Finite Group Databasebeta

← C2×#72980 #11712791

Subgroups — C2×#72980 #11712791

Order 768.

564 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C42⋊C3C211yesmaximal, characteristic
2C2×#4623C212yesmaximal
3#72980C214yesmaximal, direct product factor
4C2×#1994—31nomaximal, Sylow
5C2×A4⋊Q8—41nomaximal
6C22×C42⋊C3C2211yescharacteristic
7C4×C42⋊C3C2212yes—
8#4623C2214yes—
9C2×#533—31no—
10C2×#423—31no—
11C2×C43D311yescharacteristic
12#1994—34no—
13#1994—34no—
14C2×#339—32no—
15C2×#339—32no—
16C2×C4×A4—41no—
17C2×SL(2,ℤ/4)—42no—
18A4⋊Q8—44no—
19C2×C42⋊C3Q812yescomplemented
20C2×C42⋊C3C2311yescharacteristic
21C22×M16—32no—
22#533—34no—
23#533—34no—
24#533—62no—
25#533—62no—
26C2×C4⋊Q8—31no—
27C2×C4⋊Q8—31no—
28C2×#125—32no—
29C2×C22⋊Q8—61no—
30C2×C22⋊Q8—61no—
31C2×C4×Q8—61no—
32C2×C4×Q8—61no—
33C2×#117—32no—
34C22×C42—31no—
35C22×C42—31no—
36C22×C42C2×D311yescharacteristic
37#423—34no—
38#423—62no—
39C2×C4⋊C8—61no—
40C2×C4⋊C8—61no—
41C43—32no—
42C43C2×D312yes—
43#339—34no—
44#339—38no—
45#339—34no—
46C22×A4—41no—
47C2×Dic6—161no—
48C4×A4—42no—
49SL(2,ℤ/4)—44no—
50C42⋊C3C2×Q811yescharacteristic, complemented
51C22×Q8—61no—
52C22×Q8—61no—
53C23×C4—31no—
54C23×C4—31no—
55C23×C4S411yescharacteristic
56C2×M16—34no—
57C2×M16—62no—
58C2×M16—64no—
59C2×M16—64no—
60C22×C8—62no—
61C4⋊Q8—34no—
62C4⋊Q8—34no—
63C4⋊Q8—62no—
64C4⋊Q8—62no—
65#125—38no—
66#125—64no—
67C22⋊Q8—64no—
68C22⋊Q8—64no—
69C22⋊Q8—64no—
70C22⋊Q8—64no—
71C4×Q8—64no—
72C4×Q8—64no—
73C4×Q8—64no—
74C4×Q8—64no—
75#117—34no—
76#117—34no—
77#117—62no—
78#117—62no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—62no—
82C2×C4⋊C4—62no—
83C2×C4⋊C4—61no—
84C2×C4⋊C4—61no—
85C2×C22⋊C4—62no—
86C2×C42—31no—
87C2×C42—31no—
88C2×C42—31no—
89C2×C42—31no—
90C2×C42—32no—
91C2×C42—32no—
92C2×C42—32no—
93C2×C42—32no—
94C2×C42—31no—
95C2×C42—32no—
96C2×C42—32no—
97C2×C42—34no—
98C2×C42—61no—
99C2×C42—62no—
100C2×C42—62no—
101C2×C42Dic612yescomplemented
102C2×C42C22×D311yescharacteristic
103C4⋊C8—64no—
104C4⋊C8—64no—
105C2×A4—42no—
106C2×A4—41no—
107C2×C3×C4—161no—
108C2×Dic3—162no—
109Dic6—164no—
110C24C2×S411yescharacteristic
111C2×Q8—64no—
112C2×Q8—64no—
113C2×Q8—124no—
114C2×Q8—121no—
115C2×Q8—122no—
116C2×Q8—121no—
117C22×C4—31no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4—32no—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—32no—
131C22×C4—31no—
132C22×C4—32no—
133C22×C4—32no—
134C22×C4—64no—
135C22×C4—62no—
136C22×C4—62no—
137C22×C4C2×S412yes—
138M16—64no—
139M16—64no—
140M16—124no—
141C2×C8—64no—
142C2×C8—124no—
143C4⋊C4—34no—
144C4⋊C4—34no—
145C4⋊C4—62no—
146C4⋊C4—62no—
147C4⋊C4—68no—
148C4⋊C4—68no—
149C4⋊C4—64no—
150C4⋊C4—64no—
151C22⋊C4—64no—
152C22⋊C4—64no—
153C42—32no—
154C42—31no—
155C42—32no—
156C42—32no—
157C42—32no—
158C42—32no—
159C42—32no—
160C42—32no—
161C42—32no—
162C42—32no—
163C42—34no—
164C42—61no—
165C42—62no—
166C42—64no—
167C42—64no—
168C42—64no—
169C42C2×Dic611yescharacteristic, complemented
170C22×C3—161no—
171A4—41no—
172C3×C4—162no—
173Dic3—164no—
174C23—31no—
175C23—31no—
176C23—32no—
177C23A4⋊Q812yes—
178C23C22×S411yescharacteristic
179Q8—128no—
180Q8—124no—
181Q8—124no—
182C2×C4—31no—
183C2×C4—31no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—32no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—32no—
190C2×C4—32no—
191C2×C4—32no—
192C2×C4—32no—
193C2×C4—32no—
194C2×C4—32no—
195C2×C4—32no—
196C2×C4—32no—
197C2×C4—32no—
198C2×C4—34no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—32no—
203C2×C4—32no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—32no—
207C2×C4—61no—
208C2×C4—64no—
209C2×C4—64no—
210C2×C4—62no—
211C2×C4—64no—
212C2×C4—64no—
213C2×C4—124no—
214C2×C4—122no—
215C2×C4—122no—
216C2×C4#85711yescharacteristic
217C8—124no—
218C2×C3—162no—
219C2×C3—161no—
220C22—31no—
221C22—32no—
222C22—32no—
223C22—32no—
224C22—31no—
225C22—31no—
226C22—61no—
227C22C2×#85711yescharacteristic
228C22C2×A4⋊Q811yescharacteristic
229C4—31no—
230C4—31no—
231C4—32no—
232C4—31no—
233C4—31no—
234C4—32no—
235C4—34no—
236C4—32no—
237C4—62no—
238C4—124no—
239C4—124no—
240C4C2×#85712yes—
241C3—161noSylow
242C2—31no—
243C2—31no—
244C2—32no—
245C2#7298012yesdirect product factor
246C2C22×#85711yescharacteristic
subgroupG/H
centerC22C2×#857
commutator, cosocleC2×C42⋊C3C23
FrattiniC23C22×S4
FittingC2×C43D3
socleC24C2×S4
prime radicalC22×C42⋊C3C22
hypercenterC2×C4#857
hypocenterC42⋊C3C2×Q8
LCSG ▷ C2×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #72980C2direct (N × H)
19. C2×C42⋊C3Q8semidirect
50. C42⋊C3C2×Q8semidirect
101. C2×C42Dic6semidirect
169. C42C2×Dic6semidirect
245. C2#72980direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C42⋊C32
C2×#46232
#729802
C2×#19943
C2×A4⋊Q84