612no—
329C2×C4×Q8—1221no—
330C2×C4×D4—611no—
331C2×C4×D4—1221no—
332C2×#117—1221no—
333C22×C42C2×D6111yescharacteristic
334#476—1224no—
335#461—1224no—
336C2×#106—612no—
337C2×C4⋊C8C2×D6111yescharacteristic
338C2×C4⋊C8C2×D6111yescharacteristic
339C2×#103—612no—
340C2×#103—612no—
341C2×#103—612no—
342C2×#102—612no—
343C2×C22⋊C8C2×D6112yes—
344#390—611no—
345#389—611no—
346#388—1221no—
347#386—1221no—
348#380—611no—
349C4×C4⋊C4—611no—
350C4×C22⋊C4—1221no—
351#340—612no—
352#340—612no—
353#336C2×D6112yes—
354C61#2415111yescharacteristic, complemented, Sylow
355C22×Q8—612no—
356C22×Q8—1221no—
357C22×D4—611no—
358C22×D4—1221no—
359C23×C4—1221no—
360C23×C4D24411yescharacteristic
361C23×C4D24411yescharacteristic
362C23×C4C22×D6111yescharacteristic
363C2×SD16—1224no—
364C2×SD16—1224no—
365C2×SD16—2442no—
366C22×C8C22×D6112yes—
367C22×C8C22×D6112yes—
368C4⋊Q8—614no—
369C4⋊Q8—614no—
370C4⋊Q8—1222no—
371C4⋊Q8—1222no—
372#125—1228no—
373C22⋊Q8—1224no—
374C22⋊Q8—1224no—
375C22⋊Q8—1228no—
376#121—1222no—
377#121—1222no—
378C4×Q8—1224no—
379C4×Q8—1224no—
380C4×Q8—1228no—
381C4×D4—1222no—
382C4×D4—1222no—
383C4×D4—2444no—
384#117—1228no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—1222no—
404C2×C42—1221no—
405C2×C42D24411yescharacteristic
406C2×C42D24411yescharacteristic
407C2×C42C22×D6111yescharacteristic
408C2×C42C22×D6111yescharacteristic
409#106—1224no—
410C4⋊C8—22no—
411C4⋊C8—22no—
412#103—1224no—
413#103—1224no—
414#103—2442no—
415#102—1224no—
416C22⋊C8—42no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420C24—1221no—
421C24C2×D24411yescharacteristic
422C2×Q8—614no—
423C2×Q8—1222no—
424C2×Q8—1224no—
425C2×Q8—1222no—
426C2×Q8—1222no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—2442no—
430C2×Q8—2442no—
431C2×D4—612no—
432C2×D4—1224no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—2442no—
439C2×D4—2442no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—1221no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4#1175819014yescomplemented
460C22×C4C2×D24411yescharacteristic
461C22×C4C2×D24411yescharacteristic
462C22×C4C2×D24411yescharacteristic
463C22×C4C2×D24411yescharacteristic
464C22×C4C2×D24411yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—2448no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C4⋊C4—614no—
474C4⋊C4—612no—
475C4⋊C4—1228no—
476C4⋊C4—1228no—
477C4⋊C4—1222no—
478C4⋊C4—1221no—
479C4⋊C4—1224no—
480C4⋊C4—1224no—
481C4⋊C4—1224no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—2442no—
487C22⋊C4—1224no—
488C22⋊C4—1228no—
489C22⋊C4—2442no—
490C22⋊C4—2442no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—1228no—
498C42—2442no—
499C42D4×D6112yes—
500C42D4×D6112yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—1222no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23C2×#1175819012yes—
515C23C22×D24411yescharacteristic
516Q8—1224no—
517Q8—1224no—
518Q8—2448no—
519Q8—2442no—
520Q8—2442no—
521D4—1222no—
522D4—1222no—
523D4—2448no—
524D4—2441no—
525D4—2441no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—1224no—
543C2×C4—1224no—
544C2×C4—1222no—
545C2×C4—4884no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2442no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4#1451974712yes—
559C2×C4#1451974712yes—
560C2×C4C2×#1175819012yes—
561C2×C4C2×#1175819012yes—
562C2×C4#1451981012yes—
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C2×C4#41×D6111yescharacteristic
567C2×C4#1451982211yescharacteristic
568C2×C4#1451982511yescharacteristic
569C8—42no—
570C8—42no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22—2442no—
583C22—2442no—
584C22—2442no—
585C22C2×#1451974711yescharacteristic
586C22#1684027411yescharacteristic
587C22#1684034311yescharacteristic
588C22C22×#1175819011yescharacteristic
589C22C2×#1451981011yescharacteristic
590C22#1684056211yescharacteristic
591C22#1684056211yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—4881no—
597C4—4881no—
598C4—2444no—
599C4—2442no—
600C4—2444no—
601C4#1683990314yes—
602C4#1684040911yescharacteristic
603C4#1684041411yescharacteristic
604C4#1684042211yescharacteristic
605C4#1684043011yescharacteristic
606C2—42no—
607C2—4882no—
608C2#2144916511yescharacteristic
609C2C2×#1683990311yescharacteristic
610C2#2145506411yescharacteristic
611C2#2145582811yescharacteristic
612C2#2145598211yescharacteristic
613C2#2145713611yescharacteristic
614C2#2145719311yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1797C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1797D61
hypocenterC61#24151
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
1. C2×#16840319C2semidirect
2. C2×#16840053C2semidirect
3. C2×#16839968C2semidirect
5. #21447016C2semidirect
7. #21446766C2semidirect
8. #21446741C2semidirect
9. #21446173C2semidirect
10. #21446090C2semidirect
11. C61×#1797C2semidirect
57. #14519777D4semidirect
58. #14519777D4semidirect
125. C22×C4×C61SD16semidirect
309. #1797D61semidirect
354. C61#24151semidirect
459. C22×C4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168403192
C2×#168400532
C2×#168399682
C4×#145197512
#214470162
#214468242
#214467662
#214467412
#214461732
#214460902
C61×#17972
#2415161