Finite Group Databasebeta

← C2×#79080 #11658806

Subgroups — C2×#79080 #11658806

Order 768.

2830 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×M16C211yesmaximal, characteristic
2C23×ΓL(1,𝔽25)C211yesmaximal, characteristic
3C2×D3×#117C211yesmaximal, characteristic
4C2×C3×#424C211yesmaximal, characteristic, complemented
5C2×#5119C211yesmaximal, characteristic, complemented
6C2×#5102C211yesmaximal, characteristic, complemented
7C2×#4999C211yesmaximal, characteristic, complemented
8#79080C2116yesmaximal, direct product factor
9C2×#4839C214yesmaximal, complemented
10C2×#4837C214yesmaximal, complemented
11C2×#2905—31nomaximal, Sylow
12C23×C4×D3C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#926C2211yescharacteristic
17C22×#208C2211yescharacteristic
18C23×C3⋊C8C2211yescharacteristic
19C2×D3×M16—28no—
20C2×D3×M16C2218yes—
21C22×ΓL(1,𝔽25)—28no—
22C22×ΓL(1,𝔽25)C2212yes—
23C22×ΓL(1,𝔽25)C2212yes—
24C22×ΓL(1,𝔽25)C2218yes—
25C22×ΓL(1,𝔽25)C2212yes—
26C22×D3×C8C2212yes—
27D3×#117C418yes—
28D3×#117C2218yes—
29C2×#892C412yes—
30C2×D3×C4⋊C4C412yes—
31C2×D3×C22⋊C4C412yes—
32C2×#877C412yes—
33C2×#872C2212yes—
34C2×C42×D3C2212yes—
35C3×#424C2218yescomplemented
36C2×C3×C4⋊C8C2214yescomplemented
37#5119C2218yescomplemented
38#5102C2218yescomplemented
39C2×Dic3⋊C8C2214yescomplemented
40C2×Dic3⋊C8C2214yescomplemented
41#4999C2218yescomplemented
42C2×C4⋊(C3⋊C8)C2214yescomplemented
43#4839C22132yescomplemented
44#4837C22132yescomplemented
45C23×M16—31no—
46C23×M16—31no—
47C22×#117—31no—
48#2905—316no—
49C2×#424—34no—
50C2×#424—31no—
51C2×#424—31no—
52C2×#424—31no—
53C2×#424D311yescharacteristic, complemented
54C2×#423—34no—
55C24×D3C2×C411yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2×C411yescharacteristic
58C22×C4×D3C2×C414yes—
59C22×C4×D3C2×C411yescharacteristic
60C22×C4×D3C2×C411yescharacteristic
61C22×C4×D3C2×C414yes—
62C22×C4×D3D418yes—
63C22×C4×D3Q818yes—
64C22×C4×D3C2311yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×C4×D3C2314yes—
67C2×C3×M16—24no—
68C2×C3×M16C2314yes—
69C22×C3×C8—24no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2314yes—
72C22×C3×C8C2312yes—
73C3×#117C2×C414yes—
74C3×#117C2314yes—
75C2×C3×C4⋊C4C2×C412yes—
76C2×C3×C22⋊C4C2×C412yes—
77C2×C3×C42C2312yes—
78C2×#217C2×C412yes—
79C2×(C2×D3)⋊C4C2×C414yes—
80#926C2×C414yes—
81#926C2314yes—
82C2×C4⋊Dic3C2×C412yes—
83C2×Dic3⋊C4C2×C414yes—
84C2×C4×Dic3C2×C414yes—
85C2×C4×Dic3C2312yes—
86C2×#208—24no—
87C2×#208C2314yes—
88C22×C3⋊C8—24no—
89C22×C3⋊C8C2312yes—
90C22×C3⋊C8C2312yes—
91C22×C3⋊C8C2314yes—
92D3×M16—232no—
93C2×ΓL(1,𝔽25)—28no—
94C2×ΓL(1,𝔽25)—28no—
95C2×ΓL(1,𝔽25)—232no—
96C2×ΓL(1,𝔽25)—28no—
97C2×ΓL(1,𝔽25)C2318yes—
98C2×ΓL(1,𝔽25)C2318yes—
99C2×ΓL(1,𝔽25)C2318yes—
100C2×D3×C8—28no—
101C2×D3×C8C2318yes—
102#892C2×C4116yes—
103D3×C4⋊C4C2×C4116yes—
104D3×C22⋊C4C2×C4116yes—
105#877C2×C4116yes—
106#872C2×C418yes—
107#872C2318yes—
108C42×D3C2×C418yes—
109C42×D3C2318yes—
110C3×C4⋊C8C23116yescomplemented
111Dic3⋊C8C23116yescomplemented
112Dic3⋊C8C23116yescomplemented
113C4⋊(C3⋊C8)C23116yescomplemented
114C24×C4—31no—
115C22×M16—31no—
116C22×M16—38no—
117C22×M16—32no—
118C22×M16—32no—
119C22×M16—32no—
120C22×M16—38no—
121C22×M16—68no—
122C22×M16—68no—
123C22×M16C2×D311yescharacteristic
124C23×C8—31no—
125C23×C8—32no—
126C23×C8C2×D311yescharacteristic
127C2×#117—31no—
128C2×#117—38no—
129C2×#117—38no—
130C2×#117—32no—
131C2×#117—32no—
132C2×#117—32no—
133C2×#117C2×D311yescharacteristic
134C22×C4⋊C4—32no—
135C22×C22⋊C4—32no—
136C22×C42—32no—
137#424—38no—
138#424—38no—
139#424—38no—
140#424—332no—
141#424C2×D318yescomplemented
142#423—332no—
143C2×C4⋊C8—34no—
144C2×C4⋊C8—34no—
145C2×C4⋊C8—34no—
146C2×C4⋊C8C2×D314yescomplemented
147C24×C3C22×C411yescharacteristic
148C23×D3—44no—
149C23×D3C4⋊C418yes—
150C23×D3C22×C411yescharacteristic
151C23×D3C22×C411yescharacteristic
152C23×D3C22×C414yes—
153C22×C3×C4C22×C414yes—
154C22×C3×C4C22×C411yescharacteristic
155C22×C3×C4C22×C412yes—
156C22×C3×C4C2×D414yes—
157C22×C3×C4C2×Q814yes—
158C22×C3×C4C2412yes—
159C22×C3×C4C2411yescharacteristic
160C22×Dic3C4⋊C418yes—
161C22×Dic3C22×C414yes—
162C22×Dic3C22×C414yes—
163C22×Dic3C22×C411yescharacteristic
164C22×Dic3C22×C411yescharacteristic
165C2×C4×D3—48no—
166C2×C4×D3C4⋊C418yes—
167C2×C4×D3C4⋊C418yes—
168C2×C4×D3C4⋊C4132yes—
169C2×C4×D3C22×C4116yes—
170C2×C4×D3C22×C412yes—
171C2×C4×D3C22×C412yes—
172C2×C4×D3C22×C414yes—
173C2×C4×D3C22×C414yes—
174C2×C4×D3C2×D414yes—
175C2×C4×D3C2×D414yes—
176C2×C4×D3C2×D4116yes—
177C2×C4×D3C2×Q814yes—
178C2×C4×D3C2×Q814yes—
179C2×C4×D3C2×Q8116yes—
180C2×C4×D3C2412yes—
181C2×C4×D3C2412yes—
182C3×M16—28no—
183C2×C3×C8—24no—
184C2×C3×C8—24no—
185C2×C3×C8—28no—
186C2×C3×C8C2414yes—
187C2×C3×C8C2414yes—
188C3×C4⋊C4C22×C418yes—
189C3×C22⋊C4C22×C418yes—
190C3×C42C22×C414yes—
191C3×C42C2414yes—
192#217C22×C418yes—
193(C2×D3)⋊C4C22×C4116yes—
194C4⋊Dic3C22×C418yes—
195Dic3⋊C4C22×C4116yes—
196C4×Dic3C22×C414yes—
197C4×Dic3C22×C4116yes—
198C4×Dic3C2414yes—
199#208—28no—
200C2×C3⋊C8—24no—
201C2×C3⋊C8—24no—
202C2×C3⋊C8—28no—
203C2×C3⋊C8C2414yes—
204C2×C3⋊C8C2414yes—
205ΓL(1,𝔽25)—216no—
206ΓL(1,𝔽25)—216no—
207ΓL(1,𝔽25)—216no—
208D3×C8—216no—
209C25—31no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—31no—
213C23×C4—34no—
214C23×C4—38no—
215C23×C4—38no—
216C23×C4—34no—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—34no—
220C23×C4C22×D311yescharacteristic
221C2×M16—24no—
222C2×M16—34no—
223C2×M16—38no—
224C2×M16—38no—
225C2×M16—38no—
226C2×M16—68no—
227C2×M16—68no—
228C2×M16—632no—
229C2×M16—68no—
230C2×M16—632no—
231C2×M16—64no—
232C2×M16C22×D314yes—
233C22×C8—24no—
234C22×C8—34no—
235C22×C8—32no—
236C22×C8—32no—
237C22×C8—38no—
238C22×C8—68no—
239C22×C8—64no—
240C22×C8C22×D314yes—
241C22×C8C22×D312yes—
242C22×C8C22×D312yes—
243#117—34no—
244#117—34no—
245#117—316no—
246#117—316no—
247#117—38no—
248#117—38no—
249#117C4×D314yes—
250#117C22×D314yes—
251C2×C4⋊C4—34no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—316no—
254C2×C4⋊C4C4×D312yes—
255C2×C22⋊C4—32no—
256C2×C22⋊C4—34no—
257C2×C22⋊C4—316no—
258C2×C22⋊C4C4×D312yes—
259C2×C42—34no—
260C2×C42—32no—
261C2×C42—38no—
262C2×C42—38no—
263C2×C42C22×D312yes—
264C4⋊C8—316no—
265C4⋊C8—316no—
266C4⋊C8—316no—
267C4⋊C8C22×D3116yescomplemented
268C23×C3—24no—
269C23×C3C2×C4⋊C414yes—
270C23×C3C23×C412yes—
271C23×C3C23×C411yescharacteristic
272C22×D3—24no—
273C22×D3—24no—
274C22×D3—416no—
275C22×D3—48no—
276C22×D3C2×C4⋊C414yes—
277C22×D3C2×C4⋊C414yes—
278C22×D3C2×C4⋊C4116yes—
279C22×D3C23×C412yes—
280C22×D3C23×C412yes—
281C2×C3×C4—28no—
282C2×C3×C4C2×C4⋊C418yes—
283C2×C3×C4C2×C4⋊C414yes—
284C2×C3×C4C23×C418yes—
285C2×C3×C4C23×C411yescharacteristic
286C2×C3×C4C23×C412yes—
287C2×C3×C4C22×D412yes—
288C2×C3×C4C22×D414yes—
289C2×C3×C4C22×Q814yes—
290C2×C3×C4C22×Q812yes—
291C2×C3×C4C2511yescharacteristic
292C2×Dic3—28no—
293C2×Dic3C2×C4⋊C4116yes—
294C2×Dic3C2×C4⋊C414yes—
295C2×Dic3C2×C4⋊C414yes—
296C2×Dic3C23×C418yes—
297C2×Dic3C23×C412yes—
298C2×Dic3C23×C412yes—
299C4×D3—416no—
300C4×D3C2×C4⋊C418yes—
301C4×D3C2×C4⋊C418yes—
302C4×D3C2×C4⋊C4116yes—
303C4×D3C2×C4⋊C4116yes—
304C4×D3C22×D414yes—
305C4×D3C22×D414yes—
306C4×D3C22×Q814yes—
307C4×D3C22×Q814yes—
308C3×C8—24no—
309C3×C8—24no—
310C3⋊C8—24no—
311C3⋊C8—24no—
312C24—31no—
313C24—31no—
314C24—38no—
315C24—34no—
316C24—124no—
317C24C2×C4×D311yescharacteristic
318C22×C4—38no—
319C22×C4—31no—
320C22×C4—31no—
321C22×C4—34no—
322C22×C4—38no—
323C22×C4—38no—
324C22×C4—332no—
325C22×C4—32no—
326C22×C4—34no—
327C22×C4—34no—
328C22×C4—32no—
329C22×C4—34no—
330C22×C4—316no—
331C22×C4—34no—
332C22×C4—316no—
333C22×C4—34no—
334C22×C4—34no—
335C22×C4—32no—
336C22×C4—32no—
337C22×C4—34no—
338C22×C4—316no—
339C22×C4—128no—
340C22×C4C2×C4×D314yes—
341C22×C4C2×C4×D311yescharacteristic
342C22×C4C2×C4×D312yes—
343C22×C4D3×D414yes—
344C22×C4D3×Q814yes—
345C22×C4C23×D312yes—
346C22×C4C23×D311yescharacteristic
347M16—28no—
348M16—616no—
349M16—616no—
350M16—616no—
351M16—68no—
352C2×C8—24no—
353C2×C8—28no—
354C2×C8—24no—
355C2×C8—34no—
356C2×C8—34no—
357C2×C8—616no—
358C2×C8—64no—
359C2×C8—64no—
360C2×C8—68no—
361C2×C8C23×D314yes—
362C2×C8C23×D314yes—
363C4⋊C4—316no—
364C4⋊C4—38no—
365C4⋊C4C2×C4×D318yes—
366C22⋊C4—38no—
367C22⋊C4—316no—
368C22⋊C4C2×C4×D318yes—
369C42—316no—
370C42—34no—
371C42—34no—
372C42C2×C4×D314yes—
373C42C23×D314yes—
374C22×C3—28no—
375C22×C3—24no—
376C22×C3C22×C4⋊C414yes—
377C22×C3C22×C4⋊C412yes—
378C22×C3#56814yes—
379C22×C3C24×C411yescharacteristic
380C2×D3—28no—
381C2×D3—28no—
382C2×D3—24no—
383C2×D3—24no—
384C2×D3—416no—
385C2×D3C22×C4⋊C414yes—
386C2×D3C22×C4⋊C414yes—
387C3×C4—28no—
388C3×C4C22×C4⋊C412yes—
389C3×C4C22×C4⋊C414yes—
390C3×C4C23×D411yescharacteristic
391C3×C4C23×Q811yescharacteristic
392Dic3—28no—
393Dic3C22×C4⋊C414yes—
394Dic3C22×C4⋊C414yes—
395C23—24no—
396C23—32no—
397C23—32no—
398C23—34no—
399C23—34no—
400C23—316no—
401C23—64no—
402C23—64no—
403C23—128no—
404C23—1216no—
405C23D3×C4⋊C414yes—
406C23C22×C4×D312yes—
407C23C22×C4×D311yescharacteristic
408C2×C4—28no—
409C2×C4—32no—
410C2×C4—32no—
411C2×C4—316no—
412C2×C4—34no—
413C2×C4—34no—
414C2×C4—34no—
415C2×C4—34no—
416C2×C4—34no—
417C2×C4—34no—
418C2×C4—38no—
419C2×C4—316no—
420C2×C4—38no—
421C2×C4—316no—
422C2×C4—38no—
423C2×C4—68no—
424C2×C4—1216no—
425C2×C4D3×C4⋊C418yes—
426C2×C4D3×C4⋊C414yes—
427C2×C4C22×C4×D318yes—
428C2×C4C22×C4×D311yescharacteristic
429C2×C4C22×C4×D312yes—
430C2×C4C2×D3×D412yes—
431C2×C4C2×D3×D414yes—
432C2×C4C2×D3×Q814yes—
433C2×C4C2×D3×Q812yes—
434C2×C4C24×D311yescharacteristic
435C8—24no—
436C8—24no—
437C8—64no—
438C8—64no—
439C2×C3—24no—
440C2×C3#290514yescomplemented
441C2×C3C23×C4⋊C411yescharacteristic
442C2×C3C2×#56812yes—
443D3—24no—
444D3—24no—
445C22—24no—
446C22—28no—
447C22—34no—
448C22—34no—
449C22—64no—
450C22—64no—
451C22—68no—
452C22—68no—
453C22—1216no—
454C22C2×D3×C4⋊C414yes—
455C22C2×D3×C4⋊C412yes—
456C22#575014yes—
457C22C23×C4×D311yescharacteristic
458C4—28no—
459C4—34no—
460C4—34no—
461C4—68no—
462C4C2×D3×C4⋊C412yes—
463C4C2×D3×C4⋊C414yes—
464C4C22×D3×D411yescharacteristic
465C4C22×D3×Q811yescharacteristic
466C3C2×#290511yescharacteristic, complemented, Sylow
467C2—24no—
468C2—64no—
469C2—64no—
470C2#7908014yesdirect product factor
471C2C22×D3×C4⋊C411yescharacteristic
472C2C2×#575012yes—
subgroupG/H
centerC22×C4C23×D3
commutatorC22×C3C24×C4
FrattiniC2×C4C24×D3
FittingC2×C3×#424C2
socleC23×C3C23×C4
prime radicalC24×D3C2×C4
cosocleC2×C3×C4C25
hypercenterC2×#424D3
hypocenterC3C2×#2905
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
4. C2×C3×#424C2semidirect
5. C2×#5119C2semidirect
6. C2×#5102C2semidirect
7. C2×#4999C2semidirect
8. #79080C2direct (N × H)
9. C2×#4839C2semidirect
10. C2×#4837C2semidirect
35. C3×#424C22semidirect
36. C2×C3×C4⋊C8C22semidirect
37. #5119C22semidirect
38. #5102C22semidirect
39. C2×Dic3⋊C8C22semidirect
40. C2×Dic3⋊C8C22semidirect
41. #4999C22semidirect
42. C2×C4⋊(C3⋊C8)C22semidirect
43. #4839C22semidirect
44. #4837C22semidirect
53. C2×#424D3semidirect
110. C3×C4⋊C8C23semidirect
111. Dic3⋊C8C23semidirect
112. Dic3⋊C8C23semidirect
113. C4⋊(C3⋊C8)C23semidirect
141. #424C2×D3semidirect
146. C2×C4⋊C8C2×D3semidirect
267. C4⋊C8C22×D3semidirect
440. C2×C3#2905semidirect
466. C3C2×#2905semidirect
470. C2#79080direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×M162
C23×ΓL(1,𝔽25)2
C2×D3×#1172
C2×C3×#4242
C2×#51192
C2×#51022
C2×#49992
#790802
C2×#48392
C2×#48372
C2×#29053