Finite Group Databasebeta

← C2×#20242722 #26197448

Subgroups — C2×#20242722 #26197448

Order 12032.

894 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689533C211yesmaximal, characteristic, complemented
2C2×#15689493C211yesmaximal, characteristic, complemented
3C2×#15689482C211yesmaximal, characteristic
4#20242722C218yesmaximal, direct product factor
5C2×#15688865C211yesmaximal, characteristic, complemented
6C2×#15688865C211yesmaximal, characteristic, complemented
7C2×#15688863C211yesmaximal, characteristic
8C2×C47×#563C211yesmaximal, characteristic
9C2×#13094024C2211yescharacteristic
10#15689533—22no—
11#15689533C2214yescomplemented
12#15689493—22no—
13#15689493C2214yescomplemented
14#15689482—22no—
15#15689482C2214yes—
16C2×#13093957—21no—
17C2×#13093955—21no—
18C2×#13093948—21no—
19C2×#13093947—21no—
20C2×#13093947—21no—
21C2×#13093945—21no—
22C2×#13093935C2211yescharacteristic
23C2×#13093935C2211yescharacteristic
24C2×#13093935C2211yescharacteristic
25#15688865C2214yescomplemented
26#15688865C2214yescomplemented
27#15688863C2214yes—
28C47×#563—22no—
29C47×#563—22no—
30C47×#563—22no—
31C47×#563C2214yes—
32C2×C4⋊Q8×C47—21no—
33C2×#126×C47—21no—
34C2×#126×C47—21no—
35C2×#125×C47C2211yescharacteristic
36C2×#125×C47C2211yescharacteristic
37C2×#124×C47C2211yescharacteristic
38C2×#123×C47—21no—
39C2×C22⋊Q8×C47—21no—
40C2×C22⋊Q8×C47—21no—
41C22×Dic94—21no—
42C22×C47⋊C8—21no—
43C22×C47⋊C8—21no—
44C22×C47⋊C8—21no—
45#13094024—21no—
46#13094024—21no—
47#13094024—21no—
48#13094024C2312yes—
49C2×C4⋊Dic47—21no—
50C2×C4⋊Dic47—21no—
51#13093957—24no—
52#13093955—24no—
53#13093948—24no—
54#13093947—24no—
55#13093947—24no—
56#13093945—24no—
57#13093935—21no—
58#13093935—21no—
59#13093935—21no—
60#13093935C2312yes—
61#13093935C2312yes—
62#13093935C2312yes—
63C22×Q8×C47—21no—
64C22×Q8×C47D411yescharacteristic
65C22×D4×C47D411yescharacteristic
66C23×C4×C47—21no—
67C4⋊Q8×C47—24no—
68C4⋊Q8×C47—24no—
69#126×C47—24no—
70#126×C47—24no—
71#126×C47—24no—
72#126×C47—24no—
73#125×C47—21no—
74#125×C47—21no—
75#125×C47—21no—
76#125×C47—21no—
77#125×C47D412yes—
78#125×C47D412yes—
79#125×C47C2312yes—
80#125×C47C2312yes—
81#124×C47—21no—
82#124×C47—21no—
83#124×C47D412yes—
84#124×C47C2312yes—
85#123×C47—24no—
86#123×C47—24no—
87C22⋊Q8×C47—24no—
88C22⋊Q8×C47—24no—
89C22⋊Q8×C47—24no—
90C22⋊Q8×C47—24no—
91C2×C4⋊C4×C47—21no—
92C2×C4⋊C4×C47—21no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47D411yescharacteristic
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47D411yescharacteristic
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—21no—
103C2×C22⋊C4×C47—21no—
104C2×C22⋊C4×C47—21no—
105C2×C22⋊C4×C47—21no—
106C2×C42×C47—21no—
107C2×C42×C47C2311yescharacteristic
108C24×C47—21no—
109C2×Q8×C47—24no—
110C2×Q8×C47—21no—
111C2×Q8×C47—42no—
112C2×Q8×C47—82no—
113C2×Q8×C47C2×D412yes—
114C2×D4×C47—21no—
115C2×D4×C47—42no—
116C2×D4×C47C2×D412yes—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—24no—
123C22×C4×C47—82no—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×Dic47—41no—
128C2×Dic94—22no—
129C2×Dic94—44no—
130C2×Dic94—41no—
131C4⋊C4×C47—24no—
132C4⋊C4×C47—24no—
133C4⋊C4×C47—22no—
134C4⋊C4×C47—22no—
135C4⋊C4×C47—22no—
136C4⋊C4×C47—22no—
137C4⋊C4×C47—21no—
138C4⋊C4×C47—21no—
139C4⋊C4×C47—21no—
140C4⋊C4×C47—21no—
141C4⋊C4×C47—22no—
142C4⋊C4×C47—22no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47—22no—
145C4⋊C4×C47C2×D412yes—
146C4⋊C4×C47C2×D412yes—
147C4⋊C4×C47C2×D412yes—
148C4⋊C4×C47C2×D412yes—
149C22⋊C4×C47—24no—
150C22⋊C4×C47—24no—
151C22⋊C4×C47—24no—
152C22⋊C4×C47—22no—
153C22⋊C4×C47—22no—
154C22⋊C4×C47—22no—
155C22⋊C4×C47—22no—
156C42×C47—24no—
157C42×C47C2×D411yescharacteristic
158C42×C47C2×D411yescharacteristic
159C42×C47C2×D411yescharacteristic
160C42×C47C2411yescharacteristic
161C4⋊Dic47—22no—
162C4⋊Dic47—22no—
163C4⋊Dic47—41no—
164C4⋊Dic47—41no—
165C2×C47⋊C8—24no—
166C2×C47⋊C8—22no—
167C2×C47⋊C8—22no—
168C2×C47⋊C8—22no—
169C2×C47⋊C8—42no—
170C2×C47⋊C8—42no—
171C23×C47—22no—
172C23×C47—42no—
173C23×C47—42no—
174C23×C47—42no—
175C23×C47#12011yescharacteristic
176Q8×C47—42no—
177Q8×C47—81no—
178Q8×C47—84no—
179D4×C47—42no—
180D4×C47—81no—
181C2×C4×C47—22no—
182C2×C4×C47—22no—
183C2×C4×C47—22no—
184C2×C4×C47—22no—
185C2×C4×C47—22no—
186C2×C4×C47—22no—
187C2×C4×C47—22no—
188C2×C4×C47—22no—
189C2×C4×C47—42no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—42no—
194C2×C4×C47—84no—
195C2×C4×C47#12011yescharacteristic
196C2×C4×C47#12011yescharacteristic
197C2×C4×C47#12011yescharacteristic
198C2×C4×C47C22×D411yescharacteristic
199C2×C4×C47C22×D411yescharacteristic
200C2×C4×C47C22×D411yescharacteristic
201C2×Dic47—42no—
202C2×Dic47—82no—
203Dic94—44no—
204Dic94—82no—
205C47⋊C8—24no—
206C47⋊C8—42no—
207C47⋊C8—42no—
208C2×#1664—471nomaximal, Sylow
209C22×C47—42no—
210C22×C47—42no—
211C22×C47—42no—
212C22×C47—84no—
213C22×C47#45412yes—
214C22×C47#45512yes—
215C22×C47#45612yes—
216C22×C47C2×#12011yescharacteristic
217C4×C47—21no—
218C4×C47—21no—
219C4×C47—21no—
220C4×C47—21no—
221C4×C47—21no—
222C4×C47—21no—
223C4×C47—42no—
224C4×C47—42no—
225C4×C47—42no—
226C4×C47—42no—
227C4×C47—42no—
228Dic47—82no—
229C2×#563D4711yescharacteristic
230C2×#501—471no—
231C2×#491—471no—
232C2×#489—471no—
233#1664—478no—
234C2×#330—471no—
235C2×#330—471no—
236C2×#329—471no—
237C2×C47—82no—
238C2×C47#166414yescomplemented
239C2×C47C2×#45511yescharacteristic
240C2×C47C2×#45411yescharacteristic
241C2×C47C2×#45611yescharacteristic
242#563—22no—
243#563—22no—
244#563—22no—
245#563C2×D4714yes—
246C2×C4⋊Q8—21no—
247C2×C4⋊Q8—471no—
248C2×#126—21no—
249C2×#126—21no—
250C2×#125C2×D4711yescharacteristic
251C2×#125C2×D4711yescharacteristic
252C2×#124C2×D4711yescharacteristic
253C2×#123—21no—
254C2×C22⋊Q8—21no—
255C2×C22⋊Q8—21no—
256#501—474no—
257#501—942no—
258#491—474no—
259#491—942no—
260#489—474no—
261#489—942no—
262C2×#107—941no—
263C2×#106—941no—
264C2×#103—941no—
265C2×#103—941no—
266C2×#103—941no—
267C2×#102—941no—
268C2×#97—471no—
269C2×#97—471no—
270C2×#97—471no—
271#330—474no—
272#330—474no—
273#329—474no—
274C47C2×#166411yescharacteristic, complemented, Sylow
275C22×Q8—21no—
276C22×Q8—941no—
277C22×Q8C22⋊D4711yescharacteristic
278C22×D4C22⋊D4711yescharacteristic
279C23×C4—21no—
280C22×C8—941no—
281C22×C8—941no—
282C22×C8—941no—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—472no—
286C4⋊Q8—941no—
287C4⋊Q8—941no—
288C4⋊Q8—941no—
289#126—24no—
290#126—24no—
291#126—24no—
292#126—24no—
293#125—21no—
294#125—21no—
295#125—21no—
296#125—21no—
297#125C22⋊D4712yes—
298#125C22⋊D4712yes—
299#125C22×D4712yes—
300#125C22×D4712yes—
301#124—21no—
302#124—21no—
303#124C22⋊D4712yes—
304#124C22×D4712yes—
305#123—24no—
306#123—24no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—941no—
318C2×C4⋊C4—941no—
319C2×C4⋊C4C22⋊D4711yescharacteristic
320C2×C4⋊C4C22⋊D4711yescharacteristic
321C2×C4⋊C4C22⋊D4711yescharacteristic
322C2×C4⋊C4C22⋊D4711yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C42—21no—
329C2×C42C22×D4711yescharacteristic
330#107—944no—
331#106—944no—
332#103—944no—
333#103—944no—
334#103—944no—
335#102—944no—
336#97—472no—
337#97—472no—
338#97—472no—
339#97—941no—
340#97—941no—
341#97—941no—
342C24—21no—
343C2×Q8—24no—
344C2×Q8—21no—
345C2×Q8—42no—
346C2×Q8—82no—
347C2×Q8—942no—
348C2×Q8—1884no—
349C2×Q8—1881no—
350C2×Q8C2×C22⋊D4712yes—
351C2×D4—21no—
352C2×D4—42no—
353C2×D4C2×C22⋊D4712yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—82no—
361C22×C4—1881no—
362C22×C4C2×C22⋊D4711yescharacteristic
363C22×C4C2×C22⋊D4711yescharacteristic
364C22×C4C2×C22⋊D4711yescharacteristic
365C2×C8—944no—
366C2×C8—942no—
367C2×C8—942no—
368C2×C8—942no—
369C2×C8—1882no—
370C2×C8—1882no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—942no—
386C4⋊C4—942no—
387C4⋊C4—1881no—
388C4⋊C4—1881no—
389C4⋊C4C2×C22⋊D4712yes—
390C4⋊C4C2×C22⋊D4712yes—
391C4⋊C4C2×C22⋊D4712yes—
392C4⋊C4C2×C22⋊D4712yes—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C42—24no—
401C42C2×C22⋊D4711yescharacteristic
402C42C2×C22⋊D4711yescharacteristic
403C42C2×C22⋊D4711yescharacteristic
404C42C23×D4711yescharacteristic
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23#1309399311yescharacteristic
410Q8—42no—
411Q8—81no—
412Q8—84no—
413Q8—3762no—
414Q8—1884no—
415D4—42no—
416D4—81no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—84no—
431C2×C4—3762no—
432C2×C4—1882no—
433C2×C4#1309399311yescharacteristic
434C2×C4#1309399311yescharacteristic
435C2×C4#1309399311yescharacteristic
436C2×C4C22×C22⋊D4711yescharacteristic
437C2×C4C22×C22⋊D4711yescharacteristic
438C2×C4C22×C22⋊D4711yescharacteristic
439C8—944no—
440C8—1882no—
441C8—1882no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—84no—
446C22#1568929612yes—
447C22#1568930112yes—
448C22#1568930612yes—
449C22C2×#1309399311yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—3762no—
462C2—82no—
463C2#2024272214yesdirect product factor
464C2C2×#1568930111yescharacteristic
465C2C2×#1568929611yescharacteristic
466C2C2×#1568930611yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#563C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenterC2×#563D47
hypocenterC47C2×#1664
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. C2×#15689533C2semidirect
2. C2×#15689493C2semidirect
4. #20242722C2direct (N × H)
5. C2×#15688865C2semidirect
6. C2×#15688865C2semidirect
11. #15689533C22semidirect
13. #15689493C22semidirect
25. #15688865C22semidirect
26. #15688865C22semidirect
238. C2×C47#1664semidirect
274. C47C2×#1664semidirect
463. C2#20242722direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156895332
C2×#156894932
C2×#156894822
#202427222
C2×#156888652
C2×#156888632
C2×C47×#5632
C2×#166447