Finite Group Databasebeta

← C2×#20241717 #26182118

Subgroups — C2×#20241717 #26182118

Order 12032.

1718 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic47⋊C4C211yesmaximal, characteristic
2C2×#15689087C211yesmaximal, characteristic
3#20241717C218yesmaximal, direct product factor
4C2×#15688848C214yesmaximal, complemented
5C2×C47×#407C211yesmaximal, characteristic
6C24×Dic47C411yescharacteristic
7C24×Dic47C411yescharacteristic
8C22×#10625329—21no—
9C22×Dic47⋊C4—21no—
10C22×Dic47⋊C4—21no—
11C22×Dic47⋊C4—22no—
12C22×Dic47⋊C4—21no—
13C22×Dic47⋊C4—21no—
14C22×Dic47⋊C4—22no—
15C22×Dic47⋊C4C414yes—
16C22×Dic47⋊C4C412yes—
17C22×Dic47⋊C4C2212yes—
18C22×Dic47⋊C4C2214yes—
19#15689087—22no—
20#15689087C2214yes—
21C2×C22⋊(C47⋊C8)—21no—
22C2×C22⋊(C47⋊C8)C2212yes—
23#15688848C22116yescomplemented
24C24×C4×C47C2211yescharacteristic
25C22×M16×C47—21no—
26C47×#407—22no—
27C47×#407C2214yes—
28C2×C22⋊C8×C47—21no—
29C2×C22⋊C8×C47C2212yes—
30C23×Dic47—22no—
31C23×Dic47—21no—
32C23×Dic47—21no—
33C23×Dic47—28no—
34C23×Dic47—22no—
35C23×Dic47—21no—
36C23×Dic47—21no—
37C23×Dic47—44no—
38C23×Dic47C2×C414yes—
39C23×Dic47C2×C412yes—
40C23×Dic47C2×C414yes—
41C23×Dic47C2×C412yes—
42C2×#10625329—24no—
43C2×#10625329—44no—
44C22×C47⋊C8—22no—
45C2×Dic47⋊C4—24no—
46C2×Dic47⋊C4—24no—
47C2×Dic47⋊C4—22no—
48C2×Dic47⋊C4—22no—
49C2×Dic47⋊C4—24no—
50C2×Dic47⋊C4—24no—
51C2×Dic47⋊C4—24no—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4—24no—
54C2×Dic47⋊C4—22no—
55C2×Dic47⋊C4—24no—
56C2×Dic47⋊C4—22no—
57C2×Dic47⋊C4—24no—
58C2×Dic47⋊C4—22no—
59C2×Dic47⋊C4—24no—
60C2×Dic47⋊C4C2×C414yes—
61C2×Dic47⋊C4C2×C418yes—
62C2×Dic47⋊C4C2314yes—
63C22⋊(C47⋊C8)—24no—
64C22⋊(C47⋊C8)—22no—
65C22⋊(C47⋊C8)C2314yes—
66C25×C47C2×C411yescharacteristic
67C23×C4×C47—44no—
68C23×C4×C47C2×C412yes—
69C23×C4×C47C2×C411yescharacteristic
70C23×C4×C47D411yescharacteristic
71C23×C4×C47D411yescharacteristic
72C23×C4×C47D411yescharacteristic
73C23×C4×C47D412yes—
74C23×C4×C47D411yescharacteristic
75C23×C4×C47D412yes—
76C23×C4×C47C2311yescharacteristic
77C23×C4×C47C2312yes—
78C2×M16×C47—24no—
79C2×M16×C47—44no—
80C22×C8×C47—22no—
81C22⋊C8×C47—24no—
82C22⋊C8×C47—22no—
83C22⋊C8×C47C2314yes—
84C24×C47—24no—
85C24×C47—24no—
86C24×C47—24no—
87C24×C47C22⋊C412yes—
88C24×C47C22⋊C411yescharacteristic
89C24×C47C22⋊C411yescharacteristic
90C24×C47C22×C412yes—
91C24×C47C22×C411yescharacteristic
92C22×C4×C47—24no—
93C22×C4×C47—24no—
94C22×C4×C47—44no—
95C22×C4×C47—42no—
96C22×C4×C47—42no—
97C22×C4×C47—44no—
98C22×C4×C47—42no—
99C22×C4×C47—44no—
100C22×C4×C47—42no—
101C22×C4×C47—44no—
102C22×C4×C47C22⋊C412yes—
103C22×C4×C47C22⋊C412yes—
104C22×C4×C47C22⋊C411yescharacteristic
105C22×C4×C47C22⋊C411yescharacteristic
106C22×C4×C47C22⋊C412yes—
107C22×C4×C47C22⋊C412yes—
108C22×C4×C47C22⋊C412yes—
109C22×C4×C47C22×C411yescharacteristic
110C22×C4×C47C22×C412yes—
111C22×C4×C47C2×D411yescharacteristic
112C22×C4×C47C2×D412yes—
113C22×C4×C47C2×D411yescharacteristic
114C22×C4×C47C2×D412yes—
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D412yes—
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D412yes—
119C22×C4×C47C2411yescharacteristic
120C22×Dic47—216no—
121C22×Dic47—28no—
122C22×Dic47—28no—
123C22×Dic47—24no—
124C22×Dic47—24no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—24no—
128C22×Dic47—216no—
129C22×Dic47—22no—
130C22×Dic47—24no—
131C22×Dic47—24no—
132C22×Dic47—22no—
133C22×Dic47—48no—
134C22×Dic47—44no—
135C22×Dic47—44no—
136C22×Dic47—48no—
137C22×Dic47—44no—
138C22×Dic47M1618yescomplemented
139C22×Dic47C22×C414yes—
140C22×Dic47C22×C414yes—
141M16×C47—48no—
142C2×C8×C47—24no—
143C2×C8×C47—44no—
144Dic47⋊C4—24no—
145Dic47⋊C4—28no—
146Dic47⋊C4—24no—
147Dic47⋊C4—28no—
148Dic47⋊C4—24no—
149Dic47⋊C4—24no—
150#10625329—48no—
151C2×C47⋊C8—24no—
152C2×C47⋊C8—44no—
153C23×C47—232no—
154C23×C47—24no—
155C23×C47—24no—
156C23×C47—22no—
157C23×C47—22no—
158C23×C47—24no—
159C23×C47—22no—
160C23×C47—24no—
161C23×C47—22no—
162C23×C47—24no—
163C23×C47—22no—
164C23×C47—24no—
165C23×C47—22no—
166C23×C47#9914yes—
167C23×C47#10114yes—
168C23×C47C2×C22⋊C411yescharacteristic
169C23×C47C2×C22⋊C412yes—
170C23×C47C2×C22⋊C412yes—
171C23×C47C2×C22⋊C411yescharacteristic
172C23×C47C2×M1614yes—
173C23×C47C23×C411yescharacteristic
174C2×C4×C47—22no—
175C2×C4×C47—24no—
176C2×C4×C47—22no—
177C2×C4×C47—24no—
178C2×C4×C47—24no—
179C2×C4×C47—24no—
180C2×C4×C47—22no—
181C2×C4×C47—22no—
182C2×C4×C47—42no—
183C2×C4×C47—42no—
184C2×C4×C47—42no—
185C2×C4×C47—41no—
186C2×C4×C47—42no—
187C2×C4×C47—41no—
188C2×C4×C47—42no—
189C2×C4×C47—41no—
190C2×C4×C47—42no—
191C2×C4×C47—41no—
192C2×C4×C47C2×C22⋊C411yescharacteristic
193C2×C4×C47C2×C22⋊C412yes—
194C2×C4×C47C2×C22⋊C411yescharacteristic
195C2×C4×C47C2×C22⋊C412yes—
196C2×C4×C47C22×D411yescharacteristic
197C2×C4×C47C22×D411yescharacteristic
198C2×Dic47—216no—
199C2×Dic47—216no—
200C2×Dic47—24no—
201C2×Dic47—24no—
202C2×Dic47—28no—
203C2×Dic47—28no—
204C2×Dic47—24no—
205C2×Dic47—44no—
206C2×Dic47—48no—
207C2×Dic47—48no—
208C2×Dic47—44no—
209C2×Dic47—44no—
210C2×Dic47C2×M1618yescomplemented
211C8×C47—44no—
212C47⋊C8—44no—
213C2×#1463—471nomaximal, Sylow
214C22×C47—216no—
215C22×C47—216no—
216C22×C47—22no—
217C22×C47—22no—
218C22×C47—21no—
219C22×C47—22no—
220C22×C47—21no—
221C22×C47—21no—
222C22×C47—22no—
223C22×C47—21no—
224C22×C47—28no—
225C22×C47—28no—
226C22×C47—28no—
227C22×C47#40712yes—
228C22×C47#40712yes—
229C22×C47C2×#9914yes—
230C22×C47C2×#9912yes—
231C22×C47C2×#10114yes—
232C22×C47C2×#10112yes—
233C22×C47C22×C22⋊C411yescharacteristic
234C22×C47C22×M1612yes—
235C4×C47—21no—
236C4×C47—21no—
237C4×C47—21no—
238C4×C47—22no—
239C4×C47—21no—
240C4×C47—22no—
241Dic47—28no—
242Dic47—44no—
243C23×C4⋊C4—471no—
244C2×#407—471no—
245C2×#407D4711yescharacteristic
246#1463—478no—
247C2×#324—474no—
248C2×C47—22no—
249C2×C47—22no—
250C2×C47—24no—
251C2×C47—24no—
252C2×C47#146314yescomplemented
253C2×C47C2×#40711yescharacteristic
254C2×C47C22×#9911yescharacteristic
255C2×C47C22×#10111yescharacteristic
256C24×C4—471no—
257C24×C4—471no—
258C24×C4C2×D4711yescharacteristic
259C22×M16—21no—
260C22×M16—941no—
261C22×C4⋊C4—472no—
262C22×C4⋊C4—474no—
263C22×C4⋊C4—472no—
264C22×C4⋊C4—474no—
265C22×C4⋊C4—941no—
266C22×C4⋊C4—941no—
267C22×C4⋊C4—942no—
268C22×C4⋊C4—941no—
269C22×C4⋊C4—941no—
270C22×C4⋊C4—942no—
271#407—22no—
272#407—474no—
273#407—942no—
274#407C2×D4714yes—
275C2×C22⋊C8—21no—
276C2×C22⋊C8—472no—
277C2×C22⋊C8—941no—
278C2×C22⋊C8C2×D4712yes—
279#324—4716no—
280C47C2×#146311yescharacteristic, complemented, Sylow
281C25C4×D4711yescharacteristic
282C23×C4—44no—
283C23×C4—472no—
284C23×C4—474no—
285C23×C4—474no—
286C23×C4—472no—
287C23×C4—948no—
288C23×C4—942no—
289C23×C4—941no—
290C23×C4—941no—
291C23×C4—942no—
292C23×C4—941no—
293C23×C4—941no—
294C23×C4—1884no—
295C23×C4C4×D4712yes—
296C23×C4C4×D4711yescharacteristic
297C23×C4D18811yescharacteristic
298C23×C4D18811yescharacteristic
299C23×C4D18812yes—
300C23×C4C22⋊D4711yescharacteristic
301C23×C4C22⋊D4711yescharacteristic
302C23×C4C22⋊D4712yes—
303C23×C4C22×D4711yescharacteristic
304C23×C4C22×D4712yes—
305C2×M16—24no—
306C2×M16—44no—
307C2×M16—944no—
308C2×M16—1884no—
309C22×C8—22no—
310C22×C8—942no—
311C2×C4⋊C4—474no—
312C2×C4⋊C4—478no—
313C2×C4⋊C4—474no—
314C2×C4⋊C4—944no—
315C2×C4⋊C4—944no—
316C2×C4⋊C4—944no—
317C2×C4⋊C4—944no—
318C2×C4⋊C4—944no—
319C2×C4⋊C4—942no—
320C2×C4⋊C4—942no—
321C2×C4⋊C4—944no—
322C2×C4⋊C4—942no—
323C2×C4⋊C4—944no—
324C2×C4⋊C4—942no—
325C2×C4⋊C4—944no—
326C2×C4⋊C4—942no—
327C2×C4⋊C4—944no—
328C2×C4⋊C4—942no—
329C22⋊C8—24no—
330C22⋊C8—22no—
331C22⋊C8—474no—
332C22⋊C8—942no—
333C22⋊C8—944no—
334C22⋊C8C22×D4714yes—
335C24—24no—
336C24—24no—
337C24—24no—
338C24(C2×D47)⋊C412yes—
339C24(C2×D47)⋊C411yescharacteristic
340C24(C2×D47)⋊C411yescharacteristic
341C24C2×C4×D4712yes—
342C24C2×C4×D4711yescharacteristic
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—478no—
354C22×C4—474no—
355C22×C4—474no—
356C22×C4—9416no—
357C22×C4—9416no—
358C22×C4—948no—
359C22×C4—948no—
360C22×C4—944no—
361C22×C4—944no—
362C22×C4—942no—
363C22×C4—942no—
364C22×C4—944no—
365C22×C4—942no—
366C22×C4—944no—
367C22×C4—944no—
368C22×C4—942no—
369C22×C4—1888no—
370C22×C4—1884no—
371C22×C4—1884no—
372C22×C4—1888no—
373C22×C4—1884no—
374C22×C4(C2×D47)⋊C412yes—
375C22×C4(C2×D47)⋊C412yes—
376C22×C4(C2×D47)⋊C411yescharacteristic
377C22×C4(C2×D47)⋊C411yescharacteristic
378C22×C4(C2×D47)⋊C412yes—
379C22×C4(C2×D47)⋊C412yes—
380C22×C4(C2×D47)⋊C412yes—
381C22×C4C2×C4×D4711yescharacteristic
382C22×C4C2×C4×D4712yes—
383C22×C4C2×D18812yes—
384C22×C4C2×D18811yescharacteristic
385C22×C4C2×D18811yescharacteristic
386C22×C4C2×D18812yes—
387C22×C4C2×C22⋊D4712yes—
388C22×C4C2×C22⋊D4711yescharacteristic
389C22×C4C2×C22⋊D4711yescharacteristic
390C22×C4C2×C22⋊D4712yes—
391C22×C4C23×D4711yescharacteristic
392M16—48no—
393M16—1888no—
394C2×C8—24no—
395C2×C8—44no—
396C2×C8—944no—
397C2×C8—1884no—
398C4⋊C4—944no—
399C4⋊C4—944no—
400C4⋊C4—948no—
401C4⋊C4—948no—
402C4⋊C4—944no—
403C4⋊C4—944no—
404C23—232no—
405C23—24no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—22no—
410C23—24no—
411C23—22no—
412C23—24no—
413C23—22no—
414C23—24no—
415C23—22no—
416C23—24no—
417C23#1309393814yes—
418C23#1309394214yes—
419C23C2×(C2×D47)⋊C411yescharacteristic
420C23C2×(C2×D47)⋊C412yes—
421C23C2×(C2×D47)⋊C412yes—
422C23C2×(C2×D47)⋊C411yescharacteristic
423C23M16×D4714yes—
424C23C22×C4×D4711yescharacteristic
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—478no—
444C2×C4—9416no—
445C2×C4—9416no—
446C2×C4—944no—
447C2×C4—948no—
448C2×C4—948no—
449C2×C4—944no—
450C2×C4—944no—
451C2×C4—1884no—
452C2×C4—1888no—
453C2×C4—1888no—
454C2×C4—1884no—
455C2×C4—1884no—
456C2×C4C2×(C2×D47)⋊C411yescharacteristic
457C2×C4C2×(C2×D47)⋊C411yescharacteristic
458C2×C4C2×(C2×D47)⋊C412yes—
459C2×C4C2×(C2×D47)⋊C412yes—
460C2×C4C22×C22⋊D4711yescharacteristic
461C2×C4C22×D18811yescharacteristic
462C8—44no—
463C8—1884no—
464C22—216no—
465C22—216no—
466C22—28no—
467C22—28no—
468C22—28no—
469C22—22no—
470C22—22no—
471C22—21no—
472C22—22no—
473C22—21no—
474C22—21no—
475C22—22no—
476C22—21no—
477C22#1568908612yes—
478C22#1568908612yes—
479C22C2×#1309393814yes—
480C22C2×#1309393812yes—
481C22C2×#1309394214yes—
482C22C2×#1309394212yes—
483C22C22×(C2×D47)⋊C411yescharacteristic
484C22C2×M16×D4712yes—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—22no—
490C4—22no—
491C4—948no—
492C4—1884no—
493C2—22no—
494C2—24no—
495C2—24no—
496C2—22no—
497C2#2024171714yesdirect product factor
498C2C2×#1568908611yescharacteristic
499C2C22×#1309393811yescharacteristic
500C2C22×#1309394211yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#407C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47C2×C4
cosocleC22×C4×C47C24
hypercenterC2×#407D47
hypocenterC47C2×#1463
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
3. #20241717C2direct (N × H)
4. C2×#15688848C2semidirect
23. #15688848C22semidirect
138. C22×Dic47M16semidirect
210. C2×Dic47C2×M16semidirect
252. C2×C47#1463semidirect
280. C47C2×#1463semidirect
497. C2#20241717direct (N × H)

Maximal subgroups

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

subgroupindex
C23×Dic47⋊C42
C2×#156890872
#202417172
C2×#156888482
C2×C47×#4072
C2×#146347