Finite Group Databasebeta

← #29580216

Subgroups — #29580216

Order 15616.

1044 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839896C211yesmaximal, characteristic
2#21451561C211yesmaximal, characteristic, complemented
3#21445750C211yesmaximal, characteristic, complemented
4C2×#16839495C211yesmaximal, characteristic, complemented
5C2×#16839488C211yesmaximal, characteristic, complemented
6#21443698C211yesmaximal, characteristic, complemented
7C2×C61×#394C211yesmaximal, characteristic, complemented
8C22×#11758199—21no—
9C2×#14519760—21no—
10C2×#14519751C2211yescharacteristic
11#16839896—22no—
12#16839896—22no—
13C2×#14519698—21no—
14C2×#14519698—22no—
15C2×C22⋊#92418C2211yescharacteristic
16C2×C22⋊#92418C2211yescharacteristic
17#16839692—21no—
18#16839684—22no—
19#16839679—21no—
20#16839641—21no—
21#16839622C2211yescharacteristic
22#16839495—22no—
23#16839488—22no—
24C23×D4×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C22×C22⋊C4×C61—21no—
27C22×C22⋊C4×C61—21no—
28C61×#394—22no—
29C61×#394—22no—
30C2×#95×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C2×#11758199—24no—
34C2×#11758199—44no—
35C22×#92418—21no—
36C22×#92418—21no—
37#14519760—42no—
38#14519760—42no—
39#14519751—21no—
40#14519751—21no—
41#14519751—21no—
42#14519751—21no—
43C2×C4⋊Dic61D411yescharacteristic
44C2×C4⋊Dic61D412yes—
45C2×#11758196—22no—
46C2×C22⋊Dic61—21no—
47C2×#11758198—22no—
48C2×#11758198—21no—
49C2×#11758198—22no—
50#14519698—24no—
51#14519698—44no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55C22⋊#92418D412yes—
56#14519684—22no—
57#14519684—21no—
58C25×C61—21no—
59C22×D4×C61—21no—
60C22×D4×C61—21no—
61C22×D4×C61—21no—
62C22×D4×C61—21no—
63C22×D4×C61—21no—
64C22×D4×C61—44no—
65C22×D4×C61D412yes—
66C23×C4×C61—21no—
67C23×C4×C61D411yescharacteristic
68C23×C4×C61D411yescharacteristic
69C23×C4×C61C2311yescharacteristic
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C22⋊C4×C61—24no—
77C2×C22⋊C4×C61—24no—
78C2×C22⋊C4×C61—42no—
79C2×C22⋊C4×C61—44no—
80C2×C22⋊C4×C61—42no—
81#95×C61—21no—
82#95×C61D412yes—
83C24×C61—24no—
84C24×C61—22no—
85C24×C61—42no—
86C24×C61—42no—
87C24×C61—42no—
88C24×C61—41no—
89C24×C61—41no—
90C24×C61—41no—
91C24×C61—41no—
92C24×C61—41no—
93C24×C61—41no—
94C24×C61C2×D411yescharacteristic
95C23×D61—21no—
96C2×D4×C61—24no—
97C2×D4×C61—24no—
98C2×D4×C61—22no—
99C2×D4×C61—22no—
100C2×D4×C61—42no—
101C2×D4×C61—42no—
102C2×D4×C61—44no—
103C2×D4×C61—44no—
104C2×D4×C61—44no—
105C2×D4×C61—44no—
106C2×D4×C61—41no—
107C2×D4×C61—41no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61D812yescomplemented
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61#4111yescharacteristic
131C2×C22⋊D61—41no—
132C2×C22⋊D61—41no—
133C2×C22⋊D61—41no—
134C2×C22⋊D61—41no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—42no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C22⋊C4×C61—48no—
147C22⋊C4×C61—48no—
148C22⋊Dic61—41no—
149C22⋊Dic61—41no—
150#11758199—48no—
151#11758198—44no—
152#11758198—41no—
153#11758198—41no—
154#11758198—82no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—42no—
158#11758196—44no—
159C2×#92418—22no—
160C2×#92418—42no—
161C2×#92418—42no—
162C2×#92418—41no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61—24no—
166C23×C61—42no—
167C23×C61—44no—
168C23×C61—44no—
169C23×C61—44no—
170C23×C61—44no—
171C23×C61—44no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—44no—
178C23×C61—82no—
179C23×C61—82no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61—82no—
183C23×C61—81no—
184C23×C61—81no—
185C23×C61—82no—
186C23×C61—81no—
187C23×C61—81no—
188C23×C61#12011yescharacteristic
189C23×C61#12011yescharacteristic
190C23×C61#12111yescharacteristic
191C23×C61C2×D811yescharacteristic
192C23×C61#13511yescharacteristic
193C23×C61AGL(1,ℤ/8)11yescharacteristic
194C23×C61AGL(1,ℤ/8)11yescharacteristic
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—42no—
199C22×D61—42no—
200D4×C61—24no—
201D4×C61—24no—
202D4×C61—42no—
203D4×C61—44no—
204D4×C61—44no—
205D4×C61—42no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—82no—
226C2×C4×C61—82no—
227C2×C4×C61—81no—
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12711yescharacteristic
232C22⋊D61—81no—
233C22⋊D61—81no—
234C22⋊D61—81no—
235C22⋊D61—81no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—42no—
239C2×Dic61—41no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242D244—42no—
243D244—81no—
244#92418—42no—
245#92418—81no—
246#9915—611nomaximal, Sylow
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—44no—
256C22×C61—44no—
257C22×C61—44no—
258C22×C61—82no—
259C22×C61—82no—
260C22×C61—84no—
261C22×C61—84no—
262C22×C61—84no—
263C22×C61—84no—
264C22×C61—82no—
265C22×C61—82no—
266C22×C61#39211yescharacteristic
267C22×C61#44711yescharacteristic
268C22×C61#44711yescharacteristic
269C22×C61#44911yescharacteristic
270C22×C61#44911yescharacteristic
271C22×C61#45412yes—
272C22×C61B(4,𝔽2)12yes—
273C22×C61#46611yescharacteristic
274C22×C61#46911yescharacteristic
275C2×D61—82no—
276C2×D61—82no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—42no—
285C4×C61—82no—
286C4×C61—81no—
287Dic61—82no—
288Dic61—81no—
289C2×#447—611no—
290#2618—611no—
291C2×#394D6111yescharacteristic, complemented
292#1786—611no—
293C2×#327—611no—
294C2×#324—611no—
295#1294—611no—
296C2×C61—21no—
297C2×C61—21no—
298C2×C61—21no—
299C2×C61—21no—
300C2×C61—84no—
301C2×C61#159712yes—
302C2×C61#159812yes—
303C2×C61#199911yescharacteristic
304C2×C61#200311yescharacteristic
305C2×C61#202111yescharacteristic
306D61—161no—
307C23×D4C2×D6111yescharacteristic
308C22×D8—1221no—
309C2×#123—1221no—
310C2×#121—611no—
311C22×C4⋊C4C2×D6111yescharacteristic
312C22×C22⋊C4—21no—
313C22×C22⋊C4—21no—
314#447—1222no—
315#447—1222no—
316C2×#102—1221no—
317C2×#102—1222no—
318C2×C22⋊C8—611no—
319C2×C22⋊C8—611no—
320#398—1221no—
321#396—1222no—
322#394—22no—
323#394—22no—
324#394—1221no—
325#386—1221no—
326#380—611no—
327C2×#95C2×D6111yescharacteristic
328#327—1222no—
329#324—1222no—
330C61#991511yescharacteristic, complemented, Sylow
331C25—21no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—21no—
337C22×D4—44no—
338C22×D4—611no—
339C22×D4—1221no—
340C22×D4C22⋊D6112yes—
341C23×C4—21no—
342C23×C4C22⋊D6111yescharacteristic
343C23×C4C22⋊D6111yescharacteristic
344C23×C4C22×D6111yescharacteristic
345C2×D8—1224no—
346C2×D8—2444no—
347C22×C8—1221no—
348C22×C8—1221no—
349#123—2442no—
350#123—2442no—
351#121—1221no—
352#121—1221no—
353#121—1221no—
354#121—1221no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—1222no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—42no—
367C2×C22⋊C4—44no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373#102—1224no—
374#102—2444no—
375C22⋊C8—612no—
376C22⋊C8—1221no—
377C22⋊C8—1221no—
378C22⋊C8—1221no—
379#95—21no—
380#95—1222no—
381#95—1221no—
382#95C22⋊D6112yes—
383C24—24no—
384C24—22no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—41no—
393C24—41no—
394C24—1221no—
395C24C2×C22⋊D6111yescharacteristic
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—1221no—
409C2×D4—1221no—
410C2×D4—2442no—
411C2×D4—2441no—
412C2×D4—2441no—
413C2×D4—2441no—
414C2×D4—2441no—
415C2×D4—2441no—
416C2×D4—2441no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4#1175819912yescomplemented
436C22×C4D4×D6111yescharacteristic
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4#1175822411yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442D8—2448no—
443C2×C8—1222no—
444C2×C8—2442no—
445C2×C8—2441no—
446C2×C8—2442no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—2442no—
454C4⋊C4—2444no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—4882no—
458C22⋊C4—2444no—
459C22⋊C4—2441no—
460C22⋊C4—2441no—
461C22⋊C4—2441no—
462C22⋊C4—2441no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—82no—
488C23—2442no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23—2442no—
493C23#1451974511yescharacteristic
494C23#1451974611yescharacteristic
495C23#1451974911yescharacteristic
496C23C2×#1175819911yescharacteristic
497C23#1451979711yescharacteristic
498C23#1451980211yescharacteristic
499C23#1451980311yescharacteristic
500D4—24no—
501D4—24no—
502D4—44no—
503D4—42no—
504D4—44no—
505D4—42no—
506D4—4881no—
507D4—4881no—
508D4—4881no—
509D4—4881no—
510D4—4881no—
511D4—2442no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2442no—
540C2×C4#1451974511yescharacteristic
541C2×C4#1451974811yescharacteristic
542C2×C4#1451974911yescharacteristic
543C2×C4#1451977311yescharacteristic
544C8—4881no—
545C8—2442no—
546C22—21no—
547C22—21no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—21no—
553C22—21no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—82no—
558C22—82no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—82no—
564C22—82no—
565C22—4881no—
566C22—4882no—
567C22—4882no—
568C22—4881no—
569C22—4881no—
570C22#1683967011yescharacteristic
571C22#1683989511yescharacteristic
572C22#1683989611yescharacteristic
573C22#1683990511yescharacteristic
574C22#1683990711yescharacteristic
575C22#1683993412yes—
576C22#1683994812yes—
577C22#1683999611yescharacteristic
578C22#1684001311yescharacteristic
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—42no—
584C4—82no—
585C4—81no—
586C4—4882no—
587C4—4881no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—84no—
593C2—9761no—
594C2#2144474612yes—
595C2#2144475312yes—
596C2#2144681511yescharacteristic
597C2#2144684111yescharacteristic
598C2#2144694211yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#394C2
socleC23×C61#120
prime radicalC2×#16839896C2
hypercenterC2×#394D61
hypocenterC61#9915
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. #21451561C2semidirect
3. #21445750C2semidirect
4. C2×#16839495C2semidirect
5. C2×#16839488C2semidirect
6. #21443698C2semidirect
7. C2×C61×#394C2semidirect
124. C22×C4×C61D8semidirect
291. C2×#394D61semidirect
330. C61#9915semidirect
435. C22×C4#11758199semidirect

Maximal subgroups

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

subgroupindex
C2×#168398962
#214515612
#214457502
C2×#168394952
C2×#168394882
#214436982
C2×C61×#3942
#991561