Finite Group Databasebeta

← C2×#71521 #10948621

Subgroups — C2×#71521 #10948621

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#510C211yesmaximal, characteristic
2C2×#5197C211yesmaximal, characteristic
3C2×#5184C211yesmaximal, characteristic, complemented
4C2×#4566C211yesmaximal, characteristic
5C2×#4565C211yesmaximal, characteristic, complemented
6#71521C218yesmaximal, direct product factor
7C2×#4513C211yesmaximal, characteristic, complemented
8C2×#4512C211yesmaximal, characteristic, complemented
9C2×#2180—31nomaximal, Sylow
10C2×C3×#135C2211yescharacteristic
11C22×C3×Q16C2211yescharacteristic
12C2×#948—21no—
13C2×Q8×Dic3—21no—
14C3×#510C414yescomplemented
15C3×#510C2214yes—
16C2×C3×Q32C412yescomplemented
17C2×C3×SD32C412yescomplemented
18C2×C3×M32C2211yescharacteristic
19C2×#837—21no—
20C2×#835—21no—
21#5197—22no—
22#5197C2214yes—
23#5184—22no—
24#5184C2214yescomplemented
25C2×#819C2211yescharacteristic
26C2×#817C2211yescharacteristic
27C2×#815C2211yescharacteristic
28C2×#812C2211yescharacteristic
29#4566C2214yes—
30#4565C2214yescomplemented
31#4513C2214yescomplemented
32#4512C2214yescomplemented
33C2×#510D311yescharacteristic
34C2×#444—31no—
35C2×#441—31no—
36#2180—38no—
37C2×#365—31no—
38C2×#362—31no—
39C2×#360—31no—
40C2×#347—31no—
41C2×C3×#41—21no—
42C22×C3×Q8—21no—
43C3×#135—21no—
44C3×#135—21no—
45C3×#135C2×C412yes—
46C3×#135C2312yes—
47C2×C3×Q16—21no—
48C2×C3×Q16—21no—
49C2×C3×Q16—21no—
50C2×C3×Q16C2×C412yes—
51C2×C3×Q16C2×C411yescharacteristic
52C2×C3×Q16C2×C412yes—
53C2×C3×Q16C2312yes—
54C2×C3×SD16—21no—
55C2×C3×D8C2×C411yescharacteristic
56C22×C3×C8C2311yescharacteristic
57#948—24no—
58#948—42no—
59Q8×Dic3—24no—
60Q8×Dic3—42no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—41no—
63C2×C4×Dic3—41no—
64C2×C4×Dic3D411yescharacteristic
65C2×#208—21no—
66C2×#208—41no—
67C22×C3⋊C8—41no—
68C22×C3⋊C8D411yescharacteristic
69C3×Q32C2×C418yescomplemented
70C3×SD32C2×C418yescomplemented
71C3×M32C2×C412yes—
72C3×M32C2312yes—
73C2×C3×C16C2×C412yes—
74#837—24no—
75#835—24no—
76#819—21no—
77#819C2312yes—
78#817—21no—
79#817C2312yes—
80#815D412yes—
81#815C2312yes—
82#812—21no—
83#812C2312yes—
84C2×C3⋊C16—21no—
85C2×#135C2×D311yescharacteristic
86C22×Q16C2×D311yescharacteristic
87C2×#131—61no—
88C2×C4×Q8—61no—
89#510Dic314yescomplemented
90#510C2×D314yes—
91C2×Q32Dic312yescomplemented
92C2×SD32Dic312yescomplemented
93C2×M32—31no—
94C2×M32C2×D311yescharacteristic
95#444—34no—
96#444—62no—
97#441—34no—
98#441—62no—
99C2×#108—31no—
100C2×#106—31no—
101C2×#104—61no—
102C2×#103—61no—
103C2×#97—31no—
104#365—34no—
105#362—34no—
106#360—34no—
107#347—34no—
108C3×#41—22no—
109C3×#41—22no—
110C3×#41—41no—
111C3×#41—41no—
112C2×C3×Q8—21no—
113C2×C3×Q8—22no—
114C2×C3×Q8—22no—
115C2×C3×Q8—22no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—21no—
125C22×Dic3—41no—
126C3×Q16—22no—
127C3×Q16—22no—
128C3×Q16—21no—
129C3×Q16—22no—
130C3×Q16C22×C414yes—
131C3×Q16C22×C412yes—
132C3×SD16—22no—
133C3×SD16—22no—
134C3×D8—21no—
135C3×D8C22×C412yes—
136C2×C3×C8C22×C411yescharacteristic
137C2×C3×C8C22×C411yescharacteristic
138C2×C3×C8C2×D411yescharacteristic
139C2×C3×C8#4111yescharacteristic
140C2×C3×C8#4111yescharacteristic
141C2×C3×C8C2411yescharacteristic
142C4⋊Dic3—22no—
143C4⋊Dic3—44no—
144C4⋊Dic3—41no—
145C4×Dic3—21no—
146C4×Dic3—44no—
147C4×Dic3C2×D412yes—
148#208—22no—
149#208—44no—
150#208—41no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—44no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8C2×D412yes—
156C3×C16C22×C414yes—
157C3⋊C16—22no—
158C2×#41—21no—
159C22×Q8—21no—
160#135—21no—
161#135—21no—
162#135C2×Dic312yes—
163#135C22×D312yes—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16C2×Dic312yes—
168C2×Q16C2×Dic312yes—
169C2×Q16C2×Dic311yescharacteristic
170C2×Q16C22×D312yes—
171C2×SD16—21no—
172C2×D8C2×Dic311yescharacteristic
173#131—64no—
174#131—122no—
175C2×M16—61no—
176C2×M16—121no—
177C22×C8—31no—
178C22×C8—121no—
179C22×C8C22×D311yescharacteristic
180C4×Q8—64no—
181C4×Q8—122no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—121no—
184C2×C42—31no—
185C2×C42—121no—
186Q32C2×Dic318yescomplemented
187SD32C2×Dic318yescomplemented
188M32—32no—
189M32—61no—
190M32C2×Dic312yes—
191M32C22×D312yes—
192C2×C16—61no—
193C2×C16C2×Dic312yes—
194#108—32no—
195#108—61no—
196#106—32no—
197#106—61no—
198#104—64no—
199#103—64no—
200#97—32no—
201#97—32no—
202C23×C3—41no—
203C23×C3AGL(1,ℤ/8)11yescharacteristic
204C3×Q8—22no—
205C3×Q8—24no—
206C3×Q8—42no—
207C3×Q8—42no—
208C3×Q8—42no—
209C3×Q8—41no—
210C3×D4—22no—
211C3×D4—42no—
212C3×D4—42no—
213C3×D4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225C2×C3×C4C22×D411yescharacteristic
226C2×Dic3—22no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—82no—
230C3×C8C4×D411yescharacteristic
231C3×C8C4×D411yescharacteristic
232C3×C8C23×C411yescharacteristic
233C3×C8C2×#4111yescharacteristic
234C3⋊C8—22no—
235C3⋊C8—42no—
236#41—22no—
237#41—22no—
238#41—41no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×D4—21no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—61no—
252C22×C4—121no—
253C22×C4D3×D411yescharacteristic
254Q16—22no—
255Q16—22no—
256Q16—21no—
257Q16—22no—
258Q16C22×Dic314yes—
259Q16C22×Dic312yes—
260SD16—22no—
261SD16—22no—
262D8—21no—
263D8C22×Dic312yes—
264M16—62no—
265M16—124no—
266M16—121no—
267C2×C8—32no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—124no—
271C2×C8—121no—
272C2×C8D3×D411yescharacteristic
273C2×C8#23711yescharacteristic
274C2×C8#23711yescharacteristic
275C2×C8C22×Dic311yescharacteristic
276C2×C8C22×Dic311yescharacteristic
277C2×C8C23×D311yescharacteristic
278C4⋊C4—62no—
279C4⋊C4—124no—
280C4⋊C4—121no—
281C42—32no—
282C42—61no—
283C42—124no—
284C16—62no—
285C16C22×Dic314yes—
286C22×C3—22no—
287C22×C3—42no—
288C22×C3—82no—
289C22×C3#44211yescharacteristic
290C22×C3#44211yescharacteristic
291C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
292C3×C4—42no—
293C3×C4—44no—
294C3×C4#44211yescharacteristic
295C3×C4#44211yescharacteristic
296C3×C4C2×C4×D411yescharacteristic
297C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
298Dic3—42no—
299Dic3—82no—
300C23—41no—
301C23#91111yescharacteristic
302Q8—22no—
303Q8—24no—
304Q8—42no—
305Q8—42no—
306Q8—41no—
307Q8—42no—
308D4—22no—
309D4—42no—
310D4—42no—
311D4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—62no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—242no—
322C2×C4#91111yescharacteristic
323C2×C4#91111yescharacteristic
324C2×C4#91111yescharacteristic
325C2×C4D4×Dic311yescharacteristic
326C2×C4D4×Dic311yescharacteristic
327C2×C4C2×D3×D411yescharacteristic
328C8—62no—
329C8—122no—
330C8D4×Dic311yescharacteristic
331C8D4×Dic311yescharacteristic
332C8C2×#23711yescharacteristic
333C8C23×Dic311yescharacteristic
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—82no—
337C2×C3#218012yescomplemented
338C2×C3C2×#44211yescharacteristic
339C22—22no—
340C22—42no—
341C22—82no—
342C22#515211yescharacteristic
343C22#515211yescharacteristic
344C22C2×#91111yescharacteristic
345C4—42no—
346C4—44no—
347C4—122no—
348C4—242no—
349C4#515211yescharacteristic
350C4#515211yescharacteristic
351C4C2×#91111yescharacteristic
352C4C2×D4×Dic311yescharacteristic
353C3C2×#218011yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2—82no—
357C2#7152112yesdirect product factor
358C2C2×#515211yescharacteristic
subgroupG/H
centerC22#5152
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC2×C3×#510C2
socleC22×C3#442
prime radicalC2×C3×#135C22
cosocleC2×C3×C8C24
hypercenterC2×#510D3
hypocenterC3C2×#2180
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
3. C2×#5184C2semidirect
5. C2×#4565C2semidirect
6. #71521C2direct (N × H)
7. C2×#4513C2semidirect
8. C2×#4512C2semidirect
14. C3×#510C4semidirect
16. C2×C3×Q32C4semidirect
17. C2×C3×SD32C4semidirect
24. #5184C22semidirect
30. #4565C22semidirect
31. #4513C22semidirect
32. #4512C22semidirect
69. C3×Q32C2×C4semidirect
70. C3×SD32C2×C4semidirect
89. #510Dic3semidirect
91. C2×Q32Dic3semidirect
92. C2×SD32Dic3semidirect
186. Q32C2×Dic3semidirect
187. SD32C2×Dic3semidirect
337. C2×C3#2180semidirect
353. C3C2×#2180semidirect
357. C2#71521direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5102
C2×#51972
C2×#51842
C2×#45662
C2×#45652
#715212
C2×#45132
C2×#45122
C2×#21803