Finite Group Databasebeta

← C2×#79575 #11675617

Subgroups — C2×#79575 #11675617

Order 768.

1918 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5539C212yesmaximal, complemented
2C2×C3×C4×D8C211yesmaximal, characteristic
3C2×Dic3×D8C211yesmaximal, characteristic
4C2×Dic3×D8C211yesmaximal, characteristic
5C2×#5009C212yesmaximal, complemented
6#79575C2116yesmaximal, direct product factor
7C2×#4878C211yesmaximal, characteristic, complemented
8C2×#4874C211yesmaximal, characteristic, complemented
9C2×#4763C212yesmaximal, complemented
10C2×#4761C212yesmaximal, complemented
11C2×C8×Dic6C211yesmaximal, characteristic, complemented
12C2×#4683C211yesmaximal, characteristic, complemented
13C2×#3380—31nomaximal, Sylow
14C22×C3×D8C2211yescharacteristic
15C2×C3×C4×D4C2212yes—
16C2×D4×Dic3C2212yes—
17C2×D4×Dic3C2212yes—
18C2×#924—22no—
19C22×Dic3⋊C4—22no—
20#5539—28no—
21#5539C22116yescomplemented
22C2×#890C2212yes—
23C2×#888C2212yes—
24C2×#878—22no—
25C2×#878—22no—
26C2×#870C2212yes—
27C2×C4×Dic6C2211yescharacteristic
28C3×C4×D8C2218yes—
29C2×C3×#107C2211yescharacteristic
30C2×C3×#102C2212yes—
31C2×C3×C4×C8C2211yescharacteristic
32C2×#832C2212yes—
33C2×#832C2212yes—
34Dic3×D8C2218yes—
35Dic3×D8C2218yes—
36C2×#818C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×#817C2212yes—
39C2×Dic3⋊C8C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×C8×Dic3C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#5009C22116yescomplemented
44C2×#808C2212yes—
45C2×#807C2212yes—
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47#4878C2218yescomplemented
48#4874C2218yescomplemented
49#4763C22116yescomplemented
50#4761C22116yescomplemented
51#4683C2218yescomplemented
52C8×Dic6C2218yescomplemented
53C2×#554—32no—
54#3380—316no—
55C2×#500—31no—
56C2×#499—31no—
57C2×#499—31no—
58C2×#478—32no—
59C2×#478—32no—
60C2×#476—32no—
61C2×C8×Q8—31no—
62C2×C4×D8—31no—
63C2×C4×D8—31no—
64C2×C4×D8D311yescharacteristic
65C22×C3×D4C2312yes—
66C23×C3×C4—22no—
67C23×Dic3—22no—
68C23×Dic3—22no—
69C22×Dic6—22no—
70C22×Dic6D411yescharacteristic
71C2×C3×D8Q818yescomplemented
72C2×C3×D8C2314yes—
73C22×C3×C8C2311yescharacteristic
74C22×C3×C8C2311yescharacteristic
75C3×C4×D4—24no—
76C3×C4×D4C2318yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C42C2311yescharacteristic
80C2×#217—22no—
81C2×#217—22no—
82D4×Dic3—24no—
83D4×Dic3—24no—
84D4×Dic3C2318yes—
85D4×Dic3C2318yes—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3D411yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×C4⋊Dic3C2312yes—
90#924—216no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—216no—
96C2×Dic3⋊C4D411yescharacteristic
97C2×Dic3⋊C4D411yescharacteristic
98C2×C4×Dic3C2311yescharacteristic
99C2×C4×Dic3C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102#890—24no—
103#890C2318yes—
104#888—24no—
105#888C2318yes—
106#878—216no—
107#878—216no—
108#870—24no—
109#870C2318yes—
110C4×Dic6D414yes—
111C4×Dic6C2314yes—
112C3×#107C2314yes—
113C3×#102C2318yes—
114C3×C4×C8C2314yes—
115#832C2318yes—
116#832C2318yes—
117#818C2314yes—
118#818C2314yes—
119#817C2318yes—
120Dic3⋊C8C2314yes—
121Dic3⋊C8C2314yes—
122C8×Dic3C2314yes—
123C8×Dic3C2314yes—
124#808C2318yes—
125#807C2318yes—
126C4⋊(C3⋊C8)C2314yes—
127C22×D8C2×D311yescharacteristic
128#554—316no—
129#554—68no—
130C2×C4⋊Q8—32no—
131C2×#125—32no—
132C2×#125—32no—
133C2×C22⋊Q8—62no—
134C2×C22⋊Q8—62no—
135C2×C22⋊Q8—62no—
136C2×C4×Q8—31no—
137C2×C4×D4—32no—
138C2×C4×D4—32no—
139C2×C4×D4C2×D312yes—
140C22×C4⋊C4—62no—
141#500—38no—
142#499—38no—
143#499—38no—
144#478—316no—
145#478—316no—
146#476—316no—
147C8×Q8—38no—
148C4×D8—38no—
149C4×D8—38no—
150C4×D8C2×D318yes—
151C2×#107—31no—
152C2×#107—31no—
153C2×#107—32no—
154C2×#107C2×D311yescharacteristic
155C2×#106—32no—
156C2×#106—32no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#102—32no—
161C2×#102—32no—
162C2×#102C2×D312yes—
163C2×C4×C8—31no—
164C2×C4×C8—31no—
165C2×C4×C8C2×D311yescharacteristic
166C24×C3—22no—
167C2×C3×D4—22no—
168C2×C3×D4—24no—
169C2×C3×D4C2×Q818yes—
170C2×C3×D4C2414yes—
171C22×C3×C4—22no—
172C22×C3×C4—28no—
173C22×C3×C4—84no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—28no—
180C22×Dic3—28no—
181C22×Dic3—48no—
182C22×Dic3—84no—
183C22×Dic3C2×D411yescharacteristic
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
186C22×Dic3C2×D411yescharacteristic
187C2×Dic6—24no—
188C2×Dic6—28no—
189C2×Dic6—84no—
190C2×Dic6C2×D414yes—
191C3×D8C2×Q8116yescomplemented
192C2×C3×C8—22no—
193C2×C3×C8C2×Q814yes—
194C2×C3×C8C2412yes—
195C2×C3×C8C2412yes—
196C3×C4⋊C4—22no—
197C3×C4⋊C4C2414yes—
198C3×C22⋊C4—28no—
199C3×C42C2×D412yes—
200C3×C42C2412yes—
201#217—28no—
202#217—28no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—28no—
206C4⋊Dic3C2×D414yes—
207C4⋊Dic3C2414yes—
208C4⋊Dic3C2414yes—
209Dic3⋊C4—28no—
210Dic3⋊C4—28no—
211Dic3⋊C4—28no—
212Dic3⋊C4—28no—
213Dic3⋊C4C2×D414yes—
214Dic3⋊C4C2×D414yes—
215C4×Dic3C2×D412yes—
216C4×Dic3C2×D412yes—
217C4×Dic3C2412yes—
218C4×Dic3C2412yes—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8C2412yes—
222C2×C3⋊C8C2412yes—
223C22×Q8—31no—
224C22×Q8—62no—
225C22×D4C22×D312yes—
226C23×C4—22no—
227C23×C4—62no—
228C23×C4—62no—
229C2×D8Dic618yescomplemented
230C2×D8C22×D314yes—
231C22×C8—31no—
232C22×C8—31no—
233C22×C8C22×D311yescharacteristic
234C22×C8C22×D311yescharacteristic
235C4⋊Q8—38no—
236C4⋊Q8—64no—
237#125—38no—
238#125—38no—
239#125—64no—
240#125—64no—
241C22⋊Q8—616no—
242C22⋊Q8—616no—
243C22⋊Q8—616no—
244C4×Q8—34no—
245C4×Q8—34no—
246C4×D4—24no—
247C4×D4—38no—
248C4×D4—38no—
249C4×D4—64no—
250C4×D4—64no—
251C4×D4C22×D318yes—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—616no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4C22×D312yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C42—31no—
268C2×C42—31no—
269C2×C42C22×D311yescharacteristic
270#107—34no—
271#107—34no—
272#107—38no—
273#107C22×D314yes—
274#106—38no—
275#106—38no—
276C4⋊C8—34no—
277C4⋊C8—34no—
278C4⋊C8—34no—
279#102—38no—
280#102—38no—
281#102C22×D318yes—
282C4×C8—34no—
283C4×C8—34no—
284C4×C8C22×D314yes—
285C23×C3—28no—
286C23×C3—24no—
287C23×C3—44no—
288C23×C3—44no—
289C23×C3C22×D411yescharacteristic
290C3×D4—28no—
291C3×D4—24no—
292C3×D4C22×Q818yes—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—44no—
297C2×C3×C4—88no—
298C2×C3×C4C22×D412yes—
299C2×C3×C4C22×D412yes—
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×Q812yes—
302C2×C3×C42−1+412yes—
303C2×C3×C4C2511yescharacteristic
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—44no—
310C2×Dic3—44no—
311C2×Dic3—416no—
312C2×Dic3—88no—
313C2×Dic3#13514yes—
314C2×Dic3C22×D412yes—
315C2×Dic3C22×D412yes—
316C2×Dic3C22×D412yes—
317C2×Dic3C22×D412yes—
318Dic6—28no—
319Dic6—88no—
320C3×C8—22no—
321C3×C8C22×Q814yes—
322C3⋊C8—22no—
323C3⋊C8—22no—
324C24—22no—
325C2×Q8—34no—
326C2×Q8—64no—
327C2×Q8—68no—
328C2×Q8—244no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4C2×Dic618yes—
332C2×D4C23×D314yes—
333C22×C4—22no—
334C22×C4—28no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—68no—
340C22×C4—68no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—84no—
344C22×C4—128no—
345C22×C4—244no—
346C22×C4D3×D411yescharacteristic
347C22×C4D3×D411yescharacteristic
348C22×C4C23×D311yescharacteristic
349D8C2×Dic6116yescomplemented
350C2×C8—22no—
351C2×C8—32no—
352C2×C8—32no—
353C2×C8—62no—
354C2×C8—62no—
355C2×C8C2×Dic614yes—
356C2×C8C23×D312yes—
357C2×C8C23×D312yes—
358C4⋊C4—22no—
359C4⋊C4—34no—
360C4⋊C4—34no—
361C4⋊C4—34no—
362C4⋊C4—34no—
363C4⋊C4—34no—
364C4⋊C4—68no—
365C4⋊C4—68no—
366C4⋊C4—68no—
367C4⋊C4—62no—
368C4⋊C4—62no—
369C4⋊C4—68no—
370C4⋊C4—68no—
371C4⋊C4C23×D314yes—
372C22⋊C4—28no—
373C22⋊C4—68no—
374C22⋊C4—68no—
375C42—32no—
376C42—32no—
377C42—32no—
378C42—32no—
379C42D3×D412yes—
380C42C23×D312yes—
381C22×C3—28no—
382C22×C3—416no—
383C22×C3—44no—
384C22×C3—44no—
385C22×C3D4×Q812yes—
386C22×C3C2×#13512yes—
387C22×C3#57612yes—
388C22×C3C23×D411yescharacteristic
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—44no—
392C3×C4D4×Q812yes—
393C3×C4C23×Q811yescharacteristic
394C3×C4C2×2−1+411yescharacteristic
395Dic3—22no—
396Dic3—22no—
397Dic3—22no—
398Dic3—44no—
399Dic3—44no—
400Dic3C2×#13514yes—
401C23—24no—
402C23—28no—
403C23—44no—
404C23—44no—
405C23C2×D3×D411yescharacteristic
406Q8—68no—
407Q8—248no—
408D4—24no—
409D4—28no—
410D4C22×Dic618yes—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—32no—
415C2×C4—34no—
416C2×C4—32no—
417C2×C4—32no—
418C2×C4—32no—
419C2×C4—44no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—64no—
424C2×C4—64no—
425C2×C4—88no—
426C2×C4—1216no—
427C2×C4—124no—
428C2×C4—124no—
429C2×C4—248no—
430C2×C4C22×Dic612yes—
431C2×C4C2×D3×D412yes—
432C2×C4C2×D3×D412yes—
433C2×C4C2×D3×D411yescharacteristic
434C2×C4#101012yes—
435C2×C4C24×D311yescharacteristic
436C8—22no—
437C8—62no—
438C8—62no—
439C8C22×Dic614yes—
440C2×C3—48no—
441C2×C3#338014yescomplemented
442C2×C3C2×D4×Q811yescharacteristic
443C2×C3C22×#13511yescharacteristic
444C2×C3C2×#57611yescharacteristic
445C22—28no—
446C22—44no—
447C22—44no—
448C22—416no—
449C22D4×Dic612yes—
450C22#575712yes—
451C22D3×#13512yes—
452C22C22×D3×D411yescharacteristic
453C4—22no—
454C4—22no—
455C4—34no—
456C4—44no—
457C4—62no—
458C4—62no—
459C4—62no—
460C4—124no—
461C4—124no—
462C4D4×Dic612yes—
463C4C23×Dic611yescharacteristic
464C4C2×#101011yescharacteristic
465C3C2×#338011yescharacteristic, complemented, Sylow
466C2—48no—
467C2#7957514yesdirect product factor
468C2C2×D4×Dic611yescharacteristic
469C2C2×#575711yescharacteristic
470C2C2×D3×#13511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×C4×D8C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenterC2×C4×D8D3
hypocenterC3C2×#3380
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#5539C2semidirect
5. C2×#5009C2semidirect
6. #79575C2direct (N × H)
7. C2×#4878C2semidirect
8. C2×#4874C2semidirect
9. C2×#4763C2semidirect
10. C2×#4761C2semidirect
11. C2×C8×Dic6C2semidirect
12. C2×#4683C2semidirect
21. #5539C22semidirect
43. #5009C22semidirect
47. #4878C22semidirect
48. #4874C22semidirect
49. #4763C22semidirect
50. #4761C22semidirect
51. #4683C22semidirect
52. C8×Dic6C22semidirect
71. C2×C3×D8Q8semidirect
191. C3×D8C2×Q8semidirect
229. C2×D8Dic6semidirect
349. D8C2×Dic6semidirect
441. C2×C3#3380semidirect
465. C3C2×#3380semidirect
467. C2#79575direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55392
C2×C3×C4×D82
C2×Dic3×D82
C2×#50092
#795752
C2×#48782
C2×#48742
C2×#47632
C2×#47612
C2×C8×Dic62
C2×#46832
C2×#33803