Finite Group Databasebeta

← C2×#20243106 #26213680

Subgroups — C2×#20243106 #26213680

Order 12032.

2574 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625324C211yesmaximal, characteristic
2C22×#13093982C211yesmaximal, characteristic
3C22×C4⋊C4×D47C211yesmaximal, characteristic
4#20243106C218yesmaximal, direct product factor
5C2×#15688886C211yesmaximal, characteristic, complemented
6C2×#15688885C211yesmaximal, characteristic, complemented
7C2×#15688885C211yesmaximal, characteristic, complemented
8C2×C47×#340C211yesmaximal, characteristic, complemented
9C23×C4×D47—21no—
10C23×C4×D47C2211yescharacteristic
11C22×#10625324—24no—
12C22×#10625324—24no—
13C22×#10625324C418yescomplemented
14C22×#10625324C2214yes—
15C23×C47⋊C8C2211yescharacteristic
16C2×#13093982—24no—
17C2×#13093982—22no—
18C2×#13093982C418yescomplemented
19C2×#13093982C2214yes—
20C22×C4⋊Dic47C2211yescharacteristic
21C22×C4⋊Dic47C2211yescharacteristic
22C22×Dic47⋊C4—21no—
23C2×C4⋊C4×D47—22no—
24C2×C4⋊C4×D47—24no—
25C2×C4⋊C4×D47C418yescomplemented
26C2×C4⋊C4×D47C2214yes—
27C22×(C2×D47)⋊C4—21no—
28C22×C4×Dic47—21no—
29#15688886C2214yescomplemented
30#15688885C2214yescomplemented
31#15688885C2214yescomplemented
32C23×C8×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C4⋊C4×C47C2211yescharacteristic
35C47×#340C2214yescomplemented
36C24×D47D411yescharacteristic
37C23×Dic47—21no—
38C23×Dic47—21no—
39C23×Dic47D411yescharacteristic
40C22×C4×D47—28no—
41C22×C4×D47—24no—
42C22×C4×D47—44no—
43C22×C4×D47—44no—
44C22×C4×D47C2×C414yes—
45C22×C4×D47C2×C414yes—
46C22×C4×D47C2×C414yes—
47C22×C4×D47D412yes—
48C22×C4×D47D414yes—
49C22×C4×D47D414yes—
50C22×C4×D47Q814yes—
51C22×C4×D47C2312yes—
52C2×#10625324—232no—
53C2×#10625324—28no—
54C2×#10625324—28no—
55C2×#10625324C2×C4116yescomplemented
56C22×C47⋊C8—22no—
57C22×C47⋊C8—22no—
58C22×C47⋊C8C2×C414yescomplemented
59C22×C47⋊C8C2312yes—
60#13093982—216no—
61#13093982—28no—
62#13093982C2×C4116yescomplemented
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—22no—
67C2×C4⋊Dic47C2×C414yes—
68C2×C4⋊Dic47C2×C414yescomplemented
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2312yes—
71C2×Dic47⋊C4—28no—
72C2×Dic47⋊C4—24no—
73C4⋊C4×D47—216no—
74C4⋊C4×D47—28no—
75C4⋊C4×D47C2×C4116yescomplemented
76C2×(C2×D47)⋊C4—28no—
77C2×(C2×D47)⋊C4—24no—
78C2×C4×Dic47—28no—
79C2×C4×Dic47—24no—
80C23×C4×C47—21no—
81C23×C4×C47—21no—
82C23×C4×C47C2311yescharacteristic
83C22×C8×C47—22no—
84C22×C8×C47—22no—
85C22×C8×C47C2×C414yescomplemented
86C22×C8×C47C2312yes—
87C2×C4⋊C4×C47—21no—
88C2×C4⋊C4×C47—22no—
89C2×C4⋊C4×C47—21no—
90C2×C4⋊C4×C47—22no—
91C2×C4⋊C4×C47C2×C414yes—
92C2×C4⋊C4×C47C2×C414yescomplemented
93C2×C4⋊C4×C47C2312yes—
94C2×C4⋊C4×C47C2312yes—
95C24×C47C2×D411yescharacteristic
96C23×D47—22no—
97C23×D47—22no—
98C23×D47—22no—
99C23×D47—22no—
100C23×D47C22⋊C414yes—
101C23×D47C22⋊C414yes—
102C23×D47C4⋊C414yes—
103C23×D47C2×D412yes—
104C22×C4×C47—24no—
105C22×C4×C47—24no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—42no—
109C22×C4×C47—42no—
110C22×C4×C47—42no—
111C22×C4×C47—42no—
112C22×C4×C47C22×C412yes—
113C22×C4×C47C22×C412yes—
114C22×C4×C47C22×C412yes—
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D412yes—
117C22×C4×C47C2×D412yes—
118C22×C4×C47C2×Q812yes—
119C22×C4×C47C2411yescharacteristic
120C22×Dic47—24no—
121C22×Dic47—24no—
122C22×Dic47—22no—
123C22×Dic47—22no—
124C22×Dic47—42no—
125C22×Dic47—42no—
126C22×Dic47—42no—
127C22×Dic47—42no—
128C22×Dic47C22⋊C414yes—
129C22×Dic47C22⋊C414yes—
130C22×Dic47C4⋊C414yes—
131C22×Dic47C2×D412yes—
132C2×C4×D47—216no—
133C2×C4×D47—48no—
134C2×C4×D47—48no—
135C2×C4×D47—432no—
136C2×C4×D47C42116yes—
137C2×C4×D47C22⋊C418yes—
138C2×C4×D47C22⋊C418yes—
139C2×C4×D47C22⋊C4116yes—
140C2×C4×D47C4⋊C418yes—
141C2×C4×D47C4⋊C4116yes—
142C2×C4×D47C4⋊C4116yes—
143C2×C4×D47C22×C414yes—
144C2×C4×D47C22×C414yes—
145C2×C4×D47C22×C414yes—
146C2×C4×D47C2×D414yes—
147C2×C4×D47C2×D414yes—
148C2×C4×D47C2×Q814yes—
149C2×C8×C47—28no—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C2×C8×C47C22×C414yescomplemented
153C4⋊C4×C47—24no—
154C4⋊C4×C47—24no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47C22×C414yes—
158C4⋊C4×C47C22×C414yescomplemented
159(C2×D47)⋊C4—216no—
160C4⋊Dic47—24no—
161C4⋊Dic47—24no—
162C4⋊Dic47—22no—
163C4⋊Dic47—22no—
164C4⋊Dic47C22×C414yes—
165C4⋊Dic47C22×C414yescomplemented
166Dic47⋊C4—216no—
167C4×Dic47—216no—
168C2×C47⋊C8—28no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8C22×C414yescomplemented
172#10625324—232no—
173C23×C47C2×C22⋊C412yes—
174C23×C47C2×C22⋊C412yes—
175C23×C47C2×C4⋊C412yes—
176C23×C47AGL(1,ℤ/8)12yes—
177C23×C47AGL(1,ℤ/8)12yes—
178C23×C47#13712yes—
179C23×C47#13712yes—
180C23×C47C22×D411yescharacteristic
181C22×D47—22no—
182C22×D47—22no—
183C22×D47—22no—
184C22×D47—22no—
185C22×D47—28no—
186C22×D47—28no—
187C22×D47—28no—
188C22×D47—28no—
189C22×D47—28no—
190C22×D47—28no—
191C22×D47#95116yes—
192C22×D47C2×C22⋊C414yes—
193C22×D47C2×C22⋊C414yes—
194C22×D47C2×C4⋊C414yes—
195C2×C4×C47—24no—
196C2×C4×C47—24no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—48no—
202C2×C4×C47—48no—
203C2×C4×C47C2×C4214yes—
204C2×C4×C47C2×C22⋊C414yes—
205C2×C4×C47C2×C22⋊C412yes—
206C2×C4×C47C2×C22⋊C412yes—
207C2×C4×C47C2×C4⋊C414yes—
208C2×C4×C47C2×C4⋊C414yes—
209C2×C4×C47C2×C4⋊C412yes—
210C2×C4×C47C23×C411yescharacteristic
211C2×C4×C47C23×C411yescharacteristic
212C2×C4×C47C23×C411yescharacteristic
213C2×C4×C47C22×D411yescharacteristic
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47C22×Q811yescharacteristic
216C2×Dic47—24no—
217C2×Dic47—24no—
218C2×Dic47—42no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47—48no—
223C2×Dic47—48no—
224C2×Dic47#95116yes—
225C2×Dic47C2×C22⋊C414yes—
226C2×Dic47C2×C22⋊C414yes—
227C2×Dic47C2×C4⋊C414yes—
228C4×D47—432no—
229C4×D47#95132yes—
230C4×D47C2×C4218yes—
231C4×D47C2×C22⋊C418yes—
232C4×D47C2×C4⋊C418yes—
233C4×D47C2×C4⋊C418yes—
234C8×C47—24no—
235C47⋊C8—24no—
236C2×#1736—471nomaximal, Sylow
237C22×C47C2×#9514yes—
238C22×C47#41714yes—
239C22×C47#41714yes—
240C22×C47#41814yes—
241C22×C47#41914yes—
242C22×C47#42814yes—
243C22×C47#42814yes—
244C22×C47C22×C22⋊C411yescharacteristic
245C22×C47C22×C22⋊C411yescharacteristic
246C22×C47C22×C4⋊C411yescharacteristic
247C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
249C22×C47C2×#13711yescharacteristic
250C22×C47C2×#13711yescharacteristic
251C2×D47—24no—
252C2×D47—24no—
253C2×D47—24no—
254C2×D47—24no—
255C2×D47—24no—
256C2×D47—24no—
257C2×D47—232no—
258C2×D47C2×#9518yes—
259C4×C47—44no—
260C4×C47—44no—
261C4×C47C2×#9514yes—
262C4×C47C22×C4211yescharacteristic
263C4×C47C22×C22⋊C411yescharacteristic
264C4×C47C22×C4⋊C411yescharacteristic
265C4×C47C22×C4⋊C411yescharacteristic
266Dic47—44no—
267Dic47—44no—
268Dic47C2×#9518yes—
269C23×M16—471no—
270C22×#117—471no—
271C23×C4⋊C4—471no—
272#1736—478no—
273C2×#340—471no—
274C2×#340—471no—
275C2×#340—471no—
276C2×#340D4711yescharacteristic, complemented
277C2×C47#173618yescomplemented
278C2×C47C22×#9511yescharacteristic
279C2×C47C2×#41811yescharacteristic
280C2×C47C2×#41911yescharacteristic
281C2×C47C2×#41711yescharacteristic
282C2×C47C2×#41711yescharacteristic
283C2×C47C2×#42811yescharacteristic
284C2×C47C2×#42811yescharacteristic
285D47—28no—
286C24×C4—471no—
287C24×C4—941no—
288C22×M16—478no—
289C22×M16—474no—
290C22×M16—944no—
291C22×M16—944no—
292C23×C8—471no—
293C23×C8C2×D4711yescharacteristic
294C2×#117—478no—
295C2×#117—474no—
296C2×#117—942no—
297C2×#117—944no—
298C22×C4⋊C4—471no—
299C22×C4⋊C4—471no—
300C22×C4⋊C4—478no—
301C22×C4⋊C4—474no—
302C22×C4⋊C4—942no—
303C22×C4⋊C4—944no—
304C22×C4⋊C4—941no—
305C22×C4⋊C4C2×D4711yescharacteristic
306C22×C4⋊C4C2×D4711yescharacteristic
307C22×C22⋊C4—941no—
308C22×C42—941no—
309#340—474no—
310#340—474no—
311#340—474no—
312#340C2×D4714yescomplemented
313C47C2×#173611yescharacteristic, complemented, Sylow
314C25—471no—
315C23×C4—21no—
316C23×C4—21no—
317C23×C4—471no—
318C23×C4—474no—
319C23×C4—474no—
320C23×C4—474no—
321C23×C4—472no—
322C23×C4—474no—
323C23×C4—474no—
324C23×C4—474no—
325C23×C4—472no—
326C23×C4—948no—
327C23×C4—944no—
328C23×C4—941no—
329C23×C4—941no—
330C23×C4—1884no—
331C23×C4—1884no—
332C23×C4C22×D4711yescharacteristic
333C2×M16—4716no—
334C2×M16—9432no—
335C2×M16—948no—
336C2×M16—948no—
337C22×C8—22no—
338C22×C8—22no—
339C22×C8—474no—
340C22×C8—472no—
341C22×C8—942no—
342C22×C8—942no—
343C22×C8C4×D4714yescomplemented
344C22×C8C22×D4712yes—
345#117—4716no—
346#117—9416no—
347#117—948no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—474no—
353C2×C4⋊C4—474no—
354C2×C4⋊C4—472no—
355C2×C4⋊C4—472no—
356C2×C4⋊C4—4716no—
357C2×C4⋊C4—9416no—
358C2×C4⋊C4—948no—
359C2×C4⋊C4—948no—
360C2×C4⋊C4—944no—
361C2×C4⋊C4—941no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—941no—
364C2×C4⋊C4—942no—
365C2×C4⋊C4C4×D4714yes—
366C2×C4⋊C4C4×D4714yescomplemented
367C2×C4⋊C4C22×D4712yes—
368C2×C4⋊C4C22×D4712yes—
369C2×C22⋊C4—948no—
370C2×C22⋊C4—944no—
371C2×C42—948no—
372C2×C42—944no—
373C24—474no—
374C24—474no—
375C24—474no—
376C24—472no—
377C24—942no—
378C24—942no—
379C24—942no—
380C24—942no—
381C24D4×D4711yescharacteristic
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—474no—
391C22×C4—474no—
392C22×C4—474no—
393C22×C4—472no—
394C22×C4—4716no—
395C22×C4—478no—
396C22×C4—478no—
397C22×C4—4716no—
398C22×C4—4716no—
399C22×C4—4716no—
400C22×C4—478no—
401C22×C4—474no—
402C22×C4—474no—
403C22×C4—474no—
404C22×C4—474no—
405C22×C4—474no—
406C22×C4—474no—
407C22×C4—9416no—
408C22×C4—944no—
409C22×C4—944no—
410C22×C4—942no—
411C22×C4—942no—
412C22×C4—18832no—
413C22×C4—1888no—
414C22×C4—1888no—
415C22×C4—1882no—
416C22×C4—1882no—
417C22×C4—1882no—
418C22×C4—1882no—
419C22×C4C2×C4×D4712yes—
420C22×C4C2×C4×D4712yes—
421C22×C4C2×C4×D4712yes—
422C22×C4D4×D4711yescharacteristic
423C22×C4D4×D4712yes—
424C22×C4D4×D4712yes—
425C22×C4Q8×D4712yes—
426C22×C4C23×D4711yescharacteristic
427M16—9432no—
428C2×C8—28no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—474no—
432C2×C8—948no—
433C2×C8—942no—
434C2×C8—942no—
435C2×C8C2×C4×D4714yescomplemented
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—474no—
441C4⋊C4—474no—
442C4⋊C4—9416no—
443C4⋊C4—944no—
444C4⋊C4—944no—
445C4⋊C4—942no—
446C4⋊C4—942no—
447C4⋊C4C2×C4×D4714yes—
448C4⋊C4C2×C4×D4714yescomplemented
449C22⋊C4—9416no—
450C42—9416no—
451C23—4716no—
452C23—474no—
453C23—474no—
454C23—474no—
455C23—948no—
456C23—948no—
457C23—948no—
458C23—948no—
459C23—942no—
460C23—942no—
461C23—942no—
462C23—942no—
463C23—948no—
464C23—948no—
465C23C22⋊C4×D4712yes—
466C23C22⋊C4×D4712yes—
467C23C4⋊C4×D4712yes—
468C23#1309404812yes—
469C23#1309405112yes—
470C23#1309405312yes—
471C23#1309405612yes—
472C23C2×D4×D4711yescharacteristic
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—4716no—
482C2×C4—474no—
483C2×C4—474no—
484C2×C4—474no—
485C2×C4—4732no—
486C2×C4—478no—
487C2×C4—478no—
488C2×C4—478no—
489C2×C4—478no—
490C2×C4—944no—
491C2×C4—944no—
492C2×C4—18832no—
493C2×C4—1888no—
494C2×C4—1888no—
495C2×C4—1882no—
496C2×C4—1882no—
497C2×C4—1882no—
498C2×C4—1882no—
499C2×C4C42×D4714yes—
500C2×C4C22⋊C4×D4712yes—
501C2×C4C22⋊C4×D4712yes—
502C2×C4C22⋊C4×D4714yes—
503C2×C4C4⋊C4×D4712yes—
504C2×C4C4⋊C4×D4714yes—
505C2×C4C4⋊C4×D4714yes—
506C2×C4C22×C4×D4711yescharacteristic
507C2×C4C22×C4×D4711yescharacteristic
508C2×C4C22×C4×D4711yescharacteristic
509C2×C4C2×D4×D4711yescharacteristic
510C2×C4C2×D4×D4711yescharacteristic
511C2×C4C2×Q8×D4711yescharacteristic
512C8—24no—
513C8—944no—
514C22—478no—
515C22—944no—
516C22—944no—
517C22—9432no—
518C22—944no—
519C22—944no—
520C22—944no—
521C22—944no—
522C22#95×D4714yes—
523C22#1568912114yes—
524C22#1568912314yes—
525C22#1568912614yes—
526C22#1568913014yes—
527C22#1568916214yes—
528C22#1568916414yes—
529C22C2×C22⋊C4×D4711yescharacteristic
530C22C2×C22⋊C4×D4711yescharacteristic
531C22C2×C4⋊C4×D4711yescharacteristic
532C22C2×#1309404811yescharacteristic
533C22C2×#1309405111yescharacteristic
534C22C2×#1309405311yescharacteristic
535C22C2×#1309405611yescharacteristic
536C4—44no—
537C4—44no—
538C4—478no—
539C4—1884no—
540C4—1884no—
541C4#95×D4714yes—
542C4C2×C42×D4711yescharacteristic
543C4C2×C22⋊C4×D4711yescharacteristic
544C4C2×C4⋊C4×D4711yescharacteristic
545C4C2×C4⋊C4×D4711yescharacteristic
546C2—948no—
547C2#2024310618yesdirect product factor
548C2C2×#95×D4711yescharacteristic
549C2C2×#1568912611yescharacteristic
550C2C2×#1568913011yescharacteristic
551C2C2×#1568912111yescharacteristic
552C2C2×#1568912311yescharacteristic
553C2C2×#1568916211yescharacteristic
554C2C2×#1568916411yescharacteristic
subgroupG/H
centerC24D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC2×C47×#340C2
socleC24×C47C2×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenterC2×#340D47
hypocenterC47C2×#1736
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
4. #20243106C2direct (N × H)
5. C2×#15688886C2semidirect
6. C2×#15688885C2semidirect
7. C2×#15688885C2semidirect
8. C2×C47×#340C2semidirect
13. C22×#10625324C4semidirect
18. C2×#13093982C4semidirect
25. C2×C4⋊C4×D47C4semidirect
29. #15688886C22semidirect
30. #15688885C22semidirect
31. #15688885C22semidirect
35. C47×#340C22semidirect
55. C2×#10625324C2×C4semidirect
58. C22×C47⋊C8C2×C4semidirect
62. #13093982C2×C4semidirect
68. C2×C4⋊Dic47C2×C4semidirect
75. C4⋊C4×D47C2×C4semidirect
85. C22×C8×C47C2×C4semidirect
92. C2×C4⋊C4×C47C2×C4semidirect
152. C2×C8×C47C22×C4semidirect
158. C4⋊C4×C47C22×C4semidirect
165. C4⋊Dic47C22×C4semidirect
171. C2×C47⋊C8C22×C4semidirect
276. C2×#340D47semidirect
277. C2×C47#1736semidirect
312. #340C2×D47semidirect
313. C47C2×#1736semidirect
343. C22×C8C4×D47semidirect
366. C2×C4⋊C4C4×D47semidirect
435. C2×C8C2×C4×D47semidirect
448. C4⋊C4C2×C4×D47semidirect
547. C2#20243106direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253242
C22×#130939822
C22×C4⋊C4×D472
#202431062
C2×#156888862
C2×#156888852
C2×C47×#3402
C2×#173647