Finite Group Databasebeta

← C2×#71738 #10799697

Subgroups — C2×#71738 #10799697

Order 768.

1294 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#125C211yesmaximal, characteristic
2C2×C4×C4⋊Dic3C211yesmaximal, characteristic
3C22×C4⋊(C3⋊C8)C211yesmaximal, characteristic
4#71738C218yesmaximal, direct product factor
5C2×#4529C212yesmaximal
6C2×#4529C212yesmaximal
7C2×#1989—31nomaximal, Sylow
8C2×C3×#125—22no—
9C2×C3×#125—24no—
10C2×C3×#125C418yescomplemented
11C2×C3×#125C2214yes—
12C22×C3×C4⋊C4—21no—
13C22×C3×C4⋊C4—21no—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C4⋊Dic3C2212yes—
18C22×C4×Dic3—21no—
19C23×C3⋊C8C2212yes—
20C2×C3⋊#95—21no—
21C4×C4⋊Dic3—22no—
22C4×C4⋊Dic3C2214yes—
23C2×C4⋊(C3⋊C8)—24no—
24C2×C4⋊(C3⋊C8)C2214yes—
25#4529C2218yes—
26#4529C2218yes—
27C22×#125D311yescharacteristic
28C22×C4⋊C8—31no—
29C2×C4×C4⋊C4—31no—
30#1989—38no—
31C2×#340—32no—
32C2×#340—32no—
33C23×C3×C4—21no—
34C23×C3×C4—21no—
35C23×C3×C4D411yescharacteristic
36C23×C3×C4D411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C23×Dic3—22no—
39C3×#125—216no—
40C3×#125—28no—
41C3×#125C2×C4116yescomplemented
42C2×C3×C4⋊C4—28no—
43C2×C3×C4⋊C4—28no—
44C2×C3×C4⋊C4—24no—
45C2×C3×C4⋊C4—24no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—22no—
49C2×C3×C4⋊C4—22no—
50C2×C3×C4⋊C4C2×C414yes—
51C2×C3×C4⋊C4C2×C414yes—
52C2×C3×C4⋊C4C2312yes—
53C2×C3×C4⋊C4C2312yes—
54C2×C3×C42C2×C414yes—
55C2×C3×C42D412yes—
56C2×C3×C42D414yes—
57C2×C3×C42C2312yes—
58C2×C4⋊Dic3—22no—
59C2×C4⋊Dic3—24no—
60C2×C4⋊Dic3Q818yes—
61C2×C4⋊Dic3C2314yes—
62C2×C4×Dic3—24no—
63C2×C4×Dic3—44no—
64C22×C3⋊C8—24no—
65C22×C3⋊C8—28no—
66C22×C3⋊C8C2314yes—
67C3⋊#95—24no—
68C4⋊(C3⋊C8)—28no—
69C23×C8—32no—
70C2×#125—22no—
71C2×#125—24no—
72C2×#125Dic318yescomplemented
73C2×#125C2×D314yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—32no—
77C22×C4⋊C4C2×D311yescharacteristic
78C22×C4⋊C4C2×D311yescharacteristic
79C22×C42—61no—
80C22×C42C2×D311yescharacteristic
81C2×C4⋊C8—34no—
82C2×C4⋊C8—64no—
83C4×C4⋊C4—34no—
84C4×C4⋊C4—62no—
85C2×#95—61no—
86#340—38no—
87#340—38no—
88C24×C3C2×D411yescharacteristic
89C22×C3×C4—24no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—24no—
93C22×C3×C4—24no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—44no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4C22⋊C414yes—
100C22×C3×C4C22⋊C414yes—
101C22×C3×C4C22×C412yes—
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2×D412yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4C2×Q814yes—
107C22×C3×C4#4114yes—
108C22×C3×C4C2411yescharacteristic
109C22×Dic3—28no—
110C22×Dic3—24no—
111C22×Dic3—44no—
112C22×Dic3—44no—
113C3×C4⋊C4—216no—
114C3×C4⋊C4—216no—
115C3×C4⋊C4—24no—
116C3×C4⋊C4—24no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4C22×C414yes—
120C3×C4⋊C4C22×C414yes—
121C3×C42C22⋊C418yes—
122C3×C42C22×C414yes—
123C3×C42C2×D414yes—
124C4⋊Dic3—28no—
125C4⋊Dic3—24no—
126C4⋊Dic3C2×Q818yes—
127C4×Dic3—48no—
128C2×C3⋊C8—24no—
129C2×C3⋊C8—28no—
130C2×C3⋊C8—48no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—62no—
134C23×C4#6711yescharacteristic
135C23×C4#6711yescharacteristic
136C23×C4C22×D311yescharacteristic
137C22×C8—34no—
138C22×C8—68no—
139C22×C8—64no—
140#125—216no—
141#125—28no—
142#125C2×Dic3116yescomplemented
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—38no—
152C2×C4⋊C4—34no—
153C2×C4⋊C4—64no—
154C2×C4⋊C4—62no—
155C2×C4⋊C4C2×Dic314yes—
156C2×C4⋊C4C2×Dic314yes—
157C2×C4⋊C4C22×D312yes—
158C2×C4⋊C4C22×D312yes—
159C2×C42—64no—
160C2×C42—124no—
161C2×C42C2×Dic314yes—
162C2×C42#6712yes—
163C2×C42#6714yes—
164C2×C42C22×D312yes—
165C4⋊C8—68no—
166#95—64no—
167C23×C3C2×C22⋊C412yes—
168C23×C3C22⋊Q814yes—
169C23×C3#13514yes—
170C23×C3AGL(1,ℤ/8)12yes—
171C23×C3#13712yes—
172C23×C3C22×D411yescharacteristic
173C2×C3×C4—24no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—28no—
177C2×C3×C4—28no—
178C2×C3×C4—24no—
179C2×C3×C4—24no—
180C2×C3×C4—44no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—48no—
184C2×C3×C4—48no—
185C2×C3×C4C2×C22⋊C414yes—
186C2×C3×C4C2×C22⋊C414yes—
187C2×C3×C4C2×C22⋊C412yes—
188C2×C3×C4C4×Q818yes—
189C2×C3×C4C22⋊Q814yes—
190C2×C3×C4#12414yes—
191C2×C3×C4C4⋊Q814yes—
192C2×C3×C4C23×C411yescharacteristic
193C2×C3×C4C22×D411yescharacteristic
194C2×C3×C4C22×Q812yes—
195C2×C3×C4C2×#4112yes—
196C2×Dic3—28no—
197C2×Dic3—44no—
198C2×Dic3—44no—
199C2×Dic3—416no—
200C3⋊C8—44no—
201C24C2×#6711yescharacteristic
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—68no—
213C22×C4—64no—
214C22×C4—124no—
215C22×C4—124no—
216C22×C4#21714yes—
217C22×C4#21714yes—
218C22×C4D3×Q814yes—
219C22×C4#23914yes—
220C22×C4C22×Dic312yes—
221C22×C4C2×#6712yes—
222C22×C4C2×#6712yes—
223C22×C4C2×#6712yes—
224C22×C4C2×#6711yescharacteristic
225C22×C4C23×D311yescharacteristic
226C2×C8—68no—
227C2×C8—64no—
228C2×C8—128no—
229C4⋊C4—216no—
230C4⋊C4—216no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—38no—
236C4⋊C4—68no—
237C4⋊C4—64no—
238C4⋊C4C22×Dic314yes—
239C4⋊C4C22×Dic314yes—
240C42—128no—
241C42#21718yes—
242C42C22×Dic314yes—
243C42C2×#6714yes—
244C22×C3#39114yes—
245C22×C3#41614yes—
246C22×C3#41714yes—
247C22×C3#47818yes—
248C22×C3#47918yes—
249C22×C3C22×C22⋊C411yescharacteristic
250C22×C3C2×C22⋊Q812yes—
251C22×C3C2×#13512yes—
252C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
253C22×C3C2×#13711yescharacteristic
254C3×C4—24no—
255C3×C4—24no—
256C3×C4—44no—
257C3×C4—44no—
258C3×C4#39114yes—
259C3×C4C2×C4×Q812yes—
260C3×C4C2×#12411yescharacteristic
261C3×C4C2×C4⋊Q811yescharacteristic
262Dic3—48no—
263C23#94614yes—
264C23#94912yes—
265C23#95014yes—
266C23#95112yes—
267C23C2×#21712yes—
268C23C22×#6711yescharacteristic
269C2×C4—28no—
270C2×C4—28no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—48no—
277C2×C4—48no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—68no—
282C2×C4—124no—
283C2×C4—124no—
284C2×C4—1216no—
285C2×C4#94414yes—
286C2×C4Q8×Dic318yes—
287C2×C4#94614yes—
288C2×C4#94714yes—
289C2×C4C2×#21714yes—
290C2×C4C2×#21714yes—
291C2×C4C2×#21712yes—
292C2×C4C2×D3×Q812yes—
293C2×C4C2×#23912yes—
294C2×C4C23×Dic311yescharacteristic
295C2×C4C22×#6711yescharacteristic
296C8—124no—
297C2×C3#198918yescomplemented
298C2×C3C2×#39111yescharacteristic
299C2×C3C2×#41611yescharacteristic
300C2×C3C2×#41711yescharacteristic
301C2×C3C2×#47812yes—
302C2×C3C2×#47912yes—
303C22#506318yes—
304C22#506618yes—
305C22#522914yes—
306C22#523214yes—
307C22#523314yes—
308C22C2×#94612yes—
309C22C2×#94911yescharacteristic
310C22C2×#95012yes—
311C22C2×#95111yescharacteristic
312C22C22×#21711yescharacteristic
313C4—24no—
314C4—24no—
315C4—44no—
316C4—44no—
317C4—128no—
318C4#522914yes—
319C4C2×#94411yescharacteristic
320C4C2×Q8×Dic312yes—
321C4C2×#94711yescharacteristic
322C3C2×#198911yescharacteristic, complemented, Sylow
323C2#7173818yesdirect product factor
324C2C2×#506612yes—
325C2C2×#506312yes—
326C2C2×#522911yescharacteristic
327C2C2×#523311yescharacteristic
328C2C2×#523211yescharacteristic
subgroupG/H
centerC24C2×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC22×C3×#125C2
socle, prime radicalC24×C3C2×D4
cosocleC22×C3×C4C24
hypercenterC22×#125D3
hypocenterC3C2×#1989
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C42 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
4. #71738C2direct (N × H)
10. C2×C3×#125C4semidirect
41. C3×#125C2×C4semidirect
72. C2×#125Dic3semidirect
142. #125C2×Dic3semidirect
297. C2×C3#1989semidirect
322. C3C2×#1989semidirect
323. C2#71738direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×C4×C4⋊Dic32
C22×C4⋊(C3⋊C8)2
#717382
C2×#45292
C2×#19893