Finite Group Databasebeta

← C2×#20242016 #26184301

Subgroups — C2×#20242016 #26184301

Order 12032.

986 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689652C211yesmaximal, characteristic
2C2×#15689178C211yesmaximal, characteristic, complemented
3C2×#15689178C211yesmaximal, characteristic, complemented
4#20242016C218yesmaximal, direct product factor
5C2×#15688864C211yesmaximal, characteristic, complemented
6C2×#15688864C211yesmaximal, characteristic, complemented
7C2×#15688863C211yesmaximal, characteristic, complemented
8C2×C47×#329C211yesmaximal, characteristic, complemented
9#15689652—22no—
10#15689652—22no—
11#15689652C414yes—
12#15689652C2214yes—
13C2×#13094017—21no—
14C2×#13094016C2211yescharacteristic
15C2×#13094006—21no—
16C2×#13094000C411yescharacteristic
17C2×#13093995C411yescharacteristic
18C2×Q8×Dic47C411yescharacteristic
19C2×D4×Dic47C411yescharacteristic
20C2×(C22⋊D47)⋊C4—21no—
21C2×#13093980—21no—
22C2×C42×D47C2211yescharacteristic
23#15689178—22no—
24#15689178—22no—
25#15689178C2214yescomplemented
26#15689178C2214yescomplemented
27C2×Dic47⋊C8—21no—
28C2×Dic47⋊C8—21no—
29C2×(C2×D47)⋊C8—21no—
30C2×(C2×D47)⋊C8—21no—
31C2×#13093935C2211yescharacteristic
32C2×#13093935C2211yescharacteristic
33#15688864C2214yescomplemented
34#15688864C2214yescomplemented
35#15688863C2214yescomplemented
36C2×#124×C47C2211yescharacteristic
37C2×#97×C47C2211yescharacteristic
38C2×#97×C47C2211yescharacteristic
39C47×#329C2214yescomplemented
40C22×C22⋊D47—21no—
41C23×Dic47—21no—
42C22×C4×D47C2×C411yescharacteristic
43C22×C4×D47D411yescharacteristic
44C22×C4×D47D411yescharacteristic
45C22×C47⋊C8—21no—
46C22×C47⋊C8—21no—
47#13094017—24no—
48#13094017—24no—
49#13094016—21no—
50#13094016—21no—
51#13094016C2×C412yes—
52#13094016C2312yes—
53#13094006—24no—
54#13094006—24no—
55#13094000—22no—
56#13094000C2×C414yes—
57#13093995—22no—
58#13093995C2×C414yes—
59Q8×Dic47—22no—
60Q8×Dic47C2×C414yes—
61D4×Dic47—22no—
62D4×Dic47C2×C414yes—
63(C22⋊D47)⋊C4—24no—
64(C22⋊D47)⋊C4—24no—
65#13093980—24no—
66#13093980—24no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47C2×C411yescharacteristic
69C2×C4⋊Dic47C2×C411yescharacteristic
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×C22⋊Dic47—21no—
74C2×C22⋊Dic47—21no—
75C2×(C2×D47)⋊C4—21no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47C2×C411yescharacteristic
78C2×C4×Dic47D411yescharacteristic
79C2×C4×Dic47D411yescharacteristic
80C42×D47C2×C412yes—
81C42×D47D412yes—
82C42×D47D412yes—
83C42×D47C2312yes—
84Dic47⋊C8—24no—
85Dic47⋊C8—24no—
86(C2×D47)⋊C8—24no—
87(C2×D47)⋊C8—24no—
88#13093935—21no—
89#13093935—21no—
90#13093935C2312yes—
91#13093935C2312yes—
92C22×Q8×C47C2×C411yescharacteristic
93C22×D4×C47C2×C411yescharacteristic
94C22×C8×C47—21no—
95C22×C8×C47—21no—
96#124×C47—21no—
97#124×C47—21no—
98#124×C47C2×C412yes—
99#124×C47C2312yes—
100C2×C22⋊C4×C47—21no—
101C2×C22⋊C4×C47—21no—
102C2×C42×C47C2311yescharacteristic
103#97×C47—21no—
104#97×C47—21no—
105#97×C47C2312yes—
106#97×C47C2312yes—
107C24×C47—21no—
108C23×D47C22⋊C411yescharacteristic
109C2×Q8×C47—21no—
110C2×Q8×C47—42no—
111C2×Q8×C47C22×C412yes—
112C2×D4×C47—21no—
113C2×D4×C47—42no—
114C2×D4×C47C22×C412yes—
115C22×C4×C47—21no—
116C22×C4×C47C22×C411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D411yescharacteristic
119C2×C22⋊D47—24no—
120C2×C22⋊D47—44no—
121C22×Dic47—21no—
122C22×Dic47—21no—
123C22×Dic47—24no—
124C22×Dic47—44no—
125C22×Dic47C22⋊C411yescharacteristic
126C22×Dic47C22⋊C411yescharacteristic
127C22×Dic47C22⋊C411yescharacteristic
128C2×C4×D47—42no—
129C2×C4×D47—42no—
130C2×C4×D47—42no—
131C2×C4×D47C22⋊C412yes—
132C2×C4×D47C22⋊C412yes—
133C2×C4×D47C22⋊C412yes—
134C2×C4×D47C22×C412yes—
135C2×C4×D47C2×D412yes—
136C2×C4×D47C2×D412yes—
137C2×C8×C47—22no—
138C2×C8×C47—22no—
139C2×C8×C47—42no—
140C2×C8×C47—42no—
141C22⋊C4×C47—22no—
142C22⋊C4×C47—22no—
143C22⋊C4×C47—22no—
144C22⋊C4×C47—22no—
145C42×C47C22×C411yescharacteristic
146C42×C47C2×D411yescharacteristic
147C42×C47C2×D411yescharacteristic
148C42×C47C2411yescharacteristic
149C22⋊Dic47—24no—
150C22⋊Dic47—24no—
151(C2×D47)⋊C4—24no—
152C4⋊Dic47—21no—
153C4⋊Dic47—21no—
154C4⋊Dic47—24no—
155C4⋊Dic47C22×C412yes—
156C4⋊Dic47C22×C412yes—
157Dic47⋊C4—22no—
158Dic47⋊C4—22no—
159Dic47⋊C4—22no—
160Dic47⋊C4—22no—
161Dic47⋊C4—24no—
162C4×Dic47—24no—
163C4×Dic47C22⋊C412yes—
164C4×Dic47C22⋊C412yes—
165C4×Dic47C22⋊C412yes—
166C4×Dic47C22×C412yes—
167C4×Dic47C2×D412yes—
168C4×Dic47C2×D412yes—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—42no—
172C2×C47⋊C8—42no—
173C23×C47—22no—
174C23×C47—42no—
175C23×C47—42no—
176C23×C47—42no—
177C23×C47C2×C22⋊C411yescharacteristic
178C22×D47—22no—
179C22×D47—22no—
180C22×D47—22no—
181C22×D47C2×C22⋊C412yes—
182Q8×C47—42no—
183Q8×C47—42no—
184D4×C47—42no—
185D4×C47—42no—
186C2×C4×C47—22no—
187C2×C4×C47—22no—
188C2×C4×C47—22no—
189C2×C4×C47—22no—
190C2×C4×C47—42no—
191C2×C4×C47C2×C22⋊C411yescharacteristic
192C2×C4×C47C2×C22⋊C411yescharacteristic
193C2×C4×C47C2×C22⋊C411yescharacteristic
194C2×C4×C47C23×C411yescharacteristic
195C2×C4×C47C22×D411yescharacteristic
196C2×C4×C47C22×D411yescharacteristic
197C22⋊D47—48no—
198C2×Dic47—22no—
199C2×Dic47—22no—
200C2×Dic47—22no—
201C2×Dic47—22no—
202C2×Dic47—22no—
203C2×Dic47—48no—
204C2×Dic47—42no—
205C2×Dic47—42no—
206C2×Dic47C2×C22⋊C412yes—
207C2×Dic47C2×C22⋊C412yes—
208C2×Dic47C2×C22⋊C412yes—
209C4×D47—42no—
210C4×D47—42no—
211C4×D47—42no—
212C4×D47—42no—
213C4×D47—42no—
214C4×D47—42no—
215C8×C47—42no—
216C8×C47—42no—
217C47⋊C8—42no—
218C47⋊C8—42no—
219C2×#1535—471nomaximal, Sylow
220C22×C47—42no—
221C22×C47—42no—
222C22×C47—42no—
223C22×C47—84no—
224C22×C47#41312yes—
225C22×C47#42112yes—
226C22×C47#42112yes—
227C22×C47C22×C22⋊C411yescharacteristic
228C2×D47—22no—
229C2×D47—22no—
230C2×D47—22no—
231C2×D47—44no—
232C4×C47—21no—
233C4×C47—21no—
234C4×C47—21no—
235C4×C47—21no—
236C4×C47—21no—
237C4×C47—21no—
238C4×C47—42no—
239Dic47—22no—
240Dic47—22no—
241Dic47—22no—
242Dic47—42no—
243Dic47—42no—
244C2×#529—471no—
245C2×#432—471no—
246C2×#432—471no—
247#1535—478no—
248C2×#330—471no—
249C2×#330—471no—
250C2×#329—471no—
251C2×#329D4711yescharacteristic, complemented
252C2×C47—82no—
253C2×C47#153514yescomplemented
254C2×C47C2×#41311yescharacteristic
255C2×C47C2×#42111yescharacteristic
256C2×C47C2×#42111yescharacteristic
257D47—42no—
258#529—474no—
259#529—474no—
260#529—942no—
261#529—942no—
262C2×#126—941no—
263C2×#125—471no—
264C2×#124C2×D4711yescharacteristic
265C2×#123—941no—
266C2×C22⋊Q8—471no—
267C2×#121—471no—
268C2×C4×Q8—471no—
269C2×C4×D4—471no—
270C2×C4×D4—941no—
271C2×#117—941no—
272C22×C42—471no—
273#432—474no—
274#432—474no—
275#432—942no—
276#432—942no—
277C2×C4⋊C8—941no—
278C2×C4⋊C8—941no—
279C2×C22⋊C8—941no—
280C2×C22⋊C8—941no—
281C2×#97—471no—
282C2×#97—471no—
283C2×#97C2×D4711yescharacteristic
284C2×#97C2×D4711yescharacteristic
285#330—474no—
286#330—474no—
287#329—474no—
288#329C2×D4714yescomplemented
289C47C2×#153511yescharacteristic, complemented, Sylow
290C22×Q8C4×D4711yescharacteristic
291C22×D4—941no—
292C22×D4C4×D4711yescharacteristic
293C23×C4—471no—
294C23×C4—471no—
295C23×C4—471no—
296C23×C4—941no—
297C22×C8—21no—
298C22×C8—21no—
299C22×C8—941no—
300C22×C8—941no—
301#126—944no—
302#126—944no—
303#125—472no—
304#125—472no—
305#125—941no—
306#125—941no—
307#124—21no—
308#124—21no—
309#124C4×D4712yes—
310#124C22×D4712yes—
311#123—944no—
312#123—944no—
313C22⋊Q8—474no—
314C22⋊Q8—942no—
315#121—474no—
316#121—942no—
317C4×Q8—474no—
318C4×Q8—942no—
319C4×D4—474no—
320C4×D4—942no—
321C4×D4—944no—
322C4×D4—944no—
323#117—944no—
324#117—944no—
325C2×C4⋊C4—471no—
326C2×C4⋊C4—471no—
327C2×C4⋊C4—941no—
328C2×C4⋊C4—941no—
329C2×C4⋊C4—941no—
330C2×C4⋊C4—941no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—941no—
334C2×C22⋊C4—941no—
335C2×C22⋊C4—941no—
336C2×C42—471no—
337C2×C42—471no—
338C2×C42—471no—
339C2×C42—472no—
340C2×C42—472no—
341C2×C42—472no—
342C2×C42—472no—
343C2×C42—941no—
344C2×C42C22×D4711yescharacteristic
345C4⋊C8—944no—
346C4⋊C8—944no—
347C22⋊C8—944no—
348C22⋊C8—944no—
349#97—21no—
350#97—21no—
351#97—472no—
352#97—472no—
353#97—941no—
354#97—941no—
355#97C22×D4712yes—
356#97C22×D4712yes—
357C24—21no—
358C24—471no—
359C2×Q8—21no—
360C2×Q8—42no—
361C2×Q8C2×C4×D4712yes—
362C2×D4—21no—
363C2×D4—42no—
364C2×D4—944no—
365C2×D4—1884no—
366C2×D4C2×C4×D4712yes—
367C22×C4—21no—
368C22×C4—471no—
369C22×C4—471no—
370C22×C4—471no—
371C22×C4—472no—
372C22×C4—472no—
373C22×C4—472no—
374C22×C4—472no—
375C22×C4—472no—
376C22×C4—472no—
377C22×C4—944no—
378C22×C4—941no—
379C22×C4—941no—
380C22×C4—1882no—
381C22×C4—1882no—
382C22×C4—1882no—
383C22×C4—1884no—
384C22×C4C2×C4×D4711yescharacteristic
385C22×C4D4×D4711yescharacteristic
386C22×C4D4×D4711yescharacteristic
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—942no—
392C2×C8—942no—
393C2×C8—1882no—
394C2×C8—1882no—
395C4⋊C4—472no—
396C4⋊C4—472no—
397C4⋊C4—944no—
398C4⋊C4—944no—
399C4⋊C4—942no—
400C4⋊C4—942no—
401C4⋊C4—941no—
402C4⋊C4—941no—
403C4⋊C4—942no—
404C4⋊C4—942no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—944no—
410C22⋊C4—944no—
411C22⋊C4—944no—
412C42—472no—
413C42—472no—
414C42—472no—
415C42—472no—
416C42—472no—
417C42—472no—
418C42—944no—
419C42C2×C4×D4711yescharacteristic
420C42D4×D4711yescharacteristic
421C42D4×D4711yescharacteristic
422C42C23×D4711yescharacteristic
423C23—22no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—472no—
428C23—942no—
429C23—942no—
430C23—942no—
431C23C22⋊C4×D4711yescharacteristic
432Q8—42no—
433Q8—42no—
434D4—42no—
435D4—42no—
436D4—1888no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—472no—
443C2×C4—472no—
444C2×C4—472no—
445C2×C4—942no—
446C2×C4—942no—
447C2×C4—942no—
448C2×C4—942no—
449C2×C4—942no—
450C2×C4—1882no—
451C2×C4—1882no—
452C2×C4—1882no—
453C2×C4—1882no—
454C2×C4—1882no—
455C2×C4—1882no—
456C2×C4—1888no—
457C2×C4—1882no—
458C2×C4—1882no—
459C2×C4C22⋊C4×D4711yescharacteristic
460C2×C4C22⋊C4×D4711yescharacteristic
461C2×C4C22⋊C4×D4711yescharacteristic
462C2×C4C22×C4×D4711yescharacteristic
463C2×C4C2×D4×D4711yescharacteristic
464C2×C4C2×D4×D4711yescharacteristic
465C8—42no—
466C8—42no—
467C8—1882no—
468C8—1882no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—84no—
473C22—942no—
474C22—942no—
475C22—942no—
476C22—1884no—
477C22#1568910412yes—
478C22#1568913812yes—
479C22#1568913812yes—
480C22C2×C22⋊C4×D4711yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—42no—
488C4—942no—
489C4—942no—
490C4—942no—
491C4—1882no—
492C4—1882no—
493C2—82no—
494C2—1882no—
495C2#2024201614yesdirect product factor
496C2C2×#1568910411yescharacteristic
497C2C2×#1568913811yescharacteristic
498C2C2×#1568913811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC2×C47×#329C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093995C4
cosocleC42×C47C24
hypercenterC2×#329D47
hypocenterC47C2×#1535
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
2. C2×#15689178C2semidirect
3. C2×#15689178C2semidirect
4. #20242016C2direct (N × H)
5. C2×#15688864C2semidirect
6. C2×#15688864C2semidirect
7. C2×#15688863C2semidirect
8. C2×C47×#329C2semidirect
25. #15689178C22semidirect
26. #15689178C22semidirect
33. #15688864C22semidirect
34. #15688864C22semidirect
35. #15688863C22semidirect
39. C47×#329C22semidirect
251. C2×#329D47semidirect
253. C2×C47#1535semidirect
288. #329C2×D47semidirect
289. C47C2×#1535semidirect
495. C2#20242016direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896522
C2×#156891782
#202420162
C2×#156888642
C2×#156888632
C2×C47×#3292
C2×#153547