Finite Group Databasebeta

← C2×#20243097 #26213664

Subgroups — C2×#20243097 #26213664

Order 12032.

2270 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625329C211yesmaximal, characteristic
2C22×#13093983C211yesmaximal, characteristic
3#20243097C218yesmaximal, direct product factor
4C2×#15688882C214yesmaximal, complemented
5C22×#117×C47C211yesmaximal, characteristic
6C22×#10625329—24no—
7C22×#10625329—21no—
8C22×#10625329—22no—
9C22×#10625329C418yescomplemented
10C22×#10625329C2212yes—
11C22×#10625329C2214yes—
12C23×C47⋊C8—21no—
13C2×#13093983—24no—
14C2×#13093983—22no—
15C2×#13093983C418yescomplemented
16C2×#13093983C2214yes—
17C22×C4⋊Dic47—21no—
18C22×C22⋊Dic47—21no—
19C22×C4×Dic47C2212yes—
20#15688882C22116yescomplemented
21C24×C4×C47C2211yescharacteristic
22C2×#117×C47—21no—
23C2×#117×C47—22no—
24C2×#117×C47—21no—
25C2×#117×C47—22no—
26C2×#117×C47C412yescomplemented
27C2×#117×C47C414yescomplemented
28C2×#117×C47C2212yes—
29C2×#117×C47C2214yes—
30C22×C4⋊C4×C47C412yescomplemented
31C22×C22⋊C4×C47—21no—
32C22×C42×C47—21no—
33C23×Dic47—22no—
34C2×#10625329—28no—
35C2×#10625329—216no—
36C2×#10625329—24no—
37C2×#10625329—28no—
38C2×#10625329—24no—
39C2×#10625329—22no—
40C2×#10625329C2×C418yescomplemented
41C2×#10625329C2×C4116yescomplemented
42C2×#10625329C2314yes—
43C22×C47⋊C8—28no—
44C22×C47⋊C8—24no—
45C22×C47⋊C8—22no—
46#13093983—216no—
47#13093983—28no—
48#13093983C2×C4116yescomplemented
49C2×C4⋊Dic47—28no—
50C2×C4⋊Dic47—24no—
51C2×C22⋊Dic47—28no—
52C2×C22⋊Dic47—24no—
53C2×C4×Dic47—22no—
54C2×C4×Dic47—24no—
55C2×C4×Dic47C2×C418yescomplemented
56C2×C4×Dic47C2314yes—
57C25×C47D411yescharacteristic
58C23×C4×C47—22no—
59C23×C4×C47C2×C414yes—
60C23×C4×C47C2×C414yes—
61C23×C4×C47C2×C411yescharacteristic
62C23×C4×C47C2×C411yescharacteristic
63C23×C4×C47C2×C412yes—
64C23×C4×C47D411yescharacteristic
65C23×C4×C47D412yes—
66C23×C4×C47D414yes—
67C23×C4×C47D411yescharacteristic
68C23×C4×C47D411yescharacteristic
69C23×C4×C47D412yes—
70C23×C4×C47Q814yes—
71C23×C4×C47C2311yescharacteristic
72C23×C4×C47C2312yes—
73#117×C47—24no—
74#117×C47—24no—
75#117×C47—24no—
76#117×C47—22no—
77#117×C47—24no—
78#117×C47—22no—
79#117×C47—24no—
80#117×C47C2×C414yes—
81#117×C47C2×C418yescomplemented
82#117×C47C2314yes—
83C2×C4⋊C4×C47—24no—
84C2×C4⋊C4×C47—22no—
85C2×C4⋊C4×C47C2×C414yes—
86C2×C4⋊C4×C47C2×C418yescomplemented
87C2×C22⋊C4×C47—24no—
88C2×C22⋊C4×C47—22no—
89C2×C22⋊C4×C47—22no—
90C2×C22⋊C4×C47—24no—
91C2×C42×C47—24no—
92C2×C42×C47—22no—
93C2×C42×C47—22no—
94C2×C42×C47—24no—
95C24×C47—44no—
96C24×C47C22⋊C411yescharacteristic
97C24×C47C22⋊C411yescharacteristic
98C24×C47C22⋊C414yes—
99C24×C47C22⋊C412yes—
100C24×C47C4⋊C414yes—
101C24×C47C2×D411yescharacteristic
102C24×C47C2×D412yes—
103C22×C4×C47—24no—
104C22×C4×C47—28no—
105C22×C4×C47—48no—
106C22×C4×C47C4214yes—
107C22×C4×C47C4214yes—
108C22×C4×C47C4218yes—
109C22×C4×C47C22⋊C414yes—
110C22×C4×C47C22⋊C418yes—
111C22×C4×C47C22⋊C412yes—
112C22×C4×C47C22⋊C411yescharacteristic
113C22×C4×C47C22⋊C412yes—
114C22×C4×C47C22⋊C411yescharacteristic
115C22×C4×C47C22⋊C412yes—
116C22×C4×C47C22⋊C412yes—
117C22×C4×C47C22⋊C412yes—
118C22×C4×C47C22⋊C414yes—
119C22×C4×C47C22⋊C414yes—
120C22×C4×C47C22⋊C418yes—
121C22×C4×C47C4⋊C414yes—
122C22×C4×C47C4⋊C418yes—
123C22×C4×C47C4⋊C414yes—
124C22×C4×C47C4⋊C414yes—
125C22×C4×C47C4⋊C418yes—
126C22×C4×C47C4⋊C414yes—
127C22×C4×C47C4⋊C414yes—
128C22×C4×C47C4⋊C418yes—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C22×C414yes—
131C22×C4×C47C22×C412yes—
132C22×C4×C47C22×C414yes—
133C22×C4×C47C22×C411yescharacteristic
134C22×C4×C47C22×C412yes—
135C22×C4×C47C22×C411yescharacteristic
136C22×C4×C47C22×C412yes—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2×D414yes—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×D412yes—
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×Q812yes—
146C22×C4×C47C2×Q814yes—
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—28no—
149C22×Dic47—24no—
150C22×Dic47—48no—
151C4⋊C4×C47—24no—
152C4⋊C4×C47—28no—
153C4⋊C4×C47C22×C418yes—
154C22⋊C4×C47—24no—
155C22⋊C4×C47—28no—
156C22⋊C4×C47—24no—
157C42×C47—24no—
158C42×C47—28no—
159C42×C47—24no—
160C22⋊Dic47—216no—
161C4⋊Dic47—216no—
162C4×Dic47—28no—
163C4×Dic47—24no—
164C4×Dic47C22×C418yescomplemented
165#10625329—28no—
166#10625329—216no—
167#10625329—24no—
168#10625329C22×C418yescomplemented
169C2×C47⋊C8—216no—
170C2×C47⋊C8—28no—
171C2×C47⋊C8—24no—
172C23×C47—24no—
173C23×C47—24no—
174C23×C47—24no—
175C23×C47—48no—
176C23×C47—44no—
177C23×C47—48no—
178C23×C47—44no—
179C23×C47#9514yes—
180C23×C47#9514yes—
181C23×C47#9518yes—
182C23×C47C2×C22⋊C412yes—
183C23×C47C2×C22⋊C414yes—
184C23×C47C2×C22⋊C411yescharacteristic
185C23×C47C2×C22⋊C412yes—
186C23×C47C2×C22⋊C412yes—
187C23×C47C2×C22⋊C411yescharacteristic
188C23×C47C2×C4⋊C412yes—
189C23×C47C2×C4⋊C414yes—
190C23×C47C22×D411yescharacteristic
191C2×C4×C47—28no—
192C2×C4×C47—416no—
193C2×C4×C47—48no—
194C2×C4×C47#9518yes—
195C2×C4×C47#9518yes—
196C2×C4×C47#9518yes—
197C2×C4×C47#9518yes—
198C2×C4×C47#9518yes—
199C2×C4×C47#9514yes—
200C2×C4×C47#9514yes—
201C2×C4×C47C2×C4212yes—
202C2×C4×C47C2×C4214yes—
203C2×C4×C47C2×C4214yes—
204C2×C4×C47C2×C4212yes—
205C2×C4×C47C2×C22⋊C414yes—
206C2×C4×C47C2×C22⋊C412yes—
207C2×C4×C47C2×C22⋊C412yes—
208C2×C4×C47C2×C22⋊C414yes—
209C2×C4×C47C2×C22⋊C412yes—
210C2×C4×C47C2×C22⋊C414yes—
211C2×C4×C47C2×C22⋊C412yes—
212C2×C4×C47C2×C22⋊C411yescharacteristic
213C2×C4×C47C2×C22⋊C411yescharacteristic
214C2×C4×C47C2×C22⋊C412yes—
215C2×C4×C47C2×C4⋊C412yes—
216C2×C4×C47C2×C4⋊C414yes—
217C2×C4×C47C2×C4⋊C414yes—
218C2×C4×C47C2×C4⋊C412yes—
219C2×C4×C47C2×C4⋊C412yes—
220C2×C4×C47C2×C4⋊C414yes—
221C2×C4×C47C2×C4⋊C414yes—
222C2×C4×C47C2×C4⋊C412yes—
223C2×C4×C47C2×C4⋊C414yes—
224C2×C4×C47C2×C4⋊C412yes—
225C2×C4×C47C23×C411yescharacteristic
226C2×C4×C47C23×C411yescharacteristic
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C22×D411yescharacteristic
230C2×C4×C47C22×Q811yescharacteristic
231C2×Dic47—28no—
232C2×Dic47—48no—
233C2×Dic47—416no—
234C47⋊C8—28no—
235C2×#1733—471nomaximal, Sylow
236C22×C47—22no—
237C22×C47—24no—
238C22×C47—22no—
239C22×C47—24no—
240C22×C47—24no—
241C22×C47—28no—
242C22×C47—28no—
243C22×C47—24no—
244C22×C47—48no—
245C22×C47—44no—
246C22×C47—44no—
247C22×C47C2×#9514yes—
248C22×C47C2×#9514yes—
249C22×C47C2×#9512yes—
250C22×C47C2×#9512yes—
251C22×C47#42114yes—
252C22×C47C22×C22⋊C411yescharacteristic
253C22×C47C22×C22⋊C411yescharacteristic
254C22×C47C22×C4⋊C411yescharacteristic
255C4×C47—48no—
256C4×C47C2×#9512yes—
257C4×C47C2×#9514yes—
258C4×C47C2×#9512yes—
259C4×C47C2×#9514yes—
260C4×C47C22×C4211yescharacteristic
261C4×C47C22×C22⋊C411yescharacteristic
262C4×C47C22×C4⋊C411yescharacteristic
263C4×C47C22×C4⋊C411yescharacteristic
264Dic47—48no—
265C23×M16—471no—
266C22×#117—471no—
267C22×#117D4711yescharacteristic
268#1733—478no—
269C2×#339—474no—
270C2×C47—21no—
271C2×C47—21no—
272C2×C47—24no—
273C2×C47—22no—
274C2×C47—24no—
275C2×C47#173314yescomplemented
276C2×C47C22×#9511yescharacteristic
277C2×C47C2×#42112yes—
278C24×C4C2×D4711yescharacteristic
279C22×M16—478no—
280C22×M16—472no—
281C22×M16—474no—
282C22×M16—944no—
283C22×M16—942no—
284C22×M16—941no—
285C23×C8—941no—
286C2×#117—21no—
287C2×#117—22no—
288C2×#117—22no—
289C2×#117—21no—
290C2×#117—478no—
291C2×#117—474no—
292C2×#117—944no—
293C2×#117—942no—
294C2×#117Dic4712yescomplemented
295C2×#117Dic4714yescomplemented
296C2×#117C2×D4712yes—
297C2×#117C2×D4714yes—
298C22×C4⋊C4—941no—
299C22×C4⋊C4Dic4712yescomplemented
300C22×C22⋊C4—21no—
301C22×C22⋊C4—941no—
302C22×C42—21no—
303C22×C42—472no—
304#339—4716no—
305C47C2×#173311yescharacteristic, complemented, Sylow
306C25C22⋊D4711yescharacteristic
307C23×C4—22no—
308C23×C4—942no—
309C23×C4Dic9414yes—
310C23×C4C4×D4714yes—
311C23×C4C4×D4714yes—
312C23×C4D18814yes—
313C23×C4C2×Dic4711yescharacteristic
314C23×C4C2×Dic4711yescharacteristic
315C23×C4C2×Dic4712yes—
316C23×C4C22⋊D4711yescharacteristic
317C23×C4C22⋊D4712yes—
318C23×C4C22⋊D4712yes—
319C23×C4C22⋊D4711yescharacteristic
320C23×C4C22⋊D4711yescharacteristic
321C23×C4C22×D4712yes—
322C23×C4C22×D4711yescharacteristic
323C2×M16—478no—
324C2×M16—4716no—
325C2×M16—474no—
326C2×M16—9416no—
327C2×M16—948no—
328C2×M16—944no—
329C2×M16—948no—
330C2×M16—944no—
331C2×M16—942no—
332C22×C8—948no—
333C22×C8—944no—
334C22×C8—942no—
335#117—24no—
336#117—24no—
337#117—24no—
338#117—22no—
339#117—24no—
340#117—22no—
341#117—24no—
342#117—4716no—
343#117—9416no—
344#117—948no—
345#117C2×Dic4714yes—
346#117C2×Dic4718yescomplemented
347#117C22×D4714yes—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—948no—
351C2×C4⋊C4—944no—
352C2×C4⋊C4C2×Dic4714yes—
353C2×C4⋊C4C2×Dic4718yescomplemented
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—948no—
359C2×C22⋊C4—944no—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42—478no—
365C2×C42—474no—
366C2×C42—944no—
367C2×C42—942no—
368C24—44no—
369C24Dic47⋊C414yes—
370C24(C2×D47)⋊C414yes—
371C24C22⋊Dic4711yescharacteristic
372C24C22⋊Dic4711yescharacteristic
373C24C22⋊Dic4712yes—
374C24C2×C22⋊D4711yescharacteristic
375C24C2×C22⋊D4712yes—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—48no—
379C22×C4—948no—
380C22×C4—944no—
381C22×C4—1888no—
382C22×C4C4×Dic4714yes—
383C22×C4C4×Dic4714yes—
384C22×C4C4×Dic4718yes—
385C22×C4Dic47⋊C414yes—
386C22×C4Dic47⋊C418yes—
387C22×C4Dic47⋊C418yes—
388C22×C4Dic47⋊C414yes—
389C22×C4Dic47⋊C414yes—
390C22×C4C4⋊Dic4714yes—
391C22×C4C4⋊Dic4714yes—
392C22×C4C4⋊Dic4718yes—
393C22×C4(C2×D47)⋊C414yes—
394C22×C4(C2×D47)⋊C418yes—
395C22×C4(C2×D47)⋊C414yes—
396C22×C4(C2×D47)⋊C418yes—
397C22×C4(C2×D47)⋊C414yes—
398C22×C4C22⋊Dic4711yescharacteristic
399C22×C4C22⋊Dic4712yes—
400C22×C4C22⋊Dic4712yes—
401C22×C4C22⋊Dic4711yescharacteristic
402C22×C4C22⋊Dic4712yes—
403C22×C4C22⋊Dic4712yes—
404C22×C4C22⋊Dic4712yes—
405C22×C4C2×Dic9414yes—
406C22×C4C2×Dic9412yes—
407C22×C4C2×C4×D4714yes—
408C22×C4C2×C4×D4712yes—
409C22×C4C2×C4×D4712yes—
410C22×C4C2×C4×D4714yes—
411C22×C4C2×D18812yes—
412C22×C4C2×D18814yes—
413C22×C4C22×Dic4711yescharacteristic
414C22×C4C22×Dic4712yes—
415C22×C4C22×Dic4711yescharacteristic
416C22×C4C22×Dic4712yes—
417C22×C4C2×C22⋊D4711yescharacteristic
418C22×C4C2×C22⋊D4712yes—
419C22×C4C2×C22⋊D4712yes—
420C22×C4C2×C22⋊D4712yes—
421C22×C4C2×C22⋊D4711yescharacteristic
422C22×C4C2×C22⋊D4711yescharacteristic
423C22×C4C23×D4711yescharacteristic
424M16—478no—
425M16—948no—
426M16—9416no—
427M16—944no—
428C2×C8—9416no—
429C2×C8—948no—
430C2×C8—944no—
431C4⋊C4—24no—
432C4⋊C4—28no—
433C4⋊C4—9416no—
434C4⋊C4C22×Dic4718yes—
435C22⋊C4—24no—
436C22⋊C4—28no—
437C22⋊C4—24no—
438C22⋊C4—9416no—
439C42—24no—
440C42—28no—
441C42—24no—
442C42—478no—
443C42—948no—
444C42—944no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—48no—
449C23—48no—
450C23—44no—
451C23—44no—
452C23C47⋊#9514yes—
453C23C47⋊#9514yes—
454C23C47⋊#9518yes—
455C23C2×(C2×D47)⋊C412yes—
456C23C2×(C2×D47)⋊C414yes—
457C23C2×C22⋊Dic4711yescharacteristic
458C23C2×C22⋊Dic4712yes—
459C23C2×C22⋊Dic4712yes—
460C23C2×C22⋊Dic4711yescharacteristic
461C23C2×Dic47⋊C412yes—
462C23C2×Dic47⋊C414yes—
463C23C22×C22⋊D4711yescharacteristic
464C2×C4—28no—
465C2×C4—416no—
466C2×C4—48no—
467C2×C4—948no—
468C2×C4—1888no—
469C2×C4—18816no—
470C2×C4C47⋊#9518yes—
471C2×C4C47⋊#9518yes—
472C2×C4C47⋊#9518yes—
473C2×C4C47⋊#9518yes—
474C2×C4C47⋊#9518yes—
475C2×C4C47⋊#9514yes—
476C2×C4C47⋊#9514yes—
477C2×C4C2×C4×Dic4712yes—
478C2×C4C2×C4×Dic4714yes—
479C2×C4C2×C4×Dic4714yes—
480C2×C4C2×C4×Dic4712yes—
481C2×C4C2×(C2×D47)⋊C412yes—
482C2×C4C2×(C2×D47)⋊C414yes—
483C2×C4C2×(C2×D47)⋊C414yes—
484C2×C4C2×(C2×D47)⋊C412yes—
485C2×C4C2×(C2×D47)⋊C412yes—
486C2×C4C2×(C2×D47)⋊C414yes—
487C2×C4C2×C22⋊Dic4712yes—
488C2×C4C2×C22⋊Dic4711yescharacteristic
489C2×C4C2×C22⋊Dic4711yescharacteristic
490C2×C4C2×C22⋊Dic4712yes—
491C2×C4C2×Dic47⋊C412yes—
492C2×C4C2×Dic47⋊C414yes—
493C2×C4C2×Dic47⋊C412yes—
494C2×C4C2×Dic47⋊C414yes—
495C2×C4C2×Dic47⋊C414yes—
496C2×C4C2×Dic47⋊C412yes—
497C2×C4C2×C4⋊Dic4712yes—
498C2×C4C2×C4⋊Dic4714yes—
499C2×C4C2×C4⋊Dic4714yes—
500C2×C4C2×C4⋊Dic4712yes—
501C2×C4C22×C4×D4711yescharacteristic
502C2×C4C22×C4×D4711yescharacteristic
503C2×C4C23×Dic4711yescharacteristic
504C2×C4C22×C22⋊D4711yescharacteristic
505C2×C4C22×D18811yescharacteristic
506C2×C4C22×Dic9411yescharacteristic
507C8—948no—
508C22—24no—
509C22—22no—
510C22—22no—
511C22—24no—
512C22—24no—
513C22—28no—
514C22—28no—
515C22—24no—
516C22—48no—
517C22—44no—
518C22—44no—
519C22C2×C47⋊#9514yes—
520C22C2×C47⋊#9514yes—
521C22C2×C47⋊#9512yes—
522C22C2×C47⋊#9512yes—
523C22#1568914014yes—
524C22C22×(C2×D47)⋊C411yescharacteristic
525C22C22×C22⋊Dic4711yescharacteristic
526C22C22×Dic47⋊C411yescharacteristic
527C4—48no—
528C4—1888no—
529C4C2×C47⋊#9512yes—
530C4C2×C47⋊#9512yes—
531C4C2×C47⋊#9514yes—
532C4C2×C47⋊#9514yes—
533C4C22×C4×Dic4711yescharacteristic
534C4C22×(C2×D47)⋊C411yescharacteristic
535C4C22×Dic47⋊C411yescharacteristic
536C4C22×C4⋊Dic4711yescharacteristic
537C2—21no—
538C2—24no—
539C2—22no—
540C2—24no—
541C2—21no—
542C2#2024309714yesdirect product factor
543C2C22×C47⋊#9511yescharacteristic
544C2C2×#1568914012yes—
subgroupG/H
centerC22×C4C2×C22⋊D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC22×#117×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47D4
cosocleC22×C4×C47C24
hypercenterC22×#117D47
hypocenterC47C2×#1733
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. #20243097C2direct (N × H)
4. C2×#15688882C2semidirect
9. C22×#10625329C4semidirect
15. C2×#13093983C4semidirect
20. #15688882C22semidirect
26. C2×#117×C47C4semidirect
27. C2×#117×C47C4semidirect
30. C22×C4⋊C4×C47C4semidirect
40. C2×#10625329C2×C4semidirect
41. C2×#10625329C2×C4semidirect
48. #13093983C2×C4semidirect
55. C2×C4×Dic47C2×C4semidirect
81. #117×C47C2×C4semidirect
86. C2×C4⋊C4×C47C2×C4semidirect
164. C4×Dic47C22×C4semidirect
168. #10625329C22×C4semidirect
275. C2×C47#1733semidirect
294. C2×#117Dic47semidirect
295. C2×#117Dic47semidirect
299. C22×C4⋊C4Dic47semidirect
305. C47C2×#1733semidirect
346. #117C2×Dic47semidirect
353. C2×C4⋊C4C2×Dic47semidirect
542. C2#20243097direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253292
C22×#130939832
#202430972
C2×#156888822
C22×#117×C472
C2×#173347