Finite Group Databasebeta

← #29580272

Subgroups — #29580272

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839913C211yesmaximal, characteristic
2#21451632C211yesmaximal, characteristic, complemented
3#21445750C211yesmaximal, characteristic
4C2×#16839498C211yesmaximal, characteristic, complemented
5C2×#16839488C211yesmaximal, characteristic, complemented
6#21443698C211yesmaximal, characteristic, complemented
7C2×C61×#394C211yesmaximal, characteristic
8C22×#11758200—21no—
9C2×#14519756C2211yescharacteristic
10#16839913—22no—
11#16839913—22no—
12C2×#14519698—21no—
13C2×#14519698—22no—
14C2×C22⋊#92418C2211yescharacteristic
15C2×C22⋊#92418C2211yescharacteristic
16#16839701—21no—
17#16839695—22no—
18#16839682—21no—
19#16839635—21no—
20#16839625—21no—
21#16839622C2211yescharacteristic
22#16839498—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×Dic122D411yescharacteristic
32C2×#11758200—24no—
33C2×#11758200—44no—
34C22×#92418—21no—
35C22×#92418—21no—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40C2×C4⋊Dic61D411yescharacteristic
41C2×C4⋊Dic61D412yes—
42C2×#11758196—21no—
43C2×#11758196—22no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—21no—
46#14519698—24no—
47#14519698—44no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51C22⋊#92418D412yes—
52#14519684—22no—
53#14519684—22no—
54#14519684—22no—
55#14519684—21no—
56C25×C61—21no—
57C22×D4×C61—21no—
58C22×D4×C61—21no—
59C22×D4×C61—21no—
60C22×D4×C61—21no—
61C22×D4×C61—21no—
62C22×D4×C61—44no—
63C22×D4×C61D412yes—
64C23×C4×C61—21no—
65C23×C4×C61D411yescharacteristic
66C23×C4×C61D411yescharacteristic
67C23×C4×C61C2311yescharacteristic
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C22⋊C4×C61—24no—
75C2×C22⋊C4×C61—24no—
76C2×C22⋊C4×C61—42no—
77C2×C22⋊C4×C61—44no—
78C2×C22⋊C4×C61—42no—
79#95×C61—21no—
80#95×C61D412yes—
81C24×C61—24no—
82C24×C61—22no—
83C24×C61—42no—
84C24×C61—42no—
85C24×C61—42no—
86C24×C61—41no—
87C24×C61—41no—
88C24×C61—41no—
89C24×C61—41no—
90C24×C61—41no—
91C24×C61—41no—
92C24×C61C2×D411yescharacteristic
93C2×D4×C61—24no—
94C2×D4×C61—24no—
95C2×D4×C61—22no—
96C2×D4×C61—22no—
97C2×D4×C61—42no—
98C2×D4×C61—42no—
99C2×D4×C61—44no—
100C2×D4×C61—44no—
101C2×D4×C61—44no—
102C2×D4×C61—44no—
103C2×D4×C61—41no—
104C2×D4×C61—41no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—42no—
116C22×C4×C61—42no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61SD1612yescomplemented
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61#4111yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—42no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
138C22⋊C4×C61—48no—
139C22⋊C4×C61—48no—
140C22⋊Dic61—41no—
141C22⋊Dic61—41no—
142C22⋊Dic61—41no—
143C22⋊Dic61—41no—
144#11758200—48no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—42no—
148#11758196—41no—
149#11758196—41no—
150#11758196—82no—
151C2×#92418—22no—
152C2×#92418—42no—
153C2×#92418—42no—
154C2×#92418—41no—
155C23×C61—22no—
156C23×C61—22no—
157C23×C61—24no—
158C23×C61—42no—
159C23×C61—44no—
160C23×C61—44no—
161C23×C61—44no—
162C23×C61—44no—
163C23×C61—44no—
164C23×C61—42no—
165C23×C61—42no—
166C23×C61—42no—
167C23×C61—42no—
168C23×C61—42no—
169C23×C61—44no—
170C23×C61—82no—
171C23×C61—82no—
172C23×C61—82no—
173C23×C61—82no—
174C23×C61—82no—
175C23×C61—81no—
176C23×C61—81no—
177C23×C61—82no—
178C23×C61—81no—
179C23×C61—81no—
180C23×C61#12011yescharacteristic
181C23×C61#12011yescharacteristic
182C23×C61#12111yescharacteristic
183C23×C61C2×SD1611yescharacteristic
184C23×C61#13511yescharacteristic
185C23×C61AGL(1,ℤ/8)11yescharacteristic
186C23×C61#13711yescharacteristic
187D4×C61—24no—
188D4×C61—24no—
189D4×C61—42no—
190D4×C61—44no—
191D4×C61—44no—
192D4×C61—42no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—82no—
213C2×C4×C61—82no—
214C2×C4×C61—81no—
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12111yescharacteristic
217C2×C4×C61#12111yescharacteristic
218C2×C4×C61#12711yescharacteristic
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226Dic122—42no—
227Dic122—81no—
228#92418—42no—
229#92418—81no—
230#9923—611nomaximal, Sylow
231C22×C61—22no—
232C22×C61—22no—
233C22×C61—22no—
234C22×C61—22no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—44no—
240C22×C61—44no—
241C22×C61—44no—
242C22×C61—82no—
243C22×C61—82no—
244C22×C61—84no—
245C22×C61—84no—
246C22×C61—84no—
247C22×C61—84no—
248C22×C61—82no—
249C22×C61—82no—
250C22×C61#39211yescharacteristic
251C22×C61#44811yescharacteristic
252C22×C61#44911yescharacteristic
253C22×C61#45011yescharacteristic
254C22×C61#45211yescharacteristic
255C22×C61#45412yes—
256C22×C61B(4,𝔽2)12yes—
257C22×C61#46511yescharacteristic
258C22×C61#46811yescharacteristic
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—42no—
264C4×C61—82no—
265C4×C61—81no—
266Dic61—82no—
267Dic61—81no—
268Dic61—81no—
269C2×#450—611no—
270#2627—611no—
271C2×#394D6111yescharacteristic
272#1786—611no—
273C2×#328—611no—
274C2×#324—611no—
275#1294—611no—
276C2×C61—21no—
277C2×C61—21no—
278C2×C61—21no—
279C2×C61—21no—
280C2×C61—84no—
281C2×C61#159712yes—
282C2×C61#159912yes—
283C2×C61#200011yescharacteristic
284C2×C61#200311yescharacteristic
285C2×C61#202111yescharacteristic
286C23×D4C2×D6111yescharacteristic
287C22×SD16—1221no—
288C2×C22⋊Q8—611no—
289C22×C4⋊C4C2×D6111yescharacteristic
290C22×C22⋊C4—21no—
291C22×C22⋊C4—21no—
292#450—1222no—
293#450—1222no—
294C2×#102—1221no—
295C2×#102—1222no—
296C2×C22⋊C8—611no—
297C2×C22⋊C8—611no—
298#400—1221no—
299#398—1222no—
300#395—1221no—
301#394—22no—
302#394—22no—
303#385—1221no—
304#381—1221no—
305#380—611no—
306C2×#95C2×D6111yescharacteristic
307#328—1222no—
308#324—1222no—
309C61#992311yescharacteristic, complemented, Sylow
310C25—21no—
311C22×Q8—611no—
312C22×D4—21no—
313C22×D4—21no—
314C22×D4—21no—
315C22×D4—21no—
316C22×D4—21no—
317C22×D4—44no—
318C22×D4C22⋊D6112yes—
319C23×C4—21no—
320C23×C4C22⋊D6111yescharacteristic
321C23×C4C22⋊D6111yescharacteristic
322C23×C4C22×D6111yescharacteristic
323C2×SD16—1224no—
324C2×SD16—2444no—
325C22×C8—1221no—
326C22×C8—1221no—
327C22⋊Q8—1221no—
328C22⋊Q8—1221no—
329C22⋊Q8—1221no—
330C22⋊Q8—1221no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1221no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—42no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—42no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348#102—1224no—
349#102—2444no—
350C22⋊C8—612no—
351C22⋊C8—1221no—
352C22⋊C8—1221no—
353C22⋊C8—1221no—
354#95—21no—
355#95—1222no—
356#95—1222no—
357#95—1222no—
358#95—1221no—
359#95C22⋊D6112yes—
360C24—24no—
361C24—22no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—41no—
366C24—41no—
367C24—41no—
368C24—41no—
369C24—41no—
370C24—41no—
371C24C2×C22⋊D6111yescharacteristic
372C2×Q8—1221no—
373C2×Q8—1221no—
374C2×Q8—2442no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—41no—
386C2×D4—41no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—1222no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4#1175820012yescomplemented
407C22×C4D4×D6111yescharacteristic
408C22×C4D4×D6111yescharacteristic
409C22×C4D4×D6111yescharacteristic
410C22×C4#1175822411yescharacteristic
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413SD16—2448no—
414C2×C8—1222no—
415C2×C8—2442no—
416C2×C8—2441no—
417C2×C8—2442no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4—4882no—
425C4⋊C4—2442no—
426C4⋊C4—2441no—
427C4⋊C4—2441no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—2441no—
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C22⋊C4—2441no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—82no—
459C23#1451974511yescharacteristic
460C23#1451974611yescharacteristic
461C23#1451974911yescharacteristic
462C23C2×#1175820011yescharacteristic
463C23#1451979711yescharacteristic
464C23#1451980311yescharacteristic
465C23#1451980711yescharacteristic
466Q8—4881no—
467Q8—2442no—
468D4—24no—
469D4—24no—
470D4—44no—
471D4—42no—
472D4—44no—
473D4—42no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4#1451974511yescharacteristic
504C2×C4#1451974811yescharacteristic
505C2×C4#1451974911yescharacteristic
506C2×C4#1451977311yescharacteristic
507C8—4881no—
508C8—2442no—
509C22—21no—
510C22—21no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—21no—
516C22—21no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—82no—
521C22—82no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—82no—
527C22—82no—
528C22#1683967011yescharacteristic
529C22#1683990011yescharacteristic
530C22#1683990511yescharacteristic
531C22#1683991311yescharacteristic
532C22#1683992411yescharacteristic
533C22#1683993412yes—
534C22#1683994812yes—
535C22#1683998911yescharacteristic
536C22#1684000611yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—42no—
542C4—82no—
543C4—81no—
544C4—4881no—
545C4—4882no—
546C4—4881no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—84no—
552C2#2144474612yes—
553C2#2144475812yes—
554C2#2144682211yescharacteristic
555C2#2144684111yescharacteristic
556C2#2144694211yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#394C2
socleC23×C61#120
prime radicalC23×D4×C61C22
hypercenterC2×#394D61
hypocenterC61#9923
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. #21451632C2semidirect
4. C2×#16839498C2semidirect
5. C2×#16839488C2semidirect
6. #21443698C2semidirect
121. C22×C4×C61SD16semidirect
309. C61#9923semidirect
406. C22×C4#11758200semidirect

Maximal subgroups

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

subgroupindex
C2×#168399132
#214516322
#214457502
C2×#168394982
C2×#168394882
#214436982
C2×C61×#3942
#992361