Finite Group Databasebeta

← C2×#10591409 #11868355

Subgroups — C2×#10591409 #11868355

Order 1152.

997 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Dic3C211yesmaximal, characteristic
2C2×#63869C211yesmaximal, characteristic, complemented
3C2×#63659C211yesmaximal, characteristic
4C2×C3×#832C211yesmaximal, characteristic
5#10591409C218yesmaximal, direct product factor
6C2×#63576C211yesmaximal, characteristic, complemented
7C2×#63573C211yesmaximal, characteristic, complemented
8C2×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
9C2×#5009—31nomaximal
10C2×#4763—31nomaximal
11C22×C32×D4C2211yescharacteristic
12C23×C3×Dic3—21no—
13C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×#217—21no—
16C3×D4×Dic3—22no—
17C3×D4×Dic3C2214yes—
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×C4×Dic3C2211yescharacteristic
21C22×C3×C3⋊C8C2211yescharacteristic
22C2×#3724—21no—
23C2×Dic3⋊Dic3—21no—
24#63869—22no—
25#63869C2214yescomplemented
26#63659C2214yes—
27C3×#832C2214yes—
28#63576C2214yescomplemented
29#63573C2214yescomplemented
30Dic3⋊(C3⋊C8)C2214yescomplemented
31C2×C3×C4×D4—31no—
32C2×D4×Dic3D311yescharacteristic
33C2×#890—31no—
34C2×#870—31no—
35C2×C3×#102—31no—
36C2×#832—31no—
37C2×#832—31no—
38C2×#832—61no—
39C2×#832D311yescharacteristic
40C2×#817—31no—
41C2×Dic3⋊C8—31no—
42#5009—38no—
43C2×#808—31no—
44C2×#807—31no—
45C2×#807—31no—
46C2×C4⋊(C3⋊C8)—31no—
47#4763—38no—
48C24×C32—21no—
49C2×C32×D4—21no—
50C2×C32×D4—22no—
51C2×C32×D4Q814yes—
52C2×C32×D4C2312yes—
53C22×C32×C4C2311yescharacteristic
54C22×C3⋊Dic3—21no—
55C22×C3×Dic3—21no—
56C22×C3×Dic3—21no—
57C22×C3×Dic3—24no—
58C22×C3×Dic3—44no—
59C22×C3×Dic3D411yescharacteristic
60C22×C3×Dic3D411yescharacteristic
61C4⋊(C3⋊Dic3)—21no—
62C4⋊(C3⋊Dic3)C2312yes—
63C2×C3⋊(C3⋊C8)—22no—
64C2×C3⋊(C3⋊C8)C2312yes—
65C3×#217—24no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3C2312yes—
69C3×C4⋊Dic3C2312yes—
70C3×C4×Dic3D412yes—
71C3×C4×Dic3C2312yes—
72C2×C3×C3⋊C8—22no—
73C2×C3×C3⋊C8C2312yes—
74#3724—24no—
75Dic3⋊Dic3—24no—
76C2×#478—91noSylow
77C22×C3×D4—21no—
78C22×C3×D4C2×D311yescharacteristic
79C22×C3×D4C2×D311yescharacteristic
80C23×C3×C4—61no—
81C23×Dic3—21no—
82C22×C3×C8—31no—
83C3×C4×D4—34no—
84C3×C4×D4—62no—
85C2×C3×C4⋊C4—31no—
86C2×C3×C4⋊C4—31no—
87C2×C3×C22⋊C4—61no—
88C2×C3×C42—31no—
89C2×#217—21no—
90D4×Dic3—22no—
91D4×Dic3C2×D314yes—
92C2×C4⋊Dic3—31no—
93C2×C4⋊Dic3—31no—
94C2×C4⋊Dic3—61no—
95C2×C4⋊Dic3—61no—
96C2×C4⋊Dic3C2×D311yescharacteristic
97C2×C4⋊Dic3C2×D311yescharacteristic
98C2×Dic3⋊C4—61no—
99C2×Dic3⋊C4—61no—
100C2×Dic3⋊C4—61no—
101C2×C4×Dic3C2×D311yescharacteristic
102C22×C3⋊C8—31no—
103C22×C3⋊C8—31no—
104C22×C3⋊C8—61no—
105C22×C3⋊C8C2×D311yescharacteristic
106#890—34no—
107#890—62no—
108#870—34no—
109#870—62no—
110C3×#102—34no—
111#832—34no—
112#832—34no—
113#832—64no—
114#832C2×D314yes—
115#817—34no—
116Dic3⋊C8—34no—
117#808—34no—
118#807—34no—
119#807—34no—
120C4⋊(C3⋊C8)—34no—
121C23×C32—24no—
122C23×C32—22no—
123C23×C32—42no—
124C23×C32—42no—
125C23×C32C2×D411yescharacteristic
126C32×D4—24no—
127C32×D4—22no—
128C32×D4C2×Q814yes—
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×Q812yes—
131C2×C32×C4#4112yes—
132C2×C32×C4C2411yescharacteristic
133C2×C3⋊Dic3—22no—
134C2×C3⋊Dic3—42no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—48no—
140C2×C3×Dic3—42no—
141C2×C3×Dic3—42no—
142C2×C3×Dic3C2×D412yes—
143C2×C3×Dic3C2×D412yes—
144C3⋊(C3⋊C8)—22no—
145C3×C3⋊C8—22no—
146C2×#125—91no—
147C2×C4×D4—31no—
148#478—98no—
149C2×#107—91no—
150C2×#106—91no—
151C2×C4⋊C8—91no—
152C2×#102—31no—
153C2×#102—91no—
154C24×C3—21no—
155C24×C3—21no—
156C24×C3—41no—
157C2×C3×D4—22no—
158C2×C3×D4—21no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—42no—
164C2×C3×D4—41no—
165C2×C3×D4Dic614yes—
166C2×C3×D4Dic614yes—
167C2×C3×D4C22×D312yes—
168C2×C3×D4C22×D312yes—
169C22×C3×C4—21no—
170C22×C3×C4—31no—
171C22×C3×C4—31no—
172C22×C3×C4—64no—
173C22×C3×C4—61no—
174C22×C3×C4—61no—
175C22×C3×C4—124no—
176C22×C3×C4C22×D311yescharacteristic
177C22×C3×C4C22×D311yescharacteristic
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—24no—
181C22×Dic3—44no—
182C22×Dic3—61no—
183C22×Dic3—61no—
184C22×Dic3—121no—
185C22×Dic3#6711yescharacteristic
186C22×Dic3#6711yescharacteristic
187C2×C3×C8—32no—
188C2×C3×C8—62no—
189C3×C4⋊C4—32no—
190C3×C4⋊C4—32no—
191C3×C4⋊C4—61no—
192C3×C4⋊C4—61no—
193C3×C22⋊C4—64no—
194C3×C42—32no—
195C3×C42—32no—
196#217—24no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—32no—
200C4⋊Dic3—32no—
201C4⋊Dic3—62no—
202C4⋊Dic3—61no—
203C4⋊Dic3—61no—
204C4⋊Dic3—64no—
205C4⋊Dic3—121no—
206C4⋊Dic3C22×D312yes—
207C4⋊Dic3C22×D312yes—
208Dic3⋊C4—64no—
209Dic3⋊C4—64no—
210Dic3⋊C4—64no—
211C4×Dic3#6712yes—
212C4×Dic3C22×D312yes—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—32no—
215C2×C3⋊C8—32no—
216C2×C3⋊C8—62no—
217C2×C3⋊C8—62no—
218C2×C3⋊C8—62no—
219C2×C3⋊C8—122no—
220C2×C3⋊C8C22×D312yes—
221C22×C32—24no—
222C22×C32—42no—
223C22×C32—42no—
224C22×C32—48no—
225C22×C32C22⋊Q812yes—
226C22×C32#13512yes—
227C22×C32AGL(1,ℤ/8)12yes—
228C22×C32C22×D411yescharacteristic
229C32×C4C22⋊Q812yes—
230C32×C4C22×Q811yescharacteristic
231C32×C4C2×#4111yescharacteristic
232C3⋊Dic3—42no—
233C3×Dic3—22no—
234C3×Dic3—22no—
235C3×Dic3—42no—
236C3×Dic3—42no—
237C22×D4D3211yescharacteristic
238C23×C4—61no—
239C22×C8—31no—
240C22×C8—91no—
241#125—94no—
242#125—182no—
243C4×D4—34no—
244C4×D4—62no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—91no—
248C2×C4⋊C4—181no—
249C2×C4⋊C4—181no—
250C2×C22⋊C4—61no—
251C2×C42—31no—
252#107—94no—
253#106—94no—
254C4⋊C8—94no—
255#102—34no—
256#102—94no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—21no—
262C23×C3—42no—
263C23×C3—44no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—82no—
269C23×C3—82no—
270C23×C3D3×D411yescharacteristic
271C23×C3C2×#6711yescharacteristic
272C3×D4—24no—
273C3×D4—24no—
274C3×D4—22no—
275C3×D4—22no—
276C3×D4—24no—
277C3×D4—42no—
278C3×D4—44no—
279C3×D4C2×Dic614yes—
280C3×D4C2×Dic614yes—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—21no—
285C2×C3×C4—32no—
286C2×C3×C4—32no—
287C2×C3×C4—62no—
288C2×C3×C4—62no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—128no—
292C2×C3×C4—122no—
293C2×C3×C4—122no—
294C2×C3×C4C2×Dic612yes—
295C2×C3×C4C2×Dic612yes—
296C2×C3×C4#23512yes—
297C2×C3×C4D3×D411yescharacteristic
298C2×C3×C4#23712yes—
299C2×C3×C4C2×#6711yescharacteristic
300C2×C3×C4C23×D311yescharacteristic
301C2×C3×C4C23×D311yescharacteristic
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—48no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—62no—
310C2×Dic3—62no—
311C2×Dic3—122no—
312C2×Dic3—122no—
313C2×Dic3—122no—
314C2×Dic3—242no—
315C2×Dic3C2×#6712yes—
316C2×Dic3C2×#6712yes—
317C3×C8—62no—
318C3⋊C8—22no—
319C3⋊C8—62no—
320C3⋊C8—62no—
321C3⋊C8—122no—
322C2×C32—44no—
323C2×C32#47814yescomplemented
324C2×C32C2×C22⋊Q811yescharacteristic
325C2×C32C2×#13511yescharacteristic
326C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
327C24—21no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4#62714yes—
331C2×D4C2×D3212yes—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—64no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—124no—
338C22×C4—181no—
339C22×C4C2×D3211yescharacteristic
340C2×C8—32no—
341C2×C8—62no—
342C2×C8—92no—
343C2×C8—182no—
344C4⋊C4—32no—
345C4⋊C4—32no—
346C4⋊C4—61no—
347C4⋊C4—61no—
348C4⋊C4—92no—
349C4⋊C4—184no—
350C4⋊C4—184no—
351C4⋊C4—181no—
352C22⋊C4—64no—
353C42—32no—
354C42—32no—
355C22×C3—24no—
356C22×C3—24no—
357C22×C3—21no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—44no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—48no—
365C22×C3—48no—
366C22×C3—42no—
367C22×C3—42no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3—88no—
371C22×C3#87812yes—
372C22×C3#91112yes—
373C22×C3#91612yes—
374C22×C3#92412yes—
375C22×C3#93212yes—
376C22×C3#95012yes—
377C22×C3C2×D3×D411yescharacteristic
378C22×C3C22×#6711yescharacteristic
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—22no—
382C3×C4—62no—
383C3×C4—62no—
384C3×C4—122no—
385C3×C4—122no—
386C3×C4#87812yes—
387C3×C4#92412yes—
388C3×C4C22×Dic611yescharacteristic
389C3×C4C22×Dic611yescharacteristic
390C3×C4C2×#23511yescharacteristic
391C3×C4C2×#23711yescharacteristic
392Dic3—22no—
393Dic3—22no—
394Dic3—42no—
395Dic3—42no—
396Dic3—122no—
397Dic3—122no—
398Dic3—242no—
399C32C2×#47811yescharacteristic, complemented, Sylow
400C23—22no—
401C23—24no—
402C23—42no—
403C23—42no—
404C23D3×#6711yescharacteristic
405D4—22no—
406D4—24no—
407D4C2×#62714yes—
408C2×C4—32no—
409C2×C4—32no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—128no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—182no—
418C2×C4—362no—
419C2×C4#380412yes—
420C2×C4C2×#62712yes—
421C2×C4D3×#6711yescharacteristic
422C2×C4C22×D3211yescharacteristic
423C8—62no—
424C8—182no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—24no—
429C2×C3—44no—
430C2×C3—44no—
431C2×C3—84no—
432C2×C3#476314yescomplemented
433C2×C3#500914yescomplemented
434C2×C3C2×#87811yescharacteristic
435C2×C3C2×#91111yescharacteristic
436C2×C3C2×#91611yescharacteristic
437C2×C3C2×#92411yescharacteristic
438C2×C3C2×#93211yescharacteristic
439C2×C3C2×#95011yescharacteristic
440C22—24no—
441C22—42no—
442C22—42no—
443C22—48no—
444C22#6393212yes—
445C22#6393512yes—
446C22#6396412yes—
447C22C2×D3×#6711yescharacteristic
448C4—62no—
449C4—62no—
450C4—122no—
451C4—122no—
452C4—362no—
453C4#6396412yes—
454C4C2×#380411yescharacteristic
455C4C22×#62711yescharacteristic
456C3—21no—
457C3C2×#476311yescharacteristic, complemented
458C3C2×#500911yescharacteristic, complemented
459C2—44no—
460C2#1059140914yesdirect product factor
461C2C2×#6393211yescharacteristic
462C2C2×#6393511yescharacteristic
463C2C2×#6396411yescharacteristic
subgroupG/H
centerC23D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
Fitting, prime radicalC22×C32×D4C22
socleC23×C32C2×D4
hypercenterC22×D4D32
hypocenterC32C2×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×#63869C2semidirect
5. #10591409C2direct (N × H)
6. C2×#63576C2semidirect
7. C2×#63573C2semidirect
8. C2×Dic3⋊(C3⋊C8)C2semidirect
25. #63869C22semidirect
28. #63576C22semidirect
29. #63573C22semidirect
30. Dic3⋊(C3⋊C8)C22semidirect
323. C2×C32#478semidirect
399. C32C2×#478semidirect
432. C2×C3#4763semidirect
433. C2×C3#5009semidirect
457. C3C2×#4763semidirect
458. C3C2×#5009semidirect
460. C2#10591409direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Dic32
C2×#638692
C2×#636592
C2×C3×#8322
#105914092
C2×#635762
C2×#635732
C2×Dic3⋊(C3⋊C8)2
C2×#50093
C2×#47633