Finite Group Databasebeta

← C2×#10590271 #11868352

Subgroups — C2×#10590271 #11868352

Order 1152.

1009 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64079C211yesmaximal, characteristic, complemented
2C2×C3×C4×D12C211yesmaximal, characteristic
3C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C2×#63575C211yesmaximal, characteristic, complemented
5C2×#63573C211yesmaximal, characteristic, complemented
6C2×#63563C211yesmaximal, characteristic
7C2×#63563C211yesmaximal, characteristic
8#10590271C218yesmaximal, direct product factor
9C2×#5009—31nomaximal
10C2×#4847—31nomaximal
11C22×C3×D12C2211yescharacteristic
12C22×C3×C4×D3—21no—
13C2×C32×C42C2211yescharacteristic
14C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×#3750—21no—
17C2×#3750—21no—
18#64079—22no—
19#64079C2214yescomplemented
20C2×C3×(C2×D3)⋊C4—21no—
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C22×C3×C3⋊C8C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C3×C4×D12—22no—
25C3×C4×D12C2214yes—
26C3×C4⋊(C3⋊C8)C2214yescomplemented
27#63575C2214yescomplemented
28#63573C2214yescomplemented
29#63563C2214yes—
30#63563C2214yes—
31C2×C3×C4×D4—31no—
32C2×C4×D12D311yescharacteristic
33C2×#870—31no—
34C2×#870—31no—
35C2×#870—61no—
36C2×C3×C4⋊C8—31no—
37C2×#832—31no—
38C2×#832—31no—
39C2×#821—31no—
40C2×#821—31no—
41C2×#818—31no—
42C2×#817—31no—
43#5009—38no—
44C2×#808—31no—
45C2×#807—31no—
46C2×C4⋊(C3⋊C8)D311yescharacteristic, complemented
47#4847—38no—
48C23×C3×D3—21no—
49C22×C32×C4D411yescharacteristic
50C22×C32×C4D411yescharacteristic
51C22×C32×C4C2311yescharacteristic
52C22×C3⋊Dic3—21no—
53C22×C3⋊Dic3—21no—
54C22×C3×Dic3—21no—
55C2×C3×D12—21no—
56C2×C3×D12—22no—
57C2×C3×D12Q814yes—
58C2×C3×D12C2312yes—
59C2×C3×C4×D3—24no—
60C2×C3×C4×D3—44no—
61C32×C42D412yes—
62C32×C42C2312yes—
63C4⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊Dic3)—21no—
65C4⋊(C3⋊Dic3)C2312yes—
66C4⋊(C3⋊Dic3)C2312yes—
67#3750—24no—
68#3750—24no—
69C3×(C2×D3)⋊C4—24no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3C2312yes—
72C2×C3×C3⋊C8—22no—
73C2×C3×C3⋊C8—22no—
74C2×C3×C3⋊C8C2312yes—
75C2×C3×C3⋊C8C2312yes—
76C2×#478—91noSylow
77C22×C3×D4—31no—
78C23×C3×C4—61no—
79C22×D12C2×D311yescharacteristic
80C22×C4×D3—21no—
81C22×C3×C8—31no—
82C22×C3×C8—31no—
83C3×C4×D4—34no—
84C3×C4×D4—62no—
85C2×C3×C4⋊C4—31no—
86C2×C3×C22⋊C4—61no—
87C2×C3×C42—21no—
88C2×C3×C42C2×D311yescharacteristic
89C2×C3×C42C2×D311yescharacteristic
90C2×(C2×D3)⋊C4—21no—
91C2×C4⋊Dic3—31no—
92C2×C4⋊Dic3—31no—
93C2×C4⋊Dic3—31no—
94C2×C4⋊Dic3—31no—
95C2×C4⋊Dic3—61no—
96C2×C4⋊Dic3—61no—
97C2×C4⋊Dic3C2×D311yescharacteristic
98C2×Dic3⋊C4—62no—
99C2×Dic3⋊C4—62no—
100C2×Dic3⋊C4—61no—
101C2×Dic3⋊C4—61no—
102C2×Dic3⋊C4—61no—
103C2×Dic3⋊C4—61no—
104C22×C3⋊C8C2×D311yescharacteristic
105C22×C3⋊C8C2×D311yescharacteristic
106C4×D12—22no—
107C4×D12C2×D314yes—
108#870—34no—
109#870—34no—
110#870—64no—
111#870—64no—
112#870—62no—
113#870—62no—
114C3×C4⋊C8—34no—
115#832—34no—
116#832—34no—
117#821—34no—
118#821—34no—
119#818—34no—
120#817—34no—
121#808—34no—
122#807—34no—
123C4⋊(C3⋊C8)C2×D314yescomplemented
124C23×C32C2×D411yescharacteristic
125C22×C3×D3—24no—
126C22×C3×D3—22no—
127C22×C3×D3—42no—
128C22×C3×D3—42no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4C2×D412yes—
132C2×C32×C4C2×D412yes—
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×Q812yes—
135C2×C32×C4#4112yes—
136C2×C32×C4C2411yescharacteristic
137C2×C3⋊Dic3—22no—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3—42no—
140C2×C3⋊Dic3—42no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—42no—
143C3×D12—24no—
144C3×D12—22no—
145C3×D12C2×Q814yes—
146C3×C4×D3—48no—
147C3×C3⋊C8—22no—
148C3×C3⋊C8—22no—
149C2×#125—91no—
150C2×C4×D4—31no—
151#478—98no—
152C2×#107—91no—
153C2×#106—91no—
154C2×C4⋊C8—31no—
155C2×#102—91no—
156C2×#102—91no—
157C24×C3—61no—
158C23×D3—21no—
159C2×C3×D4—34no—
160C2×C3×D4—32no—
161C2×C3×D4—61no—
162C2×C3×D4—62no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—61no—
167C22×C3×C4—64no—
168C22×C3×C4—124no—
169C22×C3×C4D1211yescharacteristic
170C22×C3×C4D1211yescharacteristic
171C22×C3×C4#6711yescharacteristic
172C22×C3×C4#6711yescharacteristic
173C22×C3×C4C22×D311yescharacteristic
174C22×C3×C4C22×D311yescharacteristic
175C22×Dic3—21no—
176C22×Dic3—62no—
177C22×Dic3—62no—
178C22×Dic3—61no—
179C22×Dic3—61no—
180C22×Dic3—61no—
181C22×Dic3—61no—
182C2×D12—22no—
183C2×D12—21no—
184C2×D12Dic614yes—
185C2×D12C22×D312yes—
186C2×C4×D3—24no—
187C2×C4×D3—44no—
188C2×C3×C8—32no—
189C2×C3×C8—32no—
190C2×C3×C8—62no—
191C2×C3×C8—62no—
192C3×C4⋊C4—32no—
193C3×C4⋊C4—61no—
194C3×C22⋊C4—64no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42D1212yes—
198C3×C42#6712yes—
199C3×C42C22×D312yes—
200C3×C42C22×D312yes—
201(C2×D3)⋊C4—24no—
202C4⋊Dic3—21no—
203C4⋊Dic3—32no—
204C4⋊Dic3—32no—
205C4⋊Dic3—32no—
206C4⋊Dic3—32no—
207C4⋊Dic3—62no—
208C4⋊Dic3—62no—
209C4⋊Dic3—62no—
210C4⋊Dic3—62no—
211C4⋊Dic3—61no—
212C4⋊Dic3—61no—
213C4⋊Dic3—61no—
214C4⋊Dic3—61no—
215C4⋊Dic3C22×D312yes—
216Dic3⋊C4—68no—
217Dic3⋊C4—68no—
218Dic3⋊C4—64no—
219Dic3⋊C4—64no—
220Dic3⋊C4—64no—
221Dic3⋊C4—64no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8C22×D312yes—
225C2×C3⋊C8C22×D312yes—
226C22×C32C22⋊Q812yes—
227C22×C32#13512yes—
228C22×C32AGL(1,ℤ/8)12yes—
229C22×C32C22×D411yescharacteristic
230C2×C3×D3—24no—
231C2×C3×D3—42no—
232C2×C3×D3—42no—
233C2×C3×D3—48no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4C22⋊Q812yes—
237C32×C4C22×Q811yescharacteristic
238C32×C4C2×#4111yescharacteristic
239C3⋊Dic3—42no—
240C3⋊Dic3—42no—
241C3×Dic3—42no—
242C22×D4—31no—
243C23×C4—61no—
244C22×C8—31no—
245C22×C8—31no—
246#125—94no—
247#125—182no—
248C4×D4—34no—
249C4×D4—62no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—91no—
252C2×C4⋊C4—91no—
253C2×C4⋊C4—181no—
254C2×C4⋊C4—181no—
255C2×C22⋊C4—61no—
256C2×C42D3211yescharacteristic
257#107—94no—
258#106—94no—
259C4⋊C8—34no—
260#102—94no—
261#102—94no—
262C23×C3—21no—
263C23×C3—64no—
264C23×C3—62no—
265C23×C3—122no—
266C23×C3—122no—
267C23×C3C2×D1211yescharacteristic
268C23×C3C2×#6711yescharacteristic
269C22×D3—24no—
270C22×D3—22no—
271C22×D3—42no—
272C22×D3—42no—
273C3×D4—34no—
274C3×D4—64no—
275C3×D4—62no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—62no—
289C2×C3×C4—128no—
290C2×C3×C4—122no—
291C2×C3×C4C2×Dic612yes—
292C2×C3×C4C2×D1212yes—
293C2×C3×C4C2×D1212yes—
294C2×C3×C4C2×D1211yescharacteristic
295C2×C3×C4#23512yes—
296C2×C3×C4#23712yes—
297C2×C3×C4D3×Q812yes—
298C2×C3×C4C2×#6712yes—
299C2×C3×C4C2×#6712yes—
300C2×C3×C4C2×#6711yescharacteristic
301C2×C3×C4C23×D311yescharacteristic
302C2×C3×C4C23×D311yescharacteristic
303C2×Dic3—22no—
304C2×Dic3—42no—
305C2×Dic3—64no—
306C2×Dic3—64no—
307C2×Dic3—62no—
308C2×Dic3—62no—
309C2×Dic3—62no—
310C2×Dic3—62no—
311C2×Dic3—124no—
312C2×Dic3—124no—
313C2×Dic3—122no—
314C2×Dic3—122no—
315C2×Dic3—122no—
316C2×Dic3—122no—
317D12—24no—
318D12—22no—
319D12C2×Dic614yes—
320C4×D3—48no—
321C3×C8—62no—
322C3×C8—62no—
323C3⋊C8—22no—
324C3⋊C8—22no—
325C2×C32#47814yescomplemented
326C2×C32C2×C22⋊Q811yescharacteristic
327C2×C32C2×#13511yescharacteristic
328C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
329C3×D3—44no—
330C24—61no—
331C2×D4—32no—
332C2×D4—34no—
333C2×D4—62no—
334C2×D4—61no—
335C22×C4—61no—
336C22×C4—64no—
337C22×C4—124no—
338C22×C4—181no—
339C22×C4—181no—
340C22×C4#62611yescharacteristic
341C22×C4#62611yescharacteristic
342C22×C4C2×D3211yescharacteristic
343C2×C8—32no—
344C2×C8—32no—
345C2×C8—62no—
346C2×C8—62no—
347C4⋊C4—32no—
348C4⋊C4—61no—
349C4⋊C4—92no—
350C4⋊C4—92no—
351C4⋊C4—181no—
352C4⋊C4—181no—
353C4⋊C4—184no—
354C4⋊C4—184no—
355C22⋊C4—64no—
356C42#62612yes—
357C42C2×D3212yes—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—64no—
363C22×C3—122no—
364C22×C3—122no—
365C22×C3—128no—
366C22×C3#89712yes—
367C22×C3#90412yes—
368C22×C3#90812yes—
369C22×C3#92412yes—
370C22×C3#93212yes—
371C22×C3#95012yes—
372C22×C3C22×D1211yescharacteristic
373C22×C3C22×#6711yescharacteristic
374C2×D3—24no—
375C2×D3—48no—
376C2×D3—42no—
377C2×D3—42no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—22no—
385C3×C4—42no—
386C3×C4—42no—
387C3×C4—122no—
388C3×C4#89712yes—
389C3×C4#92412yes—
390C3×C4C22×Dic611yescharacteristic
391C3×C4C2×#23511yescharacteristic
392C3×C4C2×#23711yescharacteristic
393C3×C4C2×D3×Q811yescharacteristic
394Dic3—42no—
395Dic3—124no—
396Dic3—124no—
397Dic3—122no—
398Dic3—122no—
399Dic3—122no—
400Dic3—122no—
401C32C2×#47811yescharacteristic, complemented, Sylow
402C23—62no—
403C23—64no—
404C23—122no—
405C23—122no—
406C23C2×#62611yescharacteristic
407D4—34no—
408D4—62no—
409D4—64no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—62no—
413C2×C4—128no—
414C2×C4—122no—
415C2×C4—182no—
416C2×C4—182no—
417C2×C4—362no—
418C2×C4—362no—
419C2×C4D3×Dic612yes—
420C2×C4#379512yes—
421C2×C4C2×#62612yes—
422C2×C4C2×#62612yes—
423C2×C4C2×#62611yescharacteristic
424C2×C4C22×D3211yescharacteristic
425C8—62no—
426C8—62no—
427C2×C3—21no—
428C2×C3—24no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—124no—
432C2×C3#484714yescomplemented
433C2×C3#500914yescomplemented
434C2×C3C2×#89711yescharacteristic
435C2×C3C2×#90411yescharacteristic
436C2×C3C2×#90811yescharacteristic
437C2×C3C2×#92411yescharacteristic
438C2×C3C2×#93211yescharacteristic
439C2×C3C2×#95011yescharacteristic
440D3—44no—
441C22—64no—
442C22—128no—
443C22—122no—
444C22—122no—
445C22#6382912yes—
446C22#6383012yes—
447C22#6385712yes—
448C22C22×#62611yescharacteristic
449C4—22no—
450C4—22no—
451C4—122no—
452C4—362no—
453C4—362no—
454C4#6385712yes—
455C4C2×D3×Dic611yescharacteristic
456C4C2×#379511yescharacteristic
457C3—21no—
458C3C2×#484711yescharacteristic, complemented
459C3C2×#500911yescharacteristic, complemented
460C2—124no—
461C2#1059027114yesdirect product factor
462C2C2×#6382911yescharacteristic
463C2C2×#6383011yescharacteristic
464C2C2×#6385711yescharacteristic
subgroupG/H
centerC23C2×#626
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×C42C22
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
hypercenterC2×C42D32
hypocenterC32C2×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×#64079C2semidirect
3. C2×C3×C4⋊(C3⋊C8)C2semidirect
4. C2×#63575C2semidirect
5. C2×#63573C2semidirect
8. #10590271C2direct (N × H)
19. #64079C22semidirect
26. C3×C4⋊(C3⋊C8)C22semidirect
27. #63575C22semidirect
28. #63573C22semidirect
46. C2×C4⋊(C3⋊C8)D3semidirect
123. C4⋊(C3⋊C8)C2×D3semidirect
325. C2×C32#478semidirect
401. C32C2×#478semidirect
432. C2×C3#4847semidirect
433. C2×C3#5009semidirect
458. C3C2×#4847semidirect
459. C3C2×#5009semidirect
461. C2#10590271direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#640792
C2×C3×C4×D122
C2×C3×C4⋊(C3⋊C8)2
C2×#635752
C2×#635732
C2×#635632
#105902712
C2×#50093
C2×#48473